Group #11729162 — subgroups

Subgroups — C₂×(C₂×D₇)⋊C₁₆ #11729162

Order 896.

272 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₄×D₇	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×D₇	—	2	2	no	—
9	C₂×C₈×D₇	—	2	2	no	—
10	C₂×C₈×D₇	C₄	1	2	yes	—
11	C₂×C₈×D₇	C₂²	1	2	yes	—
12	C₂×C₇×C₁₆	—	2	2	no	—
13	C₂×C₇×C₁₆	C₂²	1	2	yes	complemented
14	C₂×C₇⋊C₁₆	—	2	2	no	—
15	C₂×C₇⋊C₁₆	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊C₁₆	—	7	1	no	maximal, Sylow
17	C₂³×D₇	C₈	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₇	C₈	1	1	yes	characteristic
20	C₂×C₄×D₇	—	2	2	no	—
21	C₂×C₄×D₇	—	2	2	no	—
22	C₂×C₄×D₇	C₈	1	2	yes	—
23	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
24	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇×C₈	D₄	1	2	yes	—
26	C₂×C₇×C₈	D₄	1	2	yes	—
27	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₇⋊C₈	—	2	2	no	—
29	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₈×D₇	—	2	4	no	—
31	C₈×D₇	—	2	2	no	—
32	C₈×D₇	—	2	2	no	—
33	C₇×C₁₆	—	2	2	no	—
34	C₇⋊C₁₆	—	2	2	no	—
35	C₂³×C₈	—	7	1	no	—
36	C₂²×C₁₆	—	7	1	no	—
37	C₂²×C₁₆	D₇	1	1	yes	characteristic, complemented
38	C₂²⋊C₁₆	—	7	4	no	—
39	C₂³×C₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×D₇	—	2	2	no	—
41	C₂²×D₇	—	2	2	no	—
42	C₂²×D₇	C₁₆	1	4	yes	complemented
43	C₂²×D₇	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇	M₁₆	1	2	yes	—
47	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₇	—	2	2	no	—
49	C₂×Dic₇	C₂×C₈	1	2	yes	—
50	C₄×D₇	—	2	4	no	—
51	C₄×D₇	—	2	2	no	—
52	C₄×D₇	—	2	2	no	—
53	C₇×C₈	C₂²⋊C₄	1	2	yes	—
54	C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₇×C₈	C₂×D₄	1	1	yes	characteristic
56	C₇⋊C₈	—	2	2	no	—
57	C₂³×C₄	—	7	1	no	—
58	C₂²×C₈	—	7	1	no	—
59	C₂²×C₈	—	7	2	no	—
60	C₂²×C₈	—	7	2	no	—
61	C₂²×C₈	—	14	2	no	—
62	C₂²×C₈	—	14	2	no	—
63	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	7	2	no	—
66	C₂×C₁₆	—	14	2	no	—
67	C₂×C₁₆	C₂×D₇	1	2	yes	complemented
68	C₂²×C₇	C₂²⋊C₈	1	2	yes	—
69	C₂²×C₇	C₂×C₁₆	1	2	yes	—
70	C₂²×C₇	M₃₂	1	2	yes	—
71	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×D₇	—	2	8	no	—
73	C₂×D₇	—	2	2	no	—
74	C₂×D₇	—	2	2	no	—
75	C₂×D₇	C₂×C₁₆	1	4	yes	complemented
76	C₄×C₇	C₂²⋊C₈	1	2	yes	—
77	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
79	Dic₇	—	2	2	no	—
80	C₂⁴	—	7	1	no	—
81	C₂²×C₄	—	7	1	no	—
82	C₂²×C₄	—	7	2	no	—
83	C₂²×C₄	—	7	2	no	—
84	C₂²×C₄	—	14	2	no	—
85	C₂²×C₄	—	14	2	no	—
86	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
87	C₂×C₈	—	7	2	no	—
88	C₂×C₈	—	14	2	no	—
89	C₂×C₈	—	14	4	no	—
90	C₂×C₈	—	14	2	no	—
91	C₂×C₈	—	14	2	no	—
92	C₂×C₈	C₄×D₇	1	1	yes	characteristic
93	C₂×C₈	D₂₈	1	2	yes	—
94	C₂×C₈	C₂²⋊D₇	1	2	yes	—
95	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	14	2	no	—
98	C₂×C₇	C₂²⋊C₁₆	1	4	yes	complemented
99	C₂×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₇	C₂²×C₁₆	1	1	yes	characteristic
101	C₂×C₇	C₂×M₃₂	1	1	yes	characteristic
102	D₇	—	2	4	no	—
103	C₂³	—	7	2	no	—
104	C₂³	—	7	4	no	—
105	C₂³	—	14	2	no	—
106	C₂³	—	14	2	no	—
107	C₂³	C₈×D₇	1	1	yes	characteristic
108	C₂×C₄	—	7	2	no	—
109	C₂×C₄	—	14	2	no	—
110	C₂×C₄	—	14	2	no	—
111	C₂×C₄	—	14	2	no	—
112	C₂×C₄	—	14	4	no	—
113	C₂×C₄	C₈×D₇	1	1	yes	characteristic
114	C₂×C₄	#1131	1	2	yes	—
115	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
116	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
117	C₈	—	14	2	no	—
118	C₈	(C₂×D₇)⋊C₄	1	2	yes	—
119	C₈	C₂×D₂₈	1	1	yes	characteristic
120	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
121	C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	7	4	no	—
123	C₂²	—	14	2	no	—
124	C₂²	—	14	2	no	—
125	C₂²	—	14	8	no	—
126	C₂²	D₇×C₁₆	1	2	yes	—
127	C₂²	#6372	1	2	yes	—
128	C₂²	(C₂×D₇)⋊C₈	1	2	yes	—
129	C₂²	C₂×C₈×D₇	1	1	yes	characteristic
130	C₄	—	14	2	no	—
131	C₄	(C₂×D₇)⋊C₈	1	2	yes	—
132	C₄	C₂×#1131	1	1	yes	characteristic
133	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
134	C₂	—	14	4	no	—
135	C₂	(C₂×D₇)⋊C₁₆	1	4	yes	direct product factor
136	C₂	C₂×D₇×C₁₆	1	1	yes	characteristic
137	C₂	C₂×#6372	1	1	yes	characteristic
138	C₂	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₇

C₂

yes

maximal, characteristic

C₂²×C₇×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₇)⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₄×D₇

C₄

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₄

yes

characteristic

C₂×C₈×D₇

—

C₂×C₈×D₇

—

C₂×C₈×D₇

C₄

yes

—

C₂×C₈×D₇

C₂²

yes

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

C₂²

yes

complemented

C₂×C₇⋊C₁₆

—

C₂×C₇⋊C₁₆

C₂²

yes

complemented

C₂×C₂²⋊C₁₆

—

maximal, Sylow

C₂³×D₇

C₈

yes

characteristic

C₂²×C₄×C₇

C₂×C₄

yes

characteristic

C₂²×Dic₇

C₈

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

C₈

yes

—

C₂×C₄×D₇

C₂×C₄

yes

—

C₂×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

C₂³

yes

characteristic

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂×C₄

yes

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

D₇

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂³×C₇

C₂×C₈

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₁₆

yes

complemented

C₂²×D₇

C₂×C₈

yes

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂×C₈

yes

characteristic

C₂×C₄×C₇

M₁₆

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

C₂×C₈

yes

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

C₂²⋊C₄

yes

—

C₇×C₈

C₂×D₄

yes

characteristic

C₇×C₈

C₂×D₄

yes

characteristic

C₇⋊C₈

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇

yes

complemented

C₂²×C₇

C₂²⋊C₈

yes

—

C₂²×C₇

C₂×C₁₆

yes

—

C₂²×C₇

M₃₂

yes

—

C₂²×C₇

C₂²×C₈

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×C₁₆

yes

complemented

C₄×C₇

C₂²⋊C₈

yes

—

C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇

C₂×M₁₆

yes

characteristic

Dic₇

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇

yes

characteristic

C₂×C₈

D₂₈

yes

—

C₂×C₈

C₂²⋊D₇

yes

—

C₂×C₈

C₂²×D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₇

C₂²⋊C₁₆

yes

complemented

C₂×C₇

C₂×C₂²⋊C₈

yes

characteristic

100

C₂×C₇

C₂²×C₁₆

yes

characteristic

101

C₂×C₇

C₂×M₃₂

yes

characteristic

102

D₇

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₈×D₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₈×D₇

yes

characteristic

114

C₂×C₄

#1131

yes

—

115

C₂×C₄

(C₂×D₇)⋊C₄

yes

—

116

C₂×C₄

C₂×C₄×D₇

yes

characteristic

117

C₈

—

118

C₈

(C₂×D₇)⋊C₄

yes

—

119

C₈

C₂×D₂₈

yes

characteristic

120

C₈

C₂×C₂²⋊D₇

yes

characteristic

121

C₇

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

D₇×C₁₆

yes

—

127

C₂²

#6372

yes

—

128

C₂²

(C₂×D₇)⋊C₈

yes

—

129

C₂²

C₂×C₈×D₇

yes

characteristic

130

C₄

—

131

C₄

(C₂×D₇)⋊C₈

yes

—

132

C₄

C₂×#1131

yes

characteristic

133

C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

134

C₂

—

135

C₂

(C₂×D₇)⋊C₁₆

yes

direct product factor

136

C₂

C₂×D₇×C₁₆

yes

characteristic

137

C₂

C₂×#6372

yes

characteristic

138

C₂

C₂×(C₂×D₇)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₇
commutator	C₂×C₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₂²×C₇×C₁₆	C₂
socle	C₂³×C₇	C₂×C₈
prime radical	C₂³×D₇	C₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₂²×C₁₆	D₇
hypocenter	C₇	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₇

commutator

C₂×C₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₇

Fitting

C₂²×C₇×C₁₆

C₂

socle

C₂³×C₇

C₂×C₈

prime radical

C₂³×D₇

C₈

cosocle

C₂×C₇×C₈

C₂³

hypercenter

C₂²×C₁₆

D₇

hypocenter

C₇

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂²×C₇×C₁₆	C₂	semidirect
3. C₂²×C₇⋊C₁₆	C₂	semidirect
4. (C₂×D₇)⋊C₁₆	C₂	direct (N × H)
13. C₂×C₇×C₁₆	C₂²	semidirect
15. C₂×C₇⋊C₁₆	C₂²	semidirect
37. C₂²×C₁₆	D₇	semidirect
42. C₂²×D₇	C₁₆	semidirect
67. C₂×C₁₆	C₂×D₇	semidirect
75. C₂×D₇	C₂×C₁₆	semidirect
98. C₂×C₇	C₂²⋊C₁₆	semidirect
121. C₇	C₂×C₂²⋊C₁₆	semidirect
135. C₂	(C₂×D₇)⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂²×C₇×C₁₆

C₂

semidirect

3. C₂²×C₇⋊C₁₆

C₂

semidirect

4. (C₂×D₇)⋊C₁₆

C₂

direct (N × H)

13. C₂×C₇×C₁₆

C₂²

semidirect

15. C₂×C₇⋊C₁₆

C₂²

semidirect

37. C₂²×C₁₆

D₇

semidirect

42. C₂²×D₇

C₁₆

semidirect

67. C₂×C₁₆

C₂×D₇

semidirect

75. C₂×D₇

C₂×C₁₆

semidirect

98. C₂×C₇

C₂²⋊C₁₆

semidirect

121. C₇

C₂×C₂²⋊C₁₆

semidirect

135. C₂

(C₂×D₇)⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×C₈×D₇	2
C₂²×C₇×C₁₆	2
C₂²×C₇⋊C₁₆	2
(C₂×D₇)⋊C₁₆	2
C₂×C₂²⋊C₁₆	7

subgroup

index

C₂²×C₈×D₇

C₂²×C₇×C₁₆

C₂²×C₇⋊C₁₆

(C₂×D₇)⋊C₁₆

C₂×C₂²⋊C₁₆

Subgroups — C₂×(C₂×D₇)⋊C₁₆ #11729162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×(C2×D7)⋊C16 #11729162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂×D₇)⋊C₁₆ #11729162