Subgroups — C₁₃₁×#8268 #45694152

Order 33536.

390 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1564×C₁₃₁	C₂	1	2	yes	maximal
3	#1450×C₁₃₁	C₂	1	2	yes	maximal
4	#1276×C₁₃₁	C₂	1	2	yes	maximal
5	#431×C₁₃₁	C₂²	1	4	yes	—
6	C₂×#97×C₁₃₁	C₄	1	2	yes	—
7	C₂×#97×C₁₃₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
11	#335×C₁₃₁	C₄	1	4	yes	complemented
12	#334×C₁₃₁	C₄	1	4	yes	complemented
13	#322×C₁₃₁	—	2	2	no	—
14	C₈²×C₁₃₁	—	2	2	no	—
15	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃₁	—	2	2	no	—
19	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
22	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
23	#97×C₁₃₁	—	2	4	no	—
24	#97×C₁₃₁	C₈	1	8	yes	complemented
25	C₄×C₈×C₁₃₁	—	2	4	no	—
26	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
29	C₄×C₈×C₁₃₁	Q₈	1	2	yes	—
30	C₂²×C₄×C₁₃₁	—	2	2	no	—
31	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
33	C₂×C₈×C₁₃₁	—	2	2	no	—
34	C₂×C₈×C₁₃₁	—	2	8	no	—
35	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	C₄²	1	2	yes	—
38	C₂×C₈×C₁₃₁	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₁₃₁	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₁₃₁	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₃₁	M₁₆	1	4	yes	—
42	C₄²×C₁₃₁	—	2	2	no	—
43	C₄²×C₁₃₁	—	4	4	no	—
44	C₄²×C₁₃₁	C₂×C₈	1	4	yes	—
45	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
46	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂³×C₁₃₁	#117	1	1	yes	characteristic
49	C₂×C₄×C₁₃₁	—	2	4	no	—
50	C₂×C₄×C₁₃₁	—	2	4	no	—
51	C₂×C₄×C₁₃₁	—	4	4	no	—
52	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
56	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
57	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
58	C₈×C₁₃₁	—	2	1	no	—
59	C₈×C₁₃₁	—	2	1	no	—
60	C₈×C₁₃₁	—	4	8	no	—
61	C₈×C₁₃₁	C₄×C₈	1	4	yes	—
62	C₈×C₁₃₁	C₄⋊C₈	1	8	yes	—
63	C₂²×C₁₃₁	—	2	2	no	—
64	C₂²×C₁₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₃₁	#431	1	2	yes	—
66	C₄×C₁₃₁	—	2	2	no	—
67	C₄×C₁₃₁	—	4	4	no	—
68	C₄×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₁₃₁	C₂×C₄⋊C₈	1	2	yes	—
71	C₄×C₁₃₁	#444	1	4	yes	—
72	C₂×C₁₃₁	—	2	2	no	—
73	C₂×C₁₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₁₃₁	#1773	1	2	yes	—
75	#8268	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₁₃₁	#8268	1	1	yes	characteristic, direct product factor, Sylow
77	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic
78	#1564	C₂×C₁₃₁	1	2	yes	—
79	#1450	C₂×C₁₃₁	1	2	yes	—
80	#1276	C₂×C₁₃₁	1	2	yes	—
81	#431	C₂²×C₁₃₁	1	4	yes	—
82	C₂×#97	C₄×C₁₃₁	1	2	yes	—
83	C₂×#97	C₄×C₁₃₁	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
86	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
87	#335	C₄×C₁₃₁	1	4	yes	complemented
88	#334	C₄×C₁₃₁	1	4	yes	complemented
89	#322	—	2	2	no	—
90	C₈²	—	2	2	no	—
91	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
96	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
97	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
98	C₂²⋊C₈	—	2	4	no	—
99	#97	—	2	4	no	—
100	#97	C₈×C₁₃₁	1	8	yes	complemented
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
103	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
104	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
105	C₄×C₈	Q₈×C₁₃₁	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
108	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
112	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₁₃₁	1	2	yes	—
114	C₂×C₈	C₄⋊C₄×C₁₃₁	1	4	yes	—
115	C₂×C₈	C₂×C₈×C₁₃₁	1	4	yes	—
116	C₂×C₈	C₂×C₈×C₁₃₁	1	4	yes	—
117	C₂×C₈	M₁₆×C₁₃₁	1	4	yes	—
118	C₄²	—	2	2	no	—
119	C₄²	—	4	4	no	—
120	C₄²	C₂×C₈×C₁₃₁	1	4	yes	—
121	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
122	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
123	C₄²	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
124	C₂³	#117×C₁₃₁	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
131	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
132	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
133	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	4	8	no	—
137	C₈	C₄×C₈×C₁₃₁	1	4	yes	—
138	C₈	C₄⋊C₈×C₁₃₁	1	8	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
141	C₂²	#431×C₁₃₁	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	C₂×C₄×C₈×C₁₃₁	1	1	yes	characteristic
145	C₄	C₄×M₁₆×C₁₃₁	1	1	yes	characteristic
146	C₄	C₂×C₄⋊C₈×C₁₃₁	1	2	yes	—
147	C₄	#444×C₁₃₁	1	4	yes	—
148	C₂	—	2	2	no	—
149	C₂	C₄×C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
150	C₂	#1773×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃₁	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. #335×C₁₃₁	C₄	semidirect
12. #334×C₁₃₁	C₄	semidirect
24. #97×C₁₃₁	C₈	semidirect
75. #8268	C₁₃₁	direct (N × H)
76. C₁₃₁	#8268	direct (N × H)
87. #335	C₄×C₁₃₁	semidirect
88. #334	C₄×C₁₃₁	semidirect
100. #97	C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#97×C₁₃₁	2
#1564×C₁₃₁	2
#1450×C₁₃₁	2
#1276×C₁₃₁	2
#8268	131

Subgroups — C131×#8268 #45694152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8268 #45694152