Subgroups — #30858613

Order 17152.

1106 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560817	C₂	1	2	yes	maximal, complemented
4	#22560768	C₂	1	2	yes	maximal, complemented
5	#22560502	C₂	1	2	yes	maximal
6	#22560389	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559955	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
13	#16863449	—	2	4	no	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₄×C₆₇⋊C₈	—	2	4	no	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
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	#16863151	C₂²	1	2	yes	—
26	#16863151	C₂²	1	2	yes	—
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
44	C₄³×C₆₇	C₂²	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	—	2	4	no	—
46	C₂²×C₆₇⋊C₈	—	2	4	no	—
47	C₂²×C₆₇⋊C₈	—	2	4	no	—
48	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
49	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
54	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
56	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
57	C₄×C₆₇⋊C₈	—	2	4	no	—
58	C₄×C₆₇⋊C₈	—	2	2	no	—
59	C₄×C₆₇⋊C₈	—	2	2	no	—
60	C₄×C₆₇⋊C₈	—	2	2	no	—
61	C₄×C₆₇⋊C₈	D₄	1	4	yes	complemented
62	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₆₇	—	2	4	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	C₈	1	8	yes	complemented
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	—	2	2	no	—
85	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
99	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
100	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
102	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	—	2	16	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
116	C₂×C₆₇⋊C₈	#41	1	4	yes	—
117	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	complemented
140	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	#41	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	2	2	no	—
161	C₂²⋊C₄×C₆₇	—	2	2	no	—
162	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
163	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	2	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	—	2	1	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	C₂×C₈	1	4	yes	complemented
178	C₄²×C₆₇	C₂×D₄	1	2	yes	—
179	C₄²×C₆₇	#41	1	2	yes	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	1	no	—
185	C₂³×C₆₇	—	2	1	no	—
186	C₂³×C₆₇	—	2	1	no	—
187	C₂³×C₆₇	—	2	1	no	—
188	C₂³×C₆₇	—	2	1	no	—
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	#117	1	2	yes	—
191	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
192	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
193	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
194	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
195	D₄×C₆₇	—	2	8	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	8	no	—
211	C₂×C₄×C₆₇	—	2	8	no	—
212	C₂×C₄×C₆₇	—	2	8	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	4	2	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	#117	1	2	yes	—
229	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
232	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
233	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
234	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
236	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
237	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
238	C₂×C₄×C₆₇	#131	1	2	yes	—
239	C₂×C₄×C₆₇	#131	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
242	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
244	C₆₇⋊C₈	—	2	4	no	—
245	C₆₇⋊C₈	—	2	4	no	—
246	C₆₇⋊C₈	—	4	2	no	—
247	C₆₇⋊C₈	—	4	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	2	no	—
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	—	4	1	no	—
259	C₂²×C₆₇	—	4	1	no	—
260	C₂²×C₆₇	—	4	1	no	—
261	C₂²×C₆₇	#431	1	4	yes	complemented
262	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
263	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
265	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
266	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
267	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
268	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
269	C₄×C₆₇	—	2	2	no	—
270	C₄×C₆₇	—	2	2	no	—
271	C₄×C₆₇	—	2	2	no	—
272	C₄×C₆₇	—	2	2	no	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	4	1	no	—
276	C₄×C₆₇	—	4	1	no	—
277	C₄×C₆₇	—	4	1	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	C₈×D₄	1	4	yes	—
280	C₄×C₆₇	#435	1	4	yes	—
281	C₄×C₆₇	#525	1	2	yes	—
282	C₄×C₆₇	#529	1	2	yes	—
283	#21101	—	67	1	no	maximal, Sylow
284	C₂×C₆₇	—	2	4	no	—
285	C₂×C₆₇	—	4	2	no	—
286	C₂×C₆₇	#2290	1	1	yes	characteristic
287	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
288	C₂×C₆₇	#2922	1	1	yes	characteristic
289	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
290	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
291	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
292	C₂×C₆₇	#2965	1	1	yes	characteristic
293	C₂×#431	—	67	1	no	—
294	C₂²×C₄×C₈	—	67	1	no	—
295	C₄²×D₄	D₆₇	1	1	yes	characteristic
296	#1923	—	67	2	no	—
297	#1916	—	67	2	no	—
298	#1869	—	67	2	no	—
299	#1850	—	67	2	no	—
300	C₄×C₄⋊C₈	—	67	1	no	—
301	C₄×C₂²⋊C₈	—	67	1	no	—
302	C₄×C₂²⋊C₈	—	67	1	no	—
303	#1744	—	67	1	no	—
304	C₆₇	#21101	1	1	yes	characteristic, complemented, Sylow
305	C₂³×C₈	—	67	2	no	—
306	C₂×C₄×D₄	Dic₆₇	1	1	yes	characteristic
307	C₂×C₄×D₄	Dic₆₇	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
309	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
310	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
311	#431	—	134	4	no	—
312	C₂×C₄⋊C₈	—	67	2	no	—
313	C₂×C₄⋊C₈	—	67	2	no	—
314	C₂×C₂²⋊C₈	—	67	2	no	—
315	C₂×C₂²⋊C₈	—	67	2	no	—
316	C₂×C₂²⋊C₈	—	67	2	no	—
317	C₂×C₄×C₈	—	67	1	no	—
318	C₂×C₄×C₈	—	67	2	no	—
319	C₂×C₄×C₈	—	67	1	no	—
320	C₂×C₄×C₈	—	67	1	no	—
321	C₂×C₄×C₈	—	67	1	no	—
322	C₂×C₄×C₈	—	134	4	no	—
323	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
324	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
326	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
327	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
328	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
329	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
330	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
331	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
332	C₄³	C₂×D₆₇	1	1	yes	characteristic
333	#336	—	67	2	no	—
334	#336	—	67	2	no	—
335	#336	—	67	2	no	—
336	#336	—	67	2	no	—
337	#336	—	67	2	no	—
338	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
342	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
343	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
344	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
345	C₂²×C₈	—	67	2	no	—
346	C₂²×C₈	—	67	2	no	—
347	C₂²×C₈	—	67	2	no	—
348	C₂²×C₈	—	67	2	no	—
349	C₂²×C₈	—	134	4	no	—
350	C₂²×C₈	—	134	4	no	—
351	C₂²×C₈	—	134	4	no	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	C₆₇⋊C₈	1	8	yes	complemented
355	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	2	no	—
367	C₂×C₄²	—	2	2	no	—
368	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
369	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
370	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
371	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
372	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
373	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
374	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
375	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
376	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
377	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
378	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
379	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
380	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
381	C₄⋊C₈	—	134	4	no	—
382	C₄⋊C₈	—	134	4	no	—
383	C₂²⋊C₈	—	134	4	no	—
384	C₂²⋊C₈	—	134	4	no	—
385	C₂²⋊C₈	—	134	4	no	—
386	C₄×C₈	—	67	4	no	—
387	C₄×C₈	—	134	4	no	—
388	C₄×C₈	—	134	2	no	—
389	C₄×C₈	—	134	2	no	—
390	C₄×C₈	—	134	2	no	—
391	#95	C₂×Dic₆₇	1	1	yes	characteristic
392	#95	C₂×Dic₆₇	1	1	yes	characteristic
393	#95	C₂×Dic₆₇	1	1	yes	characteristic
394	#95	C₂×Dic₆₇	1	1	yes	characteristic
395	#95	C₂×Dic₆₇	1	1	yes	characteristic
396	#95	C₂×Dic₆₇	1	1	yes	characteristic
397	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
398	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	C₂×C₆₇⋊C₈	1	4	yes	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
419	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
420	C₂²×C₄	#11761612	1	4	yes	—
421	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
428	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
433	C₂²×C₄	#11761628	1	4	yes	—
434	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
435	C₂×C₈	—	67	4	no	—
436	C₂×C₈	—	67	4	no	—
437	C₂×C₈	—	134	2	no	—
438	C₂×C₈	—	134	2	no	—
439	C₂×C₈	—	134	2	no	—
440	C₂×C₈	—	134	2	no	—
441	C₂×C₈	—	134	2	no	—
442	C₂×C₈	—	134	2	no	—
443	C₂×C₈	—	134	16	no	—
444	C₂×C₈	—	134	2	no	—
445	C₂×C₈	—	134	2	no	—
446	C₂×C₈	—	134	2	no	—
447	C₂×C₈	—	134	2	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
451	C₂²⋊C₄	—	2	2	no	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
456	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
457	C₄²	—	2	2	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	2	2	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	2	no	—
470	C₄²	C₂×C₆₇⋊C₈	1	4	yes	complemented
471	C₄²	D₄×D₆₇	1	2	yes	—
472	C₄²	#11761627	1	2	yes	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	1	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	1	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	1	no	—
483	C₂³	#14524002	1	2	yes	—
484	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
485	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
486	C₂³	C₂×#11761612	1	2	yes	—
487	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
488	D₄	—	2	8	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	#14524002	1	2	yes	—
522	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
523	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
524	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
525	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
526	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
527	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
528	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
529	C₂×C₄	C₂×#11761612	1	2	yes	—
530	C₂×C₄	C₂×#11761612	1	2	yes	—
531	C₂×C₄	#14524050	1	2	yes	—
532	C₂×C₄	#14524050	1	2	yes	—
533	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
534	C₂×C₄	#41×D₆₇	1	2	yes	—
535	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
536	C₂×C₄	C₂×#11761628	1	2	yes	—
537	C₈	—	268	2	no	—
538	C₈	—	268	2	no	—
539	C₈	—	134	4	no	—
540	C₈	—	134	4	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	#16863449	1	4	yes	complemented
555	C₂²	C₂×#14524002	1	1	yes	characteristic
556	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
557	C₂²	#41×Dic₆₇	1	1	yes	characteristic
558	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
559	C₂²	C₂²×#11761612	1	1	yes	characteristic
560	C₂²	C₂×#14524050	1	1	yes	characteristic
561	C₂²	C₂×#14524050	1	1	yes	characteristic
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	D₄×C₆₇⋊C₈	1	4	yes	—
573	C₄	#16863468	1	4	yes	—
574	C₄	#16863905	1	2	yes	—
575	C₄	#16863926	1	2	yes	—
576	C₂	—	2	4	no	—
577	C₂	—	4	2	no	—
578	C₂	#22562654	1	1	yes	characteristic
579	C₂	C₂×#16863449	1	1	yes	characteristic
580	C₂	#22568853	1	1	yes	characteristic
581	C₂	C₂×D₄×C₆₇⋊C₈	1	1	yes	characteristic
582	C₂	C₂×#16863468	1	1	yes	characteristic
583	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
584	C₂	#22569176	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄²×D₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×D₄	D₆₇
hypocenter	C₆₇	#21101
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16863449	C₂	semidirect
2. C₂²×C₄×C₆₇⋊C₈	C₂	semidirect
3. #22560817	C₂	semidirect
4. #22560768	C₂	semidirect
6. #22560389	C₂	semidirect
7. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
8. C₄×C₂²⋊(C₆₇⋊C₈)	C₂	semidirect
9. C₄×C₂²⋊(C₆₇⋊C₈)	C₂	semidirect
10. #22559955	C₂	semidirect
61. C₄×C₆₇⋊C₈	D₄	semidirect
71. C₄×D₄×C₆₇	C₈	semidirect
139. C₂²×C₄×C₆₇	C₂×C₈	semidirect
157. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
163. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
177. C₄²×C₆₇	C₂×C₈	semidirect
261. C₂²×C₆₇	#431	semidirect
304. C₆₇	#21101	semidirect
354. C₄×D₄	C₆₇⋊C₈	semidirect
419. C₂²×C₄	C₂×C₆₇⋊C₈	semidirect
450. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
456. C₂²⋊C₄	C₂×C₆₇⋊C₈	semidirect
470. C₄²	C₂×C₆₇⋊C₈	semidirect
554. C₂²	#16863449	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂²×C₄×C₆₇⋊C₈	2
#22560817	2
#22560768	2
#22560502	2
#22560389	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄×C₂²⋊(C₆₇⋊C₈)	2
#22559955	2
C₄²×D₄×C₆₇	2
#21101	67