Subgroups — C₁₃₁×#33414 #45719298

Order 33536.

622 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2916×C₁₃₁	C₂	1	2	yes	maximal
3	#2161×C₁₃₁	C₂	1	6	yes	maximal, complemented
4	C₂×#368×C₁₃₁	C₂	1	6	yes	maximal, complemented
5	#568×C₁₃₁	—	2	2	no	—
6	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
7	#504×C₁₃₁	C₄	1	8	yes	—
8	C₂×M₃₂×C₁₃₁	C₂²	1	3	yes	—
9	C₂²×C₁₆×C₁₃₁	C₂²	1	3	yes	—
10	#430×C₁₃₁	—	2	6	no	—
11	C₂×#108×C₁₃₁	C₂²	1	12	yes	—
12	#368×C₁₃₁	C₂²	1	4	yes	—
13	#368×C₁₃₁	C₂²	1	12	yes	—
14	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
15	#131×C₁₃₁	—	4	8	no	—
16	#131×C₁₃₁	C₂×C₄	1	4	yes	—
17	#131×C₁₃₁	Q₈	1	4	yes	—
18	C₂×M₁₆×C₁₃₁	—	2	12	no	—
19	C₂×M₁₆×C₁₃₁	D₄	1	3	yes	—
20	C₂²×C₈×C₁₃₁	C₂³	1	3	yes	—
21	M₃₂×C₁₃₁	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₁₃₁	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₂×C₁₆×C₁₃₁	C₂³	1	3	yes	—
25	#108×C₁₃₁	—	2	12	no	—
26	#108×C₁₃₁	C₂³	1	8	yes	—
27	#41×C₁₃₁	C₄⋊C₄	1	4	yes	—
28	#41×C₁₃₁	D₈	1	4	yes	—
29	C₂×Q₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
30	C₂×D₄×C₁₃₁	Q₁₆	1	3	yes	—
31	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
32	M₁₆×C₁₃₁	—	2	8	no	—
33	M₁₆×C₁₃₁	—	4	12	no	—
34	M₁₆×C₁₃₁	C₄⋊C₄	1	12	yes	—
35	C₂×C₈×C₁₃₁	—	4	12	no	—
36	C₂×C₈×C₁₃₁	C₂²×C₄	1	6	yes	—
37	C₂×C₈×C₁₃₁	C₂×D₄	1	3	yes	—
38	C₂×C₈×C₁₃₁	C₂×Q₈	1	6	yes	—
39	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
40	C₁₆×C₁₃₁	C₂²×C₄	1	2	yes	—
41	C₁₆×C₁₃₁	C₂²×C₄	1	6	yes	—
42	C₂³×C₁₃₁	C₂×Q₁₆	1	3	yes	—
43	Q₈×C₁₃₁	#107	1	4	yes	—
44	D₄×C₁₃₁	#107	1	12	yes	—
45	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄×C₁₃₁	C₂×D₈	1	6	yes	—
47	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	3	yes	—
48	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
49	C₈×C₁₃₁	—	4	8	no	—
50	C₈×C₁₃₁	C₂×C₄⋊C₄	1	6	yes	—
51	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
52	C₈×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₁₃₁	—	4	3	no	—
54	C₂²×C₁₃₁	C₂×#107	1	6	yes	—
55	C₂²×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
56	C₄×C₁₃₁	C₂×#107	1	6	yes	—
57	C₄×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
59	C₂×C₁₃₁	—	2	1	no	—
60	C₂×C₁₃₁	—	2	6	no	—
61	C₂×C₁₃₁	C₂²×#107	1	1	yes	characteristic
62	#33414	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃₁	#33414	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×#504	C₂×C₁₃₁	1	1	yes	characteristic
65	#2916	C₂×C₁₃₁	1	2	yes	—
66	#2161	C₂×C₁₃₁	1	6	yes	complemented
67	C₂×#368	C₂×C₁₃₁	1	6	yes	complemented
68	#568	—	2	2	no	—
69	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
70	#504	C₄×C₁₃₁	1	8	yes	—
71	C₂×M₃₂	C₂²×C₁₃₁	1	3	yes	—
72	C₂²×C₁₆	C₂²×C₁₃₁	1	3	yes	—
73	#430	—	2	6	no	—
74	C₂×#108	C₂²×C₁₃₁	1	12	yes	—
75	#368	C₂²×C₁₃₁	1	4	yes	—
76	#368	C₂²×C₁₃₁	1	12	yes	—
77	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
78	#131	—	4	8	no	—
79	#131	C₂×C₄×C₁₃₁	1	4	yes	—
80	#131	Q₈×C₁₃₁	1	4	yes	—
81	C₂×M₁₆	—	2	12	no	—
82	C₂×M₁₆	D₄×C₁₃₁	1	3	yes	—
83	C₂²×C₈	C₂³×C₁₃₁	1	3	yes	—
84	M₃₂	C₂×C₄×C₁₃₁	1	12	yes	—
85	C₂×C₁₆	C₂×C₄×C₁₃₁	1	12	yes	—
86	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
87	C₂×C₁₆	C₂³×C₁₃₁	1	3	yes	—
88	#108	—	2	12	no	—
89	#108	C₂³×C₁₃₁	1	8	yes	—
90	#41	C₄⋊C₄×C₁₃₁	1	4	yes	—
91	#41	D₈×C₁₃₁	1	4	yes	—
92	C₂×Q₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
93	C₂×D₄	Q₁₆×C₁₃₁	1	3	yes	—
94	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
95	M₁₆	—	2	8	no	—
96	M₁₆	—	4	12	no	—
97	M₁₆	C₄⋊C₄×C₁₃₁	1	12	yes	—
98	C₂×C₈	—	4	12	no	—
99	C₂×C₈	C₂²×C₄×C₁₃₁	1	6	yes	—
100	C₂×C₈	C₂×D₄×C₁₃₁	1	3	yes	—
101	C₂×C₈	C₂×Q₈×C₁₃₁	1	6	yes	—
102	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
103	C₁₆	C₂²×C₄×C₁₃₁	1	2	yes	—
104	C₁₆	C₂²×C₄×C₁₃₁	1	6	yes	—
105	C₂³	C₂×Q₁₆×C₁₃₁	1	3	yes	—
106	Q₈	#107×C₁₃₁	1	4	yes	—
107	D₄	#107×C₁₃₁	1	12	yes	—
108	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	6	yes	—
109	C₂×C₄	C₂×D₈×C₁₃₁	1	6	yes	—
110	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	3	yes	—
111	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
112	C₈	—	4	8	no	—
113	C₈	C₂×C₄⋊C₄×C₁₃₁	1	6	yes	—
114	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
115	C₈	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
116	C₂²	—	4	3	no	—
117	C₂²	C₂×#107×C₁₃₁	1	6	yes	—
118	C₂²	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
119	C₄	C₂×#107×C₁₃₁	1	6	yes	—
120	C₄	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
121	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	6	no	—
124	C₂	C₂²×#107×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂×C₁₃₁	C₂²×#107
prime radical	C₂×#41×C₁₃₁	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂×#41×C₁₃₁ ◁ C₂×#131×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2161×C₁₃₁	C₂	semidirect
4. C₂×#368×C₁₃₁	C₂	semidirect
62. #33414	C₁₃₁	direct (N × H)
63. C₁₃₁	#33414	direct (N × H)
66. #2161	C₂×C₁₃₁	semidirect
67. C₂×#368	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×#504×C₁₃₁	2
#2916×C₁₃₁	2
#2161×C₁₃₁	2
C₂×#368×C₁₃₁	2
#33414	131

Subgroups — C131×#33414 #45719298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33414 #45719298