Subgroups — #22600408

Order 9088.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986719	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986658	C₂	1	1	yes	maximal, characteristic
3	#17986359	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986341	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986324	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986295	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986245	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986240	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986222	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986163	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986157	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986107	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986009	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985996	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765865	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
20	C₂×#11765854	C₂²	1	1	yes	characteristic
21	#14556128	C₂²	1	1	yes	characteristic
22	#14556121	—	2	1	no	—
23	#14556121	—	2	1	no	—
24	#14556116	C₂²	1	1	yes	characteristic
25	#14556115	—	2	1	no	—
26	#14556114	—	2	1	no	—
27	#14556107	C₂²	1	1	yes	characteristic
28	#14556107	C₂²	1	1	yes	characteristic
29	#14556107	C₂²	1	1	yes	characteristic
30	#14556106	—	2	1	no	—
31	#14556102	C₂²	1	1	yes	characteristic
32	#14556100	—	2	1	no	—
33	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
35	#14556093	—	2	1	no	—
36	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
37	#14556087	C₂²	1	1	yes	characteristic
38	#14556087	C₂²	1	1	yes	characteristic
39	#14556062	C₂²	1	1	yes	characteristic
40	#14556062	C₂²	1	1	yes	characteristic
41	#14556060	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556053	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556053	C₂²	1	1	yes	characteristic
48	#14556053	C₂²	1	1	yes	characteristic
49	#14556050	C₂²	1	1	yes	characteristic
50	#14556050	C₂²	1	1	yes	characteristic
51	#14556050	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	#11765865	—	2	2	no	—
60	#11765865	—	4	1	no	—
61	C₂×Dic₁₄₂	—	2	2	no	—
62	C₂×Dic₁₄₂	—	2	1	no	—
63	C₂×Dic₁₄₂	—	2	1	no	—
64	C₂×Dic₁₄₂	—	4	1	no	—
65	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
68	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₇₁	—	2	1	no	—
70	C₂×C₂²⋊D₇₁	—	2	1	no	—
71	C₂²×Dic₇₁	—	2	1	no	—
72	C₂×C₄×D₇₁	—	2	1	no	—
73	Dic₂₈₄	—	2	2	no	—
74	Dic₂₈₄	—	4	1	no	—
75	#11765854	—	4	1	no	—
76	C₄⋊Dic₇₁	—	2	1	no	—
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	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	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	C₂²⋊Dic₇₁	—	2	1	no	—
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	—	2	1	no	—
98	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
101	M₁₆×C₇₁	—	4	1	no	—
102	C₂×C₈×C₇₁	—	4	1	no	—
103	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₇₁	D₄	1	1	yes	characteristic
109	C₂²⋊C₄×C₇₁	D₄	1	1	yes	characteristic
110	C₄²×C₇₁	C₂³	1	1	yes	characteristic
111	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₇₁	—	2	1	no	—
114	D₄×C₇₁	—	4	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	—	2	1	no	—
119	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	—	2	1	no	—
130	C₂×Dic₇₁	—	2	1	no	—
131	C₂×Dic₇₁	—	2	1	no	—
132	C₂×Dic₇₁	—	2	1	no	—
133	C₂×Dic₇₁	—	4	2	no	—
134	D₂₈₄	—	2	1	no	—
135	D₂₈₄	—	2	1	no	—
136	C₄×D₇₁	—	4	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	—	4	2	no	—
143	Dic₁₄₂	—	4	2	no	—
144	Dic₁₄₂	#41	1	2	yes	—
145	C₈×C₇₁	—	2	1	no	—
146	C₈×C₇₁	—	2	1	no	—
147	C₈×C₇₁	—	2	1	no	—
148	C₈×C₇₁	—	2	1	no	—
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	—	2	1	no	—
152	C₂²×C₇₁	—	2	1	no	—
153	C₂²×C₇₁	—	2	1	no	—
154	C₂²×C₇₁	—	2	1	no	—
155	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₇₁	—	4	1	no	—
157	C₂×D₇₁	—	4	1	no	—
158	C₂×D₇₁	—	4	1	no	—
159	C₄×C₇₁	—	2	1	no	—
160	C₄×C₇₁	—	2	1	no	—
161	C₄×C₇₁	—	2	1	no	—
162	C₄×C₇₁	—	2	1	no	—
163	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
164	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₇₁	—	2	2	no	—
166	Dic₇₁	—	4	1	no	—
167	Dic₇₁	—	4	1	no	—
168	Dic₇₁	—	4	1	no	—
169	Dic₇₁	—	4	1	no	—
170	Dic₇₁	—	4	1	no	—
171	Dic₇₁	—	4	1	no	—
172	C₂×C₇₁	—	4	1	no	—
173	C₂×C₇₁	—	4	1	no	—
174	C₂×C₇₁	#546	1	1	yes	characteristic
175	C₂×C₇₁	#575	1	1	yes	characteristic
176	C₂×C₇₁	#578	1	1	yes	characteristic
177	D₇₁	—	8	1	no	—
178	#3313	—	71	1	no	maximal, Sylow
179	C₇₁	#3313	1	1	yes	characteristic, complemented, Sylow
180	#556	—	71	1	no	—
181	#548	—	71	1	no	—
182	#489	—	71	1	no	—
183	#485	—	71	1	no	—
184	#482	—	71	1	no	—
185	#477	—	71	1	no	—
186	#468	—	71	1	no	—
187	#467	—	71	1	no	—
188	#464	—	71	1	no	—
189	#452	—	71	1	no	—
190	#451	—	71	1	no	—
191	#441	—	71	1	no	—
192	#435	D₇₁	1	1	yes	characteristic, complemented
193	#419	—	71	1	no	—
194	#416	—	71	1	no	—
195	C₂×#41	—	71	1	no	—
196	C₂²×Q₈	—	71	1	no	—
197	C₂×Q₁₆	—	71	1	no	—
198	C₂×Q₁₆	—	71	1	no	—
199	C₂×SD₁₆	—	71	1	no	—
200	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
201	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	71	1	no	—
203	#126	—	142	1	no	—
204	#126	—	142	1	no	—
205	#124	—	71	1	no	—
206	#124	—	142	1	no	—
207	#124	—	142	1	no	—
208	C₂²⋊Q₈	—	71	1	no	—
209	C₂²⋊Q₈	—	71	1	no	—
210	C₂²⋊Q₈	—	71	1	no	—
211	C₂²⋊Q₈	—	142	1	no	—
212	#121	—	71	1	no	—
213	#121	—	142	1	no	—
214	C₄×Q₈	—	71	1	no	—
215	C₄×Q₈	—	71	1	no	—
216	C₄×Q₈	—	142	1	no	—
217	C₄×D₄	—	142	1	no	—
218	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
219	#117	—	71	1	no	—
220	#117	—	71	1	no	—
221	#107	—	71	1	no	—
222	#107	—	71	1	no	—
223	#106	—	71	1	no	—
224	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
225	#103	—	71	1	no	—
226	#103	—	71	1	no	—
227	#103	—	71	1	no	—
228	#103	—	71	1	no	—
229	#103	—	71	1	no	—
230	#103	—	71	1	no	—
231	#103	—	71	1	no	—
232	#102	—	71	1	no	—
233	#102	—	71	1	no	—
234	#102	—	71	1	no	—
235	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
236	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
237	#97	C₂×D₇₁	1	1	yes	characteristic
238	#41	—	284	1	no	—
239	#41	—	142	2	no	—
240	C₂×Q₈	—	284	1	no	—
241	C₂×Q₈	—	71	1	no	—
242	C₂×Q₈	—	71	1	no	—
243	C₂×Q₈	—	71	1	no	—
244	C₂×Q₈	—	142	2	no	—
245	C₂×Q₈	—	142	1	no	—
246	C₂×Q₈	—	142	1	no	—
247	C₂×D₄	—	71	1	no	—
248	C₂×D₄	—	142	1	no	—
249	C₂×D₄	—	142	1	no	—
250	C₂×D₄	D₂₈₄	1	1	yes	characteristic
251	C₂²×C₄	—	142	1	no	—
252	C₂²×C₄	—	142	1	no	—
253	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
255	Q₁₆	—	284	1	no	—
256	Q₁₆	—	142	2	no	—
257	SD₁₆	—	284	1	no	—
258	M₁₆	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
264	C₄⋊C₄	—	71	1	no	—
265	C₄⋊C₄	—	71	1	no	—
266	C₄⋊C₄	—	71	1	no	—
267	C₄⋊C₄	—	71	1	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₄⋊C₄	—	142	1	no	—
270	C₄⋊C₄	—	142	1	no	—
271	C₄⋊C₄	—	142	1	no	—
272	C₄⋊C₄	—	142	1	no	—
273	C₄⋊C₄	—	142	1	no	—
274	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
275	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₂²⋊C₄	—	142	1	no	—
278	C₂²⋊C₄	—	142	1	no	—
279	C₂²⋊C₄	—	142	1	no	—
280	C₂²⋊C₄	—	142	1	no	—
281	C₂²⋊C₄	—	142	1	no	—
282	C₂²⋊C₄	—	142	1	no	—
283	C₂²⋊C₄	D₂₈₄	1	1	yes	characteristic
284	C₂²⋊C₄	D₂₈₄	1	1	yes	characteristic
285	C₄²	—	142	1	no	—
286	C₄²	—	142	1	no	—
287	C₄²	—	142	1	no	—
288	C₄²	—	142	1	no	—
289	C₄²	C₂²×D₇₁	1	1	yes	characteristic
290	C₂³	—	142	1	no	—
291	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
292	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
293	Q₈	—	284	2	no	—
294	Q₈	—	284	2	no	—
295	Q₈	—	71	2	no	—
296	Q₈	—	142	1	no	—
297	Q₈	—	142	1	no	—
298	Q₈	—	142	1	no	—
299	Q₈	—	142	1	no	—
300	Q₈	—	142	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	284	2	no	—
303	D₄	—	284	2	no	—
304	D₄	—	142	1	no	—
305	D₄	—	142	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	284	2	no	—
311	C₂×C₄	—	284	2	no	—
312	C₂×C₄	—	142	1	no	—
313	C₂×C₄	—	142	1	no	—
314	C₂×C₄	—	142	1	no	—
315	C₂×C₄	—	142	1	no	—
316	C₂×C₄	—	142	1	no	—
317	C₂×C₄	—	142	1	no	—
318	C₂×C₄	—	142	1	no	—
319	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
320	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
321	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
322	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	284	1	no	—
335	C₂²	—	284	1	no	—
336	C₂²	—	284	1	no	—
337	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	284	1	no	—
343	C₄	—	284	1	no	—
344	C₄	—	284	1	no	—
345	C₄	—	284	1	no	—
346	C₄	—	284	1	no	—
347	C₄	—	284	1	no	—
348	C₄	—	142	2	no	—
349	C₄	#41×D₇₁	1	1	yes	characteristic
350	C₄	#14556173	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	4	1	no	—
353	C₂	—	568	1	no	—
354	C₂	#17986644	1	1	yes	characteristic
355	C₂	#17986799	1	1	yes	characteristic
356	C₂	#17986816	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#435×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#435	D₇₁
hypocenter	C₇₁	#3313
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #17986719	C₂	semidirect
3. #17986359	C₂	semidirect
4. #17986341	C₂	semidirect
5. #17986324	C₂	semidirect
6. #17986295	C₂	semidirect
7. #17986245	C₂	semidirect
8. #17986240	C₂	semidirect
9. #17986222	C₂	semidirect
10. #17986163	C₂	semidirect
11. #17986157	C₂	semidirect
12. #17986107	C₂	semidirect
13. #17986009	C₂	semidirect
14. #17985996	C₂	semidirect
15. #435×C₇₁	C₂	semidirect
179. C₇₁	#3313	semidirect
192. #435	D₇₁	semidirect