Subgroups — #16847471

Order 3968.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#14521004	C₂	1	1	yes	maximal, characteristic
3	Dic₃₁⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	#11758503	C₄	1	1	yes	characteristic
7	C₂×#92474	C₄	1	1	yes	characteristic
8	C₃₁×M₃₂	—	2	2	no	—
9	C₂×C₁₆×C₃₁	C₂²	1	2	yes	—
10	#11758403	C₄	1	2	yes	—
11	C₈×Dic₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁⋊C₁₆	—	2	2	no	—
13	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₃₁	D₄	1	2	yes	—
17	C₂×C₈×C₃₁	Q₈	1	2	yes	—
18	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²⋊Dic₃₁	C₈	1	2	yes	—
20	C₄⋊Dic₃₁	C₈	1	2	yes	—
21	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
22	#92474	—	2	2	no	—
23	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₁₆×C₃₁	—	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	1	yes	characteristic
29	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
30	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
32	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
33	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
35	C₃₁⋊C₈	—	2	2	no	—
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	C₄⋊C₈	1	2	yes	—
41	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
43	Dic₃₁	—	2	4	no	—
44	C₂×M₃₂	D₃₁	1	1	yes	characteristic
45	C₂²×C₁₆	—	31	1	no	—
46	#405	—	31	1	no	—
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	—	31	1	no	—
52	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
53	#117	—	31	1	no	—
54	M₃₂	—	2	2	no	—
55	C₂×C₁₆	—	31	2	no	—
56	C₂×C₁₆	—	62	2	no	—
57	C₂×C₁₆	C₂×D₃₁	1	2	yes	—
58	#97	—	31	2	no	—
59	C₄×C₈	—	31	2	no	—
60	C₃₁	#2151	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
62	M₁₆	—	62	2	no	—
63	C₂×C₈	—	31	2	no	—
64	C₂×C₈	Dic₆₂	1	2	yes	—
65	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
67	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
68	C₄⋊C₄	—	31	2	no	—
69	C₂²⋊C₄	—	31	2	no	—
70	C₄²	—	31	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	62	2	no	—
73	C₂³	C₈×D₃₁	1	1	yes	characteristic
74	C₂×C₄	—	31	4	no	—
75	C₂×C₄	C₈×D₃₁	1	1	yes	characteristic
76	C₂×C₄	#92469	1	2	yes	—
77	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
79	C₈	—	62	2	no	—
80	C₈	Dic₃₁⋊C₄	1	2	yes	—
81	C₈	C₂×Dic₆₂	1	1	yes	characteristic
82	C₈	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	Dic₃₁⋊C₈	1	2	yes	—
85	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
86	C₄	—	62	4	no	—
87	C₄	Dic₃₁⋊C₈	1	2	yes	—
88	C₄	C₂×#92469	1	1	yes	characteristic
89	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×Dic₃₁⋊C₈	1	1	yes	characteristic
92	C₂	#14521467	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂²×C₈
Fitting	C₂×C₃₁×M₃₂	C₂
prime radical	C₂³×C₃₁	C₂×C₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	C₂×M₃₂	D₃₁
hypocenter	C₃₁	#2151
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. Dic₃₁⋊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
C₂²×C₃₁⋊C₁₆	2
#14521004	2
Dic₃₁⋊C₁₆	2
C₂×C₃₁×M₃₂	2
#2151	31