Group #11726601 — subgroups

Subgroups — #11726601

Order 896.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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	#89748	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×Dic₇⋊C₄	—	2	2	no	—
10	C₂×Dic₇⋊C₄	—	2	2	no	—
11	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₇	C₄	1	4	yes	complemented
14	#1834	—	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	Dic₇⋊C₄	—	4	4	no	—
27	Dic₇⋊C₄	—	4	4	no	—
28	C₄×Dic₇	C₂×C₄	1	2	yes	—
29	C₄×Dic₇	Q₈	1	2	yes	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	—	7	1	no	—
33	#384	—	7	2	no	—
34	#340	—	7	1	no	—
35	#340	—	7	2	no	—
36	#340	D₇	1	1	yes	characteristic
37	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇	—	4	2	no	—
39	C₂×C₄×C₇	—	4	2	no	—
40	C₂×C₄×C₇	—	4	2	no	—
41	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇	#41	1	2	yes	—
44	C₂×C₄×C₇	#41	1	2	yes	—
45	C₂×Dic₇	—	4	2	no	—
46	C₂×Dic₇	—	4	2	no	—
47	C₂×Dic₇	—	4	2	no	—
48	C₂×Dic₇	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₇	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₇	D₈	1	2	yes	—
51	C₂×Dic₇	SD₁₆	1	4	yes	—
52	C₂×Dic₇	Q₁₆	1	2	yes	—
53	C₇×C₈	—	2	2	no	—
54	C₇⋊C₈	—	2	2	no	—
55	C₂²×C₈	—	7	1	no	—
56	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	7	2	no	—
58	C₂×C₄⋊C₄	—	14	2	no	—
59	C₂×C₄⋊C₄	—	14	2	no	—
60	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
61	C₂×C₄²	—	7	1	no	—
62	C₄×C₈	—	7	4	no	—
63	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₇	#124	1	2	yes	—
65	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇	C₂×SD₁₆	1	2	yes	—
67	C₂²×C₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₄×C₇	—	8	2	no	—
69	C₄×C₇	#117	1	1	yes	characteristic
70	C₄×C₇	#117	1	1	yes	characteristic
71	C₄×C₇	#125	1	2	yes	—
72	Dic₇	—	8	2	no	—
73	Dic₇	#106	1	4	yes	—
74	Dic₇	#107	1	4	yes	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	7	1	no	—
77	C₂²×C₄	—	7	1	no	—
78	C₂²×C₄	—	14	2	no	—
79	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	7	2	no	—
82	C₂×C₈	—	14	2	no	—
83	C₂×C₈	C₄×D₇	1	2	yes	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	14	2	no	—
87	C₄⋊C₄	—	14	2	no	—
88	C₄⋊C₄	—	28	4	no	—
89	C₄⋊C₄	—	28	4	no	—
90	C₄²	—	7	2	no	—
91	C₄²	—	7	2	no	—
92	C₂×C₇	#382	1	1	yes	characteristic
93	C₂×C₇	C₂×#106	1	1	yes	characteristic
94	C₂×C₇	C₂×#107	1	1	yes	characteristic
95	C₂×C₇	#486	1	2	yes	—
96	C₂×C₇	#487	1	2	yes	—
97	C₂³	D₄×D₇	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	7	2	no	—
102	C₂×C₄	—	7	2	no	—
103	C₂×C₄	—	7	2	no	—
104	C₂×C₄	—	7	4	no	—
105	C₂×C₄	—	7	2	no	—
106	C₂×C₄	—	28	2	no	—
107	C₂×C₄	—	28	2	no	—
108	C₂×C₄	—	28	2	no	—
109	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
110	C₂×C₄	#1157	1	2	yes	—
111	C₂×C₄	#1159	1	2	yes	—
112	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	14	2	no	—
115	C₇	#1834	1	1	yes	characteristic, complemented, Sylow
116	C₂²	#6448	1	2	yes	—
117	C₂²	D₇×C₄⋊C₄	1	1	yes	characteristic
118	C₂²	D₇×D₈	1	1	yes	characteristic
119	C₂²	D₇×SD₁₆	1	2	yes	—
120	C₂²	D₇×Q₁₆	1	1	yes	characteristic
121	C₄	—	7	4	no	—
122	C₄	—	7	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	56	2	no	—
125	C₄	#6435	1	1	yes	characteristic
126	C₄	#6440	1	1	yes	characteristic
127	C₄	#6452	1	2	yes	—
128	C₂	#89422	1	1	yes	characteristic
129	C₂	#89532	1	2	yes	—
130	C₂	#89564	1	2	yes	—
131	C₂	D₇×#106	1	1	yes	characteristic
132	C₂	D₇×#107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₇

C₂

yes

maximal, characteristic

#89748

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×Dic₇⋊C₄

—

C₂×Dic₇⋊C₄

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₇

C₄

yes

complemented

#1834

—

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

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

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

C₄⋊C₄

yes

—

C₂×Dic₇

C₄⋊C₄

yes

—

C₂×Dic₇

D₈

yes

—

C₂×Dic₇

SD₁₆

yes

—

C₂×Dic₇

Q₁₆

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₄×C₈

—

C₂²×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇

#124

yes

—

C₂²×C₇

C₂×D₈

yes

characteristic

C₂²×C₇

C₂×SD₁₆

yes

—

C₂²×C₇

C₂×Q₁₆

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

Dic₇

#106

yes

—

Dic₇

#107

yes

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂×C₇

#487

yes

—

C₂³

D₄×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄×D₇

yes

characteristic

110

C₂×C₄

#1157

yes

—

111

C₂×C₄

#1159

yes

—

112

C₂×C₄

Q₈×D₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₇

#1834

yes

characteristic, complemented, Sylow

116

C₂²

#6448

yes

—

117

C₂²

D₇×C₄⋊C₄

yes

characteristic

118

C₂²

D₇×D₈

yes

characteristic

119

C₂²

D₇×SD₁₆

yes

—

120

C₂²

D₇×Q₁₆

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#6435

yes

characteristic

126

C₄

#6440

yes

characteristic

127

C₄

#6452

yes

—

128

C₂

#89422

yes

characteristic

129

C₂

#89532

yes

—

130

C₂

#89564

yes

—

131

C₂

D₇×#106

yes

characteristic

132

C₂

D₇×#107

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	#1834
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
115. C₇	#1834	semidirect

H ≅ G/N

type

13. C₈×Dic₇

C₄

semidirect

115. C₇

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₇	2
#89748	2
C₇×#340	2
#89346	2
#89324	2
#1834	7

subgroup

index

C₂×C₈×Dic₇

Subgroups — #11726601

Conjugacy classes of subgroups

Complements

Maximal subgroups