Subgroups — #30765640

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569138	C₂	1	1	yes	maximal, characteristic
2	#22561963	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558189	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863950	C₄	1	1	yes	characteristic
9	C₂×#14523982	—	2	1	no	—
10	C₂×C₁₆×D₆₇	—	2	1	no	—
11	#16863527	C₄	1	1	yes	characteristic
12	#16863525	—	2	1	no	—
13	#16863523	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₄	—	2	1	no	—
15	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
16	#16863421	—	2	1	no	—
17	#16863418	C₄	1	1	yes	characteristic
18	#16863350	—	2	1	no	—
19	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
21	Dic₆₇⋊C₁₆	—	2	1	no	—
22	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₆₇	—	2	1	no	—
24	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761611	—	2	1	no	—
28	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
29	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
30	#14524042	C₈	1	1	yes	characteristic
31	#14524041	C₈	1	1	yes	characteristic
32	#14524033	—	2	1	no	—
33	#14524023	C₈	1	1	yes	characteristic
34	#14524021	—	2	1	no	—
35	#14524019	C₈	1	1	yes	characteristic
36	C₄×Dic₁₃₄	—	2	1	no	—
37	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
38	#14524000	—	2	1	no	—
39	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
40	#14523982	—	4	2	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₂×C₆₇⋊C₁₆	—	2	1	no	—
43	C₁₆×D₆₇	—	2	4	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	Dic₆₇⋊C₈	—	2	1	no	—
47	Dic₆₇⋊C₈	—	2	1	no	—
48	Dic₆₇⋊C₈	—	2	1	no	—
49	#14523954	—	2	1	no	—
50	#14523953	—	2	1	no	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₈×Dic₆₇	—	2	1	no	—
53	C₈×Dic₆₇	D₄	1	1	yes	characteristic
54	C₈×Dic₆₇	D₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	—	2	1	no	—
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₃₄	—	2	1	no	—
63	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
67	#11761611	—	4	2	no	—
68	C₂×C₆₇⋊C₈	—	2	1	no	—
69	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₈×D₆₇	—	2	2	no	—
72	C₈×D₆₇	SD₁₆	1	2	yes	—
73	C₈×D₆₇	Q₁₆	1	2	yes	—
74	C₄⋊Dic₆₇	—	2	1	no	—
75	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
76	Dic₆₇⋊C₄	—	2	1	no	—
77	Dic₆₇⋊C₄	—	2	1	no	—
78	Dic₆₇⋊C₄	—	2	1	no	—
79	Dic₆₇⋊C₄	—	2	1	no	—
80	Dic₆₇⋊C₄	—	2	1	no	—
81	(C₂×D₆₇)⋊C₄	—	2	1	no	—
82	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₆₇	M₁₆	1	1	yes	characteristic
87	C₆₇⋊C₁₆	—	2	2	no	—
88	C₆₇⋊C₁₆	—	4	1	no	—
89	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₆₇	—	2	2	no	—
98	C₁₆×C₆₇	—	4	1	no	—
99	C₂²×D₆₇	C₂²⋊C₈	1	1	yes	characteristic
100	Q₈×C₆₇	—	2	1	no	—
101	Q₈×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	2	1	no	—
109	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
112	C₄×D₆₇	—	2	1	no	—
113	C₄×D₆₇	—	2	1	no	—
114	C₄×D₆₇	—	2	2	no	—
115	C₄×D₆₇	#103	1	2	yes	—
116	C₄×D₆₇	#104	1	2	yes	—
117	Dic₁₃₄	—	2	1	no	—
118	Dic₁₃₄	—	2	1	no	—
119	Dic₁₃₄	—	4	2	no	—
120	C₈×C₆₇	—	2	1	no	—
121	C₈×C₆₇	—	4	1	no	—
122	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₆₇⋊C₈	—	2	1	no	—
125	C₆₇⋊C₈	—	4	1	no	—
126	C₆₇⋊C₈	#103	1	2	yes	—
127	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂×D₆₇	—	2	1	no	—
129	C₂×D₆₇	—	2	1	no	—
130	C₂×D₆₇	Q₈⋊C₈	1	2	yes	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	4	1	no	—
133	C₄×C₆₇	—	4	1	no	—
134	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
136	C₄×C₆₇	#504	1	1	yes	characteristic
137	C₄×C₆₇	#504	1	1	yes	characteristic
138	Dic₆₇	—	2	1	no	—
139	Dic₆₇	—	4	1	no	—
140	Dic₆₇	—	4	1	no	—
141	Dic₆₇	Q₈⋊C₈	1	2	yes	—
142	Dic₆₇	#350	1	2	yes	—
143	#11608	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
145	C₂×C₆₇	#2113	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
147	D₆₇	—	2	2	no	—
148	#2960	—	67	1	no	—
149	#2150	—	67	1	no	—
150	C₂×C₄×C₁₆	—	67	1	no	—
151	Q₈⋊C₁₆	—	67	1	no	—
152	Q₈⋊C₁₆	—	67	1	no	—
153	Q₈⋊C₁₆	—	67	1	no	—
154	Q₈⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
155	C₆₇	#11608	1	1	yes	characteristic, complemented, Sylow
156	#533	—	67	1	no	—
157	C₂×M₃₂	—	134	1	no	—
158	C₂²×C₁₆	—	134	1	no	—
159	#446	—	67	1	no	—
160	#446	—	67	1	no	—
161	#446	—	134	1	no	—
162	C₈×Q₈	—	67	1	no	—
163	C₈×Q₈	—	134	1	no	—
164	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
165	#424	—	134	1	no	—
166	#423	—	67	1	no	—
167	#405	—	134	1	no	—
168	C₂×C₄×C₈	—	67	1	no	—
169	C₄⋊C₁₆	—	67	1	no	—
170	C₄⋊C₁₆	—	134	1	no	—
171	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
172	C₄×C₁₆	—	67	1	no	—
173	C₄×C₁₆	—	134	1	no	—
174	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	134	1	no	—
176	C₂²×C₈	—	67	1	no	—
177	C₂²×C₈	—	67	1	no	—
178	C₄⋊Q₈	—	67	1	no	—
179	C₄⋊Q₈	—	67	1	no	—
180	#125	—	134	1	no	—
181	C₂²⋊Q₈	—	67	1	no	—
182	C₂²⋊Q₈	—	67	1	no	—
183	C₂²⋊Q₈	—	134	1	no	—
184	C₄×Q₈	—	67	1	no	—
185	C₄×Q₈	—	134	1	no	—
186	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
187	#117	—	134	1	no	—
188	C₂×C₄²	—	67	1	no	—
189	M₃₂	—	268	2	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	134	4	no	—
193	C₂×C₁₆	—	134	1	no	—
194	C₂×C₁₆	—	134	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	67	1	no	—
197	C₄⋊C₈	—	134	1	no	—
198	C₄⋊C₈	—	134	1	no	—
199	C₄⋊C₈	—	134	1	no	—
200	C₄⋊C₈	—	134	1	no	—
201	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
202	#97	—	134	1	no	—
203	#97	—	134	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	67	1	no	—
206	C₄×C₈	—	67	1	no	—
207	C₄×C₈	—	67	1	no	—
208	C₄×C₈	—	134	1	no	—
209	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×Q₈	—	67	1	no	—
211	C₂×Q₈	—	134	1	no	—
212	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	—	67	1	no	—
214	C₂²×C₄	—	67	1	no	—
215	C₂²×C₄	—	67	1	no	—
216	M₁₆	—	268	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	67	1	no	—
219	C₂×C₈	—	67	2	no	—
220	C₂×C₈	—	67	2	no	—
221	C₂×C₈	—	67	1	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	—	134	1	no	—
224	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
225	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	67	1	no	—
228	C₄⋊C₄	—	134	1	no	—
229	C₄⋊C₄	—	134	1	no	—
230	C₄⋊C₄	—	134	1	no	—
231	C₄⋊C₄	—	134	1	no	—
232	C₄⋊C₄	—	134	1	no	—
233	C₄⋊C₄	—	134	1	no	—
234	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
235	C₂²⋊C₄	—	134	1	no	—
236	C₂²⋊C₄	—	134	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	67	1	no	—
239	C₄²	—	67	1	no	—
240	C₄²	—	67	1	no	—
241	C₄²	—	134	1	no	—
242	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
243	C₁₆	—	2	2	no	—
244	C₁₆	—	4	1	no	—
245	C₁₆	—	268	1	no	—
246	C₁₆	—	134	2	no	—
247	C₂³	—	67	1	no	—
248	Q₈	—	2	1	no	—
249	Q₈	—	2	1	no	—
250	Q₈	—	268	2	no	—
251	Q₈	—	134	1	no	—
252	Q₈	—	134	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	67	1	no	—
256	C₂×C₄	—	67	1	no	—
257	C₂×C₄	—	67	2	no	—
258	C₂×C₄	—	67	2	no	—
259	C₂×C₄	—	67	1	no	—
260	C₂×C₄	—	134	2	no	—
261	C₂×C₄	—	134	1	no	—
262	C₂×C₄	—	134	1	no	—
263	C₂×C₄	—	134	1	no	—
264	C₂×C₄	—	134	1	no	—
265	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
267	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	268	1	no	—
271	C₈	—	67	2	no	—
272	C₈	—	134	1	no	—
273	C₈	SD₁₆×D₆₇	1	1	yes	characteristic
274	C₈	Q₁₆×D₆₇	1	1	yes	characteristic
275	C₂²	—	67	2	no	—
276	C₂²	—	134	1	no	—
277	C₂²	—	134	1	no	—
278	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	268	1	no	—
283	C₄	—	268	1	no	—
284	C₄	—	67	2	no	—
285	C₄	—	67	2	no	—
286	C₄	—	134	1	no	—
287	C₄	#103×D₆₇	1	1	yes	characteristic
288	C₄	#104×D₆₇	1	1	yes	characteristic
289	C₄	#16863812	1	1	yes	characteristic
290	C₄	#16863814	1	1	yes	characteristic
291	C₂	—	134	2	no	—
292	C₂	Q₈⋊C₈×D₆₇	1	1	yes	characteristic
293	C₂	#22561821	1	1	yes	characteristic
294	C₂	#350×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×Q₈⋊C₁₆	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	Q₈⋊C₁₆	D₆₇
hypocenter	C₆₇	#11608
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #22558189	C₂	semidirect
5. Q₈⋊(C₆₇⋊C₁₆)	C₂	semidirect
6. #22558187	C₂	semidirect
7. C₆₇×Q₈⋊C₁₆	C₂	semidirect
154. Q₈⋊C₁₆	D₆₇	semidirect
155. C₆₇	#11608	semidirect

Maximal subgroups

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

subgroup	index
#22569138	2
#22561963	2
C₄×C₁₆×D₆₇	2
#22558189	2
Q₈⋊(C₆₇⋊C₁₆)	2
#22558187	2
C₆₇×Q₈⋊C₁₆	2
#11608	67