Subgroups — #30860664

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863457	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863352	C₂	1	1	yes	maximal, characteristic
3	#22560933	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560933	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560394	C₂	1	1	yes	maximal, characteristic
8	#22560394	C₂	1	1	yes	maximal, characteristic
9	#22560389	C₂	1	1	yes	maximal, characteristic
10	#22560389	C₂	1	1	yes	maximal, characteristic
11	#22560066	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560066	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560062	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2298	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863457	—	2	4	no	—
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×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×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊(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	#16863352	—	2	4	no	—
32	C₂×#14523954	C₂²	1	1	yes	characteristic
33	C₂×#14523954	C₂²	1	1	yes	characteristic
34	C₂×#14523954	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×#117×C₆₇	C₂²	1	1	yes	characteristic
47	#388×C₆₇	C₄	1	1	yes	characteristic
48	#388×C₆₇	C₄	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	#387×C₆₇	C₄	1	1	yes	characteristic
54	#387×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×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
59	#376×C₆₇	C₂²	1	1	yes	characteristic
60	C₄³×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	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
80	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
81	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
82	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
83	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
84	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
85	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
86	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
87	#14523954	—	2	2	no	—
88	#14523954	—	2	2	no	—
89	#14523954	—	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	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
94	#117×C₆₇	D₄	1	4	yes	—
95	#117×C₆₇	Q₈	1	4	yes	—
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×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂×C₄	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	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	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	#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	#11761612	—	2	8	no	—
131	C₂×C₆₇⋊C₈	—	2	8	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	—	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	#41	1	2	yes	—
153	C₂×C₆₇⋊C₈	#41	1	2	yes	—
154	C₂×C₆₇⋊C₈	#41	1	2	yes	—
155	C₂×C₆₇⋊C₈	#41	1	2	yes	—
156	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	4	1	no	—
180	C₄⋊C₄×C₆₇	—	4	1	no	—
181	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
182	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
183	C₂²⋊C₄×C₆₇	—	2	2	no	—
184	C₂²⋊C₄×C₆₇	—	2	2	no	—
185	C₂²⋊C₄×C₆₇	—	4	1	no	—
186	C₂²⋊C₄×C₆₇	—	4	1	no	—
187	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
188	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	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	—	4	1	no	—
200	C₄²×C₆₇	—	4	1	no	—
201	C₄²×C₆₇	C₂×D₄	1	2	yes	—
202	C₄²×C₆₇	C₂×D₄	1	2	yes	—
203	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
204	C₄²×C₆₇	C₂×Q₈	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	C₂×C₄⋊C₄	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	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	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	C₂×C₄⋊C₄	1	2	yes	—
250	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
251	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
253	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
254	C₂×C₄×C₆₇	#131	1	2	yes	—
255	C₂×C₄×C₆₇	#131	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂×#41	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	C₆₇⋊C₈	—	2	2	no	—
263	C₆₇⋊C₈	—	2	2	no	—
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	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	4	1	no	—
274	C₂²×C₆₇	—	4	1	no	—
275	C₂²×C₆₇	—	4	1	no	—
276	C₂²×C₆₇	—	4	1	no	—
277	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
278	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
279	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
281	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
282	C₂²×C₆₇	#568	1	1	yes	characteristic
283	C₂²×C₆₇	#568	1	1	yes	characteristic
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
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	—	4	1	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	#525	1	1	yes	characteristic
298	C₄×C₆₇	#525	1	1	yes	characteristic
299	C₄×C₆₇	#533	1	1	yes	characteristic
300	C₄×C₆₇	#533	1	1	yes	characteristic
301	#21357	—	67	1	no	maximal, Sylow
302	C₂×C₆₇	—	4	2	no	—
303	C₂×C₆₇	#2296	1	1	yes	characteristic
304	C₂×C₆₇	#2906	1	1	yes	characteristic
305	C₂×C₆₇	#2906	1	1	yes	characteristic
306	C₂×C₆₇	#2966	1	1	yes	characteristic
307	C₂×C₆₇	#2967	1	1	yes	characteristic
308	C₂×C₆₇	#2968	1	1	yes	characteristic
309	C₂×C₆₇	#2969	1	1	yes	characteristic
310	C₂×#433	—	67	1	no	—
311	C₂×#405	—	67	1	no	—
312	#2298	D₆₇	1	1	yes	characteristic
313	#1940	—	67	1	no	—
314	#1940	—	67	1	no	—
315	#1939	—	67	1	no	—
316	#1939	—	67	1	no	—
317	#1851	—	67	1	no	—
318	#1851	—	67	1	no	—
319	#1850	—	67	1	no	—
320	#1850	—	67	1	no	—
321	#1768	—	67	1	no	—
322	#1768	—	67	1	no	—
323	#1767	—	67	1	no	—
324	C₄×C₄⋊C₈	—	67	1	no	—
325	C₆₇	#21357	1	1	yes	characteristic, complemented, Sylow
326	C₂²×M₁₆	—	67	1	no	—
327	C₂³×C₈	—	67	1	no	—
328	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
329	#433	—	134	4	no	—
330	C₂×C₄⋊C₈	—	67	1	no	—
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×C₄⋊C₈	—	67	1	no	—
338	C₂×C₄⋊C₈	—	67	1	no	—
339	C₂×C₄⋊C₈	—	67	1	no	—
340	C₂×C₂²⋊C₈	—	67	1	no	—
341	C₂×C₂²⋊C₈	—	67	1	no	—
342	#405	—	134	4	no	—
343	C₂×#97	—	67	1	no	—
344	C₂×#97	—	67	1	no	—
345	C₂×#97	—	67	1	no	—
346	C₂×C₄×C₈	—	67	1	no	—
347	C₂×C₄×C₈	—	67	1	no	—
348	C₂×C₄×C₈	—	67	1	no	—
349	#388	Dic₆₇	1	1	yes	characteristic
350	#388	Dic₆₇	1	1	yes	characteristic
351	#388	Dic₆₇	1	1	yes	characteristic
352	#388	Dic₆₇	1	1	yes	characteristic
353	#387	Dic₆₇	1	1	yes	characteristic
354	#387	Dic₆₇	1	1	yes	characteristic
355	#387	Dic₆₇	1	1	yes	characteristic
356	#387	Dic₆₇	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	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
360	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
361	#376	C₂×D₆₇	1	1	yes	characteristic
362	C₄³	C₂×D₆₇	1	1	yes	characteristic
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	C₂²×D₆₇	1	1	yes	characteristic
372	C₂×M₁₆	—	134	2	no	—
373	C₂×M₁₆	—	134	2	no	—
374	C₂×M₁₆	—	134	2	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	C₂²×C₈	—	67	1	no	—
383	C₂²×C₈	—	134	2	no	—
384	C₂²×C₈	—	134	2	no	—
385	C₂²×C₈	—	134	2	no	—
386	#117	Dic₁₃₄	1	4	yes	—
387	#117	D₂₆₈	1	4	yes	—
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×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
397	C₂×C₂²⋊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×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
401	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
402	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
403	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
404	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
405	C₂×C₄²	C₂²×D₆₇	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	C₄⋊C₈	—	134	2	no	—
411	C₄⋊C₈	—	134	2	no	—
412	C₄⋊C₈	—	134	2	no	—
413	C₄⋊C₈	—	134	2	no	—
414	C₄⋊C₈	—	134	2	no	—
415	C₄⋊C₈	—	134	2	no	—
416	C₄⋊C₈	—	134	2	no	—
417	C₄⋊C₈	—	134	2	no	—
418	C₄⋊C₈	—	134	2	no	—
419	C₄⋊C₈	—	134	2	no	—
420	C₂²⋊C₈	—	134	2	no	—
421	C₂²⋊C₈	—	134	2	no	—
422	#97	—	134	2	no	—
423	#97	—	134	2	no	—
424	#97	—	134	2	no	—
425	C₄×C₈	—	134	2	no	—
426	C₄×C₈	—	134	2	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	#95	C₂×Dic₆₇	1	1	yes	characteristic
437	#95	C₂×Dic₆₇	1	1	yes	characteristic
438	#95	C₂×Dic₆₇	1	1	yes	characteristic
439	#95	C₂×Dic₆₇	1	1	yes	characteristic
440	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
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	C₂×D₂₆₈	1	2	yes	—
459	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
460	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
461	M₁₆	—	134	8	no	—
462	C₂×C₈	—	67	2	no	—
463	C₂×C₈	—	67	2	no	—
464	C₂×C₈	—	67	2	no	—
465	C₂×C₈	—	67	2	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	8	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	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
492	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
498	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	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	—	4	1	no	—
510	C₄²	—	4	1	no	—
511	C₄²	C₂×D₂₆₈	1	2	yes	—
512	C₄²	C₂×D₂₆₈	1	2	yes	—
513	C₄²	C₂×Dic₁₃₄	1	2	yes	—
514	C₄²	C₂×Dic₁₃₄	1	2	yes	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
522	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
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	C₂×C₄⋊Dic₆₇	1	2	yes	—
560	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
561	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
562	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
563	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
564	C₂×C₄	#14524050	1	2	yes	—
565	C₂×C₄	#14524050	1	2	yes	—
566	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
567	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
568	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
569	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
570	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
571	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
572	C₈	—	268	1	no	—
573	C₈	—	268	1	no	—
574	C₈	—	268	1	no	—
575	C₈	—	268	1	no	—
576	C₈	—	134	2	no	—
577	C₈	—	134	2	no	—
578	C₈	—	134	2	no	—
579	C₈	—	134	2	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
588	C₂²	#41×Dic₆₇	1	1	yes	characteristic
589	C₂²	#41×Dic₆₇	1	1	yes	characteristic
590	C₂²	C₂×#14524050	1	1	yes	characteristic
591	C₂²	C₂×#14524050	1	1	yes	characteristic
592	C₂²	#16864163	1	1	yes	characteristic
593	C₂²	#16864163	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	#16863908	1	1	yes	characteristic
608	C₄	#16863908	1	1	yes	characteristic
609	C₄	#16863949	1	1	yes	characteristic
610	C₄	#16863949	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	#22562693	1	1	yes	characteristic
613	C₂	#22568777	1	1	yes	characteristic
614	C₂	#22568777	1	1	yes	characteristic
615	C₂	#22569185	1	1	yes	characteristic
616	C₂	#22569196	1	1	yes	characteristic
617	C₂	#22569205	1	1	yes	characteristic
618	C₂	#22569214	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2298	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2298	D₆₇
hypocenter	C₆₇	#21357
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
3. #22560933	C₂	semidirect
4. #22560933	C₂	semidirect
5. #22560928	C₂	semidirect
6. #22560928	C₂	semidirect
11. #22560066	C₂	semidirect
12. #22560066	C₂	semidirect
13. #22560062	C₂	semidirect
14. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
325. C₆₇	#21357	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863457	2
C₂×#16863352	2
#22560933	2
#22560928	2
#22560394	2
#22560389	2
#22560066	2
#22560062	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₆₇×#2298	2
#21357	67