Subgroups — #13094271

Order 1512.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4430⋊C₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#4430⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊D₇	C₃	1	1	yes	maximal, characteristic
5	(C₂²⋊D₇)⋊C₉	C₃	1	3	yes	maximal, complemented
6	C₂×#4430	—	2	1	no	—
7	C₂×#4430	C₂²	1	1	yes	characteristic
8	#4430⋊C₂	—	2	1	no	—
9	C₂²×C₃²×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃²×D₇	C₂×C₃	1	1	yes	characteristic
11	C₃²×Dic₇	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₇⋊C₉	C₂×C₃	1	3	yes	complemented
13	C₂×D₇⋊C₉	C₂×C₃	1	3	yes	complemented
14	Dic₇⋊C₉	C₂×C₃	1	3	yes	complemented
15	D₄×3₋¹⁺²	—	7	1	no	maximal, Hall
16	#4430	D₄	1	1	yes	characteristic, complemented, Hall
17	C₃×C₂²⋊D₇	—	3	1	no	—
18	C₃×C₂²⋊D₇	C₃²	1	1	yes	characteristic
19	C₂×C₃²×C₇	—	2	1	no	—
20	C₂×C₃²×C₇	C₂²×C₃	1	1	yes	characteristic
21	C₃²×D₇	—	2	1	no	—
22	C₂×C₇⋊C₉	—	2	3	no	—
23	C₂×C₇⋊C₉	C₂²×C₃	1	3	yes	—
24	D₇⋊C₉	—	2	3	no	—
25	C₂²×3₋¹⁺²	—	7	1	no	—
26	C₂²×3₋¹⁺²	—	7	1	no	—
27	C₄×3₋¹⁺²	—	7	1	no	—
28	C₂²×C₃×C₇	—	3	1	no	—
29	C₂²×C₃×C₇	C₂×C₃²	1	1	yes	characteristic
30	C₂×C₃×D₇	—	3	1	no	—
31	C₂×C₃×D₇	C₂×C₃²	1	1	yes	characteristic
32	C₃×Dic₇	—	3	1	no	—
33	C₃×Dic₇	C₂×C₃²	1	1	yes	characteristic
34	C₃²×D₄	—	7	1	no	—
35	D₄×C₉	—	7	3	no	—
36	C₃²×C₇	C₃×D₄	1	1	yes	characteristic
37	C₇⋊C₉	C₃×D₄	1	3	yes	complemented
38	C₂²⋊D₇	3₋¹⁺²	1	1	yes	characteristic, complemented, Hall
39	C₂×3₋¹⁺²	—	7	1	no	—
40	C₂×3₋¹⁺²	—	14	1	no	—
41	C₂×3₋¹⁺²	—	14	1	no	—
42	C₂×C₃×C₇	—	2	1	no	—
43	C₂×C₃×C₇	—	3	1	no	—
44	C₂×C₃×C₇	—	6	1	no	—
45	C₂×C₃×C₇	C₂²×C₃²	1	1	yes	characteristic
46	C₃×D₇	—	2	1	no	—
47	C₃×D₇	—	6	1	no	—
48	C₂²×C₃²	—	7	1	no	—
49	C₂²×C₃²	AGL(1,𝔽₇)	1	1	yes	characteristic
50	C₃²×C₄	—	7	1	no	—
51	C₂²×C₉	—	7	3	no	—
52	C₂²×C₉	—	7	3	no	—
53	C₄×C₉	—	7	3	no	—
54	C₂²×C₇	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
55	C₂×D₇	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
56	Dic₇	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
57	3₋¹⁺²	—	7	1	no	Sylow
58	C₃×D₄	—	7	1	no	—
59	C₃×D₄	—	21	1	no	—
60	C₃×C₇	—	3	1	no	—
61	C₃×C₇	C₃²×D₄	1	1	yes	characteristic
62	C₂×C₃²	—	2	1	no	—
63	C₂×C₃²	—	14	1	no	—
64	C₂×C₃²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₂×C₉	—	7	3	no	—
66	C₂×C₉	—	14	3	no	—
67	C₂×C₉	—	14	3	no	—
68	C₂×C₇	—	2	1	no	—
69	C₂×C₇	C₂²×3₋¹⁺²	1	1	yes	characteristic
70	D₇	—	2	1	no	—
71	C₂²×C₃	—	3	1	no	—
72	C₂²×C₃	—	7	1	no	—
73	C₂²×C₃	—	21	1	no	—
74	C₂²×C₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
75	C₃×C₄	—	7	1	no	—
76	C₃×C₄	—	21	1	no	—
77	C₃²	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
78	C₉	—	7	3	no	—
79	D₄	—	7	1	no	Sylow
80	C₇	D₄×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	2	1	no	—
82	C₂×C₃	—	3	1	no	—
83	C₂×C₃	—	6	1	no	—
84	C₂×C₃	—	14	1	no	—
85	C₂×C₃	—	42	1	no	—
86	C₂×C₃	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
87	C₂²	—	7	1	no	—
88	C₂²	#4430⋊C₂	1	1	yes	characteristic, complemented
89	C₄	—	7	1	no	—
90	C₃	—	3	1	no	—
91	C₃	C₃×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	14	1	no	—
94	C₂	C₂×#4430⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃	C₂×C₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₃×C₇	C₂²×C₃²
Fitting	C₂²×C₃²×C₇	C₂×C₃
prime radical	C₃²×C₂²⋊D₇	C₃
hypercenter	C₂²×C₃²	AGL(1,𝔽₇)
hypocenter	C₇	D₄×3₋¹⁺²
LCS	G ▷ C₂×C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂²×#4430	C₂	semidirect
2. C₂×#4430⋊C₂	C₂	semidirect
3. #4430⋊C₄	C₂	semidirect
5. (C₂²⋊D₇)⋊C₉	C₃	semidirect
12. C₂²×C₇⋊C₉	C₂×C₃	semidirect
13. C₂×D₇⋊C₉	C₂×C₃	semidirect
14. Dic₇⋊C₉	C₂×C₃	semidirect
16. #4430	D₄	semidirect
37. C₇⋊C₉	C₃×D₄	semidirect
38. C₂²⋊D₇	3₋¹⁺²	semidirect
54. C₂²×C₇	C₂×3₋¹⁺²	semidirect
55. C₂×D₇	C₂×3₋¹⁺²	semidirect
56. Dic₇	C₂×3₋¹⁺²	semidirect
80. C₇	D₄×3₋¹⁺²	semidirect
88. C₂²	#4430⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#4430	2
C₂×#4430⋊C₂	2
#4430⋊C₄	2
C₃²×C₂²⋊D₇	3
(C₂²⋊D₇)⋊C₉	3
D₄×3₋¹⁺²	7