Group #28346809 — subgroups

Subgroups — #28346809

Order 13696.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20293177	C₂	1	1	yes	maximal, characteristic
2	#20292997	C₂	1	2	yes	maximal, complemented
3	#20292993	C₂	1	2	yes	maximal, complemented
4	#20292989	C₂	1	4	yes	maximal, complemented
5	#20292621	C₂	1	4	yes	maximal, complemented
6	C₂×#16812020	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133818	C₂²	1	2	yes	—
9	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
10	#16812109	C₂²	1	1	yes	characteristic
11	#16812106	C₂²	1	1	yes	characteristic
12	#16812097	C₂²	1	2	yes	—
13	#16812083	—	2	2	no	—
14	C₄×D₄₂₈	—	2	2	no	—
15	#16812032	C₂²	1	8	yes	—
16	#16812029	C₂²	1	8	yes	—
17	#16812020	C₂²	1	2	yes	—
18	#16812020	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
20	#125×C₁₀₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
22	C₄×Q₈×C₁₀₇	—	2	2	no	—
23	#117×C₁₀₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
25	#13133818	—	2	8	no	—
26	C₂×Dic₂₁₄	C₂³	1	2	yes	—
27	C₂×D₄₂₈	—	2	2	no	—
28	C₂×D₄₂₈	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₀₇	—	2	2	no	—
30	C₂×C₄×D₁₀₇	—	2	2	no	—
31	C₂×C₁₀₇⋊C₈	—	2	4	no	—
32	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₀₇	—	2	2	no	—
34	(C₂×D₁₀₇)⋊C₄	—	2	4	no	—
35	C₂×Q₈×C₁₀₇	—	2	2	no	—
36	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₇	—	2	4	no	—
39	C₄⋊C₄×C₁₀₇	—	2	2	no	—
40	C₄⋊C₄×C₁₀₇	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
42	C₄²×C₁₀₇	—	2	2	no	—
43	C₄²×C₁₀₇	D₄	1	2	yes	—
44	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
45	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
46	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₁₀₇	—	2	2	no	—
48	Q₈×C₁₀₇	—	4	4	no	—
49	C₂×C₄×C₁₀₇	—	2	4	no	—
50	C₂×C₄×C₁₀₇	—	4	2	no	—
51	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₇	#41	1	4	yes	—
55	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₁₀₇	—	4	4	no	—
57	C₂×Dic₁₀₇	—	2	2	no	—
58	D₄₂₈	—	2	4	no	—
59	D₄₂₈	—	4	4	no	—
60	C₄×D₁₀₇	—	4	4	no	—
61	Dic₂₁₄	—	2	4	no	—
62	C₁₀₇⋊C₈	—	2	4	no	—
63	C₂²×C₁₀₇	—	2	2	no	—
64	C₂²×C₁₀₇	#124	1	2	yes	—
65	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₀₇	—	4	2	no	—
67	C₂×D₁₀₇	—	4	4	no	—
68	C₄×C₁₀₇	—	2	2	no	—
69	C₄×C₁₀₇	—	2	2	no	—
70	C₄×C₁₀₇	—	4	4	no	—
71	C₄×C₁₀₇	#124	1	2	yes	—
72	C₄×C₁₀₇	C₂×#41	1	2	yes	—
73	Dic₁₀₇	—	4	2	no	—
74	C₂×C₁₀₇	—	2	2	no	—
75	C₂×C₁₀₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₁₀₇	#575	1	2	yes	—
77	D₁₀₇	—	8	2	no	—
78	#3135	—	107	1	no	maximal, Sylow
79	C₁₀₇	#3135	1	1	yes	characteristic, complemented, Sylow
80	#533	D₁₀₇	1	1	yes	characteristic, complemented
81	#532	—	107	1	no	—
82	#491	—	107	2	no	—
83	#490	—	107	2	no	—
84	#489	—	107	4	no	—
85	#416	—	107	4	no	—
86	C₂×#97	—	107	1	no	—
87	C₂×#41	—	107	2	no	—
88	C₂²×C₈	—	107	2	no	—
89	C₄⋊Q₈	—	107	1	no	—
90	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
91	#127	—	107	1	no	—
92	#125	C₂×D₁₀₇	1	2	yes	—
93	#124	—	107	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	214	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	214	2	no	—
98	#117	C₂×D₁₀₇	1	2	yes	—
99	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
100	#103	—	107	8	no	—
101	#102	—	107	8	no	—
102	#97	—	107	2	no	—
103	#97	—	107	2	no	—
104	#41	—	214	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	107	2	no	—
107	C₂×D₄	—	107	2	no	—
108	C₂×D₄	—	214	2	no	—
109	C₂×D₄	—	214	2	no	—
110	C₂²×C₄	—	214	2	no	—
111	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
113	C₂×C₈	—	107	4	no	—
114	C₂×C₈	—	214	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	214	2	no	—
118	C₄⋊C₄	C₂²×D₁₀₇	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	214	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
123	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
125	C₂³	—	214	2	no	—
126	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	214	4	no	—
129	D₄	—	428	4	no	—
130	D₄	—	428	4	no	—
131	D₄	—	214	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	428	4	no	—
135	C₂×C₄	—	214	2	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
139	C₂×C₄	#13133819	1	4	yes	—
140	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
141	C₈	—	214	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	428	2	no	—
144	C₂²	—	428	4	no	—
145	C₂²	#16812098	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	428	2	no	—
151	C₄	#16812098	1	2	yes	—
152	C₄	C₂×#13133819	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	856	2	no	—
155	C₂	C₂×#16812098	1	1	yes	characteristic
156	C₂	#20293425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293177

C₂

yes

maximal, characteristic

#20292997

C₂

yes

maximal, complemented

#20292993

C₂

yes

maximal, complemented

#20292989

C₂

yes

maximal, complemented

#20292621

C₂

yes

maximal, complemented

C₂×#16812020

C₂

yes

maximal, characteristic, complemented

#533×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#13133818

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

#16812109

C₂²

yes

characteristic

#16812106

C₂²

yes

characteristic

#16812097

C₂²

yes

—

#16812083

—

C₄×D₄₂₈

—

#16812032

C₂²

yes

—

#16812029

C₂²

yes

—

#16812020

C₂²

yes

—

#16812020

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

#125×C₁₀₇

C₂²

yes

—

C₂²⋊Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

#117×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133818

—

C₂×Dic₂₁₄

C₂³

yes

—

C₂×D₄₂₈

—

C₂×D₄₂₈

C₂³

yes

—

C₂×C₂²⋊D₁₀₇

—

C₂×C₄×D₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

—

(C₂×D₁₀₇)⋊C₄

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

—

D₄₂₈

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#124

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#124

yes

—

C₄×C₁₀₇

C₂×#41

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#124

yes

characteristic

C₂×C₁₀₇

#575

yes

—

D₁₀₇

—

#3135

—

107

maximal, Sylow

C₁₀₇

#3135

yes

characteristic, complemented, Sylow

#533

D₁₀₇

yes

characteristic, complemented

#532

—

107

—

#491

—

107

—

#490

—

107

—

#489

—

107

—

#416

—

107

—

C₂×#97

—

107

—

C₂×#41

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

#127

—

107

—

#125

C₂×D₁₀₇

yes

—

#124

—

107

—

C₂²⋊Q₈

—

#121

—

214

—

C₄×Q₈

—

C₄×D₄

—

214

—

#117

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

100

#103

—

107

—

101

#102

—

107

—

102

#97

—

107

—

103

#97

—

107

—

104

#41

—

214

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

—

107

C₂×D₄

—

107

—

108

C₂×D₄

—

214

—

109

C₂×D₄

—

214

—

110

C₂²×C₄

—

214

—

111

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

112

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

113

C₂×C₈

—

107

—

114

C₂×C₈

—

214

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

214

—

118

C₄⋊C₄

C₂²×D₁₀₇

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

214

—

121

C₄²

—

122

C₄²

C₂²⋊D₁₀₇

yes

—

123

C₄²

C₂²×D₁₀₇

yes

characteristic

124

C₄²

C₂²×D₁₀₇

yes

characteristic

125

C₂³

—

214

—

126

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

127

Q₈

—

128

Q₈

—

214

—

129

D₄

—

428

—

130

D₄

—

428

—

131

D₄

—

214

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

428

—

135

C₂×C₄

—

214

—

136

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

139

C₂×C₄

#13133819

yes

—

140

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

141

C₈

—

214

—

142

C₂²

—

143

C₂²

—

428

—

144

C₂²

—

428

—

145

C₂²

#16812098

yes

—

146

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

428

—

151

C₄

#16812098

yes

—

152

C₄

C₂×#13133819

yes

—

153

C₂

—

154

C₂

—

856

—

155

C₂

C₂×#16812098

yes

characteristic

156

C₂

#20293425

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	#533×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	#20293177	C₂
hypercenter	#533	D₁₀₇
hypocenter	C₁₀₇	#3135
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₇

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

#533×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂²×D₄

C₂

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20292997	C₂	semidirect
3. #20292993	C₂	semidirect
4. #20292989	C₂	semidirect
5. #20292621	C₂	semidirect
6. C₂×#16812020	C₂	semidirect
7. #533×C₁₀₇	C₂	semidirect
79. C₁₀₇	#3135	semidirect
80. #533	D₁₀₇	semidirect

H ≅ G/N

type

2. #20292997

C₂

semidirect

3. #20292993

C₂

semidirect

4. #20292989

C₂

semidirect

5. #20292621

C₂

semidirect

6. C₂×#16812020

C₂

semidirect

7. #533×C₁₀₇

C₂

semidirect

79. C₁₀₇

#3135

semidirect

80. #533

D₁₀₇

semidirect

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

subgroup	index
#20293177	2
#20292997	2
#20292993	2
#20292989	2
#20292621	2
C₂×#16812020	2
#533×C₁₀₇	2
#3135	107

subgroup

index

#533×C₁₀₇

107

Subgroups — #28346809

Conjugacy classes of subgroups

Complements

Maximal subgroups