Group #11744564 — subgroups

Subgroups — #11744564

Order 896.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×2₋¹⁺⁴	C₂	1	12	yes	maximal, complemented
3	#90618	C₂	1	20	yes	maximal, complemented
4	#90616	C₂	1	15	yes	maximal, complemented
5	C₂×#6554	C₂	1	15	yes	maximal, complemented
6	C₇×2₋¹⁺⁴	C₂²	1	6	yes	complemented
7	C₇×H(5,𝔽₂)	C₂²	1	10	yes	complemented
8	C₂×C₇×#41	C₂²	1	15	yes	complemented
9	#6554	C₂²	1	60	yes	complemented
10	#6554	C₂²	1	120	yes	complemented
11	#6553	C₂²	1	20	yes	complemented
12	D₇×#41	C₂²	1	120	yes	complemented
13	#6551	C₂²	1	30	yes	complemented
14	C₂×Q₈×D₇	C₂²	1	30	yes	complemented
15	#6548	C₂²	1	90	yes	complemented
16	C₂×#1159	C₂²	1	90	yes	complemented
17	C₂×#1157	C₂²	1	45	yes	complemented
18	C₂²×Dic₁₄	C₂²	1	15	yes	complemented
19	2₋¹⁺⁶	—	7	1	no	maximal, Sylow
20	C₇×#41	C₂³	1	60	yes	—
21	C₇×#41	C₂³	1	20	yes	—
22	C₂×C₇×Q₈	C₂³	1	15	yes	—
23	C₂×C₇×D₄	C₂³	1	45	yes	—
24	C₂²×C₄×C₇	C₂³	1	15	yes	—
25	C₂×C₂²⋊D₇	C₂³	1	90	yes	—
26	C₂²×Dic₇	C₂³	1	30	yes	—
27	#1161	C₂³	1	40	yes	—
28	Q₈×D₇	C₂³	1	120	yes	—
29	#1159	C₂³	1	360	yes	—
30	D₄×D₇	C₂³	1	120	yes	—
31	#1157	C₂³	1	60	yes	—
32	#1157	C₂³	1	180	yes	—
33	C₂×D₂₈	C₂³	1	15	yes	—
34	C₂×C₄×D₇	C₂³	1	90	yes	—
35	C₂×Dic₁₄	C₂³	1	45	yes	—
36	C₂×Dic₁₄	C₂³	1	90	yes	—
37	#585	—	7	20	no	—
38	#585	—	7	15	no	—
39	#585	D₇	1	1	yes	characteristic, complemented
40	C₂×2₋¹⁺⁴	—	7	12	no	—
41	C₂×2₋¹⁺⁴	—	7	15	no	—
42	C₂³×C₇	C₂⁴	1	15	yes	—
43	C₂²×D₇	C₂⁴	1	30	yes	—
44	C₇×Q₈	C₂⁴	1	20	yes	—
45	C₇×D₄	C₂⁴	1	60	yes	—
46	C₂×C₄×C₇	C₂⁴	1	15	yes	—
47	C₂×C₄×C₇	C₂⁴	1	45	yes	—
48	C₂²⋊D₇	C₂⁴	1	120	yes	—
49	C₂×Dic₇	C₂⁴	1	90	yes	—
50	D₂₈	C₂⁴	1	6	yes	—
51	D₂₈	C₂⁴	1	30	yes	—
52	C₄×D₇	C₂⁴	1	120	yes	—
53	Dic₁₄	C₂⁴	1	10	yes	—
54	Dic₁₄	C₂⁴	1	90	yes	—
55	2₋¹⁺⁴	—	7	120	no	—
56	2₋¹⁺⁴	—	7	60	no	—
57	2₋¹⁺⁴	—	7	30	no	—
58	2₋¹⁺⁴	C₂×D₇	1	6	yes	complemented
59	H(5,𝔽₂)	—	7	20	no	—
60	H(5,𝔽₂)	—	7	90	no	—
61	H(5,𝔽₂)	C₂×D₇	1	10	yes	complemented
62	C₂×#41	—	7	120	no	—
63	C₂×#41	—	7	90	no	—
64	C₂×#41	—	7	45	no	—
65	C₂×#41	C₂×D₇	1	15	yes	complemented
66	C₂²×Q₈	—	7	30	no	—
67	C₂²×Q₈	—	7	15	no	—
68	C₂²×C₇	—	4	15	no	—
69	C₂²×C₇	C₂⁵	1	15	yes	—
70	C₂×D₇	—	4	30	no	—
71	C₂×D₇	C₂⁵	1	12	yes	—
72	C₄×C₇	C₂⁵	1	1	yes	characteristic
73	C₄×C₇	C₂⁵	1	15	yes	—
74	Dic₇	C₂⁵	1	20	yes	—
75	#41	—	7	40	no	—
76	#41	—	7	360	no	—
77	#41	—	7	180	no	—
78	#41	—	7	60	no	—
79	#41	C₂²×D₇	1	60	yes	—
80	#41	C₂²×D₇	1	20	yes	—
81	C₂×Q₈	—	7	120	no	—
82	C₂×Q₈	—	7	90	no	—
83	C₂×Q₈	—	7	45	no	—
84	C₂×Q₈	C₂²×D₇	1	15	yes	—
85	C₂×D₄	—	7	90	no	—
86	C₂×D₄	—	7	120	no	—
87	C₂×D₄	—	7	15	no	—
88	C₂×D₄	C₂²×D₇	1	45	yes	—
89	C₂²×C₄	—	7	30	no	—
90	C₂²×C₄	—	7	90	no	—
91	C₂²×C₄	C₂²×D₇	1	15	yes	—
92	C₂×C₇	—	2	15	no	—
93	C₂×C₇	C₂⁶	1	1	yes	characteristic
94	D₇	—	2	12	no	—
95	C₂³	—	7	30	no	—
96	C₂³	C₂³×D₇	1	15	yes	—
97	Q₈	—	7	90	no	—
98	Q₈	—	7	10	no	—
99	Q₈	C₂³×D₇	1	20	yes	—
100	D₄	—	7	120	no	—
101	D₄	—	7	30	no	—
102	D₄	—	7	6	no	—
103	D₄	C₂³×D₇	1	60	yes	—
104	C₂×C₄	—	7	90	no	—
105	C₂×C₄	—	7	120	no	—
106	C₂×C₄	C₂³×D₇	1	45	yes	—
107	C₂×C₄	C₂³×D₇	1	15	yes	—
108	C₇	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
109	C₂²	—	4	15	no	—
110	C₂²	—	7	12	no	—
111	C₂²	—	28	30	no	—
112	C₂²	C₂⁴×D₇	1	15	yes	—
113	C₄	—	7	20	no	—
114	C₄	C₂⁴×D₇	1	15	yes	—
115	C₄	C₂⁴×D₇	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	14	12	no	—
118	C₂	C₂⁵×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#585

C₂

yes

maximal, characteristic, complemented

D₇×2₋¹⁺⁴

C₂

yes

maximal, complemented

#90618

C₂

yes

maximal, complemented

#90616

C₂

yes

maximal, complemented

C₂×#6554

C₂

yes

maximal, complemented

C₇×2₋¹⁺⁴

C₂²

yes

complemented

C₇×H(5,𝔽₂)

C₂²

yes

complemented

C₂×C₇×#41

C₂²

yes

complemented

#6554

C₂²

yes

complemented

#6554

C₂²

120

yes

complemented

#6553

C₂²

yes

complemented

D₇×#41

C₂²

120

yes

complemented

#6551

C₂²

yes

complemented

C₂×Q₈×D₇

C₂²

yes

complemented

#6548

C₂²

yes

complemented

C₂×#1159

C₂²

yes

complemented

C₂×#1157

C₂²

yes

complemented

C₂²×Dic₁₄

C₂²

yes

complemented

2₋¹⁺⁶

—

maximal, Sylow

C₇×#41

C₂³

yes

—

C₇×#41

C₂³

yes

—

C₂×C₇×Q₈

C₂³

yes

—

C₂×C₇×D₄

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂×C₂²⋊D₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

#1161

C₂³

yes

—

Q₈×D₇

C₂³

120

yes

—

#1159

C₂³

360

yes

—

D₄×D₇

C₂³

120

yes

—

#1157

C₂³

yes

—

#1157

C₂³

180

yes

—

C₂×D₂₈

C₂³

yes

—

C₂×C₄×D₇

C₂³

yes

—

C₂×Dic₁₄

C₂³

yes

—

C₂×Dic₁₄

C₂³

yes

—

#585

—

#585

—

#585

D₇

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂³×C₇

C₂⁴

yes

—

C₂²×D₇

C₂⁴

yes

—

C₇×Q₈

C₂⁴

yes

—

C₇×D₄

C₂⁴

yes

—

C₂×C₄×C₇

C₂⁴

yes

—

C₂×C₄×C₇

C₂⁴

yes

—

C₂²⋊D₇

C₂⁴

120

yes

—

C₂×Dic₇

C₂⁴

yes

—

D₂₈

C₂⁴

yes

—

D₂₈

C₂⁴

yes

—

C₄×D₇

C₂⁴

120

yes

—

Dic₁₄

C₂⁴

yes

—

Dic₁₄

C₂⁴

yes

—

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₇

yes

—

—

C₂×D₇

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₇

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₇

—

C₂²×C₇

C₂⁵

yes

—

C₂×D₇

—

C₂×D₇

C₂⁵

yes

—

C₄×C₇

C₂⁵

yes

characteristic

C₄×C₇

C₂⁵

yes

—

Dic₇

C₂⁵

yes

—

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₇

yes

—

#41

C₂²×D₇

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₇

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂×C₇

—

C₂×C₇

C₂⁶

yes

characteristic

D₇

—

C₂³

—

C₂³

C₂³×D₇

yes

—

Q₈

—

Q₈

—

Q₈

C₂³×D₇

yes

—

100

D₄

—

120

—

101

D₄

—

102

D₄

—

103

D₄

C₂³×D₇

yes

—

104

C₂×C₄

—

105

C₂×C₄

—

120

—

106

C₂×C₄

C₂³×D₇

yes

—

107

C₂×C₄

C₂³×D₇

yes

—

108

C₇

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂⁴×D₇

yes

—

113

C₄

—

114

C₄

C₂⁴×D₇

yes

—

115

C₄

C₂⁴×D₇

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

C₂⁵×D₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁶
Fitting	C₇×#585	C₂
hypercenter	#585	D₇
hypocenter	C₇	2₋¹⁺⁶
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₇

commutator, socle, cosocle

C₂×C₇

C₂⁶

C₂

D₇

C₇

2₋¹⁺⁶

LCS

G ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. C₇×#585	C₂	semidirect
2. D₇×2₋¹⁺⁴	C₂	semidirect
3. #90618	C₂	semidirect
4. #90616	C₂	semidirect
5. C₂×#6554	C₂	semidirect
6. C₇×2₋¹⁺⁴	C₂²	semidirect
7. C₇×H(5,𝔽₂)	C₂²	semidirect
8. C₂×C₇×#41	C₂²	semidirect
9. #6554	C₂²	semidirect
10. #6554	C₂²	semidirect
11. #6553	C₂²	semidirect
12. D₇×#41	C₂²	semidirect
13. #6551	C₂²	semidirect
14. C₂×Q₈×D₇	C₂²	semidirect
15. #6548	C₂²	semidirect
16. C₂×#1159	C₂²	semidirect
17. C₂×#1157	C₂²	semidirect
18. C₂²×Dic₁₄	C₂²	semidirect
39. #585	D₇	semidirect
58. 2₋¹⁺⁴	C₂×D₇	semidirect
61. H(5,𝔽₂)	C₂×D₇	semidirect
65. C₂×#41	C₂×D₇	semidirect
108. C₇	2₋¹⁺⁶	semidirect

H ≅ G/N

type

1. C₇×#585

C₂

semidirect

2. D₇×2₋¹⁺⁴

C₂

semidirect

3. #90618

C₂

semidirect

4. #90616

C₂

semidirect

5. C₂×#6554

C₂

semidirect

6. C₇×2₋¹⁺⁴

C₂²

semidirect

7. C₇×H(5,𝔽₂)

C₂²

semidirect

8. C₂×C₇×#41

C₂²

semidirect

9. #6554

C₂²

semidirect

10. #6554

C₂²

semidirect

11. #6553

C₂²

semidirect

12. D₇×#41

C₂²

semidirect

13. #6551

C₂²

semidirect

14. C₂×Q₈×D₇

C₂²

semidirect

15. #6548

C₂²

semidirect

16. C₂×#1159

C₂²

semidirect

17. C₂×#1157

C₂²

semidirect

18. C₂²×Dic₁₄

C₂²

semidirect

39. #585

D₇

semidirect

58. 2₋¹⁺⁴

C₂×D₇

semidirect

61. H(5,𝔽₂)

C₂×D₇

semidirect

65. C₂×#41

C₂×D₇

semidirect

108. C₇

2₋¹⁺⁶

semidirect

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

subgroup	index
C₇×#585	2
D₇×2₋¹⁺⁴	2
#90618	2
#90616	2
C₂×#6554	2
2₋¹⁺⁶	7

subgroup

index

D₇×2₋¹⁺⁴

2₋¹⁺⁶

Subgroups — #11744564

Conjugacy classes of subgroups

Complements

Maximal subgroups