Subgroups — #22600690

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986707	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986650	C₂	1	1	yes	maximal, characteristic
3	#17986396	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986328	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986319	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986318	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986288	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986282	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986231	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986230	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986084	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₇₁)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14556059	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#483×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765865	—	2	1	no	—
17	C₂×#11765852	C₂²	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556118	—	2	1	no	—
23	#14556113	—	2	1	no	—
24	#14556111	—	2	1	no	—
25	#14556108	—	2	1	no	—
26	#14556106	—	2	1	no	—
27	#14556105	—	2	1	no	—
28	#14556104	C₂²	1	1	yes	characteristic
29	#14556103	C₂²	1	1	yes	characteristic
30	#14556102	C₂²	1	1	yes	characteristic
31	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
32	#14556090	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
35	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₇₁	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	#14556059	C₂²	1	1	yes	characteristic
44	#14556059	C₂²	1	1	yes	characteristic
45	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556050	C₂²	1	1	yes	characteristic
48	#14556050	C₂²	1	1	yes	characteristic
49	#14556048	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
54	#106×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
59	#11765865	—	4	4	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×D₂₈₄	—	2	1	no	—
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
71	#11765852	—	2	2	no	—
72	C₂×C₇₁⋊C₈	—	2	2	no	—
73	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×D₇₁	—	2	2	no	—
76	C₄⋊Dic₇₁	—	2	1	no	—
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	Dic₇₁⋊C₄	—	2	1	no	—
86	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
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	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	—	2	1	no	—
97	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₄⋊C₄×C₇₁	—	2	1	no	—
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₇₁	—	2	1	no	—
108	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×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	—	4	2	no	—
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₇₁	—	4	2	no	—
121	C₂²⋊D₇₁	SD₁₆	1	4	yes	complemented
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	8	1	no	—
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
129	D₂₈₄	—	2	1	no	—
130	D₂₈₄	—	2	1	no	—
131	D₂₈₄	—	4	2	no	—
132	C₄×D₇₁	—	2	1	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	4	2	no	—
135	C₄×D₇₁	—	4	2	no	—
136	C₄×D₇₁	—	8	1	no	—
137	Dic₁₄₂	—	4	2	no	—
138	Dic₁₄₂	—	8	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₇₁⋊C₈	#41	1	2	yes	complemented
143	C₂²×C₇₁	—	2	1	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	C₂×SD₁₆	1	2	yes	—
146	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₇₁	—	2	1	no	—
148	C₂×D₇₁	—	2	1	no	—
149	C₂×D₇₁	—	4	1	no	—
150	C₂×D₇₁	—	4	1	no	—
151	C₂×D₇₁	—	4	1	no	—
152	C₂×D₇₁	C₂×SD₁₆	1	2	yes	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	4	1	no	—
155	C₄×C₇₁	—	4	1	no	—
156	C₄×C₇₁	—	4	1	no	—
157	C₄×C₇₁	—	4	1	no	—
158	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
159	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	—	4	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	C₂×SD₁₆	1	2	yes	—
166	C₂×C₇₁	—	2	2	no	—
167	C₂×C₇₁	#547	1	1	yes	characteristic
168	C₂×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
169	C₂×C₇₁	#577	1	1	yes	characteristic
170	D₇₁	—	2	2	no	—
171	D₇₁	—	8	1	no	—
172	#3335	—	71	1	no	maximal, Sylow
173	C₇₁	#3335	1	1	yes	characteristic, complemented, Sylow
174	#554	—	71	1	no	—
175	#547	—	71	1	no	—
176	#498	—	71	1	no	—
177	#483	—	71	1	no	—
178	#483	D₇₁	1	1	yes	characteristic, complemented
179	#481	—	71	1	no	—
180	#481	—	71	1	no	—
181	#476	—	71	1	no	—
182	#475	—	71	1	no	—
183	#465	—	71	1	no	—
184	#465	—	71	1	no	—
185	C₄×SD₁₆	—	71	1	no	—
186	C₈×D₄	—	71	1	no	—
187	C₂×#106	—	71	1	no	—
188	C₂×#106	—	71	1	no	—
189	C₂×#41	—	142	1	no	—
190	C₂×SD₁₆	—	71	1	no	—
191	C₂²×C₈	—	71	1	no	—
192	C₂²×C₈	—	71	1	no	—
193	C₄⋊Q₈	—	71	1	no	—
194	C₄⋊Q₈	—	71	1	no	—
195	#125	—	142	1	no	—
196	#123	—	142	1	no	—
197	#123	—	142	1	no	—
198	C₂²⋊Q₈	—	71	1	no	—
199	C₂²⋊Q₈	—	142	1	no	—
200	C₂²⋊Q₈	—	142	1	no	—
201	C₂²⋊Q₈	—	142	1	no	—
202	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
203	#121	—	71	1	no	—
204	#121	—	71	1	no	—
205	C₄×Q₈	—	71	1	no	—
206	C₄×D₄	—	71	1	no	—
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	142	1	no	—
209	C₂×C₄⋊C₄	—	71	1	no	—
210	C₂×C₄⋊C₄	—	71	1	no	—
211	C₂×C₄⋊C₄	—	71	1	no	—
212	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
213	#106	—	71	1	no	—
214	#106	—	71	1	no	—
215	#106	—	71	1	no	—
216	#106	—	71	1	no	—
217	#106	—	71	1	no	—
218	#106	—	71	1	no	—
219	#106	—	71	1	no	—
220	#106	C₂×D₇₁	1	1	yes	characteristic
221	#106	C₂×D₇₁	1	1	yes	characteristic
222	C₄⋊C₈	—	71	1	no	—
223	#103	—	71	1	no	—
224	#103	C₂×D₇₁	1	1	yes	characteristic
225	#103	C₂×D₇₁	1	1	yes	characteristic
226	#102	—	71	1	no	—
227	#102	—	71	1	no	—
228	#102	—	71	1	no	—
229	C₂²⋊C₈	—	71	1	no	—
230	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
231	C₄×C₈	—	71	1	no	—
232	#41	—	284	4	no	—
233	C₂×Q₈	—	142	1	no	—
234	C₂×Q₈	—	142	1	no	—
235	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₂×D₄	—	71	1	no	—
237	C₂×D₄	—	71	1	no	—
238	C₂×D₄	—	142	1	no	—
239	C₂×D₄	—	142	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	71	1	no	—
242	C₂²×C₄	—	142	1	no	—
243	C₂²×C₄	—	142	1	no	—
244	C₂²×C₄	—	142	1	no	—
245	C₂²×C₄	—	142	1	no	—
246	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
247	SD₁₆	—	142	2	no	—
248	C₂×C₈	—	71	1	no	—
249	C₂×C₈	—	71	1	no	—
250	C₂×C₈	—	142	2	no	—
251	C₂×C₈	—	142	2	no	—
252	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	71	1	no	—
257	C₄⋊C₄	—	71	1	no	—
258	C₄⋊C₄	—	71	1	no	—
259	C₄⋊C₄	—	71	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	—	142	1	no	—
262	C₄⋊C₄	—	142	1	no	—
263	C₄⋊C₄	—	142	1	no	—
264	C₄⋊C₄	—	142	1	no	—
265	C₄⋊C₄	—	142	1	no	—
266	C₄⋊C₄	—	142	1	no	—
267	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	71	1	no	—
272	C₂²⋊C₄	—	71	1	no	—
273	C₂²⋊C₄	—	142	1	no	—
274	C₂²⋊C₄	—	142	1	no	—
275	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₄²	—	71	1	no	—
278	C₄²	—	142	1	no	—
279	C₂³	—	71	1	no	—
280	C₂³	—	142	1	no	—
281	C₂³	D₄×D₇₁	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	284	2	no	—
285	Q₈	—	568	1	no	—
286	D₄	—	284	2	no	—
287	D₄	—	284	2	no	—
288	D₄	—	71	4	no	—
289	D₄	—	142	1	no	—
290	D₄	—	142	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	284	2	no	—
299	C₂×C₄	—	284	2	no	—
300	C₂×C₄	—	568	1	no	—
301	C₂×C₄	—	568	1	no	—
302	C₂×C₄	—	71	1	no	—
303	C₂×C₄	—	71	1	no	—
304	C₂×C₄	—	71	1	no	—
305	C₂×C₄	—	142	1	no	—
306	C₂×C₄	—	142	1	no	—
307	C₂×C₄	—	142	1	no	—
308	C₂×C₄	—	142	1	no	—
309	C₂×C₄	—	142	1	no	—
310	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	71	2	no	—
315	C₈	—	142	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	284	1	no	—
319	C₂²	—	284	1	no	—
320	C₂²	—	284	1	no	—
321	C₂²	—	71	2	no	—
322	C₂²	—	142	1	no	—
323	C₂²	—	142	1	no	—
324	C₂²	SD₁₆×D₇₁	1	2	yes	—
325	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	284	1	no	—
332	C₄	—	284	1	no	—
333	C₄	—	284	1	no	—
334	C₄	—	71	2	no	—
335	C₄	—	142	1	no	—
336	C₄	—	142	1	no	—
337	C₄	#41×D₇₁	1	1	yes	characteristic
338	C₄	#14556175	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	568	1	no	—
341	C₂	—	142	2	no	—
342	C₂	#17986647	1	1	yes	characteristic
343	C₂	C₂×SD₁₆×D₇₁	1	1	yes	characteristic
344	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	#483×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#483	D₇₁
hypocenter	C₇₁	#3335
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
1. #17986707	C₂	semidirect
3. #17986396	C₂	semidirect
4. #17986328	C₂	semidirect
5. #17986319	C₂	semidirect
6. #17986318	C₂	semidirect
7. #17986288	C₂	semidirect
8. #17986282	C₂	semidirect
9. #17986231	C₂	semidirect
10. #17986230	C₂	semidirect
11. #17986084	C₂	semidirect
12. (C₂²⋊D₇₁)⋊C₈	C₂	semidirect
13. C₂×#14556059	C₂	semidirect
14. #106×D₇₁	C₂	semidirect
15. #483×C₇₁	C₂	semidirect
121. C₂²⋊D₇₁	SD₁₆	semidirect
142. C₇₁⋊C₈	#41	semidirect
173. C₇₁	#3335	semidirect
178. #483	D₇₁	semidirect