Subgroups — #30746583

Order 17152.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523969	C₂	1	1	yes	maximal, characteristic
3	C₄×#14523953	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558589	C₂	1	1	yes	maximal, characteristic
5	#22558047	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558041	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761628	C₂²	1	1	yes	characteristic
9	#16863526	C₄	1	2	yes	complemented
10	#16863475	C₄	1	2	yes	complemented
11	#16863467	—	2	2	no	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863447	—	2	1	no	—
14	C₂×#14523969	C₄	1	2	yes	—
15	C₄×#11761611	—	2	1	no	—
16	C₂×#14523953	—	2	2	no	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863157	C₄	1	2	yes	—
20	#16863126	C₄	1	2	yes	complemented
21	#16863123	C₄	1	2	yes	complemented
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#322×C₆₇	—	2	1	no	—
26	C₈²×C₆₇	—	2	1	no	—
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	C₂×#11761611	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	C₄×C₂²⋊D₆₇	—	2	1	no	—
36	#14524002	C₂×C₄	1	1	yes	characteristic
37	#14524000	—	2	1	no	—
38	C₂×C₄×Dic₆₇	—	2	1	no	—
39	C₂×C₄×Dic₆₇	—	2	1	no	—
40	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
41	C₄²×D₆₇	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₆₇⋊C₈	—	2	2	no	—
44	#14523969	—	2	4	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
46	(C₂×D₆₇)⋊C₈	—	2	2	no	—
47	#14523953	—	2	8	no	—
48	#14523953	—	2	2	no	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₆₇	—	2	1	no	—
57	#97×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
63	#11761628	—	2	2	no	—
64	#11761628	—	2	2	no	—
65	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
66	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₆₇	—	2	1	no	—
68	C₂²×Dic₆₇	—	2	2	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	C₂×C₄×D₆₇	—	2	1	no	—
72	#11761611	—	4	4	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
77	C₄⋊Dic₆₇	C₄²	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	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	C₄×Dic₆₇	—	2	4	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₄×Dic₆₇	—	2	2	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₄×Dic₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₆₇	#131	1	2	yes	—
116	C₂²⋊D₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	2	4	no	—
118	C₂×Dic₆₇	—	2	1	no	—
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	C₂×Dic₆₇	—	4	2	no	—
123	D₂₆₈	—	2	1	no	—
124	D₂₆₈	—	2	1	no	—
125	C₄×D₆₇	—	4	2	no	—
126	C₄×D₆₇	—	4	2	no	—
127	C₄×D₆₇	—	4	2	no	—
128	Dic₁₃₄	—	2	1	no	—
129	Dic₁₃₄	—	2	1	no	—
130	C₈×C₆₇	—	2	2	no	—
131	C₈×C₆₇	—	2	4	no	—
132	C₈×C₆₇	—	4	2	no	—
133	C₈×C₆₇	#104	1	4	yes	complemented
134	C₆₇⋊C₈	—	2	4	no	—
135	C₆₇⋊C₈	—	4	2	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₆₇	#436	1	2	yes	—
140	C₂×D₆₇	—	4	1	no	—
141	C₂×D₆₇	—	4	1	no	—
142	C₂×D₆₇	—	4	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
146	C₄×C₆₇	#405	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#104	1	2	yes	—
148	C₄×C₆₇	#435	1	2	yes	—
149	C₄×C₆₇	#444	1	2	yes	—
150	Dic₆₇	—	4	4	no	—
151	Dic₆₇	—	4	1	no	—
152	Dic₆₇	—	4	1	no	—
153	Dic₆₇	—	4	1	no	—
154	#8196	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	#1753	1	1	yes	characteristic
157	C₂×C₆₇	C₄×#104	1	1	yes	characteristic
158	C₂×C₆₇	#1765	1	1	yes	characteristic
159	D₆₇	—	8	1	no	—
160	#2965	—	67	1	no	—
161	C₄×#104	—	67	1	no	—
162	C₄×#97	—	67	1	no	—
163	#1476	—	67	1	no	—
164	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
165	#1278	—	67	1	no	—
166	#1276	—	67	1	no	—
167	C₆₇	#8196	1	1	yes	characteristic, complemented, Sylow
168	C₄×#41	—	67	1	no	—
169	#446	—	67	2	no	—
170	#436	—	67	2	no	—
171	#435	—	134	2	no	—
172	#431	—	67	1	no	—
173	#431	—	134	1	no	—
174	#431	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#104	—	67	2	no	—
176	C₄×M₁₆	—	134	1	no	—
177	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
178	C₂×#97	—	134	2	no	—
179	C₂×C₄×C₈	—	67	1	no	—
180	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
181	C₄³	—	67	1	no	—
182	#339	—	67	2	no	—
183	Q₈⋊C₈	—	67	2	no	—
184	D₄⋊C₈	—	67	2	no	—
185	#322	—	2	1	no	—
186	C₈²	—	2	1	no	—
187	C₂×#41	—	67	1	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
190	C₂²×C₈	—	134	1	no	—
191	C₂²×C₈	D₂₆₈	1	2	yes	—
192	C₄×Q₈	—	67	1	no	—
193	C₄×Q₈	—	67	1	no	—
194	C₄×D₄	—	67	1	no	—
195	C₄×D₄	—	67	1	no	—
196	C₄×D₄	—	134	1	no	—
197	C₄×D₄	—	134	1	no	—
198	#117	—	67	1	no	—
199	#117	—	134	1	no	—
200	C₂×C₄²	—	67	2	no	—
201	C₂×C₄²	—	134	1	no	—
202	C₂×C₄²	—	134	1	no	—
203	C₂×C₄²	—	134	1	no	—
204	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	134	2	no	—
207	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
208	#104	—	134	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	134	1	no	—
211	C₂²⋊C₈	—	134	2	no	—
212	#97	—	2	1	no	—
213	#97	—	134	8	no	—
214	#97	—	134	2	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	67	2	no	—
218	C₄×C₈	—	134	1	no	—
219	C₄×C₈	—	134	1	no	—
220	C₄×C₈	C₄×D₆₇	1	2	yes	—
221	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
223	#41	—	134	2	no	—
224	#41	—	134	2	no	—
225	C₂×Q₈	—	67	1	no	—
226	C₂×D₄	—	67	1	no	—
227	C₂×D₄	—	134	1	no	—
228	C₂²×C₄	—	134	2	no	—
229	C₂²×C₄	—	134	1	no	—
230	C₂²×C₄	—	134	1	no	—
231	C₂²×C₄	—	134	1	no	—
232	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
234	C₂²×C₄	#11761628	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	268	4	no	—
237	M₁₆	#11761611	1	4	yes	complemented
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	134	2	no	—
242	C₂×C₈	—	134	4	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
245	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
246	C₄⋊C₄	—	67	1	no	—
247	C₄⋊C₄	—	67	1	no	—
248	C₄⋊C₄	—	134	1	no	—
249	C₄⋊C₄	—	134	1	no	—
250	C₂²⋊C₄	—	134	1	no	—
251	C₂²⋊C₄	—	134	1	no	—
252	C₂²⋊C₄	—	134	1	no	—
253	C₄²	—	268	2	no	—
254	C₄²	—	134	4	no	—
255	C₄²	—	134	2	no	—
256	C₄²	—	134	2	no	—
257	C₄²	—	134	1	no	—
258	C₄²	—	134	1	no	—
259	C₄²	—	134	1	no	—
260	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₄²	#11761628	1	1	yes	characteristic
264	C₂³	—	134	1	no	—
265	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
266	Q₈	—	134	1	no	—
267	Q₈	—	134	1	no	—
268	D₄	—	268	2	no	—
269	D₄	—	134	1	no	—
270	D₄	—	134	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	268	2	no	—
274	C₂×C₄	—	268	2	no	—
275	C₂×C₄	—	268	2	no	—
276	C₂×C₄	—	268	2	no	—
277	C₂×C₄	—	268	2	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	C₄²×D₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#14524000	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761611	1	2	yes	—
289	C₂×C₄	#14524049	1	2	yes	—
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	268	2	no	—
294	C₈	—	134	4	no	—
295	C₈	#14523969	1	4	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	268	1	no	—
299	C₂²	—	268	1	no	—
300	C₂²	—	268	1	no	—
301	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
302	C₂²	#16863475	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	268	4	no	—
306	C₄	—	268	1	no	—
307	C₄	—	268	1	no	—
308	C₄	—	268	1	no	—
309	C₄	C₄×#11761611	1	1	yes	characteristic
310	C₄	#16863350	1	1	yes	characteristic
311	C₄	C₂×#14523969	1	2	yes	—
312	C₄	#16863467	1	2	yes	—
313	C₄	#16863516	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	536	1	no	—
316	C₂	#22559992	1	1	yes	characteristic
317	C₂	C₄×#14523969	1	1	yes	characteristic
318	C₂	#22560055	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#8196
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#14523953	C₂	semidirect
5. #22558047	C₂	semidirect
6. #22558041	C₂	semidirect
7. C₈×M₁₆×C₆₇	C₂	semidirect
9. #16863526	C₄	semidirect
10. #16863475	C₄	semidirect
20. #16863126	C₄	semidirect
21. #16863123	C₄	semidirect
94. M₁₆×C₆₇	M₁₆	semidirect
133. C₈×C₆₇	#104	semidirect
164. C₈×M₁₆	D₆₇	semidirect
167. C₆₇	#8196	semidirect
237. M₁₆	#11761611	semidirect
295. C₈	#14523969	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
C₄×#14523969	2
C₄×#14523953	2
#22558589	2
#22558047	2
#22558041	2
C₈×M₁₆×C₆₇	2
#8196	67