Group #28336261 — subgroups

Subgroups — #28336261

Order 13696.

284 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 121 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	#20292611	C₂	1	1	yes	maximal, characteristic
2	#20292601	C₂	1	1	yes	maximal, characteristic
3	#20292582	C₂	1	2	yes	maximal, complemented
4	#20292398	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	#16812068	C₂²	1	2	yes	—
8	#16812028	C₄	1	2	yes	complemented
9	#16812026	C₄	1	2	yes	complemented
10	#16812024	C₄	1	4	yes	—
11	#16812019	—	2	2	no	—
12	C₈×Dic₁₀₇	—	2	2	no	—
13	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
14	#131×C₁₀₇	—	2	2	no	—
15	#131×C₁₀₇	—	2	2	no	—
16	C₂×M₁₆×C₁₀₇	C₄	1	2	yes	complemented
17	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₇	C₄	1	2	yes	complemented
19	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
20	#13133802	—	2	2	no	—
21	#13133802	C₂×C₄	1	4	yes	—
22	C₂×C₁₀₇⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₀₇	—	2	2	no	—
24	C₂²⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
25	C₄×Dic₁₀₇	—	2	2	no	—
26	#41×C₁₀₇	—	2	2	no	—
27	#41×C₁₀₇	—	2	2	no	—
28	C₂×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₇	—	2	2	no	—
34	M₁₆×C₁₀₇	—	2	4	no	—
35	C₂×C₈×C₁₀₇	—	2	2	no	—
36	C₂×C₈×C₁₀₇	—	2	2	no	—
37	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
41	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
42	C₂³×C₁₀₇	C₄²	1	2	yes	—
43	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₁₀₇	—	2	2	no	—
45	D₄×C₁₀₇	—	2	4	no	—
46	D₄×C₁₀₇	—	2	2	no	—
47	C₂×C₄×C₁₀₇	—	2	2	no	—
48	C₂×C₄×C₁₀₇	—	2	2	no	—
49	C₂×C₄×C₁₀₇	C₄²	1	2	yes	—
50	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₇	#41	1	2	yes	—
54	C₂×Dic₁₀₇	—	2	4	no	—
55	C₈×C₁₀₇	—	2	2	no	—
56	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
58	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₀₇⋊C₈	—	2	4	no	—
60	C₂²×C₁₀₇	—	2	2	no	—
61	C₂²×C₁₀₇	—	4	1	no	—
62	C₂²×C₁₀₇	—	4	2	no	—
63	C₂²×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₇	—	2	2	no	—
66	C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₀₇	#117	1	1	yes	characteristic
68	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
69	Dic₁₀₇	—	4	4	no	—
70	C₂×C₁₀₇	—	2	2	no	—
71	C₂×C₁₀₇	—	2	1	no	—
72	C₂×C₁₀₇	—	4	2	no	—
73	C₂×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	#1757	—	107	1	no	maximal, Sylow
75	C₁₀₇	#1757	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	D₁₀₇	1	1	yes	characteristic
77	#413	—	107	1	no	—
78	#410	—	107	1	no	—
79	#405	—	107	2	no	—
80	#344	—	107	2	no	—
81	C₂×#41	C₂×D₁₀₇	1	1	yes	characteristic
82	#131	—	2	2	no	—
83	#131	—	2	2	no	—
84	C₂×M₁₆	—	107	2	no	—
85	C₂×M₁₆	Dic₁₀₇	1	2	yes	complemented
86	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
87	C₂²×C₈	Dic₁₀₇	1	2	yes	complemented
88	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
89	#117	—	107	2	no	—
90	#101	—	107	2	no	—
91	#100	—	107	2	no	—
92	#99	—	107	4	no	—
93	#97	—	214	2	no	—
94	C₄×C₈	—	214	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₄×D₁₀₇	1	1	yes	characteristic
98	C₂×D₄	C₄×D₁₀₇	1	2	yes	—
99	C₂×D₄	C₄×D₁₀₇	1	1	yes	characteristic
100	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
101	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	4	no	—
104	M₁₆	—	107	4	no	—
105	M₁₆	—	214	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	214	2	no	—
109	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
110	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
113	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
114	C₄⋊C₄	—	214	2	no	—
115	C₂²⋊C₄	—	107	4	no	—
116	C₄²	—	214	2	no	—
117	C₂³	C₄×Dic₁₀₇	1	2	yes	—
118	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	214	4	no	—
125	C₂×C₄	C₄×Dic₁₀₇	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
127	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
129	C₂×C₄	#13133818	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	214	4	no	—
132	C₈	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
133	C₈	C₂²⋊Dic₁₀₇	1	2	yes	—
134	C₈	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₁₀₇	1	1	yes	characteristic
139	C₂²	C₄×C₂²⋊D₁₀₇	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	428	4	no	—
142	C₄	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
143	C₄	#16812068	1	1	yes	characteristic
144	C₄	C₄×C₂²⋊D₁₀₇	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₄×C₂²⋊Dic₁₀₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292611

C₂

yes

maximal, characteristic

#20292601

C₂

yes

maximal, characteristic

#20292582

C₂

yes

maximal, complemented

#20292398

C₂

yes

maximal, complemented

C₂×#131×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

#16812068

C₂²

yes

—

#16812028

C₄

yes

complemented

#16812026

C₄

yes

complemented

#16812024

C₄

yes

—

#16812019

—

C₈×Dic₁₀₇

—

C₂×#41×C₁₀₇

C₂²

yes

characteristic

#131×C₁₀₇

—

#131×C₁₀₇

—

C₂×M₁₆×C₁₀₇

C₄

yes

complemented

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₄

yes

complemented

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#13133802

—

#13133802

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₂²⋊Dic₁₀₇

C₂×C₄

yes

—

C₄×Dic₁₀₇

—

#41×C₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₄²

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄²

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₇

C₂²⋊C₄

yes

—

C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₄²

yes

characteristic

C₂²×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₇

#117

yes

characteristic

C₄×C₁₀₇

C₄×D₄

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₄×C₂²⋊C₄

yes

characteristic

#1757

—

107

maximal, Sylow

C₁₀₇

#1757

yes

characteristic, complemented, Sylow

C₂×#131

D₁₀₇

yes

characteristic

#413

—

107

—

#410

—

107

—

#405

—

107

—

#344

—

107

—

C₂×#41

C₂×D₁₀₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

107

—

C₂×M₁₆

Dic₁₀₇

yes

complemented

C₂×M₁₆

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

Dic₁₀₇

yes

complemented

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

#117

—

107

—

#101

—

107

—

#100

—

107

—

#99

—

107

—

#97

—

214

—

C₄×C₈

—

214

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₀₇

yes

characteristic

C₂×D₄

C₄×D₁₀₇

yes

—

C₂×D₄

C₄×D₁₀₇

yes

characteristic

100

C₂²×C₄

C₂×Dic₁₀₇

yes

—

101

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

107

—

105

M₁₆

—

214

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

214

—

109

C₂×C₈

C₂×Dic₁₀₇

yes

—

110

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

111

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

112

C₂×C₈

C₂²⋊D₁₀₇

yes

—

113

C₂×C₈

C₂²⋊D₁₀₇

yes

—

114

C₄⋊C₄

—

214

—

115

C₂²⋊C₄

—

107

—

116

C₄²

—

214

—

117

C₂³

C₄×Dic₁₀₇

yes

—

118

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

214

—

125

C₂×C₄

C₄×Dic₁₀₇

yes

—

126

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

127

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

129

C₂×C₄

#13133818

yes

—

130

C₈

—

131

C₈

—

214

—

132

C₈

C₂²⋊Dic₁₀₇

yes

characteristic

133

C₈

C₂²⋊Dic₁₀₇

yes

—

134

C₈

C₂²⋊Dic₁₀₇

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₁₀₇

yes

characteristic

139

C₂²

C₄×C₂²⋊D₁₀₇

yes

—

140

C₄

—

141

C₄

—

428

—

142

C₄

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

143

C₄

#16812068

yes

characteristic

144

C₄

C₄×C₂²⋊D₁₀₇

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₄×C₂²⋊Dic₁₀₇

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₁₀₇
commutator	C₂²×C₁₀₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×#131×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₇	C₂²
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×#131	D₁₀₇
hypocenter	C₁₀₇	#1757
LCS	G ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

subgroup

G/H

center

C₈

C₂²⋊Dic₁₀₇

commutator

C₂²×C₁₀₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×#131×C₁₀₇

C₂

socle

C₂×C₁₀₇

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₁₀₇

C₂²

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

N	H ≅ G/N	type
3. #20292582	C₂	semidirect
4. #20292398	C₂	semidirect
8. #16812028	C₄	semidirect
9. #16812026	C₄	semidirect
16. C₂×M₁₆×C₁₀₇	C₄	semidirect
18. C₂²×C₈×C₁₀₇	C₄	semidirect
75. C₁₀₇	#1757	semidirect
85. C₂×M₁₆	Dic₁₀₇	semidirect
87. C₂²×C₈	Dic₁₀₇	semidirect

H ≅ G/N

type

3. #20292582

C₂

semidirect

4. #20292398

C₂

semidirect

8. #16812028

C₄

semidirect

9. #16812026

C₄

semidirect

16. C₂×M₁₆×C₁₀₇

C₄

semidirect

18. C₂²×C₈×C₁₀₇

C₄

semidirect

75. C₁₀₇

#1757

semidirect

85. C₂×M₁₆

Dic₁₀₇

semidirect

87. C₂²×C₈

Dic₁₀₇

semidirect

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

subgroup	index
#20292611	2
#20292601	2
#20292582	2
#20292398	2
C₂×#131×C₁₀₇	2
#1757	107

subgroup

index

C₂×#131×C₁₀₇

107

Subgroups — #28336261

Conjugacy classes of subgroups

Complements

Maximal subgroups