Subgroups — #16848026

Order 3968.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521481	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14520888	C₂	1	2	yes	maximal, complemented
4	D₉₉₂	C₂	1	2	yes	maximal, complemented
5	C₃₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758481	—	2	1	no	—
7	C₂×D₂₄₈	—	2	1	no	—
8	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	Dic₂₄₈	C₂²	1	1	yes	characteristic, complemented
10	#11758388	—	2	1	no	—
11	D₄₉₆	—	2	1	no	—
12	D₄₉₆	C₂²	1	2	yes	—
13	D₄₉₆	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×C₃₂	C₂²	1	2	yes	complemented
15	#92495	—	4	1	no	—
16	C₂×D₁₂₄	—	4	1	no	—
17	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
18	Dic₁₂₄	—	2	1	no	—
19	D₂₄₈	—	2	2	no	—
20	D₂₄₈	—	2	1	no	—
21	D₂₄₈	—	4	1	no	—
22	#92470	—	4	1	no	—
23	C₁₆×C₃₁	D₄	1	1	yes	characteristic
24	C₁₆×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×D₃₁	—	8	1	no	—
26	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
27	C₂²⋊D₃₁	—	8	1	no	—
28	D₁₂₄	—	4	1	no	—
29	D₁₂₄	—	4	2	no	—
30	D₁₂₄	—	8	1	no	—
31	C₄×D₃₁	—	8	1	no	—
32	Dic₆₂	—	4	1	no	—
33	C₈×C₃₁	D₈	1	1	yes	characteristic
34	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
35	#2263	—	31	1	no	maximal, Sylow
36	C₂²×C₃₁	D₁₆	1	1	yes	characteristic
37	C₂×D₃₁	—	8	1	no	—
38	C₂×D₃₁	—	8	2	no	—
39	C₂×D₃₁	—	16	2	no	—
40	C₄×C₃₁	D₁₆	1	1	yes	characteristic
41	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
42	Dic₃₁	—	8	1	no	—
43	#508	—	31	1	no	—
44	C₂×D₁₆	—	31	1	no	—
45	SD₆₄	—	31	2	no	—
46	D₃₂	—	31	2	no	—
47	M₆₄	D₃₁	1	1	yes	characteristic, complemented
48	C₂×C₃₁	—	2	1	no	—
49	C₂×C₃₁	C₂×D₁₆	1	1	yes	characteristic
50	D₃₁	—	16	1	no	—
51	D₃₁	—	16	2	no	—
52	#135	—	62	1	no	—
53	C₂×D₈	—	62	1	no	—
54	Q₃₂	—	31	1	no	—
55	SD₃₂	—	62	1	no	—
56	D₁₆	—	31	2	no	—
57	D₁₆	—	31	1	no	—
58	D₁₆	—	62	1	no	—
59	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
60	C₃₂	C₂×D₃₁	1	2	yes	complemented
61	C₃₁	#2263	1	1	yes	characteristic, complemented, Sylow
62	#41	—	124	1	no	—
63	C₂×D₄	—	124	1	no	—
64	Q₁₆	—	62	1	no	—
65	SD₁₆	—	124	1	no	—
66	D₈	—	62	2	no	—
67	D₈	—	62	1	no	—
68	D₈	—	124	1	no	—
69	C₂×C₈	D₁₂₄	1	1	yes	characteristic
70	C₁₆	D₁₂₄	1	1	yes	characteristic
71	C₁₆	C₂²×D₃₁	1	1	yes	characteristic
72	C₂³	—	248	1	no	—
73	Q₈	—	124	1	no	—
74	D₄	—	124	2	no	—
75	D₄	—	124	1	no	—
76	D₄	—	248	1	no	—
77	D₄	—	248	1	no	—
78	C₂×C₄	—	248	1	no	—
79	C₂×C₄	D₂₄₈	1	1	yes	characteristic
80	C₈	D₂₄₈	1	1	yes	characteristic
81	C₈	C₂×D₁₂₄	1	1	yes	characteristic
82	C₂²	—	496	2	no	—
83	C₂²	—	248	1	no	—
84	C₂²	—	248	2	no	—
85	C₂²	D₄₉₆	1	1	yes	characteristic
86	C₄	—	248	1	no	—
87	C₄	D₄₉₆	1	1	yes	characteristic
88	C₄	C₂×D₂₄₈	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	496	2	no	—
91	C₂	—	496	1	no	—
92	C₂	C₂×D₄₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₉₆
commutator, cosocle	C₁₆×C₃₁	C₂³
Frattini	C₁₆	C₂²×D₃₁
Fitting	C₃₁×M₆₄	C₂
socle	C₂×C₃₁	C₂×D₁₆
hypercenter	M₆₄	D₃₁
hypocenter	C₃₁	#2263
LCS	G ▷ C₁₆×C₃₁ ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #14521481	C₂	semidirect
2. C₂×D₄₉₆	C₂	semidirect
3. #14520888	C₂	semidirect
4. D₉₉₂	C₂	semidirect
5. C₃₁×M₆₄	C₂	semidirect
9. Dic₂₄₈	C₂²	semidirect
13. D₄₉₆	C₂²	semidirect
14. C₃₁×C₃₂	C₂²	semidirect
47. M₆₄	D₃₁	semidirect
60. C₃₂	C₂×D₃₁	semidirect
61. C₃₁	#2263	semidirect

Maximal subgroups

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

subgroup	index
#14521481	2
C₂×D₄₉₆	2
#14520888	2
D₉₉₂	2
C₃₁×M₆₄	2
#2263	31