Subgroups — #16845848

Order 3968.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521073	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14520813	C₂	1	4	yes	maximal
4	C₃₁×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×#92474	C₂²	1	2	yes	—
8	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
9	C₃₁×C₄⋊C₈	—	2	2	no	—
10	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
11	C₄×C₃₁⋊C₈	C₄	1	2	yes	—
12	C₄×C₃₁⋊C₈	C₄	1	2	yes	—
13	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
14	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
15	M₁₆×C₃₁	—	2	4	no	—
16	C₂×C₈×C₃₁	—	2	2	no	—
17	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₃₁	D₄	1	2	yes	—
20	#92474	—	2	4	no	—
21	C₂×C₃₁⋊C₈	—	2	2	no	—
22	C₂×C₃₁⋊C₈	—	2	2	no	—
23	C₂×C₃₁⋊C₈	—	2	2	no	—
24	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
25	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₃₁	—	2	4	no	—
27	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	#41	1	2	yes	—
32	C₂×C₄×C₃₁	#41	1	2	yes	—
33	C₈×C₃₁	—	4	2	no	—
34	C₃₁⋊C₈	—	2	4	no	—
35	C₃₁⋊C₈	—	4	2	no	—
36	#1837	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	—	2	2	no	—
38	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₃₁	#125	1	2	yes	—
40	C₄×C₃₁	—	2	2	no	—
41	C₄×C₃₁	#108	1	4	yes	—
42	C₄×C₃₁	#117	1	2	yes	—
43	C₄×C₃₁	#124	1	1	yes	characteristic
44	C₄×C₃₁	#124	1	1	yes	characteristic
45	#423	—	31	1	no	—
46	#423	D₃₁	1	1	yes	characteristic
47	C₂×C₄×C₈	—	31	1	no	—
48	#341	—	31	4	no	—
49	C₂×C₃₁	—	2	2	no	—
50	C₂×C₃₁	#382	1	1	yes	characteristic
51	C₂×C₃₁	C₂×#108	1	2	yes	—
52	C₂×M₁₆	—	31	2	no	—
53	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
54	C₂²×C₈	—	31	2	no	—
55	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	62	2	no	—
58	C₄×C₈	—	31	2	no	—
59	C₄×C₈	—	31	2	no	—
60	C₃₁	#1837	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	Dic₆₂	1	2	yes	—
62	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	62	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	31	4	no	—
67	C₂×C₈	—	62	2	no	—
68	C₂×C₈	—	62	2	no	—
69	C₂×C₈	—	62	2	no	—
70	C₄²	C₄×D₃₁	1	1	yes	characteristic
71	C₄²	C₄×D₃₁	1	1	yes	characteristic
72	C₄²	C₂²⋊D₃₁	1	2	yes	—
73	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
78	C₂×C₄	#92497	1	2	yes	—
79	C₂×C₄	#92499	1	2	yes	—
80	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	62	4	no	—
83	C₈	—	124	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	#11758467	1	2	yes	—
86	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	#11758410	1	4	yes	—
89	C₄	#11758469	1	2	yes	—
90	C₄	#11758513	1	1	yes	characteristic
91	C₄	#11758524	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#14520914	1	1	yes	characteristic
94	C₂	C₂×#11758410	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#423	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#423	D₃₁
hypocenter	C₃₁	#1837
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
60. C₃₁	#1837	semidirect

Maximal subgroups

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

subgroup	index
#14521073	2
C₂×C₄×C₃₁⋊C₈	2
#14520813	2
C₃₁×#423	2
#1837	31