Group #11740639 — subgroups

Subgroups — C₂×#89853 #11740639

Order 896.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6432	C₂	1	1	yes	maximal, characteristic
3	C₂×#6409	C₂	1	4	yes	maximal
4	#89853	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	2	yes	—
7	C₂²×Dic₁₄	—	2	1	no	—
8	C₇×C₄⋊Q₈	—	2	2	no	—
9	C₇×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊C₄	—	2	1	no	—
11	C₂×C₇×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇	—	2	1	no	—
14	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
16	#6432	—	2	2	no	—
17	#6432	C₂²	1	4	yes	—
18	#6409	C₂²	1	16	yes	—
19	C₄×C₇⋊C₈	C₂²	1	4	yes	—
20	C₂×#487	—	7	1	no	maximal, Sylow
21	C₂×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₈	—	2	8	no	—
23	C₂×C₇×Q₈	C₂³	1	4	yes	—
24	C₂²×C₄×C₇	—	2	2	no	—
25	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₇	—	2	2	no	—
29	C₂×Dic₁₄	—	2	4	no	—
30	C₂×Dic₁₄	—	4	4	no	—
31	C₇×C₄⋊C₄	—	2	4	no	—
32	C₇×C₄⋊C₄	—	2	4	no	—
33	C₄²×C₇	D₄	1	2	yes	—
34	C₄²×C₇	C₂³	1	2	yes	—
35	C₄⋊Dic₇	—	2	2	no	—
36	C₄⋊Dic₇	—	2	4	no	—
37	C₄⋊Dic₇	C₂³	1	4	yes	—
38	C₂×C₇⋊C₈	—	2	4	no	—
39	C₂×C₇⋊C₈	C₂³	1	4	yes	—
40	C₂×C₄⋊Q₈	—	7	1	no	—
41	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
42	#487	—	7	8	no	—
43	C₂×#103	—	7	4	no	—
44	C₂×C₄×C₈	—	7	1	no	—
45	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
46	C₇×Q₈	—	2	8	no	—
47	C₇×Q₈	—	4	4	no	—
48	C₂×C₄×C₇	—	2	4	no	—
49	C₂×C₄×C₇	—	4	4	no	—
50	C₂×C₄×C₇	SD₁₆	1	4	yes	—
51	C₂×C₄×C₇	Q₁₆	1	4	yes	—
52	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇	#41	1	4	yes	—
56	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₇	—	2	4	no	—
58	C₂×Dic₇	—	4	4	no	—
59	Dic₁₄	—	4	8	no	—
60	C₇⋊C₈	—	2	4	no	—
61	C₂²×Q₈	—	14	1	no	—
62	C₂²×Q₈	C₂×D₇	1	2	yes	—
63	C₂²×C₈	—	7	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	7	4	no	—
66	C₄⋊Q₈	—	14	2	no	—
67	C₄⋊Q₈	C₂×D₇	1	4	yes	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	7	2	no	—
71	C₂×C₄⋊C₄	—	14	1	no	—
72	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
73	#103	—	7	16	no	—
74	C₄×C₈	—	7	4	no	—
75	C₂²×C₇	#124	1	2	yes	—
76	C₂²×C₇	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₇	C₂×Q₁₆	1	2	yes	—
78	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₇	—	4	4	no	—
80	C₄×C₇	#124	1	2	yes	—
81	C₄×C₇	C₂×SD₁₆	1	4	yes	—
82	C₄×C₇	C₂×Q₁₆	1	4	yes	—
83	C₄×C₇	C₂×#41	1	2	yes	—
84	Dic₇	—	4	4	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	—	2	8	no	—
87	C₂×Q₈	—	14	4	no	—
88	C₂×Q₈	—	28	4	no	—
89	C₂×Q₈	C₂²×D₇	1	4	yes	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	14	2	no	—
92	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
95	C₂×C₈	—	7	4	no	—
96	C₂×C₈	—	14	4	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	7	4	no	—
100	C₄⋊C₄	—	14	2	no	—
101	C₄⋊C₄	—	14	4	no	—
102	C₄²	C₂²⋊D₇	1	2	yes	—
103	C₄²	C₂²×D₇	1	2	yes	—
104	C₂×C₇	#487	1	4	yes	complemented
105	C₂×C₇	C₂×#124	1	1	yes	characteristic
106	C₂×C₇	C₂²×SD₁₆	1	1	yes	characteristic
107	C₂×C₇	C₂²×Q₁₆	1	1	yes	characteristic
108	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
109	Q₈	—	2	8	no	—
110	Q₈	—	4	4	no	—
111	Q₈	—	28	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	14	4	no	—
115	C₂×C₄	—	28	4	no	—
116	C₂×C₄	#1143	1	4	yes	—
117	C₂×C₄	#1144	1	4	yes	—
118	C₂×C₄	#1159	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
122	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
123	C₈	—	14	4	no	—
124	C₇	C₂×#487	1	1	yes	characteristic, complemented, Sylow
125	C₂²	#6499	1	2	yes	—
126	C₂²	C₂×#1143	1	2	yes	—
127	C₂²	C₂×#1144	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	28	4	no	—
131	C₄	#6499	1	2	yes	—
132	C₄	C₂×#1143	1	4	yes	—
133	C₄	C₂×#1144	1	4	yes	—
134	C₄	C₂×#1159	1	2	yes	—
135	C₂	#89853	1	4	yes	direct product factor
136	C₂	C₂×#6499	1	1	yes	characteristic
137	C₂	C₂²×#1143	1	1	yes	characteristic
138	C₂	C₂²×#1144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#6432

C₂

yes

maximal, characteristic

C₂×#6409

C₂

yes

maximal

#89853

C₂

yes

maximal, direct product factor

C₂×C₄×C₇⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂²×Dic₁₄

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

C₂²

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

—

#6432

—

#6432

C₂²

yes

—

#6409

C₂²

yes

—

C₄×C₇⋊C₈

C₂²

yes

—

C₂×#487

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

C₂³

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

characteristic

C₂²×C₄×C₇

D₄

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄²×C₇

C₂³

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂³

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂³

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇

yes

characteristic

#487

—

C₂×#103

—

C₂×C₄×C₈

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

SD₁₆

yes

—

C₂×C₄×C₇

Q₁₆

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#103

—

C₄×C₈

—

C₂²×C₇

#124

yes

—

C₂²×C₇

C₂×SD₁₆

yes

—

C₂²×C₇

C₂×Q₁₆

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

#124

yes

—

C₄×C₇

C₂×SD₁₆

yes

—

C₄×C₇

C₂×Q₁₆

yes

—

C₄×C₇

C₂×#41

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇

yes

characteristic

C₂²×C₄

C₂²⋊D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

C₂²⋊D₇

yes

—

103

C₄²

C₂²×D₇

yes

—

104

C₂×C₇

#487

yes

complemented

105

C₂×C₇

C₂×#124

yes

characteristic

106

C₂×C₇

C₂²×SD₁₆

yes

characteristic

107

C₂×C₇

C₂²×Q₁₆

yes

characteristic

108

C₂³

C₂×C₂²⋊D₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#1143

yes

—

117

C₂×C₄

#1144

yes

—

118

C₂×C₄

#1159

yes

—

119

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₇

yes

—

121

C₂×C₄

C₂×C₂²⋊D₇

yes

—

122

C₂×C₄

C₂³×D₇

yes

characteristic

123

C₈

—

124

C₇

C₂×#487

yes

characteristic, complemented, Sylow

125

C₂²

#6499

yes

—

126

C₂²

C₂×#1143

yes

—

127

C₂²

C₂×#1144

yes

—

128

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

#6499

yes

—

132

C₄

C₂×#1143

yes

—

133

C₄

C₂×#1144

yes

—

134

C₄

C₂×#1159

yes

—

135

C₂

#89853

yes

direct product factor

136

C₂

C₂×#6499

yes

characteristic

137

C₂

C₂²×#1143

yes

characteristic

138

C₂

C₂²×#1144

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	C₂×#487
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₂×C₇×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇

hypocenter

C₇

C₂×#487

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
4. #89853	C₂	direct (N × H)
104. C₂×C₇	#487	semidirect
124. C₇	C₂×#487	semidirect
135. C₂	#89853	direct (N × H)

H ≅ G/N

type

4. #89853

C₂

direct (N × H)

104. C₂×C₇

#487

semidirect

124. C₇

C₂×#487

semidirect

135. C₂

#89853

direct (N × H)

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×#6432	2
C₂×#6409	2
#89853	2
C₂×C₄×C₇⋊C₈	2
C₂×#487	7

subgroup

index

C₂×C₇×C₄⋊Q₈

C₂×C₄×C₇⋊C₈

Subgroups — C₂×#89853 #11740639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#89853 #11740639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89853 #11740639