Group #11726827 — subgroups

Subgroups — #11726827

Order 896.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇)⋊C₁₆	C₂	1	2	yes	maximal
4	Dic₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	#89256	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
8	C₄²×D₇	C₄	1	1	yes	characteristic
9	C₂×C₇×C₁₆	C₂²	1	2	yes	—
10	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₈×Dic₇	C₄	1	1	yes	characteristic
12	C₈×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₁₆	C₂²	1	2	yes	—
14	C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	#2163	—	7	1	no	maximal, Sylow
16	C₂×C₄×D₇	C₈	1	1	yes	characteristic
17	C₂×C₄×D₇	C₈	1	1	yes	characteristic
18	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₇	C₈	1	1	yes	characteristic
23	C₄×Dic₇	C₈	1	1	yes	characteristic
24	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₈×D₇	—	2	2	no	—
28	C₈×D₇	—	2	2	no	—
29	C₇×C₁₆	—	2	2	no	—
30	C₇⋊C₁₆	—	2	2	no	—
31	C₂×C₄×C₈	—	7	1	no	—
32	C₄⋊C₁₆	—	7	2	no	—
33	C₂²⋊C₁₆	—	7	2	no	—
34	#346	—	7	1	no	—
35	#346	D₇	1	1	yes	characteristic, complemented
36	C₂²×D₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₇	C₂×C₈	1	1	yes	characteristic
41	C₂×Dic₇	C₂×C₈	1	1	yes	characteristic
42	C₂×Dic₇	C₂×C₈	1	1	yes	characteristic
43	C₄×D₇	—	2	2	no	—
44	C₄×D₇	—	2	1	no	—
45	C₄×D₇	—	2	1	no	—
46	C₄×D₇	—	2	2	no	—
47	C₇×C₈	M₁₆	1	2	yes	—
48	C₇×C₈	#41	1	2	yes	—
49	C₇⋊C₈	—	2	1	no	—
50	C₇⋊C₈	—	2	1	no	—
51	C₂²×C₈	—	7	1	no	—
52	C₂²×C₈	—	7	1	no	—
53	C₂×C₄²	—	7	1	no	—
54	C₂×C₁₆	—	7	2	no	—
55	C₂×C₁₆	C₂×D₇	1	2	yes	—
56	C₄×C₈	—	7	1	no	—
57	C₄×C₈	—	7	1	no	—
58	C₄×C₈	—	7	1	no	—
59	C₄×C₈	C₂×D₇	1	1	yes	characteristic
60	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₇	—	2	1	no	—
62	C₂×D₇	—	2	1	no	—
63	C₂×D₇	M₃₂	1	2	yes	—
64	C₄×C₇	—	2	1	no	—
65	C₄×C₇	—	2	1	no	—
66	C₄×C₇	#117	1	1	yes	characteristic
67	C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
68	Dic₇	—	2	1	no	—
69	Dic₇	—	2	1	no	—
70	Dic₇	M₃₂	1	2	yes	—
71	C₂²×C₄	—	7	1	no	—
72	C₂²×C₄	—	7	1	no	—
73	C₂²×C₄	—	7	1	no	—
74	C₂×C₈	—	7	1	no	—
75	C₂×C₈	—	7	1	no	—
76	C₂×C₈	—	14	2	no	—
77	C₂×C₈	—	14	2	no	—
78	C₂×C₈	C₄×D₇	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
80	C₄²	—	7	1	no	—
81	C₄²	—	7	1	no	—
82	C₄²	—	7	1	no	—
83	C₄²	C₄×D₇	1	1	yes	characteristic
84	C₁₆	—	2	2	no	—
85	C₁₆	—	14	2	no	—
86	C₂×C₇	#431	1	1	yes	characteristic
87	C₂×C₇	C₂×M₃₂	1	1	yes	characteristic
88	C₂×C₇	C₂×M₃₂	1	1	yes	characteristic
89	D₇	—	2	2	no	—
90	C₂³	—	7	1	no	—
91	C₂×C₄	—	7	1	no	—
92	C₂×C₄	—	7	1	no	—
93	C₂×C₄	—	7	1	no	—
94	C₂×C₄	—	14	2	no	—
95	C₂×C₄	—	14	2	no	—
96	C₂×C₄	—	14	1	no	—
97	C₂×C₄	—	14	1	no	—
98	C₂×C₄	C₈×D₇	1	1	yes	characteristic
99	C₂×C₄	C₈×D₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
101	C₈	—	14	1	no	—
102	C₈	—	14	1	no	—
103	C₈	#1131	1	2	yes	—
104	C₈	#1157	1	2	yes	—
105	C₇	#2163	1	1	yes	characteristic, complemented, Sylow
106	C₂²	—	7	2	no	—
107	C₂²	—	14	1	no	—
108	C₂²	—	14	1	no	—
109	C₂²	C₂×C₈×D₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	7	2	no	—
113	C₄	—	14	1	no	—
114	C₄	—	14	1	no	—
115	C₄	#6435	1	1	yes	characteristic
116	C₄	C₂×#1131	1	1	yes	characteristic
117	C₂	—	14	2	no	—
118	C₂	#89457	1	1	yes	characteristic
119	C₂	D₇×M₃₂	1	1	yes	characteristic
120	C₂	D₇×M₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₇

C₂

yes

maximal, characteristic

C₇×#346

C₂

yes

maximal, characteristic, complemented

(C₂×D₇)⋊C₁₆

C₂

yes

maximal

Dic₇⋊C₁₆

C₂

yes

maximal, complemented

#89256

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇

C₄

yes

characteristic

C₂×C₈×D₇

C₂²

yes

characteristic

C₄²×D₇

C₄

yes

characteristic

C₂×C₇×C₁₆

C₂²

yes

—

C₄×C₇×C₈

C₂²

yes

characteristic

C₈×Dic₇

C₄

yes

characteristic

C₈×Dic₇

C₂²

yes

characteristic

C₂×C₇⋊C₁₆

C₂²

yes

—

C₄×C₇⋊C₈

C₄

yes

characteristic

#2163

—

maximal, Sylow

C₂×C₄×D₇

C₈

yes

characteristic

C₂×C₄×D₇

C₈

yes

characteristic

C₂×C₄×D₇

C₂×C₄

yes

characteristic

C₂×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₇×C₈

C₂³

yes

characteristic

C₄²×C₇

C₂×C₄

yes

characteristic

C₄×Dic₇

C₈

yes

characteristic

C₄×Dic₇

C₈

yes

characteristic

C₄×Dic₇

C₂×C₄

yes

characteristic

C₂×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

D₇

yes

characteristic, complemented

C₂²×D₇

C₂×C₈

yes

characteristic

C₂×C₄×C₇

C₂×C₈

yes

characteristic

C₂×C₄×C₇

C₂×C₈

yes

characteristic

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×Dic₇

C₂×C₈

yes

characteristic

C₂×Dic₇

C₂×C₈

yes

characteristic

C₂×Dic₇

C₂×C₈

yes

characteristic

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

M₁₆

yes

—

C₇×C₈

#41

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

C₂²×C₈

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

M₃₂

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#117

yes

characteristic

C₄×C₇

C₂×M₁₆

yes

characteristic

Dic₇

—

Dic₇

—

Dic₇

M₃₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇

yes

characteristic

C₂×C₈

C₂²×D₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₇

#431

yes

characteristic

C₂×C₇

C₂×M₃₂

yes

characteristic

C₂×C₇

C₂×M₃₂

yes

characteristic

D₇

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₇

yes

characteristic

C₂×C₄

C₈×D₇

yes

characteristic

100

C₂×C₄

C₂×C₄×D₇

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

#1131

yes

—

104

C₈

#1157

yes

—

105

C₇

#2163

yes

characteristic, complemented, Sylow

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₈×D₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#6435

yes

characteristic

116

C₄

C₂×#1131

yes

characteristic

117

C₂

—

118

C₂

#89457

yes

characteristic

119

C₂

D₇×M₃₂

yes

characteristic

120

C₂

D₇×M₃₂

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₈
Fitting	C₇×#346	C₂
prime radical	C₂²×D₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#346	D₇
hypocenter	C₇	#2163
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇

commutator, socle

C₂²×C₇

C₂²×C₈

Fitting

C₇×#346

C₂

prime radical

C₂²×D₇

C₂×C₈

cosocle

C₂×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. C₇×#346	C₂	semidirect
4. Dic₇⋊C₁₆	C₂	semidirect
5. #89256	C₂	semidirect
35. #346	D₇	semidirect
105. C₇	#2163	semidirect

H ≅ G/N

type

2. C₇×#346

C₂

semidirect

4. Dic₇⋊C₁₆

C₂

semidirect

5. #89256

C₂

semidirect

35. #346

D₇

semidirect

105. C₇

#2163

semidirect

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

subgroup	index
C₄×C₈×D₇	2
C₇×#346	2
(C₂×D₇)⋊C₁₆	2
Dic₇⋊C₁₆	2
#89256	2
#2163	7

subgroup

index

C₄×C₈×D₇

C₇×#346

(C₂×D₇)⋊C₁₆

Dic₇⋊C₁₆

#89256

#2163

Subgroups — #11726827

Conjugacy classes of subgroups

Complements

Maximal subgroups