Subgroups — #22600209

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 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
2	Q₈×C₂²⋊D₇₁	C₂	1	1	yes	maximal, characteristic
3	#17986348	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986335	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986333	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986292	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986238	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986237	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986207	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986153	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₇₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
13	C₂×#14556052	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765855	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	#14556126	—	2	1	no	—
22	#14556126	C₂²	1	1	yes	characteristic
23	#14556118	—	2	1	no	—
24	#14556108	—	2	1	no	—
25	#14556107	—	2	1	no	—
26	#14556107	C₂²	1	1	yes	characteristic
27	#14556105	—	2	1	no	—
28	#14556104	—	2	1	no	—
29	#14556104	C₂²	1	1	yes	characteristic
30	#14556104	C₂²	1	1	yes	characteristic
31	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
32	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	—	2	1	no	—
34	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
36	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
38	#14556062	C₂²	1	1	yes	characteristic
39	#14556062	C₂²	1	1	yes	characteristic
40	#14556061	C₂²	1	1	yes	characteristic
41	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
42	#14556053	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	#14556052	C₂²	1	1	yes	characteristic
47	#14556052	C₂²	1	1	yes	characteristic
48	#14556052	C₂²	1	1	yes	characteristic
49	#14556051	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×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×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	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	Q₈×D₇₁	—	2	2	no	—
63	Q₈×D₇₁	—	4	1	no	—
64	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	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	C₂×C₇₁⋊C₈	—	4	1	no	—
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₇₁	—	4	1	no	—
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	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	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	C₂²⋊Dic₇₁	—	2	1	no	—
89	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₇₁	—	2	1	no	—
97	C₂×Q₈×C₇₁	—	2	2	no	—
98	C₂×Q₈×C₇₁	—	4	1	no	—
99	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₇₁	—	2	1	no	—
103	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
104	Q₁₆×C₇₁	—	2	2	no	—
105	Q₁₆×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	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₇₁	—	2	1	no	—
114	Q₈×C₇₁	—	2	1	no	—
115	Q₈×C₇₁	—	2	1	no	—
116	Q₈×C₇₁	—	2	1	no	—
117	Q₈×C₇₁	—	2	1	no	—
118	Q₈×C₇₁	—	4	2	no	—
119	Q₈×C₇₁	#41	1	2	yes	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	2	1	no	—
122	C₂×C₄×C₇₁	—	2	1	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	—	2	1	no	—
125	C₂×C₄×C₇₁	—	2	1	no	—
126	C₂×C₄×C₇₁	—	4	2	no	—
127	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₇₁	Q₁₆	1	4	yes	complemented
130	C₂×Dic₇₁	—	2	1	no	—
131	C₂×Dic₇₁	—	2	1	no	—
132	C₂×Dic₇₁	—	2	1	no	—
133	C₂×Dic₇₁	—	2	1	no	—
134	C₂×Dic₇₁	—	8	1	no	—
135	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
138	C₄×D₇₁	—	2	1	no	—
139	C₄×D₇₁	—	2	1	no	—
140	C₄×D₇₁	—	4	2	no	—
141	C₄×D₇₁	—	8	1	no	—
142	Dic₁₄₂	—	2	1	no	—
143	Dic₁₄₂	—	2	1	no	—
144	Dic₁₄₂	—	4	2	no	—
145	Dic₁₄₂	—	8	1	no	—
146	C₈×C₇₁	—	2	1	no	—
147	C₈×C₇₁	—	2	1	no	—
148	C₇₁⋊C₈	—	2	1	no	—
149	C₇₁⋊C₈	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	—	2	1	no	—
152	C₂²×C₇₁	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₇₁	—	2	1	no	—
155	C₂×D₇₁	—	2	1	no	—
156	C₂×D₇₁	C₂×Q₁₆	1	2	yes	—
157	C₄×C₇₁	—	2	2	no	—
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	4	1	no	—
160	C₄×C₇₁	—	4	1	no	—
161	C₄×C₇₁	—	4	1	no	—
162	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
163	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₇₁	—	2	1	no	—
165	Dic₇₁	—	2	1	no	—
166	Dic₇₁	—	4	1	no	—
167	Dic₇₁	—	4	1	no	—
168	Dic₇₁	—	4	1	no	—
169	Dic₇₁	—	4	1	no	—
170	Dic₇₁	C₂×Q₁₆	1	2	yes	—
171	C₂×C₇₁	—	2	2	no	—
172	C₂×C₇₁	#546	1	1	yes	characteristic
173	C₂×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂×C₇₁	#577	1	1	yes	characteristic
175	D₇₁	—	2	2	no	—
176	#3299	—	71	1	no	maximal, Sylow
177	C₇₁	#3299	1	1	yes	characteristic, complemented, Sylow
178	#554	—	71	1	no	—
179	D₄×Q₈	—	71	1	no	—
180	#487	—	71	1	no	—
181	#484	—	71	1	no	—
182	#484	—	71	1	no	—
183	#477	—	71	1	no	—
184	#467	—	71	1	no	—
185	#467	—	71	1	no	—
186	#462	—	71	1	no	—
187	#451	—	71	1	no	—
188	#451	D₇₁	1	1	yes	characteristic, complemented
189	C₄×Q₁₆	—	71	1	no	—
190	C₈×D₄	—	71	1	no	—
191	C₂×#103	—	71	1	no	—
192	C₂×#103	—	71	1	no	—
193	C₂²×Q₈	—	71	1	no	—
194	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
195	C₂×Q₁₆	—	71	1	no	—
196	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
197	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
198	C₂²×C₈	—	71	1	no	—
199	C₂²×C₈	—	71	1	no	—
200	C₄⋊Q₈	—	71	1	no	—
201	C₄⋊Q₈	—	71	1	no	—
202	C₄⋊Q₈	—	142	1	no	—
203	#125	—	142	1	no	—
204	C₂²⋊Q₈	—	71	1	no	—
205	C₂²⋊Q₈	—	71	1	no	—
206	C₂²⋊Q₈	—	71	1	no	—
207	C₂²⋊Q₈	—	142	1	no	—
208	C₂²⋊Q₈	—	142	1	no	—
209	C₂²⋊Q₈	—	142	1	no	—
210	C₂²⋊Q₈	—	142	1	no	—
211	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
212	C₄×Q₈	—	71	1	no	—
213	C₄×Q₈	—	71	1	no	—
214	C₄×D₄	—	71	1	no	—
215	C₄×D₄	—	142	1	no	—
216	C₄×D₄	—	142	1	no	—
217	C₂×C₄⋊C₄	—	71	1	no	—
218	C₂×C₄⋊C₄	—	71	1	no	—
219	#107	—	71	1	no	—
220	#107	—	71	1	no	—
221	#107	—	71	1	no	—
222	C₄⋊C₈	—	71	1	no	—
223	#103	—	71	1	no	—
224	#103	—	71	1	no	—
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	C₂×D₇₁	1	1	yes	characteristic
232	#103	C₂×D₇₁	1	1	yes	characteristic
233	C₂²⋊C₈	—	71	1	no	—
234	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
235	C₄×C₈	—	71	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	284	1	no	—
240	C₂×Q₈	—	71	1	no	—
241	C₂×Q₈	—	142	1	no	—
242	C₂×Q₈	—	142	1	no	—
243	C₂×Q₈	—	142	2	no	—
244	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×D₄	—	71	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	71	1	no	—
250	C₂²×C₄	—	142	1	no	—
251	C₂²×C₄	—	142	1	no	—
252	C₂²×C₄	—	142	1	no	—
253	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	284	1	no	—
257	C₂×C₈	—	284	1	no	—
258	C₂×C₈	—	284	1	no	—
259	C₂×C₈	—	71	1	no	—
260	C₂×C₈	—	71	1	no	—
261	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
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	—	142	1	no	—
275	C₄⋊C₄	—	142	1	no	—
276	C₄⋊C₄	—	142	1	no	—
277	C₄⋊C₄	—	142	1	no	—
278	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	71	1	no	—
281	C₂²⋊C₄	—	71	1	no	—
282	C₂²⋊C₄	—	142	1	no	—
283	C₂²⋊C₄	—	142	1	no	—
284	C₂²⋊C₄	—	142	1	no	—
285	C₄²	—	71	1	no	—
286	C₄²	—	142	1	no	—
287	C₄²	—	142	1	no	—
288	C₂³	—	71	1	no	—
289	C₂³	D₄×D₇₁	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	284	2	no	—
297	Q₈	—	568	1	no	—
298	Q₈	—	142	1	no	—
299	Q₈	—	142	1	no	—
300	Q₈	#11765864	1	2	yes	—
301	D₄	—	71	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	284	2	no	—
310	C₂×C₄	—	568	1	no	—
311	C₂×C₄	—	568	1	no	—
312	C₂×C₄	—	71	1	no	—
313	C₂×C₄	—	71	1	no	—
314	C₂×C₄	—	71	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	—	142	1	no	—
320	C₂×C₄	—	142	1	no	—
321	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	142	1	no	—
326	C₈	—	142	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	71	2	no	—
330	C₂²	—	142	1	no	—
331	C₂²	—	142	1	no	—
332	C₂²	Q₁₆×D₇₁	1	2	yes	—
333	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	284	1	no	—
340	C₄	—	284	1	no	—
341	C₄	—	284	1	no	—
342	C₄	—	284	1	no	—
343	C₄	—	71	2	no	—
344	C₄	—	142	1	no	—
345	C₄	—	142	1	no	—
346	C₄	C₂×#11765864	1	1	yes	characteristic
347	C₄	#14556173	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	142	2	no	—
350	C₂	#17986636	1	1	yes	characteristic
351	C₂	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
352	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	#451×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#451	D₇₁
hypocenter	C₇₁	#3299
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
3. #17986348	C₂	semidirect
4. #17986335	C₂	semidirect
5. #17986333	C₂	semidirect
6. #17986292	C₂	semidirect
7. #17986238	C₂	semidirect
8. #17986237	C₂	semidirect
9. #17986207	C₂	semidirect
10. #17986153	C₂	semidirect
11. Q₁₆×Dic₇₁	C₂	semidirect
13. C₂×#14556052	C₂	semidirect
14. #103×D₇₁	C₂	semidirect
15. #451×C₇₁	C₂	semidirect
129. C₂²⋊D₇₁	Q₁₆	semidirect
177. C₇₁	#3299	semidirect
188. #451	D₇₁	semidirect