Subgroups — #22598944

Order 9088.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986553	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986536	C₂	1	1	yes	maximal, characteristic
3	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986392	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986366	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986360	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986256	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986254	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986248	C₂	1	1	yes	maximal, characteristic
10	#17986246	C₂	1	1	yes	maximal, characteristic
11	C₄×#11765855	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11765854	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#11765852	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11765848	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765855	C₂²	1	1	yes	characteristic
17	C₂×#11765854	C₂²	1	1	yes	characteristic
18	C₂×#11765852	C₂²	1	1	yes	characteristic
19	C₂×#11765848	C₂²	1	1	yes	characteristic
20	C₂×#11765848	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556126	C₂²	1	1	yes	characteristic
23	#14556123	—	2	1	no	—
24	#14556119	C₂²	1	1	yes	characteristic
25	#14556118	—	2	1	no	—
26	#14556117	C₂²	1	1	yes	characteristic
27	#14556113	—	2	1	no	—
28	#14556108	C₂²	1	1	yes	characteristic
29	#14556106	—	2	1	no	—
30	#14556104	C₂²	1	1	yes	characteristic
31	#14556104	C₂²	1	1	yes	characteristic
32	#14556103	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₄₂	—	2	1	no	—
34	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
35	C₄×D₂₈₄	—	2	1	no	—
36	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
37	#14556085	—	2	1	no	—
38	#14556085	—	2	1	no	—
39	C₄²×D₇₁	C₂²	1	1	yes	characteristic
40	#14556061	C₂²	1	1	yes	characteristic
41	#14556060	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556052	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556051	C₂²	1	1	yes	characteristic
47	#14556050	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	#14556038	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×D₂₈₄	—	2	1	no	—
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
66	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
67	#11765855	—	2	2	no	—
68	#11765854	—	2	2	no	—
69	#11765852	—	2	2	no	—
70	#11765848	—	2	2	no	—
71	#11765848	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
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	—	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	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	D₄	1	1	yes	characteristic
94	C₄×Dic₇₁	D₄	1	1	yes	characteristic
95	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₇₁	—	2	2	no	—
99	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	—	2	1	no	—
102	C₄⋊C₄×C₇₁	—	2	1	no	—
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
105	C₄²×C₇₁	—	2	1	no	—
106	C₄²×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	C₂³	1	1	yes	characteristic
108	C₂²×D₇₁	—	2	1	no	—
109	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	2	1	no	—
111	Q₈×C₇₁	—	2	1	no	—
112	Q₈×C₇₁	—	2	1	no	—
113	Q₈×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	D₂₈₄	—	2	1	no	—
128	D₂₈₄	—	2	1	no	—
129	D₂₈₄	—	4	2	no	—
130	C₄×D₇₁	—	2	2	no	—
131	C₄×D₇₁	—	2	1	no	—
132	C₄×D₇₁	—	2	1	no	—
133	C₄×D₇₁	—	4	1	no	—
134	C₄×D₇₁	—	4	2	no	—
135	Dic₁₄₂	—	2	1	no	—
136	Dic₁₄₂	—	2	1	no	—
137	Dic₁₄₂	—	4	2	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	#41	1	2	yes	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₇₁⋊C₈	#41	1	2	yes	—
142	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₇₁	—	2	1	no	—
144	C₂×D₇₁	—	2	1	no	—
145	C₂×D₇₁	—	2	1	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₂×D₇₁	—	4	1	no	—
148	C₂×D₇₁	—	4	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	#137	1	2	yes	—
155	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
157	Dic₇₁	—	2	1	no	—
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	C₂×C₇₁	#532	1	1	yes	characteristic
164	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
165	C₂×C₇₁	#575	1	1	yes	characteristic
166	D₇₁	—	4	1	no	—
167	D₇₁	—	8	1	no	—
168	#3172	—	71	1	no	maximal, Sylow
169	C₇₁	#3172	1	1	yes	characteristic, complemented, Sylow
170	#533	—	71	1	no	—
171	#529	—	71	1	no	—
172	#501	—	71	1	no	—
173	#497	—	71	1	no	—
174	#491	—	71	1	no	—
175	#489	—	71	1	no	—
176	#470	—	71	1	no	—
177	#470	—	71	1	no	—
178	#468	—	71	1	no	—
179	#468	—	71	1	no	—
180	C₄×Q₁₆	—	71	1	no	—
181	C₄×Q₁₆	D₇₁	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	71	1	no	—
183	C₄×SD₁₆	—	71	1	no	—
184	C₄×M₁₆	—	71	1	no	—
185	C₂×Q₁₆	—	71	1	no	—
186	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
187	C₂×SD₁₆	—	71	1	no	—
188	C₂×SD₁₆	—	71	1	no	—
189	C₂×M₁₆	—	71	1	no	—
190	C₂×M₁₆	—	71	1	no	—
191	C₄⋊Q₈	—	71	1	no	—
192	C₄⋊Q₈	—	71	1	no	—
193	#126	—	142	1	no	—
194	#125	—	71	1	no	—
195	#125	—	142	1	no	—
196	#124	—	71	1	no	—
197	#123	—	142	1	no	—
198	C₂²⋊Q₈	—	71	1	no	—
199	C₂²⋊Q₈	—	71	1	no	—
200	C₂²⋊Q₈	—	71	1	no	—
201	C₂²⋊Q₈	—	142	1	no	—
202	#121	—	71	1	no	—
203	C₄×Q₈	—	71	1	no	—
204	C₄×Q₈	—	142	1	no	—
205	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
206	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	142	1	no	—
209	#117	—	142	1	no	—
210	#117	—	142	1	no	—
211	C₂×C₄²	—	71	1	no	—
212	#107	—	71	1	no	—
213	#107	C₂×D₇₁	1	1	yes	characteristic
214	#106	—	71	1	no	—
215	#106	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	#103	C₂×D₇₁	1	1	yes	characteristic
221	#103	C₂×D₇₁	1	1	yes	characteristic
222	#102	—	71	1	no	—
223	#102	—	71	1	no	—
224	#97	—	71	1	no	—
225	#97	—	71	1	no	—
226	C₄×C₈	—	71	1	no	—
227	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
228	C₂×Q₈	—	71	1	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×D₄	—	71	1	no	—
233	C₂×D₄	—	142	1	no	—
234	C₂²×C₄	—	71	1	no	—
235	C₂²×C₄	—	71	1	no	—
236	C₂²×C₄	—	71	1	no	—
237	C₂²×C₄	—	142	1	no	—
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	142	2	no	—
240	SD₁₆	—	142	2	no	—
241	SD₁₆	—	142	2	no	—
242	M₁₆	—	142	2	no	—
243	M₁₆	—	142	2	no	—
244	C₂×C₈	—	71	1	no	—
245	C₂×C₈	—	71	1	no	—
246	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	71	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	—	142	1	no	—
258	C₄⋊C₄	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₂²⋊C₄	—	142	1	no	—
266	C₂²⋊C₄	—	142	1	no	—
267	C₂²⋊C₄	—	142	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₂²⋊C₄	—	142	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	71	1	no	—
273	C₄²	—	71	1	no	—
274	C₄²	—	71	1	no	—
275	C₄²	—	142	1	no	—
276	C₄²	C₂²×D₇₁	1	1	yes	characteristic
277	C₂³	—	71	1	no	—
278	C₂³	—	142	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	284	2	no	—
284	Q₈	—	142	1	no	—
285	Q₈	—	142	1	no	—
286	D₄	—	284	2	no	—
287	D₄	—	142	1	no	—
288	D₄	—	142	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	284	1	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	71	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	2	no	—
302	C₂×C₄	—	142	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
305	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	71	2	no	—
309	C₈	—	142	1	no	—
310	C₈	#11765865	1	2	yes	—
311	C₂²	—	284	1	no	—
312	C₂²	—	284	1	no	—
313	C₂²	—	284	1	no	—
314	C₂²	—	142	1	no	—
315	C₂²	—	142	1	no	—
316	C₂²	—	142	1	no	—
317	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	284	1	no	—
325	C₄	—	284	1	no	—
326	C₄	—	142	1	no	—
327	C₄	—	142	1	no	—
328	C₄	—	142	1	no	—
329	C₄	#14556160	1	2	yes	—
330	C₄	#41×D₇₁	1	1	yes	characteristic
331	C₄	C₂×#11765865	1	1	yes	characteristic
332	C₂	—	284	1	no	—
333	C₂	—	568	1	no	—
334	C₂	#17986546	1	1	yes	characteristic
335	C₂	C₂×#14556160	1	1	yes	characteristic
336	C₂	#17986796	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₄×Q₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₇₁
hypocenter	C₇₁	#3172
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #17986553	C₂	semidirect
3. #17986407	C₂	semidirect
4. #17986392	C₂	semidirect
5. #17986366	C₂	semidirect
6. #17986360	C₂	semidirect
7. #17986256	C₂	semidirect
8. #17986254	C₂	semidirect
11. C₄×#11765855	C₂	semidirect
12. C₄×#11765854	C₂	semidirect
13. C₄×#11765852	C₂	semidirect
14. C₄×#11765848	C₂	semidirect
15. C₄×Q₁₆×C₇₁	C₂	semidirect
169. C₇₁	#3172	semidirect
181. C₄×Q₁₆	D₇₁	semidirect