Subgroups — #30874393

Order 17152.

1032 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863468	C₂	1	2	yes	maximal
2	#22560933	C₂	1	2	yes	maximal, complemented
3	#22560394	C₂	1	2	yes	maximal, complemented
4	#22560389	C₂	1	2	yes	maximal, complemented
5	#22560326	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560189	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560189	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560062	C₂	1	1	yes	maximal, characteristic, complemented
9	#22559903	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863151	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11761612	C₂²	1	2	yes	—
13	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
14	#16863468	—	2	8	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
21	C₂×#14523954	C₂²	1	2	yes	—
22	#16863151	C₂²	1	1	yes	characteristic
23	#16863151	C₂²	1	1	yes	characteristic
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	1	yes	characteristic
26	#16863151	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₆₇	C₄	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	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
43	C₄³×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#11761612	—	2	4	no	—
45	C₂×#11761612	—	2	4	no	—
46	C₂×#11761612	—	4	2	no	—
47	C₂²×C₆₇⋊C₈	—	2	4	no	—
48	C₂²×C₆₇⋊C₈	—	2	4	no	—
49	C₂²×C₆₇⋊C₈	—	4	2	no	—
50	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
54	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
56	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
57	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
58	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
59	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
60	#14523954	—	2	4	no	—
61	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×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	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	—	2	4	no	—
70	C₄×D₄×C₆₇	D₄	1	4	yes	—
71	C₄×D₄×C₆₇	Q₈	1	4	yes	—
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	—	4	2	no	—
82	C₂×C₄²×C₆₇	—	4	2	no	—
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	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	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
97	#95×C₆₇	C₂×C₄	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	#11761612	—	4	8	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	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	—	4	8	no	—
115	C₂×C₆₇⋊C₈	#41	1	4	yes	—
116	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
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	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	4	1	no	—
131	C₂²×C₄×C₆₇	—	4	1	no	—
132	C₂²×C₄×C₆₇	—	4	1	no	—
133	C₂²×C₄×C₆₇	—	4	1	no	—
134	C₂²×C₄×C₆₇	—	4	1	no	—
135	C₂²×C₄×C₆₇	—	4	1	no	—
136	C₂²×C₄×C₆₇	M₁₆	1	4	yes	complemented
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
161	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
162	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	complemented
163	C₄²×C₆₇	—	2	2	no	—
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	—	4	1	no	—
171	C₄²×C₆₇	—	4	1	no	—
172	C₄²×C₆₇	—	4	1	no	—
173	C₄²×C₆₇	—	4	1	no	—
174	C₄²×C₆₇	—	4	1	no	—
175	C₄²×C₆₇	C₂×D₄	1	2	yes	—
176	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	—	2	1	no	—
179	C₂³×C₆₇	—	2	1	no	—
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	C₂×C₄⋊C₄	1	2	yes	—
188	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
189	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
190	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
191	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	D₄×C₆₇	—	2	8	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	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	1	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	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	—	4	4	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	—	4	4	no	—
227	C₂×C₄×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
233	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
234	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
235	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
236	C₂×C₄×C₆₇	#131	1	2	yes	—
237	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
240	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
241	C₆₇⋊C₈	—	2	4	no	—
242	C₆₇⋊C₈	—	4	2	no	—
243	C₆₇⋊C₈	—	4	2	no	—
244	C₆₇⋊C₈	—	4	2	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	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	—	4	1	no	—
255	C₂²×C₆₇	—	4	1	no	—
256	C₂²×C₆₇	—	4	1	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	#423	1	4	yes	—
259	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
260	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
261	C₂²×C₆₇	#518	1	1	yes	characteristic
262	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
263	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
264	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
265	C₂²×C₆₇	#568	1	1	yes	characteristic
266	C₄×C₆₇	—	2	2	no	—
267	C₄×C₆₇	—	2	2	no	—
268	C₄×C₆₇	—	2	2	no	—
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	#546	1	2	yes	—
279	C₄×C₆₇	#554	1	2	yes	—
280	#22698	—	67	1	no	maximal, Sylow
281	C₂×C₆₇	—	2	4	no	—
282	C₂×C₆₇	—	4	2	no	—
283	C₂×C₆₇	#2349	1	1	yes	characteristic
284	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
285	C₂×C₆₇	#2906	1	1	yes	characteristic
286	C₂×C₆₇	#2965	1	1	yes	characteristic
287	C₂×C₆₇	#2970	1	1	yes	characteristic
288	C₂×C₆₇	#2973	1	1	yes	characteristic
289	C₂×C₆₇	#2974	1	1	yes	characteristic
290	C₂×#435	—	67	2	no	—
291	C₄²×D₄	D₆₇	1	1	yes	characteristic
292	#1940	—	67	2	no	—
293	#1851	—	67	2	no	—
294	#1850	—	67	2	no	—
295	#1831	—	67	1	no	—
296	#1797	—	67	1	no	—
297	#1797	—	67	1	no	—
298	#1767	—	67	1	no	—
299	#1728	—	67	1	no	—
300	C₂×#336	—	67	1	no	—
301	C₆₇	#22698	1	1	yes	characteristic, complemented, Sylow
302	C₂²×M₁₆	—	67	2	no	—
303	C₂³×C₈	—	67	2	no	—
304	C₂×C₄×D₄	Dic₆₇	1	1	yes	characteristic
305	C₂×C₄×D₄	Dic₆₇	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
307	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
308	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
309	#435	—	134	8	no	—
310	C₂×C₄⋊C₈	—	67	2	no	—
311	C₂×C₄⋊C₈	—	67	2	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×#97	—	67	2	no	—
317	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
318	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
319	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
320	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
321	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
322	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
323	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
324	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
325	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
326	C₄³	C₂×D₆₇	1	1	yes	characteristic
327	#336	—	67	1	no	—
328	#336	—	67	1	no	—
329	#336	—	67	1	no	—
330	#336	—	67	2	no	—
331	#336	—	67	2	no	—
332	#336	—	67	2	no	—
333	#336	—	67	1	no	—
334	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
335	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
336	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
337	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
338	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
341	C₂×M₁₆	—	268	2	no	—
342	C₂×M₁₆	—	134	4	no	—
343	C₂×M₁₆	—	134	4	no	—
344	C₂²×C₈	—	268	2	no	—
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	C₄×D₄	—	2	4	no	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	Dic₁₃₄	1	4	yes	—
354	C₄×D₄	D₂₆₈	1	4	yes	—
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	—	4	2	no	—
365	C₂×C₄²	—	4	2	no	—
366	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
367	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
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	C₂²×D₆₇	1	1	yes	characteristic
375	C₂×C₄²	C₂²×D₆₇	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	C₄⋊C₈	—	268	2	no	—
380	C₄⋊C₈	—	268	2	no	—
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	#97	—	134	4	no	—
386	#95	C₂×Dic₆₇	1	1	yes	characteristic
387	#95	C₂×Dic₆₇	1	1	yes	characteristic
388	#95	C₂×Dic₆₇	1	1	yes	characteristic
389	#95	C₂×Dic₆₇	1	1	yes	characteristic
390	#95	C₂×Dic₆₇	1	1	yes	characteristic
391	#95	C₂×Dic₆₇	1	1	yes	characteristic
392	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
393	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	C₄⋊Dic₆₇	1	4	yes	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
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	—	4	1	no	—
407	C₂²×C₄	—	4	1	no	—
408	C₂²×C₄	—	4	1	no	—
409	C₂²×C₄	—	4	1	no	—
410	C₂²×C₄	—	4	1	no	—
411	C₂²×C₄	—	4	1	no	—
412	C₂²×C₄	#11761612	1	4	yes	complemented
413	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
414	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
416	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
418	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
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	C₂×D₂₆₈	1	2	yes	—
426	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
427	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
428	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
429	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
430	M₁₆	—	268	8	no	—
431	C₂×C₈	—	268	8	no	—
432	C₂×C₈	—	67	4	no	—
433	C₂×C₈	—	134	2	no	—
434	C₂×C₈	—	134	2	no	—
435	C₂×C₈	—	134	2	no	—
436	C₂×C₈	—	134	2	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	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
451	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
452	C₂²⋊C₄	#11761612	1	4	yes	complemented
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	2	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	4	1	no	—
461	C₄²	—	4	1	no	—
462	C₄²	—	4	1	no	—
463	C₄²	—	4	1	no	—
464	C₄²	—	4	1	no	—
465	C₄²	C₂×D₂₆₈	1	2	yes	—
466	C₄²	C₂×Dic₁₃₄	1	2	yes	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
478	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
479	C₂³	C₂×#11761612	1	2	yes	—
480	C₂³	C₂×#11761612	1	2	yes	—
481	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
482	D₄	—	2	8	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	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	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	1	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	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
519	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
520	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
521	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
522	C₂×C₄	C₂×#11761612	1	2	yes	—
523	C₂×C₄	C₂×#11761612	1	2	yes	—
524	C₂×C₄	C₂×#11761612	1	2	yes	—
525	C₂×C₄	C₂×#11761612	1	2	yes	—
526	C₂×C₄	#14524050	1	2	yes	—
527	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
528	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
529	C₂×C₄	#41×D₆₇	1	2	yes	—
530	C₂×C₄	#14524088	1	2	yes	—
531	C₈	—	268	2	no	—
532	C₈	—	268	2	no	—
533	C₈	—	268	2	no	—
534	C₈	—	134	4	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	#16863420	1	4	yes	—
549	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
550	C₂²	#41×Dic₆₇	1	1	yes	characteristic
551	C₂²	#16863871	1	1	yes	characteristic
552	C₂²	C₂²×#11761612	1	1	yes	characteristic
553	C₂²	C₂²×#11761612	1	1	yes	characteristic
554	C₂²	C₂×#14524050	1	1	yes	characteristic
555	C₂²	#16864163	1	1	yes	characteristic
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	#16864041	1	2	yes	—
569	C₄	#16864105	1	2	yes	—
570	C₂	—	2	4	no	—
571	C₂	—	4	2	no	—
572	C₂	#22563119	1	1	yes	characteristic
573	C₂	C₂×#16863420	1	1	yes	characteristic
574	C₂	#22568777	1	1	yes	characteristic
575	C₂	#22569176	1	1	yes	characteristic
576	C₂	#22569223	1	1	yes	characteristic
577	C₂	#22569238	1	1	yes	characteristic
578	C₂	#22569244	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₄²×D₄×C₆₇	C₂
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×D₄	D₆₇
hypocenter	C₆₇	#22698
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. #22560933	C₂	semidirect
3. #22560394	C₂	semidirect
4. #22560389	C₂	semidirect
5. #22560326	C₂	semidirect
6. #22560189	C₂	semidirect
7. #22560189	C₂	semidirect
8. #22560062	C₂	semidirect
9. #22559903	C₂	semidirect
10. C₂×#16863151	C₂	semidirect
136. C₂²×C₄×C₆₇	M₁₆	semidirect
162. C₂²⋊C₄×C₆₇	M₁₆	semidirect
301. C₆₇	#22698	semidirect
412. C₂²×C₄	#11761612	semidirect
452. C₂²⋊C₄	#11761612	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863468	2
#22560933	2
#22560394	2
#22560389	2
#22560326	2
#22560189	2
#22560062	2
#22559903	2
C₂×#16863151	2
C₄²×D₄×C₆₇	2
#22698	67