Subgroups — #30863768

Order 17152.

848 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863468	C₂	1	1	yes	maximal, characteristic
3	#22561146	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561004	C₂	1	1	yes	maximal, characteristic
5	#22560824	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560824	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560817	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560817	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560329	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559993	C₂	1	1	yes	maximal, characteristic, complemented
11	#22559993	C₂	1	1	yes	maximal, characteristic, complemented
12	#22559987	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#22559896	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2398	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863527	—	2	4	no	—
19	#16863468	—	2	4	no	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
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×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊(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×#14523954	C₂²	1	1	yes	characteristic
31	C₂×#14523954	C₂²	1	1	yes	characteristic
32	C₂×#14523954	C₂²	1	1	yes	characteristic
33	C₂×#14523954	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	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₆₇⋊C₈	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×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
48	#400×C₆₇	C₄	1	1	yes	characteristic
49	#400×C₆₇	C₄	1	1	yes	characteristic
50	#397×C₆₇	C₄	1	1	yes	characteristic
51	#397×C₆₇	C₄	1	1	yes	characteristic
52	#394×C₆₇	C₄	1	1	yes	characteristic
53	#394×C₆₇	C₄	1	1	yes	characteristic
54	#393×C₆₇	C₄	1	1	yes	characteristic
55	#393×C₆₇	C₄	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
57	C₄×C₂²⋊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	#376×C₆₇	C₂²	1	1	yes	characteristic
61	C₂×#11761612	—	2	2	no	—
62	C₂×#11761612	—	2	2	no	—
63	C₂×#11761612	—	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	C₂³	1	1	yes	characteristic
68	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
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	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
76	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
77	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
78	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
79	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
80	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
81	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
82	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
83	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
84	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
85	#14523954	—	2	2	no	—
86	#14523954	—	2	2	no	—
87	#14523954	—	2	2	no	—
88	#14523954	D₄	1	4	yes	—
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	—	2	2	no	—
93	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₆₇	—	2	2	no	—
98	C₄×Q₈×C₆₇	—	2	2	no	—
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×C₂²⋊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×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
121	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
122	#95×C₆₇	C₂×C₄	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	#11761612	—	4	4	no	—
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	—	4	4	no	—
152	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
153	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
154	C₂×C₆₇⋊C₈	#41	1	2	yes	—
155	C₂×C₆₇⋊C₈	#41	1	2	yes	—
156	C₂×C₆₇⋊C₈	#41	1	2	yes	—
157	C₂×C₆₇⋊C₈	#41	1	2	yes	—
158	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	4	1	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	4	1	no	—
166	C₂²×C₄×C₆₇	—	2	1	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	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	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	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	2	2	no	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂²⋊C₄×C₆₇	—	4	1	no	—
202	C₂²⋊C₄×C₆₇	—	4	1	no	—
203	C₂²⋊C₄×C₆₇	—	4	1	no	—
204	C₂²⋊C₄×C₆₇	—	4	1	no	—
205	C₄²×C₆₇	—	2	1	no	—
206	C₄²×C₆₇	—	2	1	no	—
207	C₄²×C₆₇	—	2	1	no	—
208	C₄²×C₆₇	—	2	1	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	—	2	1	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	—	2	1	no	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	#41	1	2	yes	—
215	C₂³×C₆₇	—	2	1	no	—
216	C₂³×C₆₇	—	2	1	no	—
217	C₂³×C₆₇	—	2	1	no	—
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×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	#131	1	2	yes	—
229	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
230	Q₈×C₆₇	—	4	4	no	—
231	D₄×C₆₇	—	4	4	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	1	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	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	1	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	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	1	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	1	no	—
269	C₂×C₄×C₆₇	—	4	1	no	—
270	C₂×C₄×C₆₇	—	4	1	no	—
271	C₂×C₄×C₆₇	—	4	1	no	—
272	C₂×C₄×C₆₇	—	4	1	no	—
273	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
275	C₂×C₄×C₆₇	#131	1	2	yes	—
276	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
280	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
281	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
282	C₆₇⋊C₈	—	2	2	no	—
283	C₆₇⋊C₈	—	2	2	no	—
284	C₆₇⋊C₈	—	2	2	no	—
285	C₆₇⋊C₈	—	4	1	no	—
286	C₆₇⋊C₈	—	4	1	no	—
287	C₆₇⋊C₈	—	4	1	no	—
288	C₆₇⋊C₈	—	4	1	no	—
289	C₆₇⋊C₈	#124	1	4	yes	—
290	C₂²×C₆₇	—	2	2	no	—
291	C₂²×C₆₇	—	2	2	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	4	1	no	—
294	C₂²×C₆₇	—	4	1	no	—
295	C₂²×C₆₇	—	4	1	no	—
296	C₂²×C₆₇	—	4	1	no	—
297	C₂²×C₆₇	—	4	1	no	—
298	C₂²×C₆₇	—	4	1	no	—
299	C₂²×C₆₇	—	4	1	no	—
300	C₂²×C₆₇	—	4	1	no	—
301	C₂²×C₆₇	—	4	1	no	—
302	C₂²×C₆₇	—	4	1	no	—
303	C₂²×C₆₇	—	4	1	no	—
304	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
305	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
306	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
307	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
308	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
309	C₂²×C₆₇	#568	1	1	yes	characteristic
310	C₂²×C₆₇	#568	1	1	yes	characteristic
311	C₄×C₆₇	—	2	2	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	4	1	no	—
315	C₄×C₆₇	—	4	1	no	—
316	C₄×C₆₇	—	4	1	no	—
317	C₄×C₆₇	—	4	1	no	—
318	C₄×C₆₇	—	4	1	no	—
319	C₄×C₆₇	—	4	1	no	—
320	C₄×C₆₇	—	4	1	no	—
321	C₄×C₆₇	—	4	1	no	—
322	C₄×C₆₇	—	4	1	no	—
323	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
324	C₄×C₆₇	#529	1	1	yes	characteristic
325	C₄×C₆₇	#529	1	1	yes	characteristic
326	C₄×C₆₇	#532	1	1	yes	characteristic
327	#21637	—	67	1	no	maximal, Sylow
328	C₂×C₆₇	—	4	2	no	—
329	C₂×C₆₇	—	8	1	no	—
330	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
331	C₂×C₆₇	#2933	1	1	yes	characteristic
332	C₂×C₆₇	#2933	1	1	yes	characteristic
333	C₂×C₆₇	#2966	1	1	yes	characteristic
334	C₂×C₆₇	#2967	1	1	yes	characteristic
335	C₂×C₆₇	#2968	1	1	yes	characteristic
336	C₂×C₆₇	#2969	1	1	yes	characteristic
337	C₂×#446	—	67	1	no	—
338	C₂×#435	—	67	1	no	—
339	#2398	D₆₇	1	1	yes	characteristic
340	#1984	—	67	1	no	—
341	#1954	—	67	1	no	—
342	#1924	—	67	1	no	—
343	#1924	—	67	1	no	—
344	#1923	—	67	1	no	—
345	#1923	—	67	1	no	—
346	#1832	—	67	1	no	—
347	#1753	—	67	1	no	—
348	#1753	—	67	1	no	—
349	#1752	—	67	1	no	—
350	C₈×C₂²⋊C₄	—	67	1	no	—
351	#1726	—	67	1	no	—
352	C₆₇	#21637	1	1	yes	characteristic, complemented, Sylow
353	C₂²×M₁₆	—	67	1	no	—
354	C₂³×C₈	—	67	1	no	—
355	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
356	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
357	#446	—	134	4	no	—
358	#435	—	134	4	no	—
359	C₂×C₄⋊C₈	—	67	1	no	—
360	C₂×C₄⋊C₈	—	67	1	no	—
361	C₂×C₄⋊C₈	—	67	1	no	—
362	C₂×C₄⋊C₈	—	67	1	no	—
363	C₂×C₂²⋊C₈	—	67	1	no	—
364	C₂×C₂²⋊C₈	—	67	1	no	—
365	C₂×C₂²⋊C₈	—	67	1	no	—
366	C₂×C₂²⋊C₈	—	67	1	no	—
367	C₂×C₂²⋊C₈	—	67	1	no	—
368	C₂×C₂²⋊C₈	—	67	1	no	—
369	C₂×#97	—	67	1	no	—
370	C₂×#97	—	67	1	no	—
371	C₂×#97	—	67	1	no	—
372	C₂×#97	—	67	1	no	—
373	C₂×C₄×C₈	—	67	1	no	—
374	C₂×C₄×C₈	—	67	1	no	—
375	C₂×C₄×C₈	—	67	1	no	—
376	C₂×C₄×C₈	—	67	1	no	—
377	#400	Dic₆₇	1	1	yes	characteristic
378	#400	Dic₆₇	1	1	yes	characteristic
379	#397	Dic₆₇	1	1	yes	characteristic
380	#397	Dic₆₇	1	1	yes	characteristic
381	#394	Dic₆₇	1	1	yes	characteristic
382	#394	Dic₆₇	1	1	yes	characteristic
383	#393	Dic₆₇	1	1	yes	characteristic
384	#393	Dic₆₇	1	1	yes	characteristic
385	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
386	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
387	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
388	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
389	#376	C₂×D₆₇	1	1	yes	characteristic
390	#336	—	67	1	no	—
391	#336	—	67	1	no	—
392	#336	—	67	1	no	—
393	#336	—	67	1	no	—
394	#336	—	67	1	no	—
395	#336	—	67	1	no	—
396	#336	—	67	1	no	—
397	#336	—	67	1	no	—
398	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
399	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
400	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
401	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
402	C₂×M₁₆	—	134	2	no	—
403	C₂×M₁₆	—	134	2	no	—
404	C₂×M₁₆	—	134	2	no	—
405	C₂²×C₈	—	67	1	no	—
406	C₂²×C₈	—	67	1	no	—
407	C₂²×C₈	—	67	1	no	—
408	C₂²×C₈	—	67	1	no	—
409	C₂²×C₈	—	67	1	no	—
410	C₂²×C₈	—	67	1	no	—
411	C₂²×C₈	—	67	1	no	—
412	C₂²×C₈	—	67	1	no	—
413	C₂²×C₈	—	134	2	no	—
414	C₂²×C₈	—	134	2	no	—
415	C₂²×C₈	—	134	2	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
424	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
435	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
436	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
438	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
439	C₄⋊C₈	—	134	2	no	—
440	C₄⋊C₈	—	134	2	no	—
441	C₄⋊C₈	—	134	2	no	—
442	C₄⋊C₈	—	134	2	no	—
443	C₂²⋊C₈	—	134	2	no	—
444	C₂²⋊C₈	—	134	2	no	—
445	C₂²⋊C₈	—	134	2	no	—
446	C₂²⋊C₈	—	134	2	no	—
447	C₂²⋊C₈	—	134	2	no	—
448	C₂²⋊C₈	—	134	2	no	—
449	#97	—	67	4	no	—
450	#97	—	134	2	no	—
451	#97	—	134	2	no	—
452	#97	—	134	2	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	#95	C₂×Dic₆₇	1	1	yes	characteristic
458	#95	C₂×Dic₆₇	1	1	yes	characteristic
459	#95	C₂×Dic₆₇	1	1	yes	characteristic
460	#95	C₂×Dic₆₇	1	1	yes	characteristic
461	#95	C₂×Dic₆₇	1	1	yes	characteristic
462	#95	C₂×Dic₆₇	1	1	yes	characteristic
463	#95	C₂×Dic₆₇	1	1	yes	characteristic
464	#95	C₂×Dic₆₇	1	1	yes	characteristic
465	#95	C₂×Dic₆₇	1	1	yes	characteristic
466	#95	C₂×Dic₆₇	1	1	yes	characteristic
467	#95	C₂×Dic₆₇	1	1	yes	characteristic
468	#95	C₂×Dic₆₇	1	1	yes	characteristic
469	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
470	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	4	1	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	4	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	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	1	no	—
489	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
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	C₂³×D₆₇	1	1	yes	characteristic
502	M₁₆	—	268	4	no	—
503	C₂×C₈	—	268	4	no	—
504	C₂×C₈	—	67	2	no	—
505	C₂×C₈	—	67	2	no	—
506	C₂×C₈	—	67	2	no	—
507	C₂×C₈	—	67	2	no	—
508	C₂×C₈	—	67	2	no	—
509	C₂×C₈	—	67	2	no	—
510	C₂×C₈	—	134	1	no	—
511	C₂×C₈	—	134	1	no	—
512	C₂×C₈	—	134	1	no	—
513	C₂×C₈	—	134	1	no	—
514	C₂×C₈	—	134	1	no	—
515	C₂×C₈	—	134	1	no	—
516	C₂×C₈	—	134	1	no	—
517	C₂×C₈	—	134	1	no	—
518	C₂×C₈	—	134	1	no	—
519	C₂×C₈	—	134	1	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	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	2	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	4	1	no	—
539	C₂²⋊C₄	—	4	1	no	—
540	C₂²⋊C₄	—	4	1	no	—
541	C₂²⋊C₄	—	4	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	D₄×D₆₇	1	2	yes	—
551	C₄²	#11761627	1	2	yes	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	#14524050	1	2	yes	—
566	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
567	Q₈	—	4	4	no	—
568	D₄	—	4	4	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	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	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
611	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
612	C₂×C₄	#14524050	1	2	yes	—
613	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
614	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
615	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
616	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
617	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
618	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
619	C₈	—	268	1	no	—
620	C₈	—	268	1	no	—
621	C₈	—	268	1	no	—
622	C₈	—	268	1	no	—
623	C₈	—	67	4	no	—
624	C₈	—	134	2	no	—
625	C₈	—	134	2	no	—
626	C₈	—	134	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
642	C₂²	#41×Dic₆₇	1	1	yes	characteristic
643	C₂²	#41×Dic₆₇	1	1	yes	characteristic
644	C₂²	C₂×#14524050	1	1	yes	characteristic
645	C₂²	C₂×#14524050	1	1	yes	characteristic
646	C₂²	#16864163	1	1	yes	characteristic
647	C₂²	#16864163	1	1	yes	characteristic
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	no	—
651	C₄	—	4	1	no	—
652	C₄	—	4	1	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	1	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	1	no	—
660	C₄	#124×D₆₇	1	1	yes	characteristic
661	C₄	#16863925	1	1	yes	characteristic
662	C₄	#16863925	1	1	yes	characteristic
663	C₄	#16863945	1	1	yes	characteristic
664	C₂	—	4	2	no	—
665	C₂	—	8	1	no	—
666	C₂	#124×Dic₆₇	1	1	yes	characteristic
667	C₂	#22568936	1	1	yes	characteristic
668	C₂	#22568936	1	1	yes	characteristic
669	C₂	#22569185	1	1	yes	characteristic
670	C₂	#22569196	1	1	yes	characteristic
671	C₂	#22569205	1	1	yes	characteristic
672	C₂	#22569214	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2398	C₂
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2398	D₆₇
hypocenter	C₆₇	#21637
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2398

Complements

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

N	H ≅ G/N	type
1. C₂×#16863527	C₂	semidirect
3. #22561146	C₂	semidirect
5. #22560824	C₂	semidirect
6. #22560824	C₂	semidirect
7. #22560817	C₂	semidirect
8. #22560817	C₂	semidirect
9. #22560329	C₂	semidirect
10. #22559993	C₂	semidirect
11. #22559993	C₂	semidirect
12. #22559987	C₂	semidirect
13. C₂²⋊C₄×C₆₇⋊C₈	C₂	semidirect
14. #22559896	C₂	semidirect
352. C₆₇	#21637	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
C₂×#16863468	2
#22561146	2
#22561004	2
#22560824	2
#22560817	2
#22560329	2
#22559993	2
#22559987	2
C₂²⋊C₄×C₆₇⋊C₈	2
#22559896	2
C₆₇×#2398	2
#21637	67