Subgroups — #22600560

Order 9088.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986683	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986643	C₂	1	1	yes	maximal, characteristic
3	#17986352	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986341	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986316	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986297	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986255	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986242	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986219	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986158	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986074	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986004	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985997	C₂	1	1	yes	maximal, characteristic, complemented
15	#450×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	C₂²	1	1	yes	characteristic
17	C₂×#11765855	C₂²	1	1	yes	characteristic, complemented
18	C₂×#11765849	C₂²	1	1	yes	characteristic
19	C₂×#11765848	C₂²	1	1	yes	characteristic, complemented
20	#14556121	—	2	1	no	—
21	#14556119	C₂²	1	1	yes	characteristic, complemented
22	#14556118	—	2	1	no	—
23	#14556115	C₂²	1	1	yes	characteristic
24	#14556110	—	2	1	no	—
25	#14556109	—	2	1	no	—
26	#14556107	C₂²	1	1	yes	characteristic
27	#14556104	C₂²	1	1	yes	characteristic, complemented
28	#14556100	—	2	1	no	—
29	#14556099	C₂²	1	1	yes	characteristic
30	#14556097	—	2	1	no	—
31	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
32	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
34	#14556087	C₂²	1	1	yes	characteristic
35	#14556086	—	2	1	no	—
36	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊Dic₇₁	—	2	1	no	—
38	#14556062	C₂²	1	1	yes	characteristic, complemented
39	#14556060	C₂²	1	1	yes	characteristic, complemented
40	#14556059	C₂²	1	1	yes	characteristic, complemented
41	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	#14556053	C₂²	1	1	yes	characteristic, complemented
43	#14556053	C₂²	1	1	yes	characteristic, complemented
44	#14556052	C₂²	1	1	yes	characteristic, complemented
45	#14556051	C₂²	1	1	yes	characteristic, complemented
46	#14556049	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	#14556049	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
56	#102×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	2	2	no	—
60	#11765864	—	4	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₁	—	2	1	no	—
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
69	#11765855	—	4	1	no	—
70	#11765849	—	4	1	no	—
71	#11765848	—	4	1	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	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	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	D₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
96	C₂⁴×C₇₁	—	2	1	no	—
97	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₇₁	—	2	2	no	—
99	C₂×D₄×C₇₁	—	2	1	no	—
100	C₂×D₄×C₇₁	—	4	1	no	—
101	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
104	SD₁₆×C₇₁	—	2	2	no	—
105	SD₁₆×C₇₁	—	4	1	no	—
106	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₇₁	—	2	1	no	—
109	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₇₁	—	2	1	no	—
111	C₂³×C₇₁	—	2	1	no	—
112	C₂³×C₇₁	—	2	1	no	—
113	C₂³×C₇₁	—	4	2	no	—
114	C₂³×C₇₁	—	4	2	no	—
115	C₂³×C₇₁	—	8	1	no	—
116	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₇₁	—	2	1	no	—
119	Q₈×C₇₁	—	2	1	no	—
120	D₄×C₇₁	—	2	1	no	—
121	D₄×C₇₁	—	2	1	no	—
122	D₄×C₇₁	—	2	1	no	—
123	D₄×C₇₁	—	4	2	no	—
124	D₄×C₇₁	#41	1	2	yes	—
125	C₂×C₄×C₇₁	—	2	1	no	—
126	C₂×C₄×C₇₁	—	2	1	no	—
127	C₂×C₄×C₇₁	—	2	1	no	—
128	C₂×C₄×C₇₁	—	2	1	no	—
129	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₇₁	—	4	1	no	—
132	C₂²⋊D₇₁	—	4	2	no	—
133	C₂²⋊D₇₁	—	8	1	no	—
134	C₂×Dic₇₁	—	2	1	no	—
135	C₂×Dic₇₁	—	2	1	no	—
136	C₂×Dic₇₁	—	2	1	no	—
137	C₂×Dic₇₁	—	2	1	no	—
138	C₂×Dic₇₁	—	4	2	no	—
139	C₂×Dic₇₁	—	8	1	no	—
140	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
143	C₄×D₇₁	—	2	1	no	—
144	C₄×D₇₁	—	2	1	no	—
145	C₄×D₇₁	—	4	2	no	—
146	Dic₁₄₂	—	2	1	no	—
147	Dic₁₄₂	—	2	1	no	—
148	C₈×C₇₁	—	2	1	no	—
149	C₈×C₇₁	—	2	1	no	—
150	C₇₁⋊C₈	—	2	1	no	—
151	C₇₁⋊C₈	—	2	1	no	—
152	C₂²×C₇₁	—	2	2	no	—
153	C₂²×C₇₁	—	2	1	no	—
154	C₂²×C₇₁	—	2	1	no	—
155	C₂²×C₇₁	—	2	1	no	—
156	C₂²×C₇₁	—	4	1	no	—
157	C₂²×C₇₁	—	4	1	no	—
158	C₂²×C₇₁	—	4	1	no	—
159	C₂²×C₇₁	—	4	1	no	—
160	C₂²×C₇₁	—	4	1	no	—
161	C₂²×C₇₁	—	8	4	no	—
162	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₇₁	—	2	1	no	—
164	C₂×D₇₁	—	2	1	no	—
165	C₂×D₇₁	—	2	1	no	—
166	C₄×C₇₁	—	2	1	no	—
167	C₄×C₇₁	—	4	1	no	—
168	C₄×C₇₁	—	4	1	no	—
169	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
170	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₇₁	—	2	1	no	—
172	Dic₇₁	—	2	1	no	—
173	Dic₇₁	—	2	1	no	—
174	Dic₇₁	—	4	1	no	—
175	Dic₇₁	—	4	1	no	—
176	Dic₇₁	—	4	1	no	—
177	Dic₇₁	—	4	1	no	—
178	C₂×C₇₁	—	4	1	no	—
179	C₂×C₇₁	—	4	2	no	—
180	C₂×C₇₁	—	8	1	no	—
181	C₂×C₇₁	#546	1	1	yes	characteristic
182	C₂×C₇₁	#575	1	1	yes	characteristic
183	C₂×C₇₁	#577	1	1	yes	characteristic
184	D₇₁	—	4	1	no	—
185	#3323	—	71	1	no	maximal, Sylow
186	C₇₁	#3323	1	1	yes	characteristic, complemented, Sylow
187	#552	—	71	1	no	—
188	#546	—	71	1	no	—
189	#488	—	71	1	no	—
190	#485	—	71	1	no	—
191	#481	—	71	1	no	—
192	#478	—	71	1	no	—
193	#470	—	71	1	no	—
194	#468	—	71	1	no	—
195	#464	—	71	1	no	—
196	#452	—	71	1	no	—
197	#450	D₇₁	1	1	yes	characteristic, complemented
198	C₄×SD₁₆	—	71	1	no	—
199	#436	—	71	1	no	—
200	#418	—	71	1	no	—
201	#417	—	71	1	no	—
202	C₂×#41	—	71	1	no	—
203	C₂²×D₄	C₂×D₇₁	1	1	yes	characteristic
204	C₂×Q₁₆	—	71	1	no	—
205	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
206	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
207	C₂×M₁₆	—	71	1	no	—
208	C₂×M₁₆	—	71	1	no	—
209	#126	—	142	1	no	—
210	#125	—	71	1	no	—
211	#125	—	142	1	no	—
212	#124	—	71	1	no	—
213	#123	—	142	1	no	—
214	#123	—	142	1	no	—
215	C₂²⋊Q₈	—	71	1	no	—
216	C₂²⋊Q₈	—	71	1	no	—
217	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
218	#121	—	71	1	no	—
219	#121	—	142	1	no	—
220	#120	—	142	1	no	—
221	C₄×Q₈	—	71	1	no	—
222	C₄×D₄	—	71	1	no	—
223	C₄×D₄	—	71	1	no	—
224	#117	—	71	1	no	—
225	#117	—	142	1	no	—
226	C₂×C₄⋊C₄	—	71	1	no	—
227	C₂×C₂²⋊C₄	—	142	1	no	—
228	#107	—	71	1	no	—
229	#106	—	71	1	no	—
230	#106	—	71	1	no	—
231	C₄⋊C₈	—	71	1	no	—
232	#103	—	71	1	no	—
233	#103	—	71	1	no	—
234	#103	—	71	1	no	—
235	#103	—	71	1	no	—
236	#102	—	71	1	no	—
237	#102	—	71	1	no	—
238	#102	—	71	1	no	—
239	#102	—	71	1	no	—
240	#102	C₂×D₇₁	1	1	yes	characteristic
241	#102	C₂×D₇₁	1	1	yes	characteristic
242	C₂²⋊C₈	—	71	1	no	—
243	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
244	C₄×C₈	—	71	1	no	—
245	C₂⁴	—	2	1	no	—
246	#41	—	284	1	no	—
247	#41	—	142	2	no	—
248	C₂×Q₈	—	71	1	no	—
249	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	71	1	no	—
254	C₂×D₄	—	142	1	no	—
255	C₂×D₄	—	142	1	no	—
256	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₂²×C₄	—	71	1	no	—
259	C₂²×C₄	—	142	1	no	—
260	C₂²×C₄	—	142	1	no	—
261	C₂²×C₄	—	142	1	no	—
262	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
263	Q₁₆	—	284	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	4	1	no	—
266	M₁₆	—	284	1	no	—
267	M₁₆	—	284	1	no	—
268	C₂×C₈	—	71	1	no	—
269	C₂×C₈	—	71	1	no	—
270	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	71	1	no	—
274	C₄⋊C₄	—	71	1	no	—
275	C₄⋊C₄	—	71	1	no	—
276	C₄⋊C₄	—	71	1	no	—
277	C₄⋊C₄	—	142	1	no	—
278	C₄⋊C₄	—	142	1	no	—
279	C₄⋊C₄	—	142	1	no	—
280	C₄⋊C₄	—	142	1	no	—
281	C₄⋊C₄	—	142	1	no	—
282	C₄⋊C₄	—	142	1	no	—
283	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	71	1	no	—
286	C₂²⋊C₄	—	71	1	no	—
287	C₂²⋊C₄	—	142	1	no	—
288	C₂²⋊C₄	—	142	1	no	—
289	C₂²⋊C₄	—	142	1	no	—
290	C₂²⋊C₄	—	142	1	no	—
291	C₂²⋊C₄	—	142	1	no	—
292	C₂²⋊C₄	—	142	1	no	—
293	C₂²⋊C₄	—	142	1	no	—
294	C₄²	—	71	1	no	—
295	C₄²	—	142	1	no	—
296	C₄²	—	142	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	71	1	no	—
303	C₂³	D₄×D₇₁	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	142	1	no	—
307	Q₈	—	142	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	2	no	—
312	D₄	—	284	2	no	—
313	D₄	—	284	1	no	—
314	D₄	—	568	1	no	—
315	D₄	#11765864	1	2	yes	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	284	2	no	—
321	C₂×C₄	—	284	2	no	—
322	C₂×C₄	—	568	1	no	—
323	C₂×C₄	—	71	1	no	—
324	C₂×C₄	—	71	1	no	—
325	C₂×C₄	—	71	1	no	—
326	C₂×C₄	—	142	1	no	—
327	C₂×C₄	—	142	1	no	—
328	C₂×C₄	—	142	1	no	—
329	C₂×C₄	—	142	1	no	—
330	C₂×C₄	—	142	1	no	—
331	C₂×C₄	—	142	1	no	—
332	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	142	1	no	—
337	C₈	—	142	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	8	4	no	—
348	C₂²	—	142	1	no	—
349	C₂²	—	142	1	no	—
350	C₂²	—	142	1	no	—
351	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	284	1	no	—
356	C₄	—	284	1	no	—
357	C₄	—	284	1	no	—
358	C₄	—	284	1	no	—
359	C₄	—	142	1	no	—
360	C₄	—	142	1	no	—
361	C₄	—	142	1	no	—
362	C₄	C₂×#11765864	1	1	yes	characteristic
363	C₄	#14556173	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	4	2	no	—
366	C₂	—	8	1	no	—
367	C₂	—	284	1	no	—
368	C₂	#17986636	1	1	yes	characteristic
369	C₂	#17986795	1	1	yes	characteristic
370	C₂	#17986805	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#450×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#450	D₇₁
hypocenter	C₇₁	#3323
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

Complements

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

N	H ≅ G/N	type
1. #17986683	C₂	semidirect
3. #17986352	C₂	semidirect
4. #17986341	C₂	semidirect
5. #17986316	C₂	semidirect
6. #17986297	C₂	semidirect
7. #17986255	C₂	semidirect
8. #17986242	C₂	semidirect
9. #17986219	C₂	semidirect
10. #17986158	C₂	semidirect
11. SD₁₆×Dic₇₁	C₂	semidirect
12. #17986074	C₂	semidirect
13. #17986004	C₂	semidirect
14. #17985997	C₂	semidirect
15. #450×C₇₁	C₂	semidirect
17. C₂×#11765855	C₂²	semidirect
19. C₂×#11765848	C₂²	semidirect
21. #14556119	C₂²	semidirect
27. #14556104	C₂²	semidirect
31. Q₈×Dic₇₁	C₂²	semidirect
36. C₄⋊C₄×D₇₁	C₂²	semidirect
38. #14556062	C₂²	semidirect
39. #14556060	C₂²	semidirect
40. #14556059	C₂²	semidirect
41. Dic₇₁⋊C₈	C₂²	semidirect
42. #14556053	C₂²	semidirect
43. #14556053	C₂²	semidirect
44. #14556052	C₂²	semidirect
45. #14556051	C₂²	semidirect
50. (C₂×D₇₁)⋊C₈	C₂²	semidirect
51. C₈×Dic₇₁	C₂²	semidirect
186. C₇₁	#3323	semidirect
197. #450	D₇₁	semidirect