Group #13052456 — subgroups

Subgroups — #13052456

Order 1344.

248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₇×#875	C₂	1	1	yes	maximal, characteristic
2	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#66569	C₂	1	2	yes	maximal, complemented
4	C₂×#66564	C₂	1	2	yes	maximal
5	#10620642	C₂	1	1	yes	maximal, characteristic, complemented
6	#89832	—	3	1	no	maximal, Hall
7	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
8	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
10	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₇×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₃×C₄⋊Dic₇	—	2	2	no	—
14	#66569	—	2	4	no	—
15	#66564	—	2	4	no	—
16	C₇×#124	—	3	1	no	—
17	C₂×#1144	—	3	2	no	—
18	C₂×#1142	—	3	2	no	—
19	#6432	D₃	1	1	yes	characteristic, complemented
20	#6377	—	3	1	no	—
21	#5086	—	7	1	no	maximal, Hall
22	C₂²×D₃×C₇	—	2	1	no	—
23	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
24	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₇×Dic₃	—	2	1	no	—
26	C₇×D₁₂	—	2	2	no	—
27	C₇×Dic₆	—	2	2	no	—
28	C₂×C₃×Dic₇	—	2	2	no	—
29	C₃×Dic₁₄	—	2	2	no	—
30	C₃×Dic₁₄	—	2	2	no	—
31	C₃⋊(C₇⋊C₈)	D₄	1	4	yes	—
32	C₂×C₇×Q₈	—	3	1	no	—
33	C₂×C₇×D₄	—	3	1	no	—
34	C₂×Dic₁₄	C₂×D₃	1	2	yes	—
35	C₇×C₂²⋊C₄	—	6	2	no	—
36	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
37	#1144	—	6	4	no	—
38	#1142	—	6	4	no	—
39	C₄⋊Dic₇	—	2	2	no	—
40	C₂×C₇⋊C₈	—	3	2	no	—
41	C₃×C₄⋊Q₈	—	7	1	no	—
42	C₂×#216	—	7	2	no	—
43	C₂×#215	—	7	2	no	—
44	#875	D₇	1	1	yes	characteristic
45	#803	—	7	1	no	—
46	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂×D₃×C₇	—	4	1	no	—
48	C₂×D₃×C₇	—	4	2	no	—
49	C₃×C₄×C₇	—	2	2	no	—
50	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₃×Dic₇	—	4	2	no	—
52	C₇×Dic₃	—	4	1	no	—
53	#497	—	21	1	no	Sylow
54	C₂³×C₇	—	6	1	no	—
55	C₇×Q₈	—	6	2	no	—
56	C₇×D₄	—	6	2	no	—
57	C₂×C₄×C₇	—	6	1	no	—
58	C₂×C₄×C₇	#67	1	2	yes	—
59	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
60	C₂×Dic₇	—	2	2	no	—
61	Dic₁₄	—	2	2	no	—
62	Dic₁₄	—	2	2	no	—
63	C₇⋊C₈	—	3	4	no	—
64	C₂×C₃×Q₈	—	7	2	no	—
65	C₂×D₁₂	C₂×D₇	1	1	yes	characteristic
66	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
67	C₃×C₄⋊C₄	—	14	2	no	—
68	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
69	#216	—	14	4	no	—
70	#215	—	14	4	no	—
71	(C₂×D₃)⋊C₄	—	2	2	no	—
72	C₂×C₃⋊C₈	—	7	2	no	—
73	C₂×C₃×C₇	#127	1	1	yes	characteristic
74	C₂×C₃×C₇	#137	1	2	yes	—
75	D₃×C₇	—	8	1	no	—
76	C₂×Q₁₆	—	21	2	no	—
77	C₂×SD₁₆	—	21	2	no	—
78	C₄⋊Q₈	—	7	1	no	—
79	#124	—	3	1	no	—
80	#97	—	21	1	no	—
81	C₂²×C₇	—	12	1	no	—
82	C₂²×C₇	—	12	2	no	—
83	C₂²×C₇	C₂×#67	1	1	yes	characteristic
84	C₄×C₇	—	2	2	no	—
85	C₄×C₇	—	12	1	no	—
86	C₄×C₇	D₃×D₄	1	2	yes	—
87	Dic₇	—	4	2	no	—
88	C₂²×D₃	—	2	1	no	—
89	C₃×Q₈	—	14	2	no	—
90	C₃×Q₈	—	14	2	no	—
91	C₂×C₃×C₄	—	14	2	no	—
92	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
93	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₂×Dic₃	—	2	1	no	—
95	D₁₂	—	2	2	no	—
96	Dic₆	—	2	2	no	—
97	C₃⋊C₈	—	7	4	no	—
98	C₃×C₇	#497	1	1	yes	characteristic, complemented, Hall
99	C₂×Q₈	—	3	1	no	—
100	C₂×Q₈	—	7	2	no	—
101	C₂×D₄	—	3	1	no	—
102	Q₁₆	—	42	4	no	—
103	SD₁₆	—	42	4	no	—
104	C₂×C₈	—	21	2	no	—
105	C₄⋊C₄	—	14	2	no	—
106	C₂²⋊C₄	—	6	2	no	—
107	C₄²	D₃×D₇	1	1	yes	characteristic
108	C₂×C₇	—	24	1	no	—
109	C₂×C₇	#940	1	1	yes	characteristic
110	C₂×C₇	#942	1	2	yes	—
111	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
112	C₂×D₃	—	4	1	no	—
113	C₂×D₃	—	4	2	no	—
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	—	28	2	no	—
116	C₃×C₄	D₄×D₇	1	2	yes	—
117	Dic₃	—	4	1	no	—
118	C₂³	—	6	1	no	—
119	Q₈	—	6	2	no	—
120	Q₈	—	14	2	no	—
121	Q₈	—	14	2	no	—
122	D₄	—	6	2	no	—
123	C₂×C₄	—	6	1	no	—
124	C₂×C₄	—	14	2	no	—
125	C₂×C₄	#4245	1	2	yes	—
126	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
127	C₈	—	21	4	no	—
128	C₇	#5086	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₃	#6503	1	1	yes	characteristic
130	C₂×C₃	#6514	1	2	yes	—
131	D₃	—	8	1	no	—
132	C₂²	—	12	1	no	—
133	C₂²	—	12	2	no	—
134	C₂²	C₂×#4245	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	12	1	no	—
137	C₄	—	28	2	no	—
138	C₄	#66681	1	2	yes	—
139	C₃	#89832	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	24	1	no	—
141	C₂	#10620871	1	2	yes	—
142	C₂	#10621025	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#875

C₂

yes

maximal, characteristic

C₃×#6432

C₂

yes

maximal, characteristic, complemented

C₂×#66569

C₂

yes

maximal, complemented

C₂×#66564

C₂

yes

maximal

#10620642

C₂

yes

maximal, characteristic, complemented

#89832

—

maximal, Hall

C₂×C₇×D₁₂

C₂²

yes

characteristic

C₂×C₇×Dic₆

C₂²

yes

characteristic

C₂×C₃×Dic₁₄

C₂²

yes

—

C₃×C₄²×C₇

C₂²

yes

characteristic

C₂×C₃⋊(C₇⋊C₈)

C₂²

yes

—

C₇×(C₂×D₃)⋊C₄

—

C₃×C₄⋊Dic₇

—

—

—

—

—

—

D₃

yes

characteristic, complemented

#6377

—

#5086

—

maximal, Hall

C₂²×D₃×C₇

—

C₂×C₃×C₄×C₇

D₄

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×C₇×Dic₃

—

C₇×D₁₂

—

C₇×Dic₆

—

C₂×C₃×Dic₇

—

C₃×Dic₁₄

—

C₃×Dic₁₄

—

C₃⋊(C₇⋊C₈)

D₄

yes

—

C₂×C₇×Q₈

—

C₂×C₇×D₄

—

C₂×Dic₁₄

C₂×D₃

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

C₂×D₃

yes

characteristic

#1144

—

#1142

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₃×C₄⋊Q₈

—

C₂×#216

—

C₂×#215

—

#875

D₇

yes

characteristic

#803

—

C₂²×C₃×C₇

C₂×D₄

yes

characteristic

C₂×D₃×C₇

—

C₂×D₃×C₇

—

C₃×C₄×C₇

—

C₃×C₄×C₇

C₂×D₄

yes

—

C₃×Dic₇

—

C₇×Dic₃

—

#497

—

Sylow

C₂³×C₇

—

C₇×Q₈

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#67

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇⋊C₈

—

C₂×C₃×Q₈

—

C₂×D₁₂

C₂×D₇

yes

characteristic

C₂×Dic₆

C₂×D₇

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₇

yes

characteristic

#216

—

#215

—

(C₂×D₃)⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃×C₇

#127

yes

characteristic

C₂×C₃×C₇

#137

yes

—

D₃×C₇

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#124

—

#97

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×#67

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

D₃×D₄

yes

—

Dic₇

—

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊D₇

yes

—

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

C₂×Dic₃

—

D₁₂

—

Dic₆

—

C₃⋊C₈

—

C₃×C₇

#497

yes

characteristic, complemented, Hall

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

Q₁₆

—

103

SD₁₆

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

D₃×D₇

yes

characteristic

108

C₂×C₇

—

109

C₂×C₇

#940

yes

characteristic

110

C₂×C₇

#942

yes

—

111

C₂²×C₃

C₂×C₂²⋊D₇

yes

characteristic

112

C₂×D₃

—

113

C₂×D₃

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

D₄×D₇

yes

—

117

Dic₃

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#4245

yes

—

126

C₂×C₄

C₂×D₃×D₇

yes

characteristic

127

C₈

—

128

C₇

#5086

yes

characteristic, complemented, Sylow

129

C₂×C₃

#6503

yes

characteristic

130

C₂×C₃

#6514

yes

—

131

D₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×#4245

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#66681

yes

—

139

C₃

#89832

yes

characteristic, complemented, Sylow

140

C₂

—

141

C₂

#10620871

yes

—

142

C₂

#10621025

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#4245
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂×D₃×D₇
Fitting	C₃×C₄²×C₇	C₂²
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×D₁₂	C₂²
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	#497
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#4245

commutator, cosocle

C₂×C₃×C₄×C₇

C₂³

Frattini

C₂×C₄

C₂×D₃×D₇

Fitting

C₃×C₄²×C₇

C₂²

socle

C₂²×C₃×C₇

C₂×D₄

prime radical

C₂×C₇×D₁₂

C₂²

hypercenter

C₄²

D₃×D₇

hypocenter

C₃×C₇

#497

LCS

G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₃×#6432	C₂	semidirect
3. C₂×#66569	C₂	semidirect
5. #10620642	C₂	semidirect
19. #6432	D₃	semidirect
98. C₃×C₇	#497	semidirect
128. C₇	#5086	semidirect
139. C₃	#89832	semidirect

H ≅ G/N

type

2. C₃×#6432

C₂

semidirect

3. C₂×#66569

C₂

semidirect

5. #10620642

C₂

semidirect

19. #6432

D₃

semidirect

98. C₃×C₇

#497

semidirect

128. C₇

#5086

semidirect

139. C₃

#89832

semidirect

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

subgroup	index
C₇×#875	2
C₃×#6432	2
C₂×#66569	2
C₂×#66564	2
#10620642	2
#89832	3
#5086	7

subgroup

index

Subgroups — #13052456

Conjugacy classes of subgroups

Complements

Maximal subgroups