Subgroups — #30876722

Order 17152.

980 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863449	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560817	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560817	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560502	C₂	1	1	yes	maximal, characteristic
9	#22560389	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560389	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16863151	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	D₄×C₆₇⋊C₈	—	2	4	no	—
20	#16863449	—	2	4	no	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	#16863151	C₂²	1	1	yes	characteristic
37	#16863151	C₂²	1	1	yes	characteristic
38	#16863151	C₂²	1	1	yes	characteristic
39	#16863151	C₂²	1	1	yes	characteristic
40	#16863151	C₂²	1	1	yes	characteristic
41	#16863151	C₂²	1	1	yes	characteristic
42	#16863151	C₂²	1	1	yes	characteristic
43	#16863151	C₂²	1	1	yes	characteristic
44	#16863151	C₂²	1	1	yes	characteristic
45	#16863151	C₂²	1	1	yes	characteristic
46	C₂×#123×C₆₇	C₄	1	1	yes	characteristic
47	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
49	#388×C₆₇	C₄	1	1	yes	characteristic
50	#388×C₆₇	C₄	1	1	yes	characteristic
51	#387×C₆₇	C₄	1	1	yes	characteristic
52	#387×C₆₇	C₄	1	1	yes	characteristic
53	#386×C₆₇	C₄	1	1	yes	characteristic
54	#385×C₆₇	C₄	1	1	yes	characteristic
55	#381×C₆₇	C₄	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
59	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
60	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₆₇⋊C₈	—	2	2	no	—
63	C₂²×C₆₇⋊C₈	—	2	2	no	—
64	C₂²×C₆₇⋊C₈	—	2	2	no	—
65	C₂²×C₆₇⋊C₈	—	2	2	no	—
66	C₂²×C₆₇⋊C₈	—	2	2	no	—
67	C₂²×C₆₇⋊C₈	—	4	1	no	—
68	C₂²×C₆₇⋊C₈	—	4	1	no	—
69	C₂²×C₆₇⋊C₈	—	4	1	no	—
70	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
79	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
80	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
81	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
82	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
83	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
84	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
85	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
86	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
87	C₂²⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
88	C₄×C₆₇⋊C₈	—	2	2	no	—
89	C₄×C₆₇⋊C₈	—	2	2	no	—
90	C₄×C₆₇⋊C₈	—	2	2	no	—
91	C₄×C₆₇⋊C₈	—	2	2	no	—
92	C₄×C₆₇⋊C₈	—	4	1	no	—
93	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
98	#123×C₆₇	C₈	1	8	yes	complemented
99	C₄×D₄×C₆₇	—	2	2	no	—
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	—	2	2	no	—
115	C₂×C₄²×C₆₇	—	2	2	no	—
116	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
124	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
125	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
126	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
127	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
128	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
129	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
130	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
131	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
132	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
133	C₂×C₆₇⋊C₈	—	2	1	no	—
134	C₂×C₆₇⋊C₈	—	2	1	no	—
135	C₂×C₆₇⋊C₈	—	2	1	no	—
136	C₂×C₆₇⋊C₈	—	2	1	no	—
137	C₂×C₆₇⋊C₈	—	2	1	no	—
138	C₂×C₆₇⋊C₈	—	2	1	no	—
139	C₂×C₆₇⋊C₈	—	2	1	no	—
140	C₂×C₆₇⋊C₈	—	2	1	no	—
141	C₂×C₆₇⋊C₈	—	2	1	no	—
142	C₂×C₆₇⋊C₈	—	2	1	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	—	2	1	no	—
147	C₂×C₆₇⋊C₈	—	2	1	no	—
148	C₂×C₆₇⋊C₈	—	2	1	no	—
149	C₂×C₆₇⋊C₈	—	2	1	no	—
150	C₂×C₆₇⋊C₈	—	2	1	no	—
151	C₂×C₆₇⋊C₈	—	2	1	no	—
152	C₂×C₆₇⋊C₈	—	2	1	no	—
153	C₂×C₆₇⋊C₈	—	4	4	no	—
154	C₂×C₆₇⋊C₈	—	4	4	no	—
155	C₂×C₆₇⋊C₈	—	4	4	no	—
156	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
157	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
158	C₂×C₆₇⋊C₈	#41	1	2	yes	—
159	C₂×C₆₇⋊C₈	#41	1	2	yes	—
160	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	1	no	—
169	C₂²×C₄×C₆₇	—	2	1	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	1	no	—
178	C₂²×C₄×C₆₇	—	2	1	no	—
179	C₂²×C₄×C₆₇	—	4	1	no	—
180	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	complemented
181	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₆₇	#41	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	4	1	no	—
199	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
200	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	2	2	no	—
203	C₂²⋊C₄×C₆₇	—	2	2	no	—
204	C₂²⋊C₄×C₆₇	—	2	2	no	—
205	C₂²⋊C₄×C₆₇	—	4	1	no	—
206	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
207	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
208	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
209	C₄²×C₆₇	—	2	2	no	—
210	C₄²×C₆₇	—	2	2	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	—	2	1	no	—
213	C₄²×C₆₇	—	2	1	no	—
214	C₄²×C₆₇	—	2	1	no	—
215	C₄²×C₆₇	—	2	1	no	—
216	C₄²×C₆₇	—	2	1	no	—
217	C₄²×C₆₇	—	2	1	no	—
218	C₄²×C₆₇	—	2	1	no	—
219	C₄²×C₆₇	—	2	1	no	—
220	C₄²×C₆₇	—	2	1	no	—
221	C₂³×C₆₇	—	2	1	no	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	1	no	—
224	C₂³×C₆₇	—	2	1	no	—
225	C₂³×C₆₇	—	2	1	no	—
226	C₂³×C₆₇	—	2	1	no	—
227	C₂³×C₆₇	—	2	1	no	—
228	C₂³×C₆₇	—	2	1	no	—
229	C₂³×C₆₇	—	2	1	no	—
230	C₂³×C₆₇	—	2	1	no	—
231	C₂³×C₆₇	C₄×D₄	1	2	yes	—
232	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
233	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
234	C₂³×C₆₇	#131	1	2	yes	—
235	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
236	D₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	1	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	1	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	1	no	—
267	C₂×C₄×C₆₇	—	2	1	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	1	no	—
273	C₂×C₄×C₆₇	—	4	1	no	—
274	C₂×C₄×C₆₇	—	4	1	no	—
275	C₂×C₄×C₆₇	—	4	1	no	—
276	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
277	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
278	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
279	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
280	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
282	C₂×C₄×C₆₇	#131	1	2	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
288	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
289	C₆₇⋊C₈	—	2	2	no	—
290	C₆₇⋊C₈	—	2	2	no	—
291	C₆₇⋊C₈	—	2	2	no	—
292	C₆₇⋊C₈	—	2	2	no	—
293	C₆₇⋊C₈	—	4	1	no	—
294	C₆₇⋊C₈	—	4	1	no	—
295	C₆₇⋊C₈	—	4	1	no	—
296	C₆₇⋊C₈	—	4	1	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	2	2	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	2	2	no	—
304	C₂²×C₆₇	—	2	2	no	—
305	C₂²×C₆₇	—	2	2	no	—
306	C₂²×C₆₇	—	4	1	no	—
307	C₂²×C₆₇	—	4	1	no	—
308	C₂²×C₆₇	—	4	1	no	—
309	C₂²×C₆₇	—	4	1	no	—
310	C₂²×C₆₇	C₈×D₄	1	4	yes	—
311	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
312	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
313	C₂²×C₆₇	#520	1	1	yes	characteristic
314	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
315	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
316	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
317	C₂²×C₆₇	#568	1	1	yes	characteristic
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	2	2	no	—
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	2	2	no	—
322	C₄×C₆₇	—	2	2	no	—
323	C₄×C₆₇	—	2	2	no	—
324	C₄×C₆₇	—	2	2	no	—
325	C₄×C₆₇	—	4	1	no	—
326	C₄×C₆₇	—	4	1	no	—
327	C₄×C₆₇	—	4	1	no	—
328	C₄×C₆₇	—	4	1	no	—
329	C₄×C₆₇	—	4	1	no	—
330	C₄×C₆₇	#546	1	1	yes	characteristic
331	C₄×C₆₇	#547	1	1	yes	characteristic
332	C₄×C₆₇	#551	1	1	yes	characteristic
333	C₄×C₆₇	#552	1	1	yes	characteristic
334	#22900	—	67	1	no	maximal, Sylow
335	C₂×C₆₇	—	2	4	no	—
336	C₂×C₆₇	—	4	2	no	—
337	C₂×C₆₇	#2359	1	1	yes	characteristic
338	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
339	C₂×C₆₇	#2933	1	1	yes	characteristic
340	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
341	C₂×C₆₇	#2968	1	1	yes	characteristic
342	C₂×C₆₇	#2988	1	1	yes	characteristic
343	C₂×C₆₇	#2994	1	1	yes	characteristic
344	C₂×C₈×D₄	—	67	1	no	—
345	C₂×#431	—	67	1	no	—
346	C₄×#123	D₆₇	1	1	yes	characteristic
347	#1939	—	67	1	no	—
348	#1923	—	67	1	no	—
349	#1923	—	67	1	no	—
350	#1916	—	67	1	no	—
351	#1916	—	67	1	no	—
352	#1869	—	67	1	no	—
353	#1850	—	67	1	no	—
354	#1850	—	67	1	no	—
355	C₈×C₄⋊C₄	—	67	1	no	—
356	C₈×C₂²⋊C₄	—	67	1	no	—
357	C₈×C₂²⋊C₄	—	67	1	no	—
358	C₂×#336	—	67	1	no	—
359	C₆₇	#22900	1	1	yes	characteristic, complemented, Sylow
360	C₂³×C₈	—	67	1	no	—
361	C₂³×C₈	—	67	1	no	—
362	C₂³×C₈	—	67	1	no	—
363	C₂×#123	Dic₆₇	1	1	yes	characteristic
364	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
365	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
366	C₈×D₄	—	134	4	no	—
367	#431	—	134	4	no	—
368	C₂×C₄⋊C₈	—	67	1	no	—
369	C₂×C₄⋊C₈	—	67	1	no	—
370	C₂×C₄⋊C₈	—	67	1	no	—
371	C₂×C₄⋊C₈	—	67	1	no	—
372	C₂×C₄⋊C₈	—	67	1	no	—
373	C₂×C₂²⋊C₈	—	67	1	no	—
374	C₂×C₂²⋊C₈	—	67	1	no	—
375	C₂×C₂²⋊C₈	—	67	1	no	—
376	C₂×C₂²⋊C₈	—	67	1	no	—
377	C₂×C₂²⋊C₈	—	67	1	no	—
378	C₂×C₄×C₈	—	67	1	no	—
379	C₂×C₄×C₈	—	67	1	no	—
380	C₂×C₄×C₈	—	67	1	no	—
381	C₂×C₄×C₈	—	67	1	no	—
382	C₂×C₄×C₈	—	67	1	no	—
383	#388	Dic₆₇	1	1	yes	characteristic
384	#388	Dic₆₇	1	1	yes	characteristic
385	#387	Dic₆₇	1	1	yes	characteristic
386	#387	Dic₆₇	1	1	yes	characteristic
387	#386	Dic₆₇	1	1	yes	characteristic
388	#385	Dic₆₇	1	1	yes	characteristic
389	#381	Dic₆₇	1	1	yes	characteristic
390	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
391	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
392	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
393	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
394	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
395	#336	—	67	1	no	—
396	#336	—	67	1	no	—
397	#336	—	67	1	no	—
398	#336	—	67	1	no	—
399	#336	—	67	1	no	—
400	#336	—	67	1	no	—
401	#336	—	67	1	no	—
402	#336	—	67	1	no	—
403	#336	—	67	1	no	—
404	#336	—	67	1	no	—
405	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
406	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
407	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
408	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
409	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
410	C₂²×C₈	—	268	1	no	—
411	C₂²×C₈	—	268	1	no	—
412	C₂²×C₈	—	268	1	no	—
413	C₂²×C₈	—	67	1	no	—
414	C₂²×C₈	—	67	1	no	—
415	C₂²×C₈	—	67	1	no	—
416	C₂²×C₈	—	67	1	no	—
417	C₂²×C₈	—	67	1	no	—
418	C₂²×C₈	—	67	1	no	—
419	C₂²×C₈	—	67	1	no	—
420	C₂²×C₈	—	67	1	no	—
421	C₂²×C₈	—	134	2	no	—
422	C₂²×C₈	—	134	2	no	—
423	C₂²×C₈	—	134	2	no	—
424	C₂²×C₈	—	134	2	no	—
425	C₂²×C₈	—	134	2	no	—
426	C₂²×C₈	—	134	2	no	—
427	#123	C₆₇⋊C₈	1	8	yes	complemented
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
443	C₂×C₄²	—	2	2	no	—
444	C₂×C₄²	—	2	2	no	—
445	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
447	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
448	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
449	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
450	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
451	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
452	C₄⋊C₈	—	268	1	no	—
453	C₄⋊C₈	—	134	2	no	—
454	C₄⋊C₈	—	134	2	no	—
455	C₄⋊C₈	—	134	2	no	—
456	C₄⋊C₈	—	134	2	no	—
457	C₂²⋊C₈	—	268	1	no	—
458	C₂²⋊C₈	—	67	4	no	—
459	C₂²⋊C₈	—	134	2	no	—
460	C₂²⋊C₈	—	134	2	no	—
461	C₂²⋊C₈	—	134	2	no	—
462	C₄×C₈	—	268	1	no	—
463	C₄×C₈	—	134	2	no	—
464	C₄×C₈	—	134	2	no	—
465	C₄×C₈	—	134	2	no	—
466	C₄×C₈	—	134	2	no	—
467	#95	C₂×Dic₆₇	1	1	yes	characteristic
468	#95	C₂×Dic₆₇	1	1	yes	characteristic
469	#95	C₂×Dic₆₇	1	1	yes	characteristic
470	#95	C₂×Dic₆₇	1	1	yes	characteristic
471	#95	C₂×Dic₆₇	1	1	yes	characteristic
472	#95	C₂×Dic₆₇	1	1	yes	characteristic
473	#95	C₂×Dic₆₇	1	1	yes	characteristic
474	#95	C₂×Dic₆₇	1	1	yes	characteristic
475	#95	C₂×Dic₆₇	1	1	yes	characteristic
476	#95	C₂×Dic₆₇	1	1	yes	characteristic
477	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
478	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	C₂×C₆₇⋊C₈	1	4	yes	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	4	1	no	—
497	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
498	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
499	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
501	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
502	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
503	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
504	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
505	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
506	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
507	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
508	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
509	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
510	C₂²×C₄	D₄×D₆₇	1	2	yes	—
511	C₂²×C₄	#11761627	1	2	yes	—
512	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
513	C₂×C₈	—	268	4	no	—
514	C₂×C₈	—	268	4	no	—
515	C₂×C₈	—	268	4	no	—
516	C₂×C₈	—	67	2	no	—
517	C₂×C₈	—	67	2	no	—
518	C₂×C₈	—	67	2	no	—
519	C₂×C₈	—	67	2	no	—
520	C₂×C₈	—	134	1	no	—
521	C₂×C₈	—	134	1	no	—
522	C₂×C₈	—	134	1	no	—
523	C₂×C₈	—	134	1	no	—
524	C₂×C₈	—	134	1	no	—
525	C₂×C₈	—	134	1	no	—
526	C₂×C₈	—	134	1	no	—
527	C₂×C₈	—	134	1	no	—
528	C₂×C₈	—	134	1	no	—
529	C₂×C₈	—	134	1	no	—
530	C₂×C₈	—	134	1	no	—
531	C₂×C₈	—	134	1	no	—
532	C₂×C₈	—	134	1	no	—
533	C₂×C₈	—	134	1	no	—
534	C₂×C₈	—	134	1	no	—
535	C₂×C₈	—	134	1	no	—
536	C₂×C₈	—	134	1	no	—
537	C₂×C₈	—	134	1	no	—
538	C₂×C₈	—	134	1	no	—
539	C₂×C₈	—	134	1	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	4	1	no	—
543	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
544	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	2	no	—
549	C₂²⋊C₄	—	4	1	no	—
550	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
551	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
552	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
553	C₄²	—	2	2	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	D₄×Dic₆₇	1	2	yes	—
576	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
577	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
578	C₂³	#14524050	1	2	yes	—
579	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
580	D₄	—	2	8	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
621	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
622	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
623	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
624	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
625	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
626	C₂×C₄	#14524050	1	2	yes	—
627	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
628	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
629	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
630	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
631	C₂×C₄	#14524089	1	1	yes	characteristic
632	C₂×C₄	#14524090	1	1	yes	characteristic
633	C₈	—	268	1	no	—
634	C₈	—	268	1	no	—
635	C₈	—	268	1	no	—
636	C₈	—	268	1	no	—
637	C₈	—	134	2	no	—
638	C₈	—	134	2	no	—
639	C₈	—	134	2	no	—
640	C₈	—	134	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	2	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	D₄×C₆₇⋊C₈	1	4	yes	—
655	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
656	C₂²	#41×Dic₆₇	1	1	yes	characteristic
657	C₂²	#16863879	1	1	yes	characteristic
658	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
659	C₂²	C₂×#14524050	1	1	yes	characteristic
660	C₂²	C₂×#14524050	1	1	yes	characteristic
661	C₂²	#16864163	1	1	yes	characteristic
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	2	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	1	no	—
674	C₄	#16864038	1	1	yes	characteristic
675	C₄	#16864044	1	1	yes	characteristic
676	C₄	#16864069	1	1	yes	characteristic
677	C₄	#16864083	1	1	yes	characteristic
678	C₂	—	2	4	no	—
679	C₂	—	4	2	no	—
680	C₂	#22563205	1	1	yes	characteristic
681	C₂	C₂×D₄×C₆₇⋊C₈	1	1	yes	characteristic
682	C₂	#22568936	1	1	yes	characteristic
683	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
684	C₂	#22569205	1	1	yes	characteristic
685	C₂	#22569320	1	1	yes	characteristic
686	C₂	#22569369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×#123×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#123	D₆₇
hypocenter	C₆₇	#22900
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#123

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#16863449	C₂	semidirect
3. #22560928	C₂	semidirect
4. #22560817	C₂	semidirect
5. #22560817	C₂	semidirect
6. #22560768	C₂	semidirect
7. #22560768	C₂	semidirect
9. #22560389	C₂	semidirect
10. #22560389	C₂	semidirect
11. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
12. C₂²⋊C₄×C₆₇⋊C₈	C₂	semidirect
13. C₂²⋊C₄×C₆₇⋊C₈	C₂	semidirect
14. C₂×#16863151	C₂	semidirect
98. #123×C₆₇	C₈	semidirect
180. C₂²×C₄×C₆₇	C₂×C₈	semidirect
199. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
200. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
206. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
207. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
208. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
359. C₆₇	#22900	semidirect
427. #123	C₆₇⋊C₈	semidirect
497. C₂²×C₄	C₂×C₆₇⋊C₈	semidirect
543. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
544. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
550. C₂²⋊C₄	C₂×C₆₇⋊C₈	semidirect
551. C₂²⋊C₄	C₂×C₆₇⋊C₈	semidirect
552. C₂²⋊C₄	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×C₆₇⋊C₈	2
C₂×#16863449	2
#22560928	2
#22560817	2
#22560768	2
#22560502	2
#22560389	2
C₄⋊C₄×C₆₇⋊C₈	2
C₂²⋊C₄×C₆₇⋊C₈	2
C₂×#16863151	2
C₄×#123×C₆₇	2
#22900	67