Subgroups — #16847470

Order 3968.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758400	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₃₁×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#117	C₄	1	1	yes	characteristic
8	C₃₁×#97	C₄	1	2	yes	—
9	C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	#11758400	—	2	2	no	—
11	C₂×C₃₁⋊C₁₆	—	2	2	no	—
12	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₃₁	—	2	2	no	—
16	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₃₁	D₄	1	2	yes	—
19	C₂×C₈×C₃₁	Q₈	1	2	yes	—
20	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₃₁	C₈	1	2	yes	—
22	C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
23	C₄²×C₃₁	C₂×C₄	1	2	yes	—
24	C₃₁⋊C₁₆	—	2	2	no	—
25	C₃₁⋊C₁₆	—	2	2	no	—
26	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
31	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₃₁	—	2	2	no	—
33	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
34	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
36	#2151	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	—	2	2	no	—
38	C₂²×C₃₁	C₄⋊C₈	1	2	yes	—
39	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₃₁	—	2	4	no	—
41	C₄×C₃₁	C₄⋊C₈	1	2	yes	—
42	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
44	C₂×M₃₂	—	31	1	no	—
45	C₂²×C₁₆	—	31	1	no	—
46	#405	D₃₁	1	1	yes	characteristic
47	C₄⋊C₁₆	—	31	4	no	—
48	C₂×C₃₁	—	2	2	no	—
49	C₂×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₃₁	#504	1	2	yes	—
51	C₂×M₁₆	Dic₃₁	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
53	#117	Dic₃₁	1	1	yes	characteristic
54	M₃₂	—	62	2	no	—
55	C₂×C₁₆	—	31	2	no	—
56	C₂×C₁₆	—	31	2	no	—
57	C₂×C₁₆	—	62	2	no	—
58	#97	Dic₃₁	1	2	yes	—
59	C₄×C₈	C₂×D₃₁	1	2	yes	—
60	C₃₁	#2151	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	Dic₆₂	1	2	yes	—
64	C₂×C₈	D₁₂₄	1	2	yes	—
65	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
66	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
67	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
68	C₄⋊C₄	C₃₁⋊C₈	1	2	yes	—
69	C₂²⋊C₄	C₃₁⋊C₈	1	2	yes	—
70	C₄²	C₂×Dic₃₁	1	2	yes	—
71	C₁₆	—	62	2	no	—
72	C₁₆	—	62	2	no	—
73	C₂³	C₂×C₃₁⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
75	C₂×C₄	C₂×C₃₁⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#92474	1	2	yes	—
77	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
78	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊Dic₃₁	1	2	yes	—
81	C₈	C₂×Dic₆₂	1	1	yes	characteristic
82	C₈	C₂×D₁₂₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊(C₃₁⋊C₈)	1	2	yes	—
85	C₂²	C₂²×C₃₁⋊C₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊(C₃₁⋊C₈)	1	2	yes	—
88	C₄	C₂×#92474	1	1	yes	characteristic
89	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₃₁⋊C₈)	1	1	yes	characteristic
92	C₂	#14521466	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂²×C₈
Fitting	C₃₁×#405	C₂
prime radical	C₂³×C₃₁	C₂×C₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	#405	D₃₁
hypocenter	C₃₁	#2151
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃₁⋊C₁₆)	C₂	semidirect
60. C₃₁	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758400	2
C₂²×C₃₁⋊C₁₆	2
C₄⋊(C₃₁⋊C₁₆)	2
C₃₁×#405	2
#2151	31