Subgroups — #15750731

Order 3328.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117129	C₂	1	1	yes	maximal, characteristic
2	#13113047	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718562	C₄	1	1	yes	characteristic
5	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
6	#11718091	C₄	1	1	yes	characteristic
7	#11718039	—	2	1	no	—
8	D₁₃×C₃₂	—	2	1	no	—
9	C₁₃×C₆₄	—	2	1	no	—
10	#11718035	—	2	1	no	—
11	#88155	C₈	1	1	yes	characteristic
12	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
13	#88052	C₈	1	1	yes	characteristic
14	#88038	—	2	1	no	—
15	C₁₆×D₁₃	—	2	1	no	—
16	C₁₃×C₃₂	D₄	1	1	yes	characteristic
17	C₁₃×C₃₂	D₄	1	1	yes	characteristic
18	#88034	—	2	1	no	—
19	#7514	—	13	1	no	maximal, Sylow
20	#6127	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
22	#6099	C₁₆	1	1	yes	characteristic
23	#6094	—	2	1	no	—
24	C₈×D₁₃	—	2	1	no	—
25	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₃×C₁₆	M₁₆	1	1	yes	characteristic
27	#6090	—	2	1	no	—
28	#2258	—	13	1	no	—
29	M₁₂₈	—	13	1	no	—
30	C₂×C₆₄	D₁₃	1	1	yes	characteristic, complemented
31	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
32	C₂²⋊D₁₃	—	2	1	no	—
33	D₅₂	C₃₂	1	1	yes	characteristic
34	C₄×D₁₃	—	2	1	no	—
35	Dic₂₆	C₃₂	1	1	yes	characteristic
36	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₈×C₁₃	M₃₂	1	1	yes	characteristic
38	#1055	—	2	1	no	—
39	#504	—	13	1	no	—
40	M₆₄	—	13	1	no	—
41	M₆₄	—	26	1	no	—
42	C₂×C₃₂	—	26	1	no	—
43	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
44	C₆₄	—	2	1	no	—
45	C₆₄	—	26	1	no	—
46	C₂²×C₁₃	M₆₄	1	1	yes	characteristic
47	C₂×D₁₃	—	2	1	no	—
48	C₄×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
49	C₄×C₁₃	C₂×C₃₂	1	1	yes	characteristic
50	Dic₁₃	—	2	1	no	—
51	#131	—	13	1	no	—
52	M₃₂	—	13	1	no	—
53	M₃₂	—	26	1	no	—
54	C₂×C₁₆	—	26	1	no	—
55	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
56	C₃₂	—	26	1	no	—
57	C₃₂	D₅₂	1	1	yes	characteristic
58	C₃₂	C₂²⋊D₁₃	1	1	yes	characteristic
59	C₂×C₁₃	—	2	1	no	—
60	C₂×C₁₃	C₂²⋊C₃₂	1	1	yes	characteristic
61	D₁₃	—	4	1	no	—
62	#41	—	13	1	no	—
63	M₁₆	—	13	1	no	—
64	M₁₆	—	26	1	no	—
65	C₂×C₈	—	26	1	no	—
66	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
67	C₁₆	—	26	1	no	—
68	C₁₆	#6094	1	1	yes	characteristic
69	C₁₆	#6103	1	1	yes	characteristic
70	C₁₃	#7514	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	13	1	no	—
72	D₄	—	13	1	no	—
73	D₄	—	26	1	no	—
74	C₂×C₄	—	26	1	no	—
75	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
76	C₈	—	26	1	no	—
77	C₈	#88038	1	1	yes	characteristic
78	C₈	#88060	1	1	yes	characteristic
79	C₂²	—	26	1	no	—
80	C₂²	#11718039	1	1	yes	characteristic
81	C₄	—	26	1	no	—
82	C₄	D₁₃×C₃₂	1	1	yes	characteristic
83	C₄	#11718099	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	52	1	no	—
86	C₂	#13112973	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₅₂
commutator	C₄×C₁₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₁₃
Fitting	C₂×C₁₃×C₆₄	C₂
socle	C₂×C₁₃	C₂²⋊C₃₂
prime radical	#6127	C₁₆
cosocle	C₂×C₁₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₁₃
hypocenter	C₁₃	#7514
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #13113047	C₂	semidirect
3. C₂×C₁₃×C₆₄	C₂	semidirect
30. C₂×C₆₄	D₁₃	semidirect
70. C₁₃	#7514	semidirect

Maximal subgroups

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

subgroup	index
#13117129	2
#13113047	2
C₂×C₁₃×C₆₄	2
#7514	13