Subgroups — #22600992

Order 9088.

350 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	#17986721	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986705	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986356	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986355	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986348	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986288	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986217	C₂	1	1	yes	maximal, characteristic
9	#17986215	C₂	1	1	yes	maximal, characteristic
10	#17986203	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986123	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11765852	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986086	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986081	C₂	1	1	yes	maximal, characteristic
15	#479×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765854	C₂²	1	1	yes	characteristic
17	C₂×#11765852	C₂²	1	1	yes	characteristic
18	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556126	C₂²	1	1	yes	characteristic
21	#14556123	—	2	1	no	—
22	#14556122	—	2	1	no	—
23	#14556119	C₂²	1	1	yes	characteristic
24	#14556117	C₂²	1	1	yes	characteristic
25	#14556116	C₂²	1	1	yes	characteristic
26	#14556108	—	2	1	no	—
27	#14556106	—	2	1	no	—
28	#14556106	—	2	1	no	—
29	#14556106	—	2	1	no	—
30	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
31	#14556093	C₂²	1	1	yes	characteristic
32	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
33	#14556090	C₂²	1	1	yes	characteristic
34	#14556090	C₂²	1	1	yes	characteristic
35	#14556090	C₂²	1	1	yes	characteristic
36	#14556085	—	2	1	no	—
37	C₄⋊C₄×D₇₁	—	2	1	no	—
38	#14556059	C₂²	1	1	yes	characteristic
39	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	#14556053	C₂²	1	1	yes	characteristic
42	#14556052	C₂²	1	1	yes	characteristic
43	#14556051	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556048	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	#125×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×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	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#103×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	#11765854	—	2	2	no	—
69	#11765852	—	2	2	no	—
70	D₅₆₈	—	2	2	no	—
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	Dic₇₁⋊C₄	—	2	1	no	—
77	Dic₇₁⋊C₄	—	2	1	no	—
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
82	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
83	(C₂×D₇₁)⋊C₄	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	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	—	2	1	no	—
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
100	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄²×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	C₂³	1	1	yes	characteristic
105	C₂²×D₇₁	—	2	1	no	—
106	C₂²×D₇₁	—	2	1	no	—
107	Q₈×C₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	D₂₈₄	—	2	1	no	—
123	D₂₈₄	—	2	1	no	—
124	D₂₈₄	—	2	1	no	—
125	D₂₈₄	#41	1	2	yes	—
126	C₄×D₇₁	—	4	2	no	—
127	C₄×D₇₁	—	4	2	no	—
128	C₄×D₇₁	—	8	1	no	—
129	C₄×D₇₁	—	8	1	no	—
130	Dic₁₄₂	—	2	2	no	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	2	1	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	—	2	1	no	—
138	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₇₁	—	2	2	no	—
140	C₂×D₇₁	—	4	1	no	—
141	C₂×D₇₁	—	4	1	no	—
142	C₂×D₇₁	—	4	1	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	—	4	1	no	—
151	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
152	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	#135	1	2	yes	—
159	C₂×C₇₁	#548	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
161	C₂×C₇₁	#577	1	1	yes	characteristic
162	D₇₁	—	4	2	no	—
163	D₇₁	—	8	1	no	—
164	#3360	—	71	1	no	maximal, Sylow
165	C₇₁	#3360	1	1	yes	characteristic, complemented, Sylow
166	#556	—	71	1	no	—
167	#554	—	71	1	no	—
168	#488	—	71	1	no	—
169	#488	—	71	1	no	—
170	#487	—	71	1	no	—
171	#479	—	71	1	no	—
172	#479	D₇₁	1	1	yes	characteristic, complemented
173	#476	—	71	1	no	—
174	#463	—	71	1	no	—
175	#463	—	71	1	no	—
176	#461	—	71	1	no	—
177	C₈×Q₈	—	71	1	no	—
178	C₄×SD₁₆	—	71	1	no	—
179	C₄×SD₁₆	—	71	1	no	—
180	C₄×D₈	—	71	1	no	—
181	C₂×SD₁₆	—	71	1	no	—
182	C₂×SD₁₆	—	71	1	no	—
183	C₂×D₈	—	71	1	no	—
184	C₄⋊Q₈	—	71	1	no	—
185	C₄⋊Q₈	—	71	1	no	—
186	#126	—	142	1	no	—
187	#126	—	142	1	no	—
188	#125	—	71	1	no	—
189	#125	C₂×D₇₁	1	1	yes	characteristic
190	#124	—	71	1	no	—
191	#124	—	71	1	no	—
192	C₂²⋊Q₈	—	142	1	no	—
193	C₂²⋊Q₈	—	142	1	no	—
194	C₂²⋊Q₈	—	142	1	no	—
195	C₂²⋊Q₈	—	142	1	no	—
196	C₄×Q₈	—	71	1	no	—
197	C₄×Q₈	—	71	1	no	—
198	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
199	C₄×D₄	—	71	1	no	—
200	C₄×D₄	—	71	1	no	—
201	C₄×D₄	—	71	1	no	—
202	C₄×D₄	—	71	1	no	—
203	#117	—	142	1	no	—
204	C₂×C₄⋊C₄	—	142	1	no	—
205	#107	C₂×D₇₁	1	1	yes	characteristic
206	#106	—	71	1	no	—
207	#106	C₂×D₇₁	1	1	yes	characteristic
208	C₄⋊C₈	—	71	1	no	—
209	C₄⋊C₈	—	71	1	no	—
210	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
211	#103	—	71	1	no	—
212	#103	—	71	1	no	—
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	C₂×D₇₁	1	1	yes	characteristic
216	#103	C₂×D₇₁	1	1	yes	characteristic
217	#102	—	71	1	no	—
218	#102	—	71	1	no	—
219	#102	—	71	1	no	—
220	#102	—	71	1	no	—
221	C₄×C₈	—	71	1	no	—
222	C₄×C₈	—	71	1	no	—
223	C₄×C₈	—	71	1	no	—
224	C₂×Q₈	—	71	1	no	—
225	C₂×Q₈	—	71	1	no	—
226	C₂×Q₈	—	142	1	no	—
227	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
228	C₂×D₄	—	71	1	no	—
229	C₂×D₄	—	71	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	—	142	1	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	SD₁₆	—	142	2	no	—
235	SD₁₆	—	142	2	no	—
236	D₈	—	142	2	no	—
237	C₂×C₈	—	71	1	no	—
238	C₂×C₈	—	71	1	no	—
239	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	71	1	no	—
245	C₄⋊C₄	—	71	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₄⋊C₄	—	71	1	no	—
248	C₄⋊C₄	—	142	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	142	1	no	—
258	C₂²⋊C₄	—	142	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	142	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	C₄²	—	2	1	no	—
266	C₄²	—	71	1	no	—
267	C₄²	—	71	1	no	—
268	C₄²	—	142	1	no	—
269	C₄²	C₂²×D₇₁	1	1	yes	characteristic
270	C₂³	—	142	1	no	—
271	C₂³	—	142	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	568	1	no	—
275	Q₈	—	142	2	no	—
276	Q₈	—	142	1	no	—
277	Q₈	—	142	1	no	—
278	D₄	—	71	2	no	—
279	D₄	—	142	1	no	—
280	D₄	—	142	1	no	—
281	D₄	—	142	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	284	2	no	—
287	C₂×C₄	—	284	2	no	—
288	C₂×C₄	—	568	1	no	—
289	C₂×C₄	—	568	1	no	—
290	C₂×C₄	—	71	1	no	—
291	C₂×C₄	—	71	1	no	—
292	C₂×C₄	—	71	1	no	—
293	C₂×C₄	—	71	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	—	142	1	no	—
302	C₈	—	142	1	no	—
303	C₂²	—	284	1	no	—
304	C₂²	—	284	1	no	—
305	C₂²	—	284	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	—	4	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	#41×D₇₁	1	1	yes	characteristic
323	C₄	#14556176	1	1	yes	characteristic
324	C₂	—	284	2	no	—
325	C₂	—	568	1	no	—
326	C₂	#17986659	1	1	yes	characteristic
327	C₂	#135×D₇₁	1	1	yes	characteristic
328	C₂	#17986808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#479×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#479	D₇₁
hypocenter	C₇₁	#3360
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #17986721	C₂	semidirect
2. #17986705	C₂	semidirect
3. #17986356	C₂	semidirect
4. #17986355	C₂	semidirect
5. #17986348	C₂	semidirect
6. #17986305	C₂	semidirect
7. #17986288	C₂	semidirect
10. #17986203	C₂	semidirect
11. #17986123	C₂	semidirect
12. C₄×#11765852	C₂	semidirect
13. #17986086	C₂	semidirect
15. #479×C₇₁	C₂	semidirect
165. C₇₁	#3360	semidirect
172. #479	D₇₁	semidirect