Subgroups — #22601394

Order 9088.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986732	C₂	1	1	yes	maximal, characteristic
2	#17986727	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986393	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986390	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986370	C₂	1	1	yes	maximal, characteristic
6	#17986221	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986220	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986208	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986204	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986203	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986198	C₂	1	1	yes	maximal, characteristic
12	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986098	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986093	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
18	C₂×#11765855	C₂²	1	1	yes	characteristic
19	C₂×#11765854	C₂²	1	1	yes	characteristic
20	C₂×#11765854	C₂²	1	1	yes	characteristic
21	C₂×#11765853	C₂²	1	1	yes	characteristic
22	C₂×#11765853	C₂²	1	1	yes	characteristic
23	C₂×#11765852	C₂²	1	1	yes	characteristic
24	#14556128	C₂²	1	1	yes	characteristic
25	#14556126	C₂²	1	1	yes	characteristic
26	#14556126	C₂²	1	1	yes	characteristic
27	#14556124	C₂²	1	1	yes	characteristic
28	#14556120	—	2	1	no	—
29	#14556118	—	2	1	no	—
30	#14556118	—	2	1	no	—
31	#14556116	C₂²	1	1	yes	characteristic
32	#14556115	C₂²	1	1	yes	characteristic
33	#14556114	—	2	1	no	—
34	#14556111	—	2	1	no	—
35	#14556110	—	2	1	no	—
36	C₄×Dic₁₄₂	—	2	1	no	—
37	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
38	#14556093	C₂²	1	1	yes	characteristic
39	#14556093	C₂²	1	1	yes	characteristic
40	#14556093	C₂²	1	1	yes	characteristic
41	#14556090	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊D₇₁	—	2	1	no	—
43	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
46	#14556051	C₂²	1	1	yes	characteristic
47	#14556051	C₂²	1	1	yes	characteristic
48	#14556051	C₂²	1	1	yes	characteristic
49	#14556048	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂×C₂²⋊D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	Dic₂₈₄	—	2	2	no	—
68	Dic₂₈₄	—	4	1	no	—
69	#11765855	—	4	1	no	—
70	#11765854	—	2	2	no	—
71	#11765854	—	4	1	no	—
72	#11765853	—	4	1	no	—
73	#11765853	—	4	1	no	—
74	#11765852	—	4	1	no	—
75	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	—	2	1	no	—
79	C₄⋊Dic₇₁	D₄	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	—	2	1	no	—
85	Dic₇₁⋊C₄	—	2	1	no	—
86	Dic₇₁⋊C₄	—	2	1	no	—
87	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
88	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₇₁	—	2	1	no	—
90	C₂²⋊Dic₇₁	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	(C₂×D₇₁)⋊C₄	—	2	1	no	—
93	(C₂×D₇₁)⋊C₄	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	—	2	1	no	—
97	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
98	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₇₁	—	2	1	no	—
102	SD₁₆×C₇₁	—	2	2	no	—
103	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₇₁	—	2	1	no	—
106	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₇₁	—	2	1	no	—
109	C₄²×C₇₁	—	2	1	no	—
110	C₄²×C₇₁	C₂³	1	1	yes	characteristic
111	C₂³×C₇₁	—	2	1	no	—
112	C₂²×D₇₁	—	2	1	no	—
113	Q₈×C₇₁	—	2	1	no	—
114	Q₈×C₇₁	—	2	1	no	—
115	D₄×C₇₁	—	2	1	no	—
116	D₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	8	1	no	—
121	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₇₁	—	4	2	no	—
125	C₂²⋊D₇₁	—	8	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
133	D₂₈₄	—	2	1	no	—
134	D₂₈₄	—	2	1	no	—
135	C₄×D₇₁	—	8	1	no	—
136	Dic₁₄₂	—	2	2	no	—
137	Dic₁₄₂	—	2	1	no	—
138	Dic₁₄₂	—	2	1	no	—
139	Dic₁₄₂	—	2	1	no	—
140	Dic₁₄₂	—	2	1	no	—
141	Dic₁₄₂	—	2	1	no	—
142	Dic₁₄₂	—	2	1	no	—
143	C₈×C₇₁	—	2	1	no	—
144	C₈×C₇₁	#41	1	2	yes	—
145	C₇₁⋊C₈	—	2	1	no	—
146	C₇₁⋊C₈	—	2	1	no	—
147	C₂²×C₇₁	—	4	1	no	—
148	C₂²×C₇₁	—	4	1	no	—
149	C₂²×C₇₁	—	4	1	no	—
150	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₇₁	—	4	1	no	—
152	C₂×D₇₁	—	4	1	no	—
153	C₂×D₇₁	—	4	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	4	1	no	—
157	C₄×C₇₁	—	4	1	no	—
158	C₄×C₇₁	—	4	1	no	—
159	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
160	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	2	1	no	—
163	Dic₇₁	—	2	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	—	4	1	no	—
166	Dic₇₁	—	4	1	no	—
167	Dic₇₁	#137	1	2	yes	—
168	C₂×C₇₁	—	8	1	no	—
169	C₂×C₇₁	#550	1	1	yes	characteristic
170	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
171	C₂×C₇₁	#577	1	1	yes	characteristic
172	D₇₁	—	8	1	no	—
173	#3400	—	71	1	no	maximal, Sylow
174	C₇₁	#3400	1	1	yes	characteristic, complemented, Sylow
175	#559	—	71	1	no	—
176	#557	—	71	1	no	—
177	#497	—	71	1	no	—
178	#497	—	71	1	no	—
179	#492	—	71	1	no	—
180	#464	—	71	1	no	—
181	#464	—	71	1	no	—
182	#462	—	71	1	no	—
183	#461	—	71	1	no	—
184	#461	—	71	1	no	—
185	#460	—	71	1	no	—
186	#446	—	71	1	no	—
187	#440	—	71	1	no	—
188	#440	D₇₁	1	1	yes	characteristic, complemented
189	C₄×Q₁₆	—	71	1	no	—
190	C₂×Q₁₆	—	71	1	no	—
191	C₂×Q₁₆	—	71	1	no	—
192	C₂×Q₁₆	—	71	1	no	—
193	C₂×SD₁₆	—	71	1	no	—
194	C₂×SD₁₆	—	71	1	no	—
195	C₂×SD₁₆	—	71	1	no	—
196	C₂×SD₁₆	—	71	1	no	—
197	C₂×SD₁₆	—	71	1	no	—
198	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
199	C₄⋊Q₈	—	71	1	no	—
200	C₄⋊Q₈	—	71	1	no	—
201	C₄⋊Q₈	—	71	1	no	—
202	#127	—	71	1	no	—
203	#125	—	142	1	no	—
204	#125	—	142	1	no	—
205	#125	—	142	1	no	—
206	#124	—	71	1	no	—
207	#124	—	71	1	no	—
208	#124	—	142	1	no	—
209	#123	—	142	1	no	—
210	#123	—	142	1	no	—
211	C₄×Q₈	—	71	1	no	—
212	C₄×Q₈	—	71	1	no	—
213	C₄×Q₈	—	71	1	no	—
214	C₄×Q₈	—	71	1	no	—
215	C₄×Q₈	—	142	1	no	—
216	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
217	C₄×D₄	—	71	1	no	—
218	C₄×D₄	—	142	1	no	—
219	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
220	#107	C₂×D₇₁	1	1	yes	characteristic
221	C₄⋊C₈	—	71	1	no	—
222	C₄⋊C₈	—	71	1	no	—
223	C₄⋊C₈	—	71	1	no	—
224	#103	—	71	1	no	—
225	#103	—	71	1	no	—
226	#103	—	71	1	no	—
227	#103	C₂×D₇₁	1	1	yes	characteristic
228	#102	—	71	1	no	—
229	#102	C₂×D₇₁	1	1	yes	characteristic
230	#97	—	71	1	no	—
231	#97	C₂×D₇₁	1	1	yes	characteristic
232	C₄×C₈	—	71	1	no	—
233	C₂×Q₈	—	71	1	no	—
234	C₂×Q₈	—	71	1	no	—
235	C₂×Q₈	—	71	1	no	—
236	C₂×Q₈	—	71	1	no	—
237	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×D₄	—	71	1	no	—
239	C₂×D₄	—	142	1	no	—
240	C₂×D₄	—	142	1	no	—
241	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	142	1	no	—
244	Q₁₆	—	284	1	no	—
245	Q₁₆	—	284	1	no	—
246	Q₁₆	—	142	2	no	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	284	1	no	—
249	SD₁₆	—	284	1	no	—
250	SD₁₆	—	284	1	no	—
251	SD₁₆	—	284	1	no	—
252	SD₁₆	—	142	2	no	—
253	C₂×C₈	—	71	1	no	—
254	C₂×C₈	—	71	1	no	—
255	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	71	1	no	—
259	C₄⋊C₄	—	71	1	no	—
260	C₄⋊C₄	—	71	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	—	142	1	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₄⋊C₄	—	142	1	no	—
270	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	142	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	71	1	no	—
280	C₄²	—	71	1	no	—
281	C₄²	—	142	1	no	—
282	C₄²	—	142	1	no	—
283	C₄²	—	142	1	no	—
284	C₄²	C₂²×D₇₁	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	142	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	142	2	no	—
290	Q₈	—	142	1	no	—
291	Q₈	—	142	1	no	—
292	Q₈	—	142	1	no	—
293	Q₈	—	142	1	no	—
294	Q₈	—	142	1	no	—
295	Q₈	—	142	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	284	2	no	—
299	D₄	—	568	1	no	—
300	D₄	—	142	1	no	—
301	D₄	—	142	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	568	1	no	—
307	C₂×C₄	—	71	1	no	—
308	C₂×C₄	—	71	1	no	—
309	C₂×C₄	—	71	1	no	—
310	C₂×C₄	—	71	1	no	—
311	C₂×C₄	—	142	1	no	—
312	C₂×C₄	—	142	1	no	—
313	C₂×C₄	—	142	1	no	—
314	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
315	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	142	1	no	—
319	C₈	—	142	1	no	—
320	C₈	#11765865	1	2	yes	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	284	1	no	—
325	C₂²	—	284	1	no	—
326	C₂²	—	284	1	no	—
327	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	284	1	no	—
334	C₄	—	284	1	no	—
335	C₄	—	284	1	no	—
336	C₄	—	71	2	no	—
337	C₄	—	142	1	no	—
338	C₄	—	142	1	no	—
339	C₄	—	142	1	no	—
340	C₄	C₂×#11765865	1	1	yes	characteristic
341	C₄	#14556174	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	568	1	no	—
344	C₂	#17986670	1	1	yes	characteristic
345	C₂	#137×D₇₁	1	1	yes	characteristic
346	C₂	#17986809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#440×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#440	D₇₁
hypocenter	C₇₁	#3400
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #17986727	C₂	semidirect
3. #17986393	C₂	semidirect
4. #17986390	C₂	semidirect
6. #17986221	C₂	semidirect
7. #17986220	C₂	semidirect
8. #17986208	C₂	semidirect
9. #17986204	C₂	semidirect
10. #17986203	C₂	semidirect
12. #17986127	C₂	semidirect
13. #17986098	C₂	semidirect
14. #17986093	C₂	semidirect
15. #440×C₇₁	C₂	semidirect
174. C₇₁	#3400	semidirect
188. #440	D₇₁	semidirect