Subgroups — C₁₃₁×#13270 #45699154

Order 33536.

316 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2153×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2110×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1378×C₁₃₁	C₂	1	2	yes	maximal
5	#1376×C₁₃₁	C₂	1	2	yes	maximal
6	#533×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
8	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
9	#446×C₁₃₁	—	2	2	no	—
10	C₈×Q₈×C₁₃₁	—	2	2	no	—
11	#424×C₁₃₁	C₄	1	1	yes	characteristic
12	#424×C₁₃₁	C₄	1	1	yes	characteristic
13	#423×C₁₃₁	C₄	1	1	yes	characteristic
14	#405×C₁₃₁	C₂²	1	1	yes	characteristic
15	#405×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	#364×C₁₃₁	—	2	2	no	—
18	#347×C₁₃₁	—	2	2	no	—
19	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
24	#125×C₁₃₁	—	2	1	no	—
25	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
26	C₄×Q₈×C₁₃₁	—	2	2	no	—
27	#117×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	#117×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₁₃₁	—	2	2	no	—
31	M₃₂×C₁₃₁	—	2	2	no	—
32	M₃₂×C₁₃₁	—	2	2	no	—
33	M₃₂×C₁₃₁	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₁₃₁	—	2	2	no	—
35	C₂×C₁₆×C₁₃₁	—	2	2	no	—
36	C₄⋊C₈×C₁₃₁	—	2	2	no	—
37	C₄⋊C₈×C₁₃₁	—	2	2	no	—
38	C₄⋊C₈×C₁₃₁	—	2	2	no	—
39	#97×C₁₃₁	—	2	1	no	—
40	#97×C₁₃₁	D₄	1	2	yes	—
41	C₄×C₈×C₁₃₁	—	2	1	no	—
42	C₄×C₈×C₁₃₁	—	2	1	no	—
43	C₄×C₈×C₁₃₁	—	2	1	no	—
44	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
45	C₂×Q₈×C₁₃₁	—	2	2	no	—
46	C₂²×C₄×C₁₃₁	—	2	1	no	—
47	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₁₃₁	—	2	2	no	—
49	M₁₆×C₁₃₁	—	2	2	no	—
50	C₂×C₈×C₁₃₁	—	2	1	no	—
51	C₂×C₈×C₁₃₁	—	2	1	no	—
52	C₂×C₈×C₁₃₁	—	2	2	no	—
53	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₁₃₁	#41	1	2	yes	—
58	C₄⋊C₄×C₁₃₁	—	2	4	no	—
59	C₄⋊C₄×C₁₃₁	—	2	2	no	—
60	C₄⋊C₄×C₁₃₁	—	2	1	no	—
61	C₄⋊C₄×C₁₃₁	—	2	1	no	—
62	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
63	C₂²⋊C₄×C₁₃₁	M₁₆	1	2	yes	—
64	C₄²×C₁₃₁	—	2	1	no	—
65	C₄²×C₁₃₁	—	2	1	no	—
66	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
68	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
69	C₁₆×C₁₃₁	—	2	4	no	—
70	C₁₆×C₁₃₁	—	4	2	no	—
71	C₂³×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
72	Q₈×C₁₃₁	—	4	4	no	—
73	C₂×C₄×C₁₃₁	—	2	1	no	—
74	C₂×C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄×C₁₃₁	—	2	2	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	4	2	no	—
78	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃₁	#131	1	2	yes	—
82	C₈×C₁₃₁	—	2	2	no	—
83	C₈×C₁₃₁	—	4	1	no	—
84	C₈×C₁₃₁	—	4	1	no	—
85	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₁₃₁	#124	1	1	yes	characteristic
87	C₈×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₁₃₁	—	4	1	no	—
89	C₂²×C₁₃₁	#407	1	1	yes	characteristic
90	C₂²×C₁₃₁	#446	1	2	yes	—
91	C₄×C₁₃₁	—	2	2	no	—
92	C₄×C₁₃₁	—	4	1	no	—
93	C₄×C₁₃₁	—	4	2	no	—
94	C₄×C₁₃₁	—	4	2	no	—
95	C₄×C₁₃₁	#391	1	1	yes	characteristic
96	C₄×C₁₃₁	#408	1	1	yes	characteristic
97	C₄×C₁₃₁	#446	1	2	yes	—
98	C₂×C₁₃₁	—	2	1	no	—
99	C₂×C₁₃₁	—	2	2	no	—
100	C₂×C₁₃₁	#1985	1	1	yes	characteristic
101	#13270	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₃₁	#13270	1	1	yes	characteristic, direct product factor, Sylow
103	#2960	C₂×C₁₃₁	1	1	yes	characteristic
104	#2153	C₂×C₁₃₁	1	1	yes	characteristic
105	#2110	C₂×C₁₃₁	1	1	yes	characteristic
106	#1378	C₂×C₁₃₁	1	2	yes	—
107	#1376	C₂×C₁₃₁	1	2	yes	—
108	#533	C₄×C₁₃₁	1	1	yes	characteristic
109	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
110	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
111	#446	—	2	2	no	—
112	C₈×Q₈	—	2	2	no	—
113	#424	C₄×C₁₃₁	1	1	yes	characteristic
114	#424	C₄×C₁₃₁	1	1	yes	characteristic
115	#423	C₄×C₁₃₁	1	1	yes	characteristic
116	#405	C₂²×C₁₃₁	1	1	yes	characteristic
117	#405	C₂²×C₁₃₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
119	#364	—	2	2	no	—
120	#347	—	2	2	no	—
121	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
122	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	1	no	—
126	#125	—	2	1	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	C₂×C₄×C₁₃₁	1	1	yes	characteristic
130	#117	C₂×C₄×C₁₃₁	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	Q₈×C₁₃₁	1	4	yes	complemented
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	#97	—	2	1	no	—
142	#97	D₄×C₁₃₁	1	2	yes	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
147	C₂×Q₈	—	2	2	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
157	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
159	C₂×C₈	#41×C₁₃₁	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	M₁₆×C₁₃₁	1	2	yes	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
169	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
170	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	4	2	no	—
173	C₂³	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
182	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
183	C₂×C₄	#131×C₁₃₁	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	1	no	—
186	C₈	—	4	1	no	—
187	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
188	C₈	#124×C₁₃₁	1	1	yes	characteristic
189	C₈	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
190	C₂²	—	4	1	no	—
191	C₂²	#407×C₁₃₁	1	1	yes	characteristic
192	C₂²	#446×C₁₃₁	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	#391×C₁₃₁	1	1	yes	characteristic
198	C₄	#408×C₁₃₁	1	1	yes	characteristic
199	C₄	#446×C₁₃₁	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	#1985×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₁	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#1985
prime radical	C₂³×C₁₃₁	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
33. M₃₂×C₁₃₁	Q₈	semidirect
101. #13270	C₁₃₁	direct (N × H)
102. C₁₃₁	#13270	direct (N × H)
135. M₃₂	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2960×C₁₃₁	2
#2153×C₁₃₁	2
#2110×C₁₃₁	2
#1378×C₁₃₁	2
#1376×C₁₃₁	2
#13270	131

Subgroups — C131×#13270 #45699154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13270 #45699154