Subgroups — #11928966

Order 1248.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10597790	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#10597658	C₂	1	2	yes	maximal, complemented
6	#88158	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	#64625	—	2	2	no	—
9	D₁₅₆	C₂²	1	2	yes	—
10	C₄×D₃₉	—	2	2	no	—
11	Dic₇₈	C₂²	1	2	yes	—
12	C₃×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₃×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#6127	—	3	2	no	—
15	C₂×C₈×C₁₃	D₃	1	1	yes	characteristic, complemented
16	Dic₅₂	—	3	1	no	—
17	D₁₀₄	—	3	1	no	—
18	#6095	—	3	2	no	—
19	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
20	C₂×D₃₉	—	2	2	no	—
21	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₃×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	Dic₃₉	—	2	2	no	—
24	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
25	C₂²⋊D₁₃	—	6	2	no	—
26	D₅₂	—	3	2	no	—
27	C₄×D₁₃	—	6	2	no	—
28	Dic₂₆	—	3	2	no	—
29	C₈×C₁₃	C₂×D₃	1	1	yes	characteristic
30	C₈×C₁₃	C₂×D₃	1	1	yes	characteristic
31	#904	—	13	1	no	maximal, Hall
32	C₂×C₃×C₁₃	—	2	1	no	—
33	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
34	D₃₉	—	4	2	no	—
35	C₂²×C₁₃	D₁₂	1	1	yes	characteristic
36	C₂×D₁₃	—	6	2	no	—
37	C₄×C₁₃	D₁₂	1	1	yes	characteristic
38	C₄×C₁₃	C₂²×D₃	1	1	yes	characteristic
39	Dic₁₃	—	6	2	no	—
40	#235	—	13	2	no	—
41	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic, complemented
42	Dic₁₂	—	13	1	no	—
43	D₂₄	—	13	1	no	—
44	#204	—	13	2	no	—
45	C₃×C₁₃	#135	1	1	yes	characteristic, complemented, Hall
46	#135	—	39	1	no	Sylow
47	C₂×C₁₃	—	2	1	no	—
48	C₂×C₁₃	C₂×D₁₂	1	1	yes	characteristic
49	D₁₃	—	12	2	no	—
50	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
51	#67	—	26	2	no	—
52	D₁₂	—	13	2	no	—
53	C₄×D₃	—	26	2	no	—
54	Dic₆	—	13	2	no	—
55	C₃×C₈	C₂×D₁₃	1	1	yes	characteristic
56	C₃×C₈	C₂×D₁₃	1	1	yes	characteristic
57	#41	—	39	2	no	—
58	Q₁₆	—	39	1	no	—
59	SD₁₆	—	39	2	no	—
60	D₈	—	39	1	no	—
61	C₂×C₈	D₃₉	1	1	yes	characteristic, complemented
62	C₁₃	#904	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₃	D₅₂	1	1	yes	characteristic
64	C₂×D₃	—	26	2	no	—
65	C₃×C₄	D₅₂	1	1	yes	characteristic
66	C₃×C₄	C₂²×D₁₃	1	1	yes	characteristic
67	Dic₃	—	26	2	no	—
68	Q₈	—	39	2	no	—
69	D₄	—	39	2	no	—
70	D₄	—	78	2	no	—
71	C₂×C₄	—	78	2	no	—
72	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
73	C₈	C₂×D₃₉	1	1	yes	characteristic
74	C₈	C₂×D₃₉	1	1	yes	characteristic
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
77	D₃	—	52	2	no	—
78	C₂²	—	78	2	no	—
79	C₂²	D₁₅₆	1	1	yes	characteristic
80	C₄	—	78	2	no	—
81	C₄	D₁₅₆	1	1	yes	characteristic
82	C₄	C₂²×D₃₉	1	1	yes	characteristic
83	C₃	#88158	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	1	no	—
85	C₂	—	156	2	no	—
86	C₂	C₂×D₁₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₅₆
commutator, cosocle	C₃×C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₃₉
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₂×C₈	D₃₉
hypocenter	C₃×C₁₃	#135
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #10597790	C₂	semidirect
2. C₂×C₃×C₈×C₁₃	C₂	semidirect
3. Dic₁₅₆	C₂	semidirect
4. D₃₁₂	C₂	semidirect
5. #10597658	C₂	semidirect
15. C₂×C₈×C₁₃	D₃	semidirect
41. C₂×C₃×C₈	D₁₃	semidirect
45. C₃×C₁₃	#135	semidirect
61. C₂×C₈	D₃₉	semidirect
62. C₁₃	#904	semidirect
83. C₃	#88158	semidirect

Maximal subgroups

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

subgroup	index
#10597790	2
C₂×C₃×C₈×C₁₃	2
Dic₁₅₆	2
D₃₁₂	2
#10597658	2
#88158	3
#904	13