Group #28337607 — subgroups

Subgroups — #28337607

Order 13696.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#16812101	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812030	C₂	1	2	yes	maximal
3	#20292389	C₂	1	2	yes	maximal, complemented
4	#20292382	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	#16812101	—	2	4	no	—
8	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	C₂×Dic₁₀₇⋊C₄	—	2	2	no	—
10	#16812030	—	2	4	no	—
11	C₂²×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₇	—	2	2	no	—
16	C₂×C₁₀₇⋊C₈	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₀₇	—	2	2	no	—
20	C₄⋊Dic₁₀₇	D₄	1	4	yes	—
21	Dic₁₀₇⋊C₄	—	2	8	no	—
22	C₂⁴×C₁₀₇	—	2	1	no	—
23	C₂×D₄×C₁₀₇	—	2	2	no	—
24	C₂×D₄×C₁₀₇	—	2	2	no	—
25	C₂×D₄×C₁₀₇	—	2	1	no	—
26	C₂×D₄×C₁₀₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₀₇	—	2	1	no	—
28	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₀₇	—	2	2	no	—
31	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
32	C₄²×C₁₀₇	—	2	2	no	—
33	C₂³×C₁₀₇	—	2	2	no	—
34	C₂³×C₁₀₇	—	4	1	no	—
35	C₂³×C₁₀₇	—	4	1	no	—
36	C₂³×C₁₀₇	—	4	2	no	—
37	C₂³×C₁₀₇	—	4	2	no	—
38	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₁₀₇	—	2	4	no	—
40	D₄×C₁₀₇	—	4	2	no	—
41	C₂×C₄×C₁₀₇	—	2	2	no	—
42	C₂×C₄×C₁₀₇	—	2	2	no	—
43	C₂×C₄×C₁₀₇	—	2	2	no	—
44	C₂×C₄×C₁₀₇	—	4	1	no	—
45	C₂×C₄×C₁₀₇	—	4	2	no	—
46	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	#41	1	2	yes	—
49	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
50	C₂×Dic₁₀₇	—	2	4	no	—
51	C₂×Dic₁₀₇	—	4	2	no	—
52	C₂×Dic₁₀₇	—	4	2	no	—
53	C₁₀₇⋊C₈	—	4	2	no	—
54	C₂²×C₁₀₇	—	4	2	no	—
55	C₂²×C₁₀₇	—	4	4	no	—
56	C₂²×C₁₀₇	—	8	2	no	—
57	C₂²×C₁₀₇	—	8	1	no	—
58	C₂²×C₁₀₇	—	8	1	no	—
59	C₂²×C₁₀₇	#120	1	1	yes	characteristic
60	C₂²×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₀₇	#124	1	1	yes	characteristic
62	C₂²×C₁₀₇	#135	1	2	yes	—
63	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₁₀₇	—	4	2	no	—
65	C₄×C₁₀₇	—	8	1	no	—
66	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₀₇	#124	1	2	yes	—
68	Dic₁₀₇	—	4	4	no	—
69	C₂×C₁₀₇	—	8	2	no	—
70	C₂×C₁₀₇	#393	1	1	yes	characteristic
71	C₂×C₁₀₇	#449	1	2	yes	—
72	C₂×C₁₀₇	#478	1	2	yes	—
73	C₂×C₁₀₇	#488	1	1	yes	characteristic
74	C₂×C₁₀₇	#490	1	1	yes	characteristic
75	#2025	—	107	1	no	maximal, Sylow
76	C₁₀₇	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	—	107	1	no	—
78	C₂×#102	—	107	2	no	—
79	#390	D₁₀₇	1	1	yes	characteristic
80	#340	—	107	2	no	—
81	#336	—	107	1	no	—
82	C₂²×D₄	C₂×D₁₀₇	1	1	yes	characteristic
83	C₂²×C₈	—	107	2	no	—
84	#125	—	214	4	no	—
85	C₂×C₄⋊C₄	—	107	2	no	—
86	C₂×C₄⋊C₄	—	214	2	no	—
87	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
90	#102	—	214	4	no	—
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	Dic₂₁₄	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	214	2	no	—
98	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
100	C₂×C₈	—	214	2	no	—
101	C₂×C₈	—	214	2	no	—
102	C₂×C₈	—	214	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	107	4	no	—
105	C₄⋊C₄	—	214	8	no	—
106	C₄⋊C₄	—	214	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	428	2	no	—
123	C₂×C₄	—	428	2	no	—
124	C₂×C₄	—	214	4	no	—
125	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
126	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
127	C₂×C₄	#13133817	1	2	yes	—
128	C₂×C₄	#13133818	1	1	yes	characteristic
129	C₈	—	428	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#16812077	1	1	yes	characteristic
136	C₂²	#16812088	1	1	yes	characteristic
137	C₂²	#16812096	1	1	yes	characteristic
138	C₂²	#16812129	1	2	yes	—
139	C₂²	#16812135	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	428	4	no	—
143	C₄	#16812086	1	2	yes	—
144	C₄	#16812095	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#20292539	1	1	yes	characteristic
147	C₂	#20292770	1	2	yes	—
148	C₂	#20292927	1	2	yes	—
149	C₂	#20292981	1	1	yes	characteristic
150	C₂	#20292991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812101

C₂

yes

maximal, characteristic, complemented

C₂×#16812030

C₂

yes

maximal

#20292389

C₂

yes

maximal, complemented

#20292382

C₂

yes

maximal, characteristic, complemented

#390×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

#16812101

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×Dic₁₀₇⋊C₄

—

#16812030

—

C₂²×D₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

D₄

yes

—

Dic₁₀₇⋊C₄

—

C₂⁴×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#120

yes

characteristic

C₂²×C₁₀₇

C₂²⋊Q₈

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

—

C₂²×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

#124

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#393

yes

characteristic

C₂×C₁₀₇

#449

yes

—

C₂×C₁₀₇

#478

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#2025

yes

characteristic, complemented, Sylow

C₂×#125

—

107

—

C₂×#102

—

107

—

#390

D₁₀₇

yes

characteristic

#340

—

107

—

#336

—

107

—

C₂²×D₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

—

107

—

#125

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#102

—

214

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₂₁₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

100

C₂×C₈

—

214

—

101

C₂×C₈

—

214

—

102

C₂×C₈

—

214

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

107

—

105

C₄⋊C₄

—

214

—

106

C₄⋊C₄

—

214

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

428

—

123

C₂×C₄

—

428

—

124

C₂×C₄

—

214

—

125

C₂×C₄

D₄×D₁₀₇

yes

—

126

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

127

C₂×C₄

#13133817

yes

—

128

C₂×C₄

#13133818

yes

characteristic

129

C₈

—

428

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#16812077

yes

characteristic

136

C₂²

#16812088

yes

characteristic

137

C₂²

#16812096

yes

characteristic

138

C₂²

#16812129

yes

—

139

C₂²

#16812135

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

428

—

143

C₄

#16812086

yes

—

144

C₄

#16812095

yes

—

145

C₂

—

146

C₂

#20292539

yes

characteristic

147

C₂

#20292770

yes

—

148

C₂

#20292927

yes

—

149

C₂

#20292981

yes

characteristic

150

C₂

#20292991

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₂²×C₄×C₁₀₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#390×C₁₀₇	C₂
socle	C₂³×C₁₀₇	C₂×D₄
prime radical	C₂²×D₄×C₁₀₇	C₂²
hypercenter	#390	D₁₀₇
hypocenter	C₁₀₇	#2025
LCS	G ▷ C₂²×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator, cosocle

C₂²×C₄×C₁₀₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#390×C₁₀₇

C₂

socle

C₂³×C₁₀₇

C₂×D₄

prime radical

C₂²×D₄×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₂²×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#16812101	C₂	semidirect
3. #20292389	C₂	semidirect
4. #20292382	C₂	semidirect
76. C₁₀₇	#2025	semidirect

H ≅ G/N

type

1. C₂×#16812101

C₂

semidirect

3. #20292389

C₂

semidirect

4. #20292382

C₂

semidirect

76. C₁₀₇

#2025

semidirect

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

subgroup	index
C₂×#16812101	2
C₂×#16812030	2
#20292389	2
#20292382	2
#390×C₁₀₇	2
#2025	107

subgroup

index

#390×C₁₀₇

107

Subgroups — #28337607

Conjugacy classes of subgroups

Complements

Maximal subgroups