Subgroups — #30765700

Order 17152.

362 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761627	C₂	1	1	yes	maximal, characteristic
2	#22562006	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558189	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558187	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761627	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₆₇	—	2	1	no	—
10	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#16863520	C₄	1	1	yes	characteristic
12	D₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₆₇	—	2	1	no	—
14	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
15	#16863448	C₄	1	1	yes	characteristic
16	#16863446	—	2	1	no	—
17	C₂×C₈×Dic₆₇	—	2	1	no	—
18	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
20	Dic₆₇⋊C₁₆	—	2	1	no	—
21	Dic₆₇⋊C₁₆	—	2	1	no	—
22	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
23	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₈×D₄×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×#11761627	C₈	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
30	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
32	#14524008	C₈	1	1	yes	characteristic
33	D₄×Dic₆₇	C₈	1	1	yes	characteristic
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	(C₂²⋊D₆₇)⋊C₄	—	2	1	no	—
36	#14524001	C₈	1	1	yes	characteristic
37	#14523999	—	2	1	no	—
38	C₂×C₄×Dic₆₇	—	2	1	no	—
39	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₁₆×D₆₇	—	4	2	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	Dic₆₇⋊C₈	—	2	1	no	—
45	Dic₆₇⋊C₈	—	2	1	no	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
47	(C₂×D₆₇)⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₆₇	—	2	1	no	—
50	C₈×Dic₆₇	—	2	1	no	—
51	C₈×Dic₆₇	—	2	1	no	—
52	C₈×Dic₆₇	D₄	1	1	yes	characteristic
53	C₈×Dic₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₄×D₄×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	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	#11761627	—	2	2	no	—
62	#11761627	C₁₆	1	4	yes	complemented
63	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₇	—	2	1	no	—
65	C₂²×Dic₆₇	—	2	1	no	—
66	C₂²×Dic₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
70	C₂×C₆₇⋊C₈	—	2	1	no	—
71	C₂×C₆₇⋊C₈	—	4	2	no	—
72	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₈×D₆₇	—	2	1	no	—
75	C₈×D₆₇	—	2	1	no	—
76	C₈×D₆₇	—	4	1	no	—
77	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
78	Dic₆₇⋊C₄	—	2	1	no	—
79	Dic₆₇⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₆₇	—	2	1	no	—
81	(C₂×D₆₇)⋊C₄	—	2	1	no	—
82	C₄×Dic₆₇	—	2	1	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₆₇	M₁₆	1	1	yes	characteristic
88	C₆₇⋊C₁₆	—	4	1	no	—
89	C₆₇⋊C₁₆	—	4	1	no	—
90	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	4	1	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₆₇	—	4	1	no	—
101	C₁₆×C₆₇	—	4	1	no	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂²×D₆₇	C₂²⋊C₈	1	1	yes	characteristic
104	D₄×C₆₇	—	2	1	no	—
105	D₄×C₆₇	C₂×C₁₆	1	2	yes	complemented
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂²⋊D₆₇	—	2	4	no	—
113	C₂×Dic₆₇	—	2	1	no	—
114	C₂×Dic₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	2	1	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×D₆₇	—	2	1	no	—
121	C₄×D₆₇	—	2	1	no	—
122	C₄×D₆₇	—	4	1	no	—
123	C₄×D₆₇	C₂×C₁₆	1	2	yes	—
124	C₄×D₆₇	M₃₂	1	2	yes	—
125	Dic₁₃₄	—	2	1	no	—
126	Dic₁₃₄	C₂×C₁₆	1	2	yes	complemented
127	C₈×C₆₇	—	2	1	no	—
128	C₈×C₆₇	—	4	1	no	—
129	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₈×C₆₇	#137	1	1	yes	characteristic
131	C₆₇⋊C₈	—	2	1	no	—
132	C₆₇⋊C₈	—	2	1	no	—
133	C₆₇⋊C₈	—	4	1	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×D₆₇	—	2	1	no	—
139	C₂×D₆₇	—	2	1	no	—
140	C₂×D₆₇	C₂²⋊C₁₆	1	2	yes	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	4	1	no	—
143	C₄×C₆₇	#417	1	1	yes	characteristic
144	C₄×C₆₇	#421	1	1	yes	characteristic
145	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
146	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
147	Dic₆₇	—	2	2	no	—
148	Dic₆₇	—	2	1	no	—
149	Dic₆₇	—	2	1	no	—
150	Dic₆₇	—	4	1	no	—
151	Dic₆₇	C₂²⋊C₁₆	1	2	yes	—
152	#11621	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	4	2	no	—
154	C₂×C₆₇	#1477	1	1	yes	characteristic
155	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
156	C₂×C₆₇	#2117	1	1	yes	characteristic
157	D₆₇	—	2	2	no	—
158	C₈×#41	—	67	1	no	—
159	#2162	—	67	1	no	—
160	C₂×C₄⋊C₁₆	—	67	1	no	—
161	Q₈⋊C₁₆	—	67	1	no	—
162	Q₈⋊C₁₆	—	67	1	no	—
163	D₄⋊C₁₆	—	67	1	no	—
164	D₄⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
165	C₆₇	#11621	1	1	yes	characteristic, complemented, Sylow
166	C₄×#41	—	67	1	no	—
167	C₂²×C₁₆	—	134	1	no	—
168	C₈×Q₈	—	67	1	no	—
169	C₈×Q₈	—	67	1	no	—
170	C₈×D₄	—	67	1	no	—
171	C₈×D₄	—	134	1	no	—
172	C₈×D₄	—	134	1	no	—
173	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
174	#431	—	67	1	no	—
175	#431	—	134	1	no	—
176	C₂×C₄×C₈	—	67	1	no	—
177	C₂×C₄×C₈	—	134	1	no	—
178	C₄⋊C₁₆	—	67	1	no	—
179	C₄⋊C₁₆	—	134	1	no	—
180	C₄⋊C₁₆	—	134	1	no	—
181	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
182	C₂²⋊C₁₆	—	134	1	no	—
183	C₄×C₁₆	—	67	1	no	—
184	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
185	C₂×#41	—	67	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	67	1	no	—
188	C₂²×C₈	—	67	1	no	—
189	C₂²×C₈	—	134	1	no	—
190	C₄×Q₈	—	67	1	no	—
191	C₄×Q₈	—	67	1	no	—
192	C₄×D₄	—	67	1	no	—
193	C₄×D₄	—	134	1	no	—
194	C₄×D₄	—	134	1	no	—
195	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
196	#117	—	67	1	no	—
197	#117	—	134	1	no	—
198	C₂×C₄²	—	67	1	no	—
199	C₂×C₄²	—	134	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	268	2	no	—
203	C₂×C₁₆	—	134	1	no	—
204	C₂×C₁₆	—	134	1	no	—
205	C₄⋊C₈	—	67	1	no	—
206	C₄⋊C₈	—	134	1	no	—
207	C₄⋊C₈	—	134	1	no	—
208	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	134	1	no	—
211	C₂²⋊C₈	—	134	1	no	—
212	C₄×C₈	—	67	1	no	—
213	C₄×C₈	—	67	1	no	—
214	C₄×C₈	—	67	1	no	—
215	C₄×C₈	—	134	1	no	—
216	C₄×C₈	—	134	1	no	—
217	C₄×C₈	—	134	1	no	—
218	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	#41	—	67	4	no	—
220	#41	—	134	2	no	—
221	C₂×Q₈	—	67	1	no	—
222	C₂×D₄	—	134	1	no	—
223	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	67	1	no	—
226	C₂²×C₄	—	67	1	no	—
227	C₂²×C₄	—	67	1	no	—
228	C₂²×C₄	—	134	1	no	—
229	C₂²×C₄	—	134	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	268	1	no	—
234	C₂×C₈	—	268	2	no	—
235	C₂×C₈	—	67	1	no	—
236	C₂×C₈	—	67	1	no	—
237	C₂×C₈	—	134	1	no	—
238	C₂×C₈	—	134	1	no	—
239	C₂×C₈	—	134	1	no	—
240	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
241	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	67	1	no	—
243	C₄⋊C₄	—	134	1	no	—
244	C₄⋊C₄	—	134	1	no	—
245	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	134	1	no	—
248	C₂²⋊C₄	—	134	1	no	—
249	C₄²	—	67	1	no	—
250	C₄²	—	67	1	no	—
251	C₄²	—	67	1	no	—
252	C₄²	—	134	1	no	—
253	C₄²	—	134	1	no	—
254	C₄²	—	134	1	no	—
255	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₁₆	—	4	1	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	—	268	1	no	—
259	C₁₆	—	268	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	67	1	no	—
262	Q₈	—	67	2	no	—
263	Q₈	—	134	1	no	—
264	D₄	—	2	1	no	—
265	D₄	—	134	4	no	—
266	D₄	C₁₆×D₆₇	1	2	yes	complemented
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	268	1	no	—
271	C₂×C₄	—	268	2	no	—
272	C₂×C₄	—	67	1	no	—
273	C₂×C₄	—	67	2	no	—
274	C₂×C₄	—	67	2	no	—
275	C₂×C₄	—	67	1	no	—
276	C₂×C₄	—	67	1	no	—
277	C₂×C₄	—	134	1	no	—
278	C₂×C₄	—	134	4	no	—
279	C₂×C₄	—	134	1	no	—
280	C₂×C₄	—	134	1	no	—
281	C₂×C₄	—	134	1	no	—
282	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	268	1	no	—
288	C₈	—	134	1	no	—
289	C₈	—	134	1	no	—
290	C₈	#14524067	1	1	yes	characteristic
291	C₈	#14524072	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	67	2	no	—
296	C₂²	—	134	1	no	—
297	C₂²	—	134	1	no	—
298	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	268	1	no	—
302	C₄	—	67	2	no	—
303	C₄	—	134	2	no	—
304	C₄	—	134	1	no	—
305	C₄	—	134	1	no	—
306	C₄	#16863394	1	1	yes	characteristic
307	C₄	#16863411	1	1	yes	characteristic
308	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
309	C₄	M₃₂×D₆₇	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	134	2	no	—
312	C₂	#22558593	1	1	yes	characteristic
313	C₂	C₂²⋊C₁₆×D₆₇	1	1	yes	characteristic
314	C₂	#22561835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×D₄⋊C₁₆	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂×#11761627	C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	D₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#11621
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22562006	C₂	semidirect
3. C₄⋊C₁₆×D₆₇	C₂	semidirect
4. #22558189	C₂	semidirect
5. #22558187	C₂	semidirect
6. D₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
7. C₆₇×D₄⋊C₁₆	C₂	semidirect
62. #11761627	C₁₆	semidirect
105. D₄×C₆₇	C₂×C₁₆	semidirect
126. Dic₁₃₄	C₂×C₁₆	semidirect
164. D₄⋊C₁₆	D₆₇	semidirect
165. C₆₇	#11621	semidirect
266. D₄	C₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761627	2
#22562006	2
C₄⋊C₁₆×D₆₇	2
#22558189	2
#22558187	2
D₄⋊(C₆₇⋊C₁₆)	2
C₆₇×D₄⋊C₁₆	2
#11621	67