Group #11728294 — subgroups

Subgroups — #11728294

Order 896.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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×D₅₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#6397	C₂	1	2	yes	maximal, complemented
4	#89773	C₂	1	2	yes	maximal, complemented
5	#89347	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₈	C₂²	1	2	yes	—
7	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#117	C₂²	1	1	yes	characteristic
10	C₂×#1136	C₂²	1	2	yes	—
11	C₂×D₅₆	—	2	4	no	—
12	C₂×D₅₆	—	2	2	no	—
13	C₇×C₄⋊C₈	—	2	2	no	—
14	#6397	—	2	4	no	—
15	#6383	—	2	4	no	—
16	#2045	—	7	1	no	maximal, Sylow
17	C₂³×D₇	—	2	2	no	—
18	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×D₂₈	—	2	2	no	—
20	C₂×D₂₈	—	2	2	no	—
21	C₂×D₂₈	—	2	2	no	—
22	C₂×D₂₈	—	2	2	no	—
23	C₂×D₂₈	—	2	2	no	—
24	C₂×D₂₈	D₄	1	4	yes	—
25	C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₂×C₇×C₈	—	2	1	no	—
28	C₂×C₇×C₈	D₄	1	2	yes	—
29	C₇×C₄⋊C₄	D₄	1	2	yes	—
30	C₇×C₂²⋊C₄	—	2	1	no	—
31	C₄²×C₇	—	2	1	no	—
32	#1136	—	2	2	no	—
33	#1136	—	2	2	no	—
34	C₂×C₇⋊C₈	—	2	2	no	—
35	D₅₆	—	2	8	no	—
36	C₂²×D₈	—	7	1	no	—
37	#424	D₇	1	1	yes	characteristic, complemented
38	#418	—	7	2	no	—
39	C₂×#100	—	7	2	no	—
40	#341	—	7	1	no	—
41	C₂³×C₇	#41	1	1	yes	characteristic
42	C₂²×D₇	—	2	4	no	—
43	C₂²×D₇	—	4	2	no	—
44	C₂²×D₇	—	4	2	no	—
45	C₂²×D₇	—	8	4	no	—
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇	#41	1	2	yes	—
51	D₂₈	—	2	4	no	—
52	D₂₈	—	2	4	no	—
53	D₂₈	—	4	2	no	—
54	D₂₈	—	4	2	no	—
55	C₇×C₈	—	2	2	no	—
56	C₇×C₈	—	4	1	no	—
57	C₇⋊C₈	—	4	2	no	—
58	C₂²×D₄	—	7	2	no	—
59	C₂×D₈	—	14	4	no	—
60	C₂×D₈	—	14	2	no	—
61	C₂×M₁₆	—	7	2	no	—
62	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
64	#117	C₂×D₇	1	1	yes	characteristic
65	C₄⋊C₈	—	2	2	no	—
66	#102	—	14	4	no	—
67	#100	—	14	4	no	—
68	C₂²×C₇	—	2	2	no	—
69	C₂²×C₇	#121	1	1	yes	characteristic
70	C₂²×C₇	#123	1	2	yes	—
71	C₂×D₇	—	4	4	no	—
72	C₂×D₇	—	8	4	no	—
73	C₂×D₇	—	8	4	no	—
74	C₂×D₇	—	8	4	no	—
75	C₄×C₇	—	4	2	no	—
76	C₄×C₇	#120	1	1	yes	characteristic
77	C₄×C₇	#121	1	2	yes	—
78	C₄×C₇	#124	1	1	yes	characteristic
79	C₂⁴	—	14	2	no	—
80	C₂×D₄	—	7	4	no	—
81	C₂×D₄	—	14	2	no	—
82	C₂×D₄	—	14	2	no	—
83	C₂×D₄	—	14	2	no	—
84	C₂×D₄	—	14	2	no	—
85	C₂×D₄	—	14	2	no	—
86	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
87	D₈	—	14	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	14	2	no	—
90	M₁₆	—	14	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	14	2	no	—
94	C₂×C₈	C₂²⋊D₇	1	2	yes	—
95	C₄⋊C₄	D₂₈	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂×C₇	—	2	2	no	—
99	C₂×C₇	#394	1	1	yes	characteristic
100	C₂×C₇	#472	1	2	yes	—
101	D₇	—	8	4	no	—
102	C₂³	—	14	4	no	—
103	C₂³	—	28	2	no	—
104	C₂³	—	28	2	no	—
105	C₂³	—	56	4	no	—
106	C₂³	#1157	1	1	yes	characteristic
107	D₄	—	14	4	no	—
108	D₄	—	14	4	no	—
109	D₄	—	28	2	no	—
110	D₄	—	28	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
113	C₂×C₄	D₄×D₇	1	2	yes	—
114	C₂×C₄	#1161	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	4	1	no	—
118	C₈	—	28	2	no	—
119	C₇	#2045	1	1	yes	characteristic, complemented, Sylow
120	C₂²	—	2	2	no	—
121	C₂²	—	28	4	no	—
122	C₂²	—	56	4	no	—
123	C₂²	—	56	4	no	—
124	C₂²	—	56	4	no	—
125	C₂²	#6457	1	2	yes	—
126	C₂²	#6493	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	#6458	1	2	yes	—
129	C₄	#6500	1	1	yes	characteristic
130	C₄	#6510	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	56	4	no	—
133	C₂	#89763	1	1	yes	characteristic
134	C₂	#89914	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₆

C₂

yes

maximal, characteristic

C₇×#424

C₂

yes

maximal, characteristic, complemented

C₂×#6397

C₂

yes

maximal, complemented

#89773

C₂

yes

maximal, complemented

#89347

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₈

C₂²

yes

—

C₂×C₇×M₁₆

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₇×#117

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂×D₅₆

—

C₂×D₅₆

—

C₇×C₄⋊C₈

—

—

—

—

C₂³×D₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

D₄

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

D₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

#1136

—

#1136

—

C₂×C₇⋊C₈

—

D₅₆

—

C₂²×D₈

—

#424

D₇

yes

characteristic, complemented

#418

—

C₂×#100

—

#341

—

C₂³×C₇

#41

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

characteristic

C₂²×C₈

C₂×D₇

yes

characteristic

#117

C₂×D₇

yes

characteristic

C₄⋊C₈

—

#102

—

#100

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#121

yes

—

C₄×C₇

#124

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

C₄⋊C₄

D₂₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

#394

yes

characteristic

100

C₂×C₇

#472

yes

—

101

D₇

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

#1157

yes

characteristic

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₂₈

yes

characteristic

113

C₂×C₄

D₄×D₇

yes

—

114

C₂×C₄

#1161

yes

—

115

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₇

#2045

yes

characteristic, complemented, Sylow

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#6457

yes

—

126

C₂²

#6493

yes

characteristic

127

C₄

—

128

C₄

#6458

yes

—

129

C₄

#6500

yes

characteristic

130

C₄

#6510

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#89763

yes

characteristic

134

C₂

#89914

yes

—

	subgroup	G/H
center	C₂²	#6493
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#424	C₂
socle	C₂²×C₇	#121
prime radical	C₂²×D₅₆	C₂
hypercenter	#424	D₇
hypocenter	C₇	#2045
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#6493

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

C₂

C₂²×C₇

C₂²×D₅₆

C₂

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. C₇×#424	C₂	semidirect
3. C₂×#6397	C₂	semidirect
4. #89773	C₂	semidirect
5. #89347	C₂	semidirect
37. #424	D₇	semidirect
119. C₇	#2045	semidirect

H ≅ G/N

type

2. C₇×#424

C₂

semidirect

3. C₂×#6397

C₂

semidirect

4. #89773

C₂

semidirect

5. #89347

C₂

semidirect

37. #424

D₇

semidirect

119. C₇

#2045

semidirect

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

subgroup	index
C₂²×D₅₆	2
C₇×#424	2
C₂×#6397	2
#89773	2
#89347	2
#2045	7

subgroup

index

C₂²×D₅₆

Subgroups — #11728294

Conjugacy classes of subgroups

Complements

Maximal subgroups