Group #11726602 — subgroups

Subgroups — #11726602

Order 896.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
2	#89749	C₂	1	2	yes	maximal
3	C₇×#340	C₂	1	1	yes	maximal, characteristic
4	#89346	C₂	1	1	yes	maximal, characteristic
5	#89324	C₂	1	2	yes	maximal
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₇⋊#95	—	2	2	no	—
12	C₇⋊#95	—	2	2	no	—
13	C₈×Dic₇	C₄	1	4	yes	complemented
14	#1835	—	7	1	no	maximal, Sylow
15	C₂²×C₄×C₇	—	2	2	no	—
16	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₇	—	2	2	no	—
18	C₂²×Dic₇	D₄	1	1	yes	characteristic
19	C₂²×Dic₇	D₄	1	1	yes	characteristic
20	C₂×C₇×C₈	—	2	2	no	—
21	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₇×C₄⋊C₄	—	2	2	no	—
23	C₇×C₄⋊C₄	—	2	2	no	—
24	C₄⋊Dic₇	—	2	2	no	—
25	C₄⋊Dic₇	—	2	2	no	—
26	C₄×Dic₇	C₂×C₄	1	2	yes	—
27	C₄×Dic₇	Q₈	1	2	yes	—
28	C₂×C₇⋊C₈	—	2	2	no	—
29	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	—	7	1	no	—
31	#382	—	7	2	no	—
32	#340	—	7	1	no	—
33	#340	—	7	2	no	—
34	#340	D₇	1	1	yes	characteristic
35	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇	—	4	2	no	—
37	C₂×C₄×C₇	—	4	2	no	—
38	C₂×C₄×C₇	—	4	2	no	—
39	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇	#41	1	2	yes	—
42	C₂×C₄×C₇	#41	1	2	yes	—
43	C₂×Dic₇	—	2	1	no	—
44	C₂×Dic₇	—	2	2	no	—
45	C₂×Dic₇	—	2	1	no	—
46	C₂×Dic₇	—	4	2	no	—
47	C₂×Dic₇	—	4	2	no	—
48	C₂×Dic₇	—	4	2	no	—
49	C₂×Dic₇	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₇	C₄⋊C₄	1	2	yes	—
51	C₇×C₈	—	2	2	no	—
52	C₇⋊C₈	—	2	2	no	—
53	C₂²×C₈	—	7	1	no	—
54	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
55	C₂×C₄⋊C₄	—	7	2	no	—
56	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
57	C₂×C₄²	—	7	1	no	—
58	C₄×C₈	—	7	4	no	—
59	#95	—	14	2	no	—
60	#95	—	14	2	no	—
61	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₇	#124	1	2	yes	—
63	C₂²×C₇	#135	1	1	yes	characteristic
64	C₂²×C₇	#135	1	2	yes	—
65	C₂²×C₇	#135	1	1	yes	characteristic
66	C₄×C₇	—	8	2	no	—
67	C₄×C₇	#117	1	1	yes	characteristic
68	C₄×C₇	#117	1	1	yes	characteristic
69	C₄×C₇	#125	1	2	yes	—
70	Dic₇	—	2	2	no	—
71	Dic₇	—	2	2	no	—
72	Dic₇	—	8	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	7	1	no	—
75	C₂²×C₄	—	7	1	no	—
76	C₂²×C₄	—	14	2	no	—
77	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	7	2	no	—
80	C₂×C₈	—	14	2	no	—
81	C₂×C₈	C₄×D₇	1	2	yes	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	14	2	no	—
85	C₄⋊C₄	—	14	2	no	—
86	C₄²	—	7	2	no	—
87	C₄²	—	7	2	no	—
88	C₂×C₇	#382	1	1	yes	characteristic
89	C₂×C₇	#427	1	1	yes	characteristic
90	C₂×C₇	#427	1	1	yes	characteristic
91	C₂×C₇	#488	1	2	yes	—
92	C₂×C₇	#488	1	2	yes	—
93	C₂³	D₄×D₇	1	1	yes	characteristic
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	7	2	no	—
98	C₂×C₄	—	7	2	no	—
99	C₂×C₄	—	14	1	no	—
100	C₂×C₄	—	14	2	no	—
101	C₂×C₄	—	14	1	no	—
102	C₂×C₄	—	28	2	no	—
103	C₂×C₄	—	28	2	no	—
104	C₂×C₄	—	28	2	no	—
105	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
106	C₂×C₄	#1157	1	2	yes	—
107	C₂×C₄	#1159	1	2	yes	—
108	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	14	2	no	—
111	C₇	#1835	1	1	yes	characteristic, complemented, Sylow
112	C₂²	#6448	1	2	yes	—
113	C₂²	D₇×C₄⋊C₄	1	1	yes	characteristic
114	C₂²	#6475	1	1	yes	characteristic
115	C₂²	#6479	1	2	yes	—
116	C₂²	#6482	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	14	2	no	—
119	C₄	—	14	2	no	—
120	C₄	—	56	2	no	—
121	C₄	#6435	1	1	yes	characteristic
122	C₄	#6440	1	1	yes	characteristic
123	C₄	#6452	1	2	yes	—
124	C₂	#89422	1	1	yes	characteristic
125	C₂	#89540	1	2	yes	—
126	C₂	#89572	1	2	yes	—
127	C₂	#89632	1	1	yes	characteristic
128	C₂	#89652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₇

C₂

yes

maximal, characteristic

#89749

C₂

yes

maximal

C₇×#340

C₂

yes

maximal, characteristic

#89346

C₂

yes

maximal, characteristic

#89324

C₂

yes

maximal

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₇⋊#95

—

C₇⋊#95

—

C₈×Dic₇

C₄

yes

complemented

#1835

—

maximal, Sylow

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

D₄

yes

characteristic

C₂²×Dic₇

D₄

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×C₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

C₂×C₄

yes

—

C₄×Dic₇

Q₈

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₄⋊C₄

yes

—

C₂×Dic₇

C₄⋊C₄

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₄×C₈

—

#95

—

#95

—

C₂²×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇

#124

yes

—

C₂²×C₇

#135

yes

characteristic

C₂²×C₇

#135

yes

—

C₂²×C₇

#135

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂×C₇

#488

yes

—

C₂³

D₄×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₄×D₇

yes

characteristic

106

C₂×C₄

#1157

yes

—

107

C₂×C₄

#1159

yes

—

108

C₂×C₄

Q₈×D₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₇

#1835

yes

characteristic, complemented, Sylow

112

C₂²

#6448

yes

—

113

C₂²

D₇×C₄⋊C₄

yes

characteristic

114

C₂²

#6475

yes

characteristic

115

C₂²

#6479

yes

—

116

C₂²

#6482

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#6435

yes

characteristic

122

C₄

#6440

yes

characteristic

123

C₄

#6452

yes

—

124

C₂

#89422

yes

characteristic

125

C₂

#89540

yes

—

126

C₂

#89572

yes

—

127

C₂

#89632

yes

characteristic

128

C₂

#89652

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#340	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#340	D₇
hypocenter	C₇	#1835
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇

commutator

C₂×C₄×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#340

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
13. C₈×Dic₇	C₄	semidirect
111. C₇	#1835	semidirect

H ≅ G/N

type

13. C₈×Dic₇

C₄

semidirect

111. C₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₇	2
#89749	2
C₇×#340	2
#89346	2
#89324	2
#1835	7

subgroup

index

C₂×C₈×Dic₇

Subgroups — #11726602

Conjugacy classes of subgroups

Complements

Maximal subgroups