Subgroups — #22601242

Order 9088.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986705	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986409	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986396	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986301	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986290	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986288	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986287	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986278	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986128	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986109	C₂	1	1	yes	maximal, characteristic
13	#17986108	C₂	1	1	yes	maximal, characteristic
14	#17986079	C₂	1	1	yes	maximal, characteristic
15	#478×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₄₂	—	2	1	no	—
17	C₂×#11765850	C₂²	1	1	yes	characteristic
18	#14556128	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556119	C₂²	1	1	yes	characteristic
23	#14556108	—	2	1	no	—
24	#14556107	—	2	1	no	—
25	#14556106	—	2	1	no	—
26	#14556106	—	2	1	no	—
27	#14556105	—	2	1	no	—
28	#14556105	—	2	1	no	—
29	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
30	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
31	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
32	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
33	#14556090	C₂²	1	1	yes	characteristic
34	#14556090	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₇₁	—	2	1	no	—
36	#14556061	C₂²	1	1	yes	characteristic
37	#14556061	C₂²	1	1	yes	characteristic
38	#14556060	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556050	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	#14556039	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	#125×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×Dic₁₄₂	—	2	1	no	—
63	C₂×Dic₁₄₂	—	4	4	no	—
64	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
65	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂²×Dic₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	—	2	1	no	—
71	#11765850	Q₈	1	4	yes	complemented
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	1	no	—
76	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	Dic₇₁⋊C₄	—	2	1	no	—
84	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
85	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	—	2	1	no	—
95	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	—	2	1	no	—
98	C₄⋊C₄×C₇₁	—	2	1	no	—
99	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	C₂³	1	1	yes	characteristic
104	C₂³×C₇₁	—	2	1	no	—
105	C₂²×D₇₁	—	2	1	no	—
106	D₄×C₇₁	—	2	1	no	—
107	D₄×C₇₁	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	4	2	no	—
112	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	4	2	no	—
119	C₂×Dic₇₁	—	4	2	no	—
120	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	D₂₈₄	—	2	1	no	—
125	D₂₈₄	C₂×Q₈	1	2	yes	complemented
126	C₄×D₇₁	—	4	2	no	—
127	C₄×D₇₁	—	8	1	no	—
128	C₄×D₇₁	—	8	1	no	—
129	Dic₁₄₂	—	2	2	no	—
130	Dic₁₄₂	—	4	2	no	—
131	Dic₁₄₂	—	4	2	no	—
132	Dic₁₄₂	—	4	2	no	—
133	Dic₁₄₂	—	8	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₇₁⋊C₈	C₂×Q₈	1	2	yes	complemented
138	C₂²×C₇₁	—	2	2	no	—
139	C₂²×C₇₁	—	4	1	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₇₁	—	2	2	no	—
143	C₂×D₇₁	—	4	1	no	—
144	C₂×D₇₁	—	4	1	no	—
145	C₄×C₇₁	—	2	1	no	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	4	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	—	4	1	no	—
150	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
151	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₇₁	—	2	1	no	—
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₇₁	—	4	2	no	—
160	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
161	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₇₁	#577	1	1	yes	characteristic
163	D₇₁	—	4	2	no	—
164	#3384	—	71	1	no	maximal, Sylow
165	C₇₁	#3384	1	1	yes	characteristic, complemented, Sylow
166	#554	—	71	1	no	—
167	D₄×Q₈	—	71	1	no	—
168	#501	—	71	1	no	—
169	#501	—	71	1	no	—
170	#498	—	71	1	no	—
171	#478	—	71	1	no	—
172	#478	D₇₁	1	1	yes	characteristic, complemented
173	#476	—	71	1	no	—
174	#476	—	71	1	no	—
175	#476	—	71	1	no	—
176	#474	—	71	1	no	—
177	#446	—	71	1	no	—
178	#442	—	71	1	no	—
179	#442	—	71	1	no	—
180	C₄×D₈	—	71	1	no	—
181	C₂²×Q₈	—	142	1	no	—
182	C₂×D₈	—	71	1	no	—
183	C₄⋊Q₈	—	71	1	no	—
184	C₄⋊Q₈	—	71	1	no	—
185	C₄⋊Q₈	—	71	1	no	—
186	C₄⋊Q₈	—	71	1	no	—
187	#125	—	71	1	no	—
188	#125	C₂×D₇₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	142	1	no	—
190	C₂²⋊Q₈	—	142	1	no	—
191	C₂²⋊Q₈	—	142	1	no	—
192	C₂²⋊Q₈	—	142	1	no	—
193	C₂²⋊Q₈	—	142	1	no	—
194	C₂²⋊Q₈	—	142	1	no	—
195	C₄×Q₈	—	71	1	no	—
196	C₄×D₄	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	C₄×D₄	—	71	1	no	—
199	C₄×D₄	—	71	1	no	—
200	C₄×D₄	—	71	1	no	—
201	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	142	1	no	—
203	#107	—	71	1	no	—
204	#107	—	71	1	no	—
205	#107	C₂×D₇₁	1	1	yes	characteristic
206	#106	—	71	1	no	—
207	#106	—	71	1	no	—
208	#106	—	71	1	no	—
209	#106	—	71	1	no	—
210	#106	—	71	1	no	—
211	#106	C₂×D₇₁	1	1	yes	characteristic
212	C₄⋊C₈	—	71	1	no	—
213	C₄⋊C₈	—	71	1	no	—
214	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
215	#102	—	71	1	no	—
216	#102	—	71	1	no	—
217	#102	—	71	1	no	—
218	#102	—	71	1	no	—
219	#102	C₂×D₇₁	1	1	yes	characteristic
220	#102	C₂×D₇₁	1	1	yes	characteristic
221	#97	—	71	1	no	—
222	#97	—	71	1	no	—
223	C₄×C₈	—	71	1	no	—
224	C₂×Q₈	—	284	4	no	—
225	C₂×Q₈	—	71	1	no	—
226	C₂×Q₈	—	142	1	no	—
227	C₂×Q₈	—	142	1	no	—
228	C₂×Q₈	—	142	1	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×D₄	—	71	1	no	—
231	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	C₂²×C₄	—	142	1	no	—
235	C₂²×C₄	—	142	1	no	—
236	C₂²×C₄	—	142	1	no	—
237	C₂²×C₄	—	142	1	no	—
238	D₈	—	71	4	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	71	1	no	—
241	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	71	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₄⋊C₄	—	71	1	no	—
248	C₄⋊C₄	—	71	1	no	—
249	C₄⋊C₄	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	—	142	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	142	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₂²⋊C₄	—	142	1	no	—
266	C₄²	—	71	1	no	—
267	C₄²	—	71	1	no	—
268	C₄²	C₂²×D₇₁	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	C₂³	—	142	1	no	—
271	Q₈	—	284	2	no	—
272	Q₈	—	284	2	no	—
273	Q₈	—	284	2	no	—
274	Q₈	—	568	1	no	—
275	Q₈	—	142	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	71	2	no	—
278	D₄	—	142	1	no	—
279	D₄	Q₈×D₇₁	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	284	2	no	—
285	C₂×C₄	—	284	2	no	—
286	C₂×C₄	—	284	2	no	—
287	C₂×C₄	—	568	1	no	—
288	C₂×C₄	—	568	1	no	—
289	C₂×C₄	—	71	1	no	—
290	C₂×C₄	—	71	1	no	—
291	C₂×C₄	—	71	1	no	—
292	C₂×C₄	—	71	1	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	—	142	1	no	—
296	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	71	2	no	—
302	C₈	—	142	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	284	1	no	—
307	C₂²	—	284	1	no	—
308	C₂²	—	142	2	no	—
309	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	284	1	no	—
316	C₄	—	284	1	no	—
317	C₄	—	284	1	no	—
318	C₄	—	71	2	no	—
319	C₄	—	142	1	no	—
320	C₄	—	142	1	no	—
321	C₄	—	142	1	no	—
322	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
323	C₄	#14556175	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	284	2	no	—
326	C₂	#17986662	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₇₁	1	1	yes	characteristic
328	C₂	#17986806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#478×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#478	D₇₁
hypocenter	C₇₁	#3384
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #17986705	C₂	semidirect
2. D₄×Dic₁₄₂	C₂	semidirect
3. #17986409	C₂	semidirect
4. #17986407	C₂	semidirect
5. #17986396	C₂	semidirect
6. #17986301	C₂	semidirect
7. #17986290	C₂	semidirect
8. #17986288	C₂	semidirect
9. #17986287	C₂	semidirect
10. #17986278	C₂	semidirect
11. #17986128	C₂	semidirect
15. #478×C₇₁	C₂	semidirect
71. #11765850	Q₈	semidirect
125. D₂₈₄	C₂×Q₈	semidirect
137. C₇₁⋊C₈	C₂×Q₈	semidirect
165. C₇₁	#3384	semidirect
172. #478	D₇₁	semidirect