Subgroups — #16848023

Order 3968.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₉₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14520886	C₂	1	2	yes	maximal, complemented
3	#14520884	C₂	1	2	yes	maximal, complemented
4	#14520883	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#135	—	2	1	no	—
7	C₂×D₂₄₈	—	2	1	no	—
8	C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
9	C₃₁×SD₃₂	—	2	1	no	—
10	C₃₁×D₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	D₄₉₆	—	2	1	no	—
13	D₄₉₆	C₂²	1	2	yes	—
14	C₃₁⋊C₃₂	C₂²	1	2	yes	complemented
15	#41×C₃₁	—	4	1	no	—
16	C₂×D₁₂₄	—	4	1	no	—
17	Q₁₆×C₃₁	—	2	1	no	—
18	SD₁₆×C₃₁	—	4	1	no	—
19	D₈×C₃₁	—	2	1	no	—
20	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
21	D₂₄₈	—	2	2	no	—
22	D₂₄₈	—	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	Q₈×C₃₁	—	4	1	no	—
27	D₄×C₃₁	—	4	1	no	—
28	D₄×C₃₁	—	8	1	no	—
29	C₂×C₄×C₃₁	—	8	1	no	—
30	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
31	D₁₂₄	—	4	2	no	—
32	D₁₂₄	—	8	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	—	8	1	no	—
37	C₂²×C₃₁	D₁₆	1	1	yes	characteristic
38	C₂×D₃₁	—	8	2	no	—
39	C₂×D₃₁	—	16	2	no	—
40	C₄×C₃₁	—	8	1	no	—
41	C₄×C₃₁	D₁₆	1	1	yes	characteristic
42	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
43	#508	D₃₁	1	1	yes	characteristic, complemented
44	C₂×D₁₆	—	31	1	no	—
45	SD₆₄	—	31	2	no	—
46	D₃₂	—	31	2	no	—
47	M₆₄	—	31	1	no	—
48	C₂×C₃₁	—	2	1	no	—
49	C₂×C₃₁	—	16	1	no	—
50	C₂×C₃₁	C₂×D₁₆	1	1	yes	characteristic
51	D₃₁	—	16	2	no	—
52	#135	—	2	1	no	—
53	C₂×D₈	—	62	1	no	—
54	Q₃₂	C₂×D₃₁	1	1	yes	characteristic, complemented
55	SD₃₂	—	2	1	no	—
56	D₁₆	—	31	2	no	—
57	D₁₆	—	62	1	no	—
58	D₁₆	C₂×D₃₁	1	1	yes	characteristic, complemented
59	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
60	C₃₂	—	31	2	no	—
61	C₃₁	#2263	1	1	yes	characteristic, complemented, Sylow
62	#41	—	4	1	no	—
63	C₂×D₄	—	124	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	D₈	—	62	2	no	—
68	D₈	—	124	1	no	—
69	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
70	C₁₆	C₂²⋊D₃₁	1	1	yes	characteristic
71	C₁₆	C₂²×D₃₁	1	1	yes	characteristic
72	C₂³	—	248	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	8	1	no	—
76	D₄	—	124	2	no	—
77	D₄	—	248	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#92479	1	1	yes	characteristic
80	C₈	#92479	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	496	2	no	—
84	C₂²	—	248	2	no	—
85	C₂²	#11758414	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	#11758414	1	1	yes	characteristic
88	C₄	C₂×#92479	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	1	no	—
91	C₂	—	496	2	no	—
92	C₂	C₂×#11758414	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11758414
commutator, cosocle	C₁₆×C₃₁	C₂³
Frattini	C₁₆	C₂²×D₃₁
Fitting	C₃₁×#508	C₂
socle	C₂×C₃₁	C₂×D₁₆
hypercenter	#508	D₃₁
hypocenter	C₃₁	#2263
LCS	G ▷ C₁₆×C₃₁ ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₉₆	C₂	semidirect
2. #14520886	C₂	semidirect
3. #14520884	C₂	semidirect
4. #14520883	C₂	semidirect
5. C₃₁×#508	C₂	semidirect
8. C₃₁×Q₃₂	C₂²	semidirect
10. C₃₁×D₁₆	C₂²	semidirect
14. C₃₁⋊C₃₂	C₂²	semidirect
43. #508	D₃₁	semidirect
54. Q₃₂	C₂×D₃₁	semidirect
58. D₁₆	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
C₂×D₄₉₆	2
#14520886	2
#14520884	2
#14520883	2
C₃₁×#508	2
#2263	31