Subgroups — #31891968

Order 18176.

356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597442	C₂	1	1	yes	maximal, characteristic
2	#22589318	C₂	1	1	yes	maximal, characteristic
3	#22587129	C₂	1	1	yes	maximal, characteristic
4	#22587129	C₂	1	1	yes	maximal, characteristic
5	#22586172	C₂	1	1	yes	maximal, characteristic
6	#22586172	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1544	C₂	1	1	yes	maximal, characteristic
8	#17986130	—	2	1	no	—
9	#17986130	—	2	1	no	—
10	Q₈×C₇₁⋊C₈	—	2	1	no	—
11	Q₈×C₇₁⋊C₈	—	2	1	no	—
12	#17986060	—	2	2	no	—
13	#17986052	C₂²	1	1	yes	characteristic
14	#17986052	C₂²	1	1	yes	characteristic
15	#17986023	C₂²	1	1	yes	characteristic
16	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
17	#17985758	—	2	2	no	—
18	#17985752	—	2	1	no	—
19	#17985752	—	2	1	no	—
20	#17985750	—	2	1	no	—
21	#17985750	—	2	1	no	—
22	#533×C₇₁	C₂²	1	1	yes	characteristic
23	#431×C₇₁	C₂²	1	1	yes	characteristic
24	#431×C₇₁	C₂²	1	1	yes	characteristic
25	#332×C₇₁	C₄	1	1	yes	characteristic, complemented
26	#332×C₇₁	C₄	1	1	yes	characteristic, complemented
27	#332×C₇₁	C₄	1	1	yes	characteristic, complemented
28	#332×C₇₁	C₄	1	1	yes	characteristic, complemented
29	C₂×#11765849	—	2	1	no	—
30	C₂×C₄×Dic₇₁	—	2	1	no	—
31	C₂×C₄×Dic₇₁	—	2	1	no	—
32	C₂×C₄×Dic₇₁	—	2	1	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
36	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
37	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
38	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
39	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
40	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
41	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
42	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
43	#14556039	—	2	2	no	—
44	C₄×C₇₁⋊C₈	—	2	2	no	—
45	C₄×C₇₁⋊C₈	—	2	1	no	—
46	C₄×C₇₁⋊C₈	—	2	1	no	—
47	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
49	#125×C₇₁	Q₈	1	2	yes	—
50	C₂²⋊Q₈×C₇₁	—	2	1	no	—
51	C₂²⋊Q₈×C₇₁	—	2	1	no	—
52	C₄×Q₈×C₇₁	—	2	1	no	—
53	C₄×Q₈×C₇₁	—	2	1	no	—
54	#117×C₇₁	D₄	1	2	yes	—
55	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₈×C₇₁	—	2	1	no	—
61	C₂²⋊C₈×C₇₁	—	2	1	no	—
62	C₄×C₈×C₇₁	—	2	1	no	—
63	C₄×C₈×C₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	4	1	no	—
65	#11765849	—	4	2	no	—
66	C₂×C₇₁⋊C₈	—	2	1	no	—
67	C₂×C₇₁⋊C₈	—	2	1	no	—
68	C₂×C₇₁⋊C₈	—	2	1	no	—
69	C₂×C₇₁⋊C₈	—	2	1	no	—
70	C₂×C₇₁⋊C₈	—	4	1	no	—
71	C₄×Dic₇₁	—	2	2	no	—
72	C₄×Dic₇₁	—	2	2	no	—
73	C₄×Dic₇₁	—	2	2	no	—
74	C₄×Dic₇₁	—	4	1	no	—
75	C₄×Dic₇₁	—	4	1	no	—
76	C₄×Dic₇₁	—	4	1	no	—
77	C₂×Q₈×C₇₁	—	2	1	no	—
78	C₂×Q₈×C₇₁	—	2	1	no	—
79	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	2	1	no	—
86	C₄⋊C₄×C₇₁	—	2	1	no	—
87	C₄⋊C₄×C₇₁	—	2	1	no	—
88	C₄⋊C₄×C₇₁	—	2	2	no	—
89	C₄⋊C₄×C₇₁	—	2	2	no	—
90	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
92	C₂²⋊C₄×C₇₁	—	2	2	no	—
93	C₄²×C₇₁	—	2	2	no	—
94	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
98	C₂³×C₇₁	#123	1	1	yes	characteristic
99	Q₈×C₇₁	—	4	1	no	—
100	Q₈×C₇₁	—	4	1	no	—
101	Q₈×C₇₁	—	4	1	no	—
102	Q₈×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₇₁	—	4	2	no	—
118	C₂×Dic₇₁	—	4	2	no	—
119	C₂×Dic₇₁	—	4	2	no	—
120	C₈×C₇₁	—	4	1	no	—
121	C₈×C₇₁	—	4	1	no	—
122	C₈×C₇₁	—	4	1	no	—
123	C₈×C₇₁	—	4	1	no	—
124	C₇₁⋊C₈	—	4	1	no	—
125	C₇₁⋊C₈	—	4	1	no	—
126	C₇₁⋊C₈	—	4	1	no	—
127	C₇₁⋊C₈	—	4	1	no	—
128	C₇₁⋊C₈	—	8	1	no	—
129	C₂²×C₇₁	—	2	1	no	—
130	C₂²×C₇₁	—	2	1	no	—
131	C₂²×C₇₁	—	2	1	no	—
132	C₂²×C₇₁	#385	1	1	yes	characteristic
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	2	1	no	—
136	C₄×C₇₁	—	4	2	no	—
137	C₄×C₇₁	—	4	2	no	—
138	C₄×C₇₁	#443	1	1	yes	characteristic
139	C₄×C₇₁	#443	1	1	yes	characteristic
140	C₄×C₇₁	#444	1	1	yes	characteristic
141	C₄×C₇₁	#444	1	1	yes	characteristic
142	C₄×C₇₁	#454	1	1	yes	characteristic
143	C₄×C₇₁	#456	1	1	yes	characteristic
144	Dic₇₁	—	4	4	no	—
145	#8936	—	71	1	no	maximal, Sylow
146	C₂×C₇₁	—	4	1	no	—
147	C₂×C₇₁	#1857	1	1	yes	characteristic
148	C₂×C₇₁	#1866	1	1	yes	characteristic
149	C₂×C₇₁	#1866	1	1	yes	characteristic
150	#2986	—	71	1	no	—
151	#1994	—	71	1	no	—
152	#1574	—	71	1	no	—
153	#1574	—	71	1	no	—
154	#1544	D₇₁	1	1	yes	characteristic
155	#1276	—	71	1	no	—
156	#1276	—	71	1	no	—
157	C₇₁	#8936	1	1	yes	characteristic, complemented, Sylow
158	#533	C₂×D₇₁	1	1	yes	characteristic
159	#446	—	142	1	no	—
160	#446	—	142	1	no	—
161	C₈×Q₈	—	142	1	no	—
162	C₈×Q₈	—	142	1	no	—
163	#433	—	142	2	no	—
164	#431	—	71	1	no	—
165	#431	—	71	1	no	—
166	#431	C₂×D₇₁	1	1	yes	characteristic
167	#431	C₂×D₇₁	1	1	yes	characteristic
168	#423	—	71	1	no	—
169	C₄³	—	71	1	no	—
170	#339	—	142	2	no	—
171	#335	—	142	1	no	—
172	#335	—	142	1	no	—
173	#334	—	142	1	no	—
174	#334	—	142	1	no	—
175	#332	Dic₇₁	1	1	yes	characteristic, complemented
176	#332	Dic₇₁	1	1	yes	characteristic, complemented
177	#332	Dic₇₁	1	1	yes	characteristic, complemented
178	#332	Dic₇₁	1	1	yes	characteristic, complemented
179	C₂×M₁₆	—	142	1	no	—
180	C₄⋊Q₈	C₂×Dic₇₁	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×Dic₇₁	1	1	yes	characteristic
182	#125	Dic₁₄₂	1	2	yes	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	—	2	1	no	—
187	#117	D₂₈₄	1	2	yes	—
188	C₂×C₄²	—	142	1	no	—
189	C₂×C₄²	—	142	1	no	—
190	C₂×C₄²	—	142	1	no	—
191	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
192	C₄⋊C₈	—	71	1	no	—
193	C₄⋊C₈	—	71	1	no	—
194	C₄⋊C₈	—	71	1	no	—
195	C₄⋊C₈	—	71	1	no	—
196	C₄⋊C₈	—	142	2	no	—
197	C₄⋊C₈	—	142	2	no	—
198	C₄⋊C₈	—	142	1	no	—
199	C₄⋊C₈	—	142	1	no	—
200	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
203	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	142	1	no	—
207	C₂²⋊C₈	—	142	1	no	—
208	#97	—	142	2	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	142	2	no	—
212	C₄×C₈	—	142	1	no	—
213	C₄×C₈	—	142	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂²×C₄	—	284	1	no	—
217	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
218	C₂²×C₄	#11765864	1	1	yes	characteristic
219	C₂²×C₄	#11765864	1	1	yes	characteristic
220	M₁₆	—	284	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	284	1	no	—
226	C₂×C₈	—	142	1	no	—
227	C₂×C₈	—	142	1	no	—
228	C₂×C₈	—	142	1	no	—
229	C₂×C₈	—	142	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
235	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	284	1	no	—
239	C₄²	—	284	1	no	—
240	C₄²	—	284	1	no	—
241	C₄²	—	142	2	no	—
242	C₄²	—	142	2	no	—
243	C₄²	—	142	2	no	—
244	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
245	C₄²	D₄×D₇₁	1	1	yes	characteristic
246	C₄²	D₄×D₇₁	1	1	yes	characteristic
247	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
248	C₂³	#14556112	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	1	no	—
252	Q₈	—	4	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	284	2	no	—
262	C₂×C₄	—	284	2	no	—
263	C₂×C₄	—	284	2	no	—
264	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
265	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
267	C₂×C₄	#14556098	1	1	yes	characteristic
268	C₂×C₄	#14556105	1	1	yes	characteristic
269	C₂×C₄	#14556105	1	1	yes	characteristic
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	284	1	no	—
275	C₈	—	284	1	no	—
276	C₈	—	284	1	no	—
277	C₈	—	284	1	no	—
278	C₈	—	568	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#17985874	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	284	4	no	—
289	C₄	#17986116	1	1	yes	characteristic
290	C₄	#17986116	1	1	yes	characteristic
291	C₄	#17986121	1	1	yes	characteristic
292	C₄	#17986121	1	1	yes	characteristic
293	C₄	#17986174	1	1	yes	characteristic
294	C₄	#17986184	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	#22588557	1	1	yes	characteristic
297	C₂	#22588617	1	1	yes	characteristic
298	C₂	#22588617	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556098
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1544	C₂
socle	C₂²×C₇₁	#385
prime radical	C₂³×C₇₁	#123
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1544	D₇₁
hypocenter	C₇₁	#8936
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

Complements

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

N	H ≅ G/N	type
25. #332×C₇₁	C₄	semidirect
26. #332×C₇₁	C₄	semidirect
27. #332×C₇₁	C₄	semidirect
28. #332×C₇₁	C₄	semidirect
157. C₇₁	#8936	semidirect
175. #332	Dic₇₁	semidirect
176. #332	Dic₇₁	semidirect
177. #332	Dic₇₁	semidirect
178. #332	Dic₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22597442	2
#22589318	2
#22587129	2
#22586172	2
C₇₁×#1544	2
#8936	71