Subgroups — #30873670

Order 17152.

744 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560933	C₂	1	1	yes	maximal, characteristic, complemented
2	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560775	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560193	C₂	1	1	yes	maximal, characteristic
8	#22560193	C₂	1	1	yes	maximal, characteristic
9	#22560189	C₂	1	1	yes	maximal, characteristic
10	#22560084	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#22559975	C₂	1	1	yes	maximal, characteristic
13	#22559975	C₂	1	1	yes	maximal, characteristic
14	C₄×C₂²⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊(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×#14523954	C₂²	1	1	yes	characteristic
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	#16863151	C₂²	1	1	yes	characteristic
33	#16863151	C₂²	1	1	yes	characteristic
34	#16863151	C₂²	1	1	yes	characteristic
35	#16863151	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	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
46	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
47	#389×C₆₇	C₄	1	1	yes	characteristic
48	#389×C₆₇	C₄	1	1	yes	characteristic
49	#389×C₆₇	C₄	1	1	yes	characteristic
50	#389×C₆₇	C₄	1	1	yes	characteristic
51	#385×C₆₇	C₄	1	1	yes	characteristic
52	#385×C₆₇	C₄	1	1	yes	characteristic
53	#385×C₆₇	C₄	1	1	yes	characteristic
54	#385×C₆₇	C₄	1	1	yes	characteristic
55	C₄×C₄⋊C₄×C₆₇	C₂²	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×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
59	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
68	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
69	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
70	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
71	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
72	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
73	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
74	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
75	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
76	C₂²⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
77	#14523954	—	2	2	no	—
78	#14523954	—	4	1	no	—
79	C₄×C₆₇⋊C₈	—	2	2	no	—
80	C₄×C₆₇⋊C₈	—	4	1	no	—
81	C₄×C₆₇⋊C₈	—	4	1	no	—
82	C₄×C₆₇⋊C₈	—	4	1	no	—
83	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	#117×C₆₇	—	2	2	no	—
87	#117×C₆₇	—	2	2	no	—
88	#117×C₆₇	—	4	1	no	—
89	#117×C₆₇	—	4	1	no	—
90	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	—	4	1	no	—
107	C₂×C₄²×C₆₇	—	4	1	no	—
108	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
117	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
118	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
119	#95×C₆₇	C₂×C₄	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	C₂×C₆₇⋊C₈	—	2	1	no	—
125	C₂×C₆₇⋊C₈	—	2	1	no	—
126	C₂×C₆₇⋊C₈	—	2	1	no	—
127	C₂×C₆₇⋊C₈	—	2	1	no	—
128	C₂×C₆₇⋊C₈	—	2	1	no	—
129	C₂×C₆₇⋊C₈	—	2	1	no	—
130	C₂×C₆₇⋊C₈	—	2	1	no	—
131	C₂×C₆₇⋊C₈	—	2	1	no	—
132	C₂×C₆₇⋊C₈	—	2	1	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	C₂×Q₈	1	2	yes	—
147	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
148	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	4	1	no	—
157	C₂²×C₄×C₆₇	—	4	1	no	—
158	C₂²×C₄×C₆₇	—	4	1	no	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
173	C₂²×C₄×C₆₇	#41	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	4	1	no	—
184	C₄⋊C₄×C₆₇	—	4	1	no	—
185	C₄⋊C₄×C₆₇	—	4	1	no	—
186	C₄⋊C₄×C₆₇	—	4	1	no	—
187	C₂²⋊C₄×C₆₇	—	2	2	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	4	1	no	—
190	C₂²⋊C₄×C₆₇	—	4	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	—	4	1	no	—
202	C₄²×C₆₇	—	4	1	no	—
203	C₄²×C₆₇	#41	1	2	yes	—
204	C₄²×C₆₇	#41	1	2	yes	—
205	C₂³×C₆₇	—	2	1	no	—
206	C₂³×C₆₇	—	2	1	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	1	no	—
210	C₂³×C₆₇	—	2	1	no	—
211	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
212	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
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	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	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	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	#117	1	2	yes	—
253	C₂×C₄×C₆₇	#117	1	2	yes	—
254	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
255	C₂×C₄×C₆₇	#131	1	2	yes	—
256	C₂×C₄×C₆₇	#131	1	2	yes	—
257	C₂×C₄×C₆₇	#131	1	2	yes	—
258	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
263	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
264	C₆₇⋊C₈	—	2	2	no	—
265	C₆₇⋊C₈	—	2	2	no	—
266	C₆₇⋊C₈	—	4	1	no	—
267	C₆₇⋊C₈	—	4	1	no	—
268	C₆₇⋊C₈	—	4	1	no	—
269	C₆₇⋊C₈	—	4	1	no	—
270	C₆₇⋊C₈	—	4	1	no	—
271	C₆₇⋊C₈	—	4	1	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	4	1	no	—
276	C₂²×C₆₇	—	4	1	no	—
277	C₂²×C₆₇	—	4	1	no	—
278	C₂²×C₆₇	—	4	1	no	—
279	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
281	C₂²×C₆₇	#518	1	1	yes	characteristic
282	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
283	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
284	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
285	C₂²×C₆₇	#568	1	1	yes	characteristic
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	2	no	—
292	C₄×C₆₇	—	4	1	no	—
293	C₄×C₆₇	—	4	1	no	—
294	C₄×C₆₇	—	4	1	no	—
295	C₄×C₆₇	—	4	1	no	—
296	C₄×C₆₇	—	4	1	no	—
297	C₄×C₆₇	—	4	1	no	—
298	C₄×C₆₇	—	4	1	no	—
299	C₄×C₆₇	#553	1	1	yes	characteristic
300	C₄×C₆₇	#553	1	1	yes	characteristic
301	C₄×C₆₇	#554	1	1	yes	characteristic
302	C₄×C₆₇	#554	1	1	yes	characteristic
303	#22630	—	67	1	no	maximal, Sylow
304	C₂×C₆₇	—	4	2	no	—
305	C₂×C₆₇	#2345	1	1	yes	characteristic
306	C₂×C₆₇	#2922	1	1	yes	characteristic
307	C₂×C₆₇	#2925	1	1	yes	characteristic
308	C₂×C₆₇	#2960	1	1	yes	characteristic
309	C₂×C₆₇	#2962	1	1	yes	characteristic
310	C₂×C₆₇	#2976	1	1	yes	characteristic
311	C₂×C₆₇	#2977	1	1	yes	characteristic
312	#2296	D₆₇	1	1	yes	characteristic
313	#1940	—	67	1	no	—
314	#1939	—	67	1	no	—
315	#1917	—	67	1	no	—
316	#1916	—	67	1	no	—
317	#1916	—	67	1	no	—
318	#1916	—	67	1	no	—
319	#1798	—	67	1	no	—
320	#1798	—	67	1	no	—
321	#1797	—	67	1	no	—
322	#1771	—	67	1	no	—
323	C₈×C₄⋊C₄	—	67	1	no	—
324	#1750	—	67	1	no	—
325	#1750	—	67	1	no	—
326	C₄×C₂²⋊C₈	—	67	1	no	—
327	C₆₇	#22630	1	1	yes	characteristic, complemented, Sylow
328	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
329	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
330	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
331	C₂×C₄⋊C₈	—	67	1	no	—
332	C₂×C₄⋊C₈	—	67	1	no	—
333	C₂×C₄⋊C₈	—	67	1	no	—
334	C₂×C₄⋊C₈	—	67	1	no	—
335	C₂×C₄⋊C₈	—	67	1	no	—
336	C₂×C₄⋊C₈	—	67	1	no	—
337	C₂×C₂²⋊C₈	—	67	1	no	—
338	C₂×C₂²⋊C₈	—	67	1	no	—
339	C₂×C₂²⋊C₈	—	67	1	no	—
340	C₂×C₂²⋊C₈	—	67	1	no	—
341	C₂×#97	—	67	1	no	—
342	C₂×#97	—	67	1	no	—
343	C₂×C₄×C₈	—	67	1	no	—
344	C₂×C₄×C₈	—	67	1	no	—
345	C₂×C₄×C₈	—	67	1	no	—
346	C₂×C₄×C₈	—	67	1	no	—
347	#389	Dic₆₇	1	1	yes	characteristic
348	#389	Dic₆₇	1	1	yes	characteristic
349	#389	Dic₆₇	1	1	yes	characteristic
350	#389	Dic₆₇	1	1	yes	characteristic
351	#385	Dic₆₇	1	1	yes	characteristic
352	#385	Dic₆₇	1	1	yes	characteristic
353	#385	Dic₆₇	1	1	yes	characteristic
354	#385	Dic₆₇	1	1	yes	characteristic
355	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
356	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
357	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
358	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
359	#336	—	67	1	no	—
360	#336	—	67	1	no	—
361	#336	—	67	1	no	—
362	#336	—	67	1	no	—
363	#336	—	67	1	no	—
364	#336	—	67	1	no	—
365	#336	—	67	1	no	—
366	#336	—	67	1	no	—
367	#336	—	67	1	no	—
368	#336	—	67	1	no	—
369	#336	—	67	1	no	—
370	#336	—	67	1	no	—
371	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
372	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
373	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
374	C₂²×C₈	—	67	1	no	—
375	C₂²×C₈	—	67	1	no	—
376	C₂²×C₈	—	67	1	no	—
377	C₂²×C₈	—	67	1	no	—
378	C₂²×C₈	—	67	1	no	—
379	C₂²×C₈	—	67	1	no	—
380	C₂²×C₈	—	67	1	no	—
381	C₂²×C₈	—	67	1	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	#117	—	4	1	no	—
385	#117	—	4	1	no	—
386	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
402	C₂×C₄²	—	4	1	no	—
403	C₂×C₄²	—	4	1	no	—
404	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
405	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
406	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
407	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
408	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
409	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
410	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
411	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
412	C₄⋊C₈	—	268	1	no	—
413	C₄⋊C₈	—	268	1	no	—
414	C₄⋊C₈	—	268	1	no	—
415	C₄⋊C₈	—	268	1	no	—
416	C₄⋊C₈	—	134	2	no	—
417	C₄⋊C₈	—	134	2	no	—
418	C₂²⋊C₈	—	67	4	no	—
419	C₂²⋊C₈	—	134	2	no	—
420	C₂²⋊C₈	—	134	2	no	—
421	C₂²⋊C₈	—	134	2	no	—
422	#97	—	268	1	no	—
423	#97	—	134	2	no	—
424	C₄×C₈	—	268	1	no	—
425	C₄×C₈	—	268	1	no	—
426	C₄×C₈	—	268	1	no	—
427	C₄×C₈	—	134	2	no	—
428	#95	C₂×Dic₆₇	1	1	yes	characteristic
429	#95	C₂×Dic₆₇	1	1	yes	characteristic
430	#95	C₂×Dic₆₇	1	1	yes	characteristic
431	#95	C₂×Dic₆₇	1	1	yes	characteristic
432	#95	C₂×Dic₆₇	1	1	yes	characteristic
433	#95	C₂×Dic₆₇	1	1	yes	characteristic
434	#95	C₂×Dic₆₇	1	1	yes	characteristic
435	#95	C₂×Dic₆₇	1	1	yes	characteristic
436	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
449	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
460	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
461	C₂²×C₄	#11761629	1	2	yes	—
462	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
463	C₂×C₈	—	67	2	no	—
464	C₂×C₈	—	67	2	no	—
465	C₂×C₈	—	134	1	no	—
466	C₂×C₈	—	134	1	no	—
467	C₂×C₈	—	134	1	no	—
468	C₂×C₈	—	134	1	no	—
469	C₂×C₈	—	134	1	no	—
470	C₂×C₈	—	134	1	no	—
471	C₂×C₈	—	134	1	no	—
472	C₂×C₈	—	134	1	no	—
473	C₂×C₈	—	134	1	no	—
474	C₂×C₈	—	134	1	no	—
475	C₂×C₈	—	134	1	no	—
476	C₂×C₈	—	134	1	no	—
477	C₂×C₈	—	134	1	no	—
478	C₂×C₈	—	134	1	no	—
479	C₂×C₈	—	134	1	no	—
480	C₂×C₈	—	134	1	no	—
481	C₂×C₈	—	134	1	no	—
482	C₂×C₈	—	134	1	no	—
483	C₂×C₈	—	134	1	no	—
484	C₂×C₈	—	134	1	no	—
485	C₂×C₈	—	134	1	no	—
486	C₂×C₈	—	134	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	4	1	no	—
502	C₂²⋊C₄	—	4	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	4	1	no	—
514	C₄²	—	4	1	no	—
515	C₄²	#11761628	1	2	yes	—
516	C₄²	#11761628	1	2	yes	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	Q₈×Dic₆₇	1	2	yes	—
524	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	4	4	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	#14524002	1	2	yes	—
565	C₂×C₄	#14524002	1	2	yes	—
566	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
567	C₂×C₄	#14524050	1	2	yes	—
568	C₂×C₄	#14524050	1	2	yes	—
569	C₂×C₄	#14524050	1	2	yes	—
570	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
571	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
572	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
573	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
574	C₂×C₄	#14524088	1	1	yes	characteristic
575	C₂×C₄	#14524088	1	1	yes	characteristic
576	C₈	—	268	1	no	—
577	C₈	—	268	1	no	—
578	C₈	—	268	1	no	—
579	C₈	—	268	1	no	—
580	C₈	—	268	1	no	—
581	C₈	—	268	1	no	—
582	C₈	—	134	2	no	—
583	C₈	—	134	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	C₂×#14524002	1	1	yes	characteristic
592	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
593	C₂²	#16863871	1	1	yes	characteristic
594	C₂²	C₂×#14524050	1	1	yes	characteristic
595	C₂²	C₂×#14524050	1	1	yes	characteristic
596	C₂²	C₂×#14524050	1	1	yes	characteristic
597	C₂²	#16864163	1	1	yes	characteristic
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	1	no	—
611	C₄	#16864094	1	1	yes	characteristic
612	C₄	#16864094	1	1	yes	characteristic
613	C₄	#16864101	1	1	yes	characteristic
614	C₄	#16864101	1	1	yes	characteristic
615	C₂	—	4	2	no	—
616	C₂	#22563086	1	1	yes	characteristic
617	C₂	#22568853	1	1	yes	characteristic
618	C₂	#22568875	1	1	yes	characteristic
619	C₂	#22569144	1	1	yes	characteristic
620	C₂	#22569158	1	1	yes	characteristic
621	C₂	#22569257	1	1	yes	characteristic
622	C₂	#22569265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2296	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2296	D₆₇
hypocenter	C₆₇	#22630
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
1. #22560933	C₂	semidirect
2. #22560928	C₂	semidirect
3. #22560775	C₂	semidirect
4. #22560768	C₂	semidirect
5. #22560768	C₂	semidirect
6. #22560768	C₂	semidirect
10. #22560084	C₂	semidirect
11. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
327. C₆₇	#22630	semidirect

Maximal subgroups

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

subgroup	index
#22560933	2
#22560928	2
#22560775	2
#22560768	2
#22560193	2
#22560189	2
#22560084	2
C₄⋊C₄×C₆₇⋊C₈	2
#22559975	2
C₄×C₂²⋊(C₆₇⋊C₈)	2
C₆₇×#2296	2
#22630	67