Subgroups — #72259293

Order 8112.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#6099	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	#72229164	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×#92788	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92788	C₄	1	1	yes	characteristic
5	C₄×#92788	C₄	1	1	yes	characteristic
6	C₄×#92788	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₃²	C₂×C₃	1	1	yes	characteristic
8	C₁₃×#1055	C₂×C₃	1	2	yes	complemented
9	C₂×#92788	—	2	1	no	—
10	C₂×#92788	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₁₃²	C₃×C₄	1	1	yes	characteristic
12	C₄×C₁₃²	C₃×C₄	1	1	yes	characteristic
13	C₄×C₁₃²	C₂²×C₃	1	1	yes	characteristic
14	M₁₆×C₁₃⋊C₃	—	13	1	no	maximal
15	#6099⋊C₃	—	13	1	no	maximal
16	#92788	M₁₆	1	1	yes	characteristic, complemented, Hall
17	C₂×C₁₃²	—	2	1	no	—
18	C₂×C₁₃²	C₂×C₃×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₃⋊C₃	—	13	1	no	—
20	C₂×C₄×C₁₃⋊C₃	—	26	6	no	—
21	C₂×C₄×C₁₃⋊C₃	—	26	1	no	—
22	C₈×C₁₃⋊C₃	—	13	2	no	—
23	#1055⋊C₃	—	13	2	no	—
24	C₁₃×M₁₆	—	13	1	no	—
25	#6099	C₁₃⋊C₃	1	1	yes	characteristic, complemented
26	C₁₃²	C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂²×C₁₃⋊C₃	—	13	1	no	—
28	C₂²×C₁₃⋊C₃	—	26	6	no	—
29	C₂²×C₁₃⋊C₃	—	26	1	no	—
30	C₄×C₁₃⋊C₃	—	13	1	no	—
31	C₄×C₁₃⋊C₃	—	13	1	no	—
32	C₄×C₁₃⋊C₃	—	26	6	no	—
33	C₄×C₁₃⋊C₃	—	26	6	no	—
34	C₄×C₁₃⋊C₃	—	26	1	no	—
35	C₄×C₁₃⋊C₃	—	26	1	no	—
36	C₂×C₄×C₁₃	—	2	6	no	—
37	C₂×C₄×C₁₃	D₁₃⋊C₃	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂×C₁₃⋊C₃	1	1	yes	characteristic
39	C₈×C₁₃	—	13	2	no	—
40	#1055	C₂×C₁₃⋊C₃	1	2	yes	complemented
41	C₂×C₁₃⋊C₃	—	13	1	no	—
42	C₂×C₁₃⋊C₃	—	26	1	no	—
43	C₂×C₁₃⋊C₃	—	26	6	no	—
44	C₂×C₁₃⋊C₃	—	26	12	no	—
45	C₂×C₁₃⋊C₃	—	26	1	no	—
46	C₂×C₁₃⋊C₃	—	26	1	no	—
47	C₂²×C₁₃	—	2	6	no	—
48	C₂²×C₁₃	Dic₁₃⋊C₃	1	1	yes	characteristic
49	C₂²×C₁₃	C₄×C₁₃⋊C₃	1	1	yes	characteristic
50	C₄×C₁₃	—	2	6	no	—
51	C₄×C₁₃	—	2	6	no	—
52	C₄×C₁₃	Dic₁₃⋊C₃	1	1	yes	characteristic
53	C₄×C₁₃	C₄×C₁₃⋊C₃	1	1	yes	characteristic
54	C₄×C₁₃	C₂×D₁₃⋊C₃	1	1	yes	characteristic
55	C₄×C₁₃	C₂²×C₁₃⋊C₃	1	1	yes	characteristic
56	C₃×M₁₆	—	169	1	no	Hall
57	C₁₃⋊C₃	—	13	1	no	—
58	C₁₃⋊C₃	—	26	6	no	—
59	C₁₃⋊C₃	—	26	1	no	—
60	C₂×C₁₃	—	2	12	no	—
61	C₂×C₁₃	—	2	1	no	—
62	C₂×C₁₃	—	2	1	no	—
63	C₂×C₁₃	—	2	6	no	—
64	C₂×C₁₃	C₂×Dic₁₃⋊C₃	1	1	yes	characteristic
65	C₂×C₁₃	C₂×C₄×C₁₃⋊C₃	1	1	yes	characteristic
66	C₂×C₃×C₄	—	338	1	no	—
67	C₃×C₈	—	169	2	no	—
68	M₁₆	—	13	1	no	Sylow
69	C₁₃	—	2	6	no	—
70	C₁₃	#6099⋊C₃	1	1	yes	characteristic, complemented
71	C₁₃	M₁₆×C₁₃⋊C₃	1	1	yes	characteristic, complemented
72	C₂²×C₃	—	338	1	no	—
73	C₃×C₄	—	338	1	no	—
74	C₃×C₄	—	338	1	no	—
75	C₂×C₄	#11759817	1	1	yes	characteristic
76	C₈	—	13	2	no	—
77	C₂×C₃	—	338	1	no	—
78	C₂×C₃	—	338	1	no	—
79	C₂²	#14523130	1	1	yes	characteristic
80	C₄	#14523130	1	1	yes	characteristic
81	C₄	C₂×#11759817	1	1	yes	characteristic
82	C₃	—	338	1	no	Sylow
83	C₂	—	2	1	no	—
84	C₂	C₂×#14523130	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×#11759817
commutator, socle	C₂×C₁₃²	C₂×C₃×C₄
Fitting	C₂×C₄×C₁₃²	C₂×C₃
prime radical	C₂²×#92788	C₄
cosocle	C₄×C₁₃²	C₂²×C₃
hypercenter	C₂×C₄	#11759817
hypocenter	C₁₃²	C₃×M₁₆
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₁₃×#6099	C₃	semidirect
2. #72229164	C₂	semidirect
8. C₁₃×#1055	C₂×C₃	semidirect
16. #92788	M₁₆	semidirect
25. #6099	C₁₃⋊C₃	semidirect
26. C₁₃²	C₃×M₁₆	semidirect
40. #1055	C₂×C₁₃⋊C₃	semidirect
70. C₁₃	#6099⋊C₃	semidirect
71. C₁₃	M₁₆×C₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#6099	3
#72229164	2
C₂×C₄×#92788	2
M₁₆×C₁₃⋊C₃	13
#6099⋊C₃	13