Subgroups — #31912646

Order 18176.

460 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590025	C₂	1	1	yes	maximal, characteristic
2	#22590005	C₂	1	1	yes	maximal, characteristic
3	#22588140	C₂	1	2	yes	maximal, complemented
4	#22586469	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
6	#17985983	C₂²	1	2	yes	—
7	#17985973	C₂²	1	2	yes	—
8	#17985954	—	2	2	no	—
9	#17985798	C₄	1	2	yes	complemented
10	#17985796	C₄	1	2	yes	complemented
11	#17985794	C₄	1	2	yes	—
12	#17985792	C₄	1	2	yes	—
13	#17985770	—	2	2	no	—
14	C₂×#131×C₇₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
16	#532×C₇₁	C₂²	1	1	yes	characteristic
17	#436×C₇₁	—	2	2	no	—
18	C₈×D₄×C₇₁	—	2	2	no	—
19	#423×C₇₁	C₄	1	1	yes	characteristic, complemented
20	#408×C₇₁	C₄	1	1	yes	characteristic, complemented
21	#408×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic, complemented
23	C₂×#11765849	—	2	2	no	—
24	#14556087	—	2	2	no	—
25	#14556047	C₂×C₄	1	4	yes	—
26	#14556045	—	2	2	no	—
27	#14556043	—	2	2	no	—
28	#14556043	C₂×C₄	1	4	yes	—
29	#14556038	—	4	2	no	—
30	C₈×Dic₇₁	—	4	2	no	—
31	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
33	#131×C₇₁	—	4	1	no	—
34	#131×C₇₁	—	4	1	no	—
35	#131×C₇₁	—	4	1	no	—
36	#131×C₇₁	—	4	1	no	—
37	C₂×M₁₆×C₇₁	—	2	1	no	—
38	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
41	C₂²×C₈×C₇₁	—	2	1	no	—
42	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	D₄	1	2	yes	—
45	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
46	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
47	#124×C₇₁	C₂×C₄	1	1	yes	characteristic
48	#124×C₇₁	C₂×C₄	1	1	yes	characteristic
49	#121×C₇₁	—	2	2	no	—
50	C₄×D₄×C₇₁	—	2	2	no	—
51	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₇₁	—	2	2	no	—
53	C₂²⋊C₈×C₇₁	—	2	2	no	—
54	C₂²⋊C₈×C₇₁	—	2	2	no	—
55	C₄×C₈×C₇₁	—	2	1	no	—
56	C₄×C₈×C₇₁	—	2	1	no	—
57	#11765849	—	2	4	no	—
58	#11765849	—	4	2	no	—
59	C₂×C₇₁⋊C₈	—	4	2	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₂²⋊Dic₇₁	—	2	4	no	—
62	C₄×Dic₇₁	—	4	2	no	—
63	#41×C₇₁	—	4	1	no	—
64	#41×C₇₁	—	4	1	no	—
65	#41×C₇₁	—	4	1	no	—
66	#41×C₇₁	—	4	1	no	—
67	C₂×Q₈×C₇₁	C₄²	1	1	yes	characteristic
68	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×D₄×C₇₁	—	2	1	no	—
70	C₂×D₄×C₇₁	—	2	2	no	—
71	C₂×D₄×C₇₁	C₄²	1	1	yes	characteristic
72	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×D₄×C₇₁	#41	1	2	yes	—
74	C₂²×C₄×C₇₁	—	2	1	no	—
75	C₂²×C₄×C₇₁	—	2	1	no	—
76	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
78	M₁₆×C₇₁	—	4	1	no	—
79	M₁₆×C₇₁	—	4	2	no	—
80	M₁₆×C₇₁	—	4	2	no	—
81	M₁₆×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	—	2	1	no	—
87	C₂×C₈×C₇₁	—	2	1	no	—
88	C₂×C₈×C₇₁	—	4	1	no	—
89	C₂×C₈×C₇₁	—	4	1	no	—
90	C₂×C₈×C₇₁	—	4	1	no	—
91	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₇₁	—	2	2	no	—
95	C₂²⋊C₄×C₇₁	—	2	2	no	—
96	C₂²⋊C₄×C₇₁	—	2	2	no	—
97	C₄²×C₇₁	—	2	1	no	—
98	C₄²×C₇₁	C₄²	1	1	yes	characteristic
99	C₄²×C₇₁	C₄²	1	1	yes	characteristic
100	C₂³×C₇₁	—	2	1	no	—
101	C₂³×C₇₁	#117	1	1	yes	characteristic
102	C₂³×C₇₁	C₄×D₄	1	2	yes	—
103	Q₈×C₇₁	—	4	1	no	—
104	Q₈×C₇₁	—	4	1	no	—
105	D₄×C₇₁	—	4	1	no	—
106	D₄×C₇₁	—	4	2	no	—
107	D₄×C₇₁	—	4	1	no	—
108	D₄×C₇₁	—	4	4	no	—
109	D₄×C₇₁	—	4	2	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	4	1	no	—
119	C₂×C₄×C₇₁	—	4	1	no	—
120	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
123	C₂×Dic₇₁	—	4	4	no	—
124	C₈×C₇₁	—	2	2	no	—
125	C₈×C₇₁	—	2	1	no	—
126	C₈×C₇₁	—	4	1	no	—
127	C₈×C₇₁	—	4	1	no	—
128	C₈×C₇₁	#99	1	1	yes	characteristic
129	C₈×C₇₁	#99	1	1	yes	characteristic
130	C₇₁⋊C₈	—	4	4	no	—
131	C₂²×C₇₁	—	2	2	no	—
132	C₂²×C₇₁	—	2	1	no	—
133	C₂²×C₇₁	—	4	1	no	—
134	C₂²×C₇₁	—	4	2	no	—
135	C₂²×C₇₁	—	4	2	no	—
136	C₂²×C₇₁	—	4	1	no	—
137	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₇₁	—	2	2	no	—
139	C₄×C₇₁	—	2	2	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	4	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	C₂×#99	1	1	yes	characteristic
144	C₄×C₇₁	#410	1	1	yes	characteristic
145	Dic₇₁	—	8	4	no	—
146	#12633	—	71	1	no	maximal, Sylow
147	C₂×C₇₁	—	2	1	no	—
148	C₂×C₇₁	—	4	2	no	—
149	C₂×C₇₁	—	4	1	no	—
150	C₂×C₇₁	—	8	1	no	—
151	C₂×C₇₁	C₄×#99	1	1	yes	characteristic
152	#2931	D₇₁	1	1	yes	characteristic
153	#2132	—	71	1	no	—
154	#2126	—	71	1	no	—
155	#1757	—	71	2	no	—
156	#1395	—	71	2	no	—
157	C₇₁	#12633	1	1	yes	characteristic, complemented, Sylow
158	C₂×#131	Dic₇₁	1	1	yes	characteristic, complemented
159	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
160	#532	C₂×D₇₁	1	1	yes	characteristic
161	#436	—	2	2	no	—
162	C₈×D₄	—	2	2	no	—
163	#423	Dic₇₁	1	1	yes	characteristic, complemented
164	#413	—	71	2	no	—
165	#410	—	71	2	no	—
166	#408	Dic₇₁	1	1	yes	characteristic, complemented
167	#408	C₂×D₇₁	1	1	yes	characteristic
168	#405	—	142	2	no	—
169	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic, complemented
170	#356	—	71	2	no	—
171	#355	—	71	2	no	—
172	#354	—	71	2	no	—
173	#353	—	71	2	no	—
174	#344	—	142	2	no	—
175	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
176	C₂×#41	C₂²×D₇₁	1	1	yes	characteristic
177	#131	—	4	1	no	—
178	#131	—	4	1	no	—
179	#131	—	4	1	no	—
180	#131	—	4	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	142	2	no	—
183	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
184	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
185	C₂×M₁₆	C₂²⋊D₇₁	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
190	C₄⋊Q₈	C₄×D₇₁	1	1	yes	characteristic
191	#127	C₄×D₇₁	1	1	yes	characteristic
192	#124	C₄×D₇₁	1	1	yes	characteristic
193	#124	C₄×D₇₁	1	1	yes	characteristic
194	#121	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	—	142	2	no	—
197	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	#101	—	71	4	no	—
200	#100	—	142	2	no	—
201	#99	—	71	4	no	—
202	#99	—	142	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	#97	—	284	2	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	284	2	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	C₄×Dic₇₁	1	1	yes	characteristic
214	C₂×Q₈	C₂×C₄×D₇₁	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₄×Dic₇₁	1	1	yes	characteristic
218	C₂×D₄	C₂×C₄×D₇₁	1	1	yes	characteristic
219	C₂×D₄	#11765865	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	284	2	no	—
229	M₁₆	—	142	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	284	2	no	—
240	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	284	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	142	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	284	2	no	—
250	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
251	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#14556087	1	1	yes	characteristic
254	C₂³	C₄×C₂²⋊D₇₁	1	2	yes	—
255	Q₈	—	4	1	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	284	4	no	—
273	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
275	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
276	C₈	—	2	1	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	284	4	no	—
281	C₈	#14556043	1	1	yes	characteristic
282	C₈	#14556043	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊Dic₇₁	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	568	4	no	—
296	C₄	C₂×#14556043	1	1	yes	characteristic
297	C₄	#17985973	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	4	2	no	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₄×#14556043	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556043
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×#41	C₂²×D₇₁
Fitting	C₇₁×#2931	C₂
socle	C₂×C₇₁	C₄×#99
prime radical	#532×C₇₁	C₂²
cosocle	C₂×#41×C₇₁	C₂³
hypercenter	#2931	D₇₁
hypocenter	C₇₁	#12633
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2931

Complements

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

N	H ≅ G/N	type
3. #22588140	C₂	semidirect
4. #22586469	C₂	semidirect
9. #17985798	C₄	semidirect
10. #17985796	C₄	semidirect
14. C₂×#131×C₇₁	C₄	semidirect
19. #423×C₇₁	C₄	semidirect
20. #408×C₇₁	C₄	semidirect
22. C₂×C₄×C₈×C₇₁	C₄	semidirect
157. C₇₁	#12633	semidirect
158. C₂×#131	Dic₇₁	semidirect
163. #423	Dic₇₁	semidirect
166. #408	Dic₇₁	semidirect
169. C₂×C₄×C₈	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22590025	2
#22590005	2
#22588140	2
#22586469	2
C₇₁×#2931	2
#12633	71