Group #11728939 — subgroups

Subgroups — #11728939

Order 896.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	#89857	C₂	1	1	yes	maximal, characteristic, complemented
3	#89851	C₂	1	1	yes	maximal, characteristic, complemented
4	#89845	C₂	1	1	yes	maximal, characteristic
5	#89842	C₂	1	1	yes	maximal, characteristic, complemented
6	#89465	C₂	1	1	yes	maximal, characteristic, complemented
7	#89249	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₇×#127	C₂²	1	1	yes	characteristic
10	C₂×#1143	—	2	2	no	—
11	C₂×#1141	—	2	2	no	—
12	C₂×#1132	—	2	2	no	—
13	#6437	C₂²	1	1	yes	characteristic
14	#6432	C₂²	1	1	yes	characteristic
15	C₇×#102	—	2	2	no	—
16	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#6406	—	2	2	no	—
18	#6382	—	2	2	no	—
19	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	#1715	—	7	1	no	maximal, Sylow
22	C₂×C₇×Q₈	—	2	1	no	—
23	C₂×C₇×D₄	—	2	1	no	—
24	C₂×C₇×D₄	—	2	1	no	—
25	C₂×C₇×D₄	—	2	1	no	—
26	C₂×D₂₈	—	2	1	no	—
27	C₂×D₂₈	—	2	1	no	—
28	C₂×D₂₈	—	2	1	no	—
29	C₂×Dic₁₄	—	2	1	no	—
30	C₂×C₇×C₈	D₄	1	2	yes	—
31	C₇×C₄⋊C₄	—	2	1	no	—
32	C₇×C₄⋊C₄	—	2	1	no	—
33	C₄²×C₇	C₂³	1	1	yes	characteristic
34	#1143	—	4	4	no	—
35	#1141	—	4	4	no	—
36	C₄⋊Dic₇	—	2	1	no	—
37	C₄⋊Dic₇	—	2	1	no	—
38	C₂×C₇⋊C₈	D₄	1	2	yes	—
39	C₂×C₇⋊C₈	D₄	1	2	yes	—
40	#1132	—	4	4	no	—
41	#498	—	7	1	no	—
42	#493	—	7	1	no	—
43	#492	—	7	1	no	—
44	#492	—	7	1	no	—
45	#486	—	7	1	no	—
46	#486	D₇	1	1	yes	characteristic, complemented
47	#322	—	7	1	no	—
48	C₂³×C₇	—	4	1	no	—
49	C₂²×D₇	—	4	1	no	—
50	C₇×Q₈	—	4	2	no	—
51	C₇×D₄	—	4	2	no	—
52	C₇×D₄	—	4	2	no	—
53	C₇×D₄	—	4	2	no	—
54	C₂×C₄×C₇	—	4	1	no	—
55	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₇	—	4	1	no	—
59	D₂₈	—	4	2	no	—
60	D₂₈	—	4	2	no	—
61	D₂₈	—	4	2	no	—
62	Dic₁₄	—	4	2	no	—
63	C₇×C₈	—	2	2	no	—
64	C₇⋊C₈	—	2	2	no	—
65	C₇⋊C₈	SD₁₆	1	4	yes	complemented
66	C₂×SD₁₆	—	14	2	no	—
67	C₂×SD₁₆	—	14	2	no	—
68	C₂×D₈	—	14	2	no	—
69	C₄⋊Q₈	—	7	1	no	—
70	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
71	#127	—	7	1	no	—
72	#127	C₂×D₇	1	1	yes	characteristic
73	#106	—	14	2	no	—
74	#102	—	2	2	no	—
75	#102	—	14	2	no	—
76	C₄×C₈	—	7	1	no	—
77	C₄×C₈	—	7	1	no	—
78	C₄×C₈	C₂×D₇	1	1	yes	characteristic
79	C₂²×C₇	—	8	1	no	—
80	C₂²×C₇	—	8	1	no	—
81	C₂²×C₇	—	8	1	no	—
82	C₂²×C₇	#127	1	1	yes	characteristic
83	C₂×D₇	—	8	1	no	—
84	C₂×D₇	—	8	1	no	—
85	C₂×D₇	—	8	1	no	—
86	C₄×C₇	—	8	1	no	—
87	C₄×C₇	C₂×SD₁₆	1	2	yes	—
88	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
90	Dic₇	—	8	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	14	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	14	1	no	—
97	C₂×D₄	—	14	1	no	—
98	C₂×D₄	—	14	1	no	—
99	SD₁₆	—	28	4	no	—
100	SD₁₆	—	28	4	no	—
101	D₈	—	28	4	no	—
102	C₂×C₈	—	7	2	no	—
103	C₂×C₈	—	7	2	no	—
104	C₂×C₈	C₂²⋊D₇	1	2	yes	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	14	1	no	—
108	C₄⋊C₄	—	14	1	no	—
109	C₄²	C₂²×D₇	1	1	yes	characteristic
110	C₂×C₇	—	16	1	no	—
111	C₂×C₇	#492	1	1	yes	characteristic
112	C₂×C₇	#496	1	1	yes	characteristic
113	C₂×C₇	#496	1	1	yes	characteristic
114	D₇	—	16	1	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	28	1	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	28	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	28	2	no	—
123	D₄	—	28	2	no	—
124	D₄	—	28	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	28	1	no	—
127	C₂×C₄	D₄×D₇	1	1	yes	characteristic
128	C₂×C₄	D₄×D₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	—	7	4	no	—
132	C₈	—	14	2	no	—
133	C₇	#1715	1	1	yes	characteristic, complemented, Sylow
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	56	1	no	—
138	C₂²	—	56	1	no	—
139	C₂²	—	56	1	no	—
140	C₂²	#6503	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	56	1	no	—
143	C₄	#6474	1	2	yes	—
144	C₄	D₇×SD₁₆	1	2	yes	—
145	C₄	#6512	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	112	1	no	—
148	C₂	#89830	1	1	yes	characteristic
149	C₂	#89926	1	1	yes	characteristic
150	C₂	#89947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#486

C₂

yes

maximal, characteristic, complemented

#89857

C₂

yes

maximal, characteristic, complemented

#89851

C₂

yes

maximal, characteristic, complemented

#89845

C₂

yes

maximal, characteristic

#89842

C₂

yes

maximal, characteristic, complemented

#89465

C₂

yes

maximal, characteristic, complemented

#89249

C₂

yes

maximal, characteristic, complemented

C₇×C₄⋊Q₈

C₂²

yes

characteristic

C₇×#127

C₂²

yes

—

—

—

C₂²

yes

characteristic

#6432

C₂²

yes

characteristic

C₇×#102

—

C₄×C₇×C₈

C₂²

yes

characteristic

#6406

—

#6382

—

C₄×C₇⋊C₈

C₂²

yes

characteristic

C₄×C₇⋊C₈

C₂²

yes

characteristic

#1715

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂³

yes

characteristic

#1143

—

#1141

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

D₄

yes

—

C₂×C₇⋊C₈

D₄

yes

—

—

—

—

—

—

—

D₇

yes

characteristic, complemented

#322

—

C₂³×C₇

—

C₂²×D₇

—

C₇×Q₈

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

SD₁₆

yes

complemented

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

characteristic

#127

—

#127

C₂×D₇

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#127

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₂×SD₁₆

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

100

SD₁₆

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²⋊D₇

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

C₂²×D₇

yes

characteristic

110

C₂×C₇

—

111

C₂×C₇

#492

yes

characteristic

112

C₂×C₇

#496

yes

characteristic

113

C₂×C₇

#496

yes

characteristic

114

D₇

—

115

C₂³

—

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×D₇

yes

characteristic

128

C₂×C₄

D₄×D₇

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₇

#1715

yes

characteristic, complemented, Sylow

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#6503

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#6474

yes

—

144

C₄

D₇×SD₁₆

yes

—

145

C₄

#6512

yes

—

146

C₂

—

147

C₂

—

112

—

148

C₂

#89830

yes

characteristic

149

C₂

#89926

yes

characteristic

150

C₂

#89947

yes

characteristic

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#486	C₂
socle	C₂²×C₇	#127
prime radical	#89845	C₂
hypercenter	#486	D₇
hypocenter	C₇	#1715
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#6503

commutator, cosocle

C₄²×C₇

C₂³

Frattini

C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. C₇×#486	C₂	semidirect
2. #89857	C₂	semidirect
3. #89851	C₂	semidirect
5. #89842	C₂	semidirect
6. #89465	C₂	semidirect
7. #89249	C₂	semidirect
46. #486	D₇	semidirect
65. C₇⋊C₈	SD₁₆	semidirect
133. C₇	#1715	semidirect

H ≅ G/N

type

1. C₇×#486

C₂

semidirect

2. #89857

C₂

semidirect

3. #89851

C₂

semidirect

5. #89842

C₂

semidirect

6. #89465

C₂

semidirect

7. #89249

C₂

semidirect

46. #486

D₇

semidirect

65. C₇⋊C₈

SD₁₆

semidirect

133. C₇

#1715

semidirect

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

subgroup	index
C₇×#486	2
#89857	2
#89851	2
#89845	2
#89842	2
#89465	2
#89249	2
#1715	7

subgroup

index

Subgroups — #11728939

Conjugacy classes of subgroups

Complements

Maximal subgroups