Group #11740512 — subgroups

Subgroups — #11740512

Order 896.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6437	C₂	1	1	yes	maximal, characteristic
3	#89763	C₂	1	8	yes	maximal, complemented
4	#89760	C₂	1	4	yes	maximal, complemented
5	#89704	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₈	C₂²	1	2	yes	—
7	C₂²×D₂₈	C₂²	1	4	yes	—
8	C₇×#125	—	2	4	no	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₇)⋊C₄	C₂²	1	16	yes	—
13	C₂×C₄×Dic₇	C₂²	1	2	yes	—
14	#6437	—	2	4	no	—
15	C₇⋊#95	C₂²	1	4	yes	—
16	#2641	—	7	1	no	maximal, Sylow
17	C₂³×D₇	C₂³	1	4	yes	—
18	C₂²×C₄×C₇	C₂³	1	4	yes	—
19	C₂²×C₄×C₇	C₂³	1	2	yes	—
20	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₇	C₂³	1	4	yes	—
22	C₂×D₂₈	—	2	8	no	—
23	C₂×D₂₈	—	2	4	no	—
24	C₂×D₂₈	—	2	8	no	—
25	C₂×D₂₈	—	4	8	no	—
26	C₇×C₄⋊C₄	—	2	8	no	—
27	C₇×C₄⋊C₄	—	2	4	no	—
28	C₄²×C₇	D₄	1	4	yes	—
29	(C₂×D₇)⋊C₄	—	4	16	no	—
30	C₄×Dic₇	—	2	4	no	—
31	C₂×#127	—	7	1	no	—
32	C₂×#125	D₇	1	1	yes	characteristic, complemented
33	#394	—	7	8	no	—
34	#390	—	7	4	no	—
35	#376	—	7	1	no	—
36	C₂³×C₇	C₂⁴	1	1	yes	characteristic
37	C₂²×D₇	—	2	4	no	—
38	C₂²×D₇	—	2	8	no	—
39	C₂²×D₇	—	2	16	no	—
40	C₂×C₄×C₇	—	2	4	no	—
41	C₂×C₄×C₇	—	2	4	no	—
42	C₂×C₄×C₇	—	2	8	no	—
43	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₇	#41	1	4	yes	—
46	C₂×Dic₇	—	2	4	no	—
47	C₂×Dic₇	—	2	8	no	—
48	D₂₈	—	4	16	no	—
49	D₂₈	—	4	8	no	—
50	C₂²×D₄	—	7	2	no	—
51	C₂²×D₄	—	7	4	no	—
52	#127	—	14	4	no	—
53	#125	—	2	4	no	—
54	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
55	C₂×C₄⋊C₄	C₂×D₇	1	4	yes	—
56	C₂×C₂²⋊C₄	—	7	16	no	—
57	C₂×C₄²	—	7	2	no	—
58	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
59	#95	—	7	4	no	—
60	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₇	C₂×#41	1	2	yes	—
62	C₂²×C₇	H(5,𝔽₂)	1	4	yes	—
63	C₂×D₇	—	4	4	no	—
64	C₂×D₇	—	4	8	no	—
65	C₂×D₇	—	4	16	no	—
66	C₄×C₇	—	2	4	no	—
67	C₄×C₇	—	4	4	no	—
68	C₄×C₇	#124	1	4	yes	—
69	Dic₇	—	4	4	no	—
70	C₂⁴	—	7	4	no	—
71	C₂×D₄	—	14	8	no	—
72	C₂×D₄	—	14	8	no	—
73	C₂×D₄	—	14	4	no	—
74	C₂×D₄	—	28	8	no	—
75	C₂²×C₄	—	7	4	no	—
76	C₂²×C₄	C₂²×D₇	1	4	yes	—
77	C₂²×C₄	C₂²×D₇	1	2	yes	—
78	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₂²⋊C₄	—	28	16	no	—
82	C₄²	—	14	4	no	—
83	C₄²	C₂²⋊D₇	1	4	yes	—
84	C₂×C₇	C₂×#124	1	1	yes	characteristic
85	C₂×C₇	#535	1	2	yes	—
86	C₂×C₇	#540	1	4	yes	—
87	D₇	—	8	4	no	—
88	C₂³	—	14	8	no	—
89	C₂³	—	14	16	no	—
90	C₂³	—	14	4	no	—
91	C₂³	C₂³×D₇	1	1	yes	characteristic
92	D₄	—	28	16	no	—
93	D₄	—	28	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	14	4	no	—
98	C₂×C₄	—	14	8	no	—
99	C₂×C₄	#1161	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₇	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
102	C₇	#2641	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	28	16	no	—
104	C₂²	—	28	4	no	—
105	C₂²	—	28	8	no	—
106	C₂²	C₂×#1161	1	2	yes	—
107	C₂²	#6553	1	4	yes	—
108	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	28	4	no	—
112	C₄	#6510	1	4	yes	—
113	C₂	—	56	4	no	—
114	C₂	#90389	1	4	yes	—
115	C₂	C₂×#6510	1	1	yes	characteristic
116	C₂	#90521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#125

C₂

yes

maximal, characteristic, complemented

C₂×#6437

C₂

yes

maximal, characteristic

#89763

C₂

yes

maximal, complemented

#89760

C₂

yes

maximal, complemented

#89704

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₈

C₂²

yes

—

C₂²×D₂₈

C₂²

yes

—

C₇×#125

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

#6437

—

C₇⋊#95

C₂²

yes

—

#2641

—

maximal, Sylow

C₂³×D₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

(C₂×D₇)⋊C₄

—

C₄×Dic₇

—

C₂×#127

—

C₂×#125

D₇

yes

characteristic, complemented

#394

—

#390

—

#376

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

#125

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#124

yes

—

Dic₇

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₇

yes

—

C₂×C₇

C₂×#124

yes

characteristic

C₂×C₇

#535

yes

—

C₂×C₇

#540

yes

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₇

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1161

yes

—

100

C₂×C₄

C₂×C₂²⋊D₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₇

yes

—

102

C₇

#2641

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×#1161

yes

—

107

C₂²

#6553

yes

—

108

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#6510

yes

—

113

C₂

—

114

C₂

#90389

yes

—

115

C₂

C₂×#6510

yes

characteristic

116

C₂

#90521

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₇×#125	C₂
prime radical	C₂×#6437	C₂
hypercenter	C₂×#125	D₇
hypocenter	C₇	#2641
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₂×C₇×#125

C₂

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. C₂×C₇×#125	C₂	semidirect
3. #89763	C₂	semidirect
4. #89760	C₂	semidirect
5. #89704	C₂	semidirect
32. C₂×#125	D₇	semidirect
102. C₇	#2641	semidirect

H ≅ G/N

type

1. C₂×C₇×#125

C₂

semidirect

3. #89763

C₂

semidirect

4. #89760

C₂

semidirect

5. #89704

C₂

semidirect

32. C₂×#125

D₇

semidirect

102. C₇

#2641

semidirect

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

subgroup	index
C₂×C₇×#125	2
C₂×#6437	2
#89763	2
#89760	2
#89704	2
#2641	7

subgroup

index

C₂×C₇×#125

Subgroups — #11740512

Conjugacy classes of subgroups

Complements

Maximal subgroups