Subgroups — #11928121

Order 1248.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#88049	—	3	1	no	maximal, Hall
5	C₂²×D₃₉	—	2	1	no	—
6	C₂×C₃×C₄×C₁₃	—	2	1	no	—
7	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	D₁₅₆	—	2	1	no	—
9	D₁₅₆	C₄	1	2	yes	complemented
10	C₃⋊#1055	—	2	1	no	—
11	C₂×D₅₂	—	3	1	no	—
12	C₁₃×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
13	C₂×#1055	—	3	1	no	—
14	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
15	C₂×D₃₉	—	2	2	no	—
16	C₂×D₃₉	—	4	1	no	—
17	C₂×D₃₉	—	4	1	no	—
18	C₃×C₄×C₁₃	—	4	1	no	—
19	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×D₁₃	—	6	1	no	—
22	C₂×C₄×C₁₃	—	2	1	no	—
23	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
24	D₅₂	—	3	2	no	—
25	D₅₂	—	6	1	no	—
26	#1055	—	6	1	no	—
27	#809	—	13	1	no	maximal, Hall
28	C₂×C₃×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃×C₁₃	D₈	1	1	yes	characteristic
30	C₂×C₃×C₁₃	SD₁₆	1	1	yes	characteristic
31	D₃₉	—	4	2	no	—
32	C₂²×C₁₃	#67	1	1	yes	characteristic
33	C₂×D₁₃	—	6	2	no	—
34	C₂×D₁₃	—	12	1	no	—
35	C₂×D₁₃	—	12	1	no	—
36	C₄×C₁₃	—	4	1	no	—
37	C₄×C₁₃	C₄×D₃	1	1	yes	characteristic
38	C₄×C₁₃	D₁₂	1	1	yes	characteristic
39	C₂×D₁₂	—	13	1	no	—
40	C₃×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
41	C₂×C₃⋊C₈	—	13	1	no	—
42	C₃×C₁₃	#102	1	1	yes	characteristic, complemented, Hall
43	#102	—	39	1	no	Sylow
44	C₂×C₁₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
45	C₂×C₁₃	#213	1	1	yes	characteristic
46	C₂×C₁₃	#215	1	1	yes	characteristic
47	D₁₃	—	12	2	no	—
48	C₂²×D₃	—	26	1	no	—
49	C₂×C₃×C₄	—	2	1	no	—
50	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
51	D₁₂	—	13	2	no	—
52	D₁₂	—	26	1	no	—
53	C₃⋊C₈	—	26	1	no	—
54	C₂×D₄	—	39	1	no	—
55	C₂×C₈	—	39	1	no	—
56	C₄⋊C₄	D₃₉	1	1	yes	characteristic, complemented
57	C₁₃	#809	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₃	C₂²⋊D₁₃	1	1	yes	characteristic
59	C₂×D₃	—	26	2	no	—
60	C₂×D₃	—	52	1	no	—
61	C₂×D₃	—	52	1	no	—
62	C₃×C₄	—	4	1	no	—
63	C₃×C₄	C₄×D₁₃	1	1	yes	characteristic
64	C₃×C₄	D₅₂	1	1	yes	characteristic
65	C₂³	—	78	1	no	—
66	D₄	—	39	2	no	—
67	D₄	—	78	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
70	C₈	—	78	1	no	—
71	C₂×C₃	#6103	1	1	yes	characteristic
72	C₂×C₃	#6104	1	1	yes	characteristic
73	C₂×C₃	#6106	1	1	yes	characteristic
74	D₃	—	52	2	no	—
75	C₂²	—	78	2	no	—
76	C₂²	—	156	1	no	—
77	C₂²	—	156	1	no	—
78	C₂²	#64625	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	C₄×D₃₉	1	1	yes	characteristic
81	C₄	D₁₅₆	1	1	yes	characteristic
82	C₃	#88049	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	156	2	no	—
84	C₂	#10597666	1	1	yes	characteristic
85	C₂	#10597667	1	1	yes	characteristic
86	C₂	#10597669	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64625
commutator	C₃×C₄×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₃₉
Fitting	C₃×C₁₃×C₄⋊C₄	C₂
socle	C₂²×C₃×C₁₃	D₄
prime radical	C₂×D₁₅₆	C₂
cosocle	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₄⋊C₄	D₃₉
hypocenter	C₃×C₁₃	#102
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₃×C₄⋊C₄	C₂	semidirect
3. C₂×C₃⋊#1055	C₂	semidirect
9. D₁₅₆	C₄	semidirect
12. C₁₃×C₄⋊C₄	D₃	semidirect
40. C₃×C₄⋊C₄	D₁₃	semidirect
42. C₃×C₁₃	#102	semidirect
56. C₄⋊C₄	D₃₉	semidirect
57. C₁₃	#809	semidirect
82. C₃	#88049	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₆	2
C₃×C₁₃×C₄⋊C₄	2
C₂×C₃⋊#1055	2
#88049	3
#809	13