Subgroups — #16847569

Order 3968.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄₈	C₂	1	1	yes	maximal, characteristic
2	#14521143	C₂	1	2	yes	maximal
3	#14520852	C₂	1	2	yes	maximal
4	C₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#366	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	C₂²	1	2	yes	—
7	#11758464	—	2	2	no	—
8	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#107	C₂²	1	2	yes	—
10	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#11758398	—	2	2	no	—
13	Dic₂₄₈	C₄	1	4	yes	complemented
14	C₂×Dic₆₂	—	2	2	no	—
15	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₃₁	—	2	2	no	—
17	Dic₃₁⋊C₄	—	4	2	no	—
18	C₄×Dic₃₁	—	4	2	no	—
19	C₄×Dic₃₁	D₄	1	1	yes	characteristic
20	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
21	Dic₁₂₄	—	2	2	no	—
22	Dic₁₂₄	C₂×C₄	1	4	yes	—
23	C₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₃₁⋊C₁₆	—	2	1	no	—
25	C₂×C₄×C₃₁	—	4	2	no	—
26	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₃₁	—	4	2	no	—
28	C₂×Dic₃₁	D₈	1	1	yes	characteristic
29	C₂×Dic₃₁	D₈	1	1	yes	characteristic
30	Dic₆₂	—	2	4	no	—
31	Dic₆₂	—	4	2	no	—
32	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₃₁	#41	1	1	yes	characteristic
34	C₃₁⋊C₈	—	2	1	no	—
35	C₄×Q₃₂	—	31	1	no	maximal, Sylow
36	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
37	C₄×C₃₁	—	8	2	no	—
38	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
39	C₄×C₃₁	#135	1	1	yes	characteristic
40	Dic₃₁	—	2	1	no	—
41	Dic₃₁	—	4	4	no	—
42	Dic₃₁	Q₃₂	1	2	yes	complemented
43	C₂×Q₃₂	—	31	1	no	—
44	C₄×Q₁₆	—	31	2	no	—
45	#366	D₃₁	1	1	yes	characteristic
46	#358	—	31	2	no	—
47	C₄×C₁₆	—	31	1	no	—
48	C₂×C₃₁	C₄×D₈	1	1	yes	characteristic
49	C₂×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×C₃₁	#508	1	1	yes	characteristic
51	C₂×Q₁₆	—	31	2	no	—
52	C₄×Q₈	—	62	2	no	—
53	Q₃₂	—	31	4	no	—
54	C₂×C₁₆	—	31	1	no	—
55	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
56	#107	C₂×D₃₁	1	2	yes	—
57	#103	—	62	2	no	—
58	C₄×C₈	—	31	1	no	—
59	C₃₁	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	62	2	no	—
61	Q₁₆	—	31	4	no	—
62	Q₁₆	—	62	2	no	—
63	C₂×C₈	—	31	1	no	—
64	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	124	2	no	—
67	C₄²	—	31	1	no	—
68	C₄²	—	124	2	no	—
69	C₁₆	—	62	1	no	—
70	C₁₆	C₄×D₃₁	1	2	yes	—
71	Q₈	—	62	4	no	—
72	Q₈	—	124	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	31	1	no	—
75	C₂×C₄	—	31	1	no	—
76	C₂×C₄	—	124	2	no	—
77	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
78	C₈	—	62	1	no	—
79	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
80	C₈	#92499	1	1	yes	characteristic
81	C₂²	D₈×D₃₁	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	31	2	no	—
84	C₄	—	62	1	no	—
85	C₄	—	124	4	no	—
86	C₄	#11758470	1	1	yes	characteristic
87	C₄	#11758496	1	1	yes	characteristic
88	C₂	#14521131	1	1	yes	characteristic
89	C₂	Q₃₂×D₃₁	1	1	yes	characteristic
90	C₂	#14521478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₁
commutator	C₈×C₃₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#366	C₂
socle, prime radical	C₂²×C₃₁	C₂×D₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	#366	D₃₁
hypocenter	C₃₁	C₄×Q₃₂
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
13. Dic₂₄₈	C₄	semidirect
42. Dic₃₁	Q₃₂	semidirect
59. C₃₁	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄₈	2
#14521143	2
#14520852	2
C₁₆×Dic₃₁	2
C₃₁×#366	2
C₄×Q₃₂	31