Subgroups — C₁₃₁×#29888 #45715772

Order 33536.

588 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3080×C₁₃₁	C₂	1	2	yes	maximal
3	#2922×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2090×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2089×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2079×C₁₃₁	C₂	1	2	yes	maximal
7	#2078×C₁₃₁	C₂	1	2	yes	maximal
8	#1998×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1837×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
11	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
12	#499×C₁₃₁	—	2	2	no	—
13	#499×C₁₃₁	—	2	2	no	—
14	#479×C₁₃₁	—	2	4	no	—
15	#478×C₁₃₁	—	2	4	no	—
16	#433×C₁₃₁	—	2	2	no	—
17	#431×C₁₃₁	—	2	2	no	—
18	#428×C₁₃₁	C₂²	1	4	yes	—
19	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
21	#424×C₁₃₁	C₂²	1	1	yes	characteristic
22	#424×C₁₃₁	C₂²	1	1	yes	characteristic
23	#423×C₁₃₁	C₂²	1	2	yes	—
24	#417×C₁₃₁	C₂²	1	4	yes	—
25	#408×C₁₃₁	C₂²	1	1	yes	characteristic
26	#408×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
29	#343×C₁₃₁	C₂²	1	4	yes	—
30	#343×C₁₃₁	C₂²	1	4	yes	—
31	#341×C₁₃₁	C₂²	1	2	yes	—
32	#341×C₁₃₁	C₂²	1	2	yes	—
33	C₂×#41×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	#125×C₁₃₁	—	2	4	no	—
40	#125×C₁₃₁	—	2	2	no	—
41	#125×C₁₃₁	—	2	2	no	—
42	C₄×Q₈×C₁₃₁	D₄	1	2	yes	—
43	C₄×D₄×C₁₃₁	—	2	2	no	—
44	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
45	#117×C₁₃₁	—	2	1	no	—
46	#117×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₃₁	—	2	1	no	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	#107×C₁₃₁	—	2	4	no	—
52	#107×C₁₃₁	—	2	4	no	—
53	#106×C₁₃₁	—	2	4	no	—
54	#106×C₁₃₁	—	2	4	no	—
55	C₄⋊C₈×C₁₃₁	—	2	2	no	—
56	C₄⋊C₈×C₁₃₁	—	2	2	no	—
57	C₄⋊C₈×C₁₃₁	—	2	4	no	—
58	#103×C₁₃₁	—	2	4	no	—
59	#102×C₁₃₁	—	2	4	no	—
60	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
61	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
62	#97×C₁₃₁	—	2	1	no	—
63	C₄×C₈×C₁₃₁	—	2	1	no	—
64	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
65	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
66	#41×C₁₃₁	—	2	2	no	—
67	#41×C₁₃₁	—	2	2	no	—
68	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×D₄×C₁₃₁	—	2	1	no	—
70	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₁	—	2	2	no	—
72	C₂²×C₄×C₁₃₁	—	2	1	no	—
73	C₂²×C₄×C₁₃₁	—	2	4	no	—
74	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
75	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₁₃₁	—	2	2	no	—
77	M₁₆×C₁₃₁	—	2	2	no	—
78	M₁₆×C₁₃₁	—	2	4	no	—
79	M₁₆×C₁₃₁	—	2	4	no	—
80	C₂×C₈×C₁₃₁	—	2	2	no	—
81	C₂×C₈×C₁₃₁	—	2	2	no	—
82	C₂×C₈×C₁₃₁	—	2	1	no	—
83	C₂×C₈×C₁₃₁	—	2	1	no	—
84	C₂×C₈×C₁₃₁	—	2	4	no	—
85	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₁₃₁	—	2	2	no	—
88	C₄⋊C₄×C₁₃₁	—	2	4	no	—
89	C₄⋊C₄×C₁₃₁	—	2	4	no	—
90	C₄⋊C₄×C₁₃₁	—	4	4	no	—
91	C₄⋊C₄×C₁₃₁	—	4	4	no	—
92	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
94	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
95	C₄²×C₁₃₁	—	2	2	no	—
96	C₄²×C₁₃₁	—	2	1	no	—
97	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
100	C₂³×C₁₃₁	—	2	1	no	—
101	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
102	Q₈×C₁₃₁	—	2	2	no	—
103	D₄×C₁₃₁	—	2	2	no	—
104	D₄×C₁₃₁	—	4	2	no	—
105	C₂×C₄×C₁₃₁	—	2	2	no	—
106	C₂×C₄×C₁₃₁	—	2	1	no	—
107	C₂×C₄×C₁₃₁	—	4	2	no	—
108	C₂×C₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	4	1	no	—
110	C₂×C₄×C₁₃₁	—	4	1	no	—
111	C₂×C₄×C₁₃₁	—	4	4	no	—
112	C₂×C₄×C₁₃₁	—	8	4	no	—
113	C₂×C₄×C₁₃₁	#121	1	2	yes	—
114	C₂×C₄×C₁₃₁	#121	1	2	yes	—
115	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
118	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
119	C₈×C₁₃₁	—	2	2	no	—
120	C₈×C₁₃₁	—	2	2	no	—
121	C₈×C₁₃₁	—	4	1	no	—
122	C₈×C₁₃₁	—	4	1	no	—
123	C₈×C₁₃₁	—	4	4	no	—
124	C₂²×C₁₃₁	—	2	2	no	—
125	C₂²×C₁₃₁	—	4	1	no	—
126	C₂²×C₁₃₁	—	4	2	no	—
127	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
128	C₂²×C₁₃₁	#541	1	2	yes	—
129	C₄×C₁₃₁	—	2	2	no	—
130	C₄×C₁₃₁	—	2	2	no	—
131	C₄×C₁₃₁	—	4	1	no	—
132	C₄×C₁₃₁	—	4	2	no	—
133	C₄×C₁₃₁	—	8	4	no	—
134	C₄×C₁₃₁	#536	1	1	yes	characteristic
135	C₄×C₁₃₁	#536	1	1	yes	characteristic
136	C₄×C₁₃₁	#536	1	2	yes	—
137	C₂×C₁₃₁	—	2	2	no	—
138	C₂×C₁₃₁	—	8	1	no	—
139	C₂×C₁₃₁	#2621	1	1	yes	characteristic
140	C₂×C₁₃₁	#3067	1	2	yes	—
141	#29888	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₁	#29888	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic, complemented
144	#3080	C₂×C₁₃₁	1	2	yes	—
145	#2922	C₂×C₁₃₁	1	1	yes	characteristic
146	#2090	C₂×C₁₃₁	1	2	yes	complemented
147	#2089	C₂×C₁₃₁	1	2	yes	complemented
148	#2079	C₂×C₁₃₁	1	2	yes	—
149	#2078	C₂×C₁₃₁	1	2	yes	—
150	#1998	C₂×C₁₃₁	1	2	yes	complemented
151	#1837	C₂×C₁₃₁	1	1	yes	characteristic, complemented
152	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
153	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
154	#499	—	2	2	no	—
155	#499	—	2	2	no	—
156	#479	—	2	4	no	—
157	#478	—	2	4	no	—
158	#433	—	2	2	no	—
159	#431	—	2	2	no	—
160	#428	C₂²×C₁₃₁	1	4	yes	—
161	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
162	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
163	#424	C₂²×C₁₃₁	1	1	yes	characteristic
164	#424	C₂²×C₁₃₁	1	1	yes	characteristic
165	#423	C₂²×C₁₃₁	1	2	yes	—
166	#417	C₂²×C₁₃₁	1	4	yes	—
167	#408	C₂²×C₁₃₁	1	1	yes	characteristic
168	#408	C₂²×C₁₃₁	1	1	yes	characteristic
169	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
171	#343	C₂²×C₁₃₁	1	4	yes	—
172	#343	C₂²×C₁₃₁	1	4	yes	—
173	#341	C₂²×C₁₃₁	1	2	yes	—
174	#341	C₂²×C₁₃₁	1	2	yes	—
175	C₂×#41	C₂³×C₁₃₁	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
177	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
178	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
181	#125	—	2	4	no	—
182	#125	—	2	2	no	—
183	#125	—	2	2	no	—
184	C₄×Q₈	D₄×C₁₃₁	1	2	yes	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
187	#117	—	2	1	no	—
188	#117	C₂³×C₁₃₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
193	#107	—	2	4	no	—
194	#107	—	2	4	no	—
195	#106	—	2	4	no	—
196	#106	—	2	4	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	4	no	—
200	#103	—	2	4	no	—
201	#102	—	2	4	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
207	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
235	C₂²⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	#121×C₁₃₁	1	2	yes	—
256	C₂×C₄	#121×C₁₃₁	1	2	yes	—
257	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
259	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
260	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
270	C₂²	#541×C₁₃₁	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	8	4	no	—
276	C₄	#536×C₁₃₁	1	1	yes	characteristic
277	C₄	#536×C₁₃₁	1	1	yes	characteristic
278	C₄	#536×C₁₃₁	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	1	no	—
281	C₂	#2621×C₁₃₁	1	1	yes	characteristic
282	C₂	#3067×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₁	C₂	semidirect
4. #2090×C₁₃₁	C₂	semidirect
5. #2089×C₁₃₁	C₂	semidirect
8. #1998×C₁₃₁	C₂	semidirect
9. #1837×C₁₃₁	C₂	semidirect
141. #29888	C₁₃₁	direct (N × H)
142. C₁₃₁	#29888	direct (N × H)
143. C₂×#499	C₂×C₁₃₁	semidirect
146. #2090	C₂×C₁₃₁	semidirect
147. #2089	C₂×C₁₃₁	semidirect
150. #1998	C₂×C₁₃₁	semidirect
151. #1837	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₁	2
#3080×C₁₃₁	2
#2922×C₁₃₁	2
#2090×C₁₃₁	2
#2089×C₁₃₁	2
#2079×C₁₃₁	2
#2078×C₁₃₁	2
#1998×C₁₃₁	2
#1837×C₁₃₁	2
#29888	131

Subgroups — C131×#29888 #45715772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29888 #45715772