Group #28338510 — subgroups

Subgroups — #28338510

Order 13696.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₈₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812055	C₂	1	1	yes	maximal, characteristic
3	#20293005	C₂	1	1	yes	maximal, characteristic
4	#20292721	C₂	1	1	yes	maximal, characteristic
5	Dic₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#20292430	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic
10	C₂×#13133808	—	2	1	no	—
11	#16812107	—	2	1	no	—
12	#16812074	—	2	1	no	—
13	Dic₈₅₆	—	2	2	no	—
14	#16812055	—	2	2	no	—
15	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16812032	—	2	1	no	—
17	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
20	#107×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₄	—	2	1	no	—
22	C₂×Dic₂₁₄	—	2	1	no	—
23	Dic₄₂₈	—	2	1	no	—
24	Dic₄₂₈	—	2	1	no	—
25	Dic₄₂₈	—	2	1	no	—
26	Dic₄₂₈	D₄	1	2	yes	—
27	#13133808	—	4	2	no	—
28	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
29	Dic₁₀₇⋊C₄	—	4	1	no	—
30	Dic₁₀₇⋊C₄	—	4	1	no	—
31	Dic₁₀₇⋊C₄	—	4	1	no	—
32	C₄×Dic₁₀₇	—	4	1	no	—
33	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
34	C₁₀₇⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₁₀₇	—	2	1	no	—
36	Q₁₆×C₁₀₇	—	2	1	no	—
37	Q₁₆×C₁₀₇	—	2	1	no	—
38	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₇	—	2	1	no	—
40	C₁₆×C₁₀₇	—	2	1	no	—
41	Q₈×C₁₀₇	—	4	1	no	—
42	Q₈×C₁₀₇	—	4	1	no	—
43	C₂×C₄×C₁₀₇	—	4	1	no	—
44	C₂×C₄×C₁₀₇	—	4	1	no	—
45	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₁₀₇	—	4	1	no	—
47	C₂×Dic₁₀₇	—	4	1	no	—
48	C₂×Dic₁₀₇	D₈	1	1	yes	characteristic
49	C₂×Dic₁₀₇	D₈	1	1	yes	characteristic
50	Dic₂₁₄	—	2	2	no	—
51	Dic₂₁₄	—	4	1	no	—
52	Dic₂₁₄	—	4	1	no	—
53	Dic₂₁₄	—	4	1	no	—
54	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₀₇	#41	1	1	yes	characteristic
56	C₁₀₇⋊C₈	—	2	1	no	—
57	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
58	C₄×C₁₀₇	—	8	1	no	—
59	C₄×C₁₀₇	—	8	1	no	—
60	C₄×C₁₀₇	#121	1	1	yes	characteristic
61	C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₀₇	—	2	1	no	—
63	Dic₁₀₇	—	4	2	no	—
64	Dic₁₀₇	—	8	1	no	—
65	Dic₁₀₇	Q₃₂	1	2	yes	complemented
66	C₂×C₁₀₇	#459	1	1	yes	characteristic
67	C₂×C₁₀₇	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₁₀₇	#510	1	1	yes	characteristic
69	#2209	—	107	1	no	maximal, Sylow
70	C₁₀₇	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	107	1	no	—
72	C₂×Q₃₂	—	107	1	no	—
73	#494	—	107	1	no	—
74	C₄×Q₁₆	—	107	1	no	—
75	C₄⋊C₁₆	—	107	1	no	—
76	#358	—	107	1	no	—
77	#358	D₁₀₇	1	1	yes	characteristic
78	C₂×Q₁₆	—	107	1	no	—
79	C₂×Q₁₆	—	107	1	no	—
80	C₂×Q₁₆	—	214	1	no	—
81	C₂×Q₁₆	C₂×D₁₀₇	1	1	yes	characteristic
82	C₄⋊Q₈	—	214	1	no	—
83	C₄×Q₈	—	214	1	no	—
84	Q₃₂	—	214	2	no	—
85	Q₃₂	—	214	2	no	—
86	C₂×C₁₆	—	107	1	no	—
87	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
88	#107	C₂×D₁₀₇	1	1	yes	characteristic
89	#103	—	214	1	no	—
90	C₄×C₈	—	107	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	214	1	no	—
93	C₂×Q₈	—	214	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	107	2	no	—
97	Q₁₆	—	428	2	no	—
98	Q₁₆	—	214	1	no	—
99	Q₁₆	—	214	1	no	—
100	Q₁₆	—	214	1	no	—
101	C₂×C₈	—	107	1	no	—
102	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	428	1	no	—
105	C₄⋊C₄	—	428	1	no	—
106	C₄⋊C₄	—	428	1	no	—
107	C₄²	—	107	1	no	—
108	C₄²	—	428	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	214	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	428	1	no	—
114	Q₈	—	428	1	no	—
115	Q₈	—	428	1	no	—
116	Q₈	—	214	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	107	1	no	—
120	C₂×C₄	—	107	1	no	—
121	C₂×C₄	—	428	1	no	—
122	C₂×C₄	—	428	1	no	—
123	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
124	C₈	—	214	1	no	—
125	C₈	D₄×D₁₀₇	1	1	yes	characteristic
126	C₈	#13133818	1	1	yes	characteristic
127	C₂²	D₈×D₁₀₇	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	856	1	no	—
131	C₄	—	107	2	no	—
132	C₄	—	428	2	no	—
133	C₄	—	214	1	no	—
134	C₄	#16812082	1	1	yes	characteristic
135	C₄	#16812136	1	1	yes	characteristic
136	C₂	#20292821	1	1	yes	characteristic
137	C₂	Q₃₂×D₁₀₇	1	1	yes	characteristic
138	C₂	#20293066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₅₆

C₂

yes

maximal, characteristic

C₂×#16812055

C₂

yes

maximal, characteristic

#20293005

C₂

yes

maximal, characteristic

#20292721

C₂

yes

maximal, characteristic

Dic₁₀₇⋊C₁₆

C₂

yes

maximal, characteristic

#20292430

C₂

yes

maximal, characteristic

#358×C₁₀₇

C₂

yes

maximal, characteristic

C₂×Dic₄₂₈

C₂²

yes

characteristic

C₂×Dic₄₂₈

C₂²

yes

—

—

—

Dic₈₅₆

—

#16812055

—

C₂×C₁₀₇⋊C₁₆

C₂²

yes

characteristic

#16812032

—

C₈×Dic₁₀₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₇

C₂²

yes

characteristic

#107×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

Dic₄₂₈

—

Dic₄₂₈

—

Dic₄₂₈

—

Dic₄₂₈

D₄

yes

—

#13133808

—

C₂×C₁₀₇⋊C₈

D₄

yes

characteristic

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

D₄

yes

characteristic

C₁₀₇⋊C₁₆

—

C₂×Q₈×C₁₀₇

—

Q₁₆×C₁₀₇

—

Q₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₁₆×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

D₈

yes

characteristic

C₂×Dic₁₀₇

D₈

yes

characteristic

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

C₂×D₄

yes

characteristic

C₈×C₁₀₇

#41

yes

characteristic

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×D₈

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

Q₃₂

yes

complemented

C₂×C₁₀₇

#459

yes

characteristic

C₂×C₁₀₇

C₂×Q₃₂

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

107

—

C₂×Q₃₂

—

107

—

#494

—

107

—

C₄×Q₁₆

—

107

—

C₄⋊C₁₆

—

107

—

#358

—

107

—

#358

D₁₀₇

yes

characteristic

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

214

—

C₂×Q₁₆

C₂×D₁₀₇

yes

characteristic

C₄⋊Q₈

—

214

—

C₄×Q₈

—

214

—

Q₃₂

—

214

—

Q₃₂

—

214

—

C₂×C₁₆

—

107

—

C₂×C₁₆

C₂×D₁₀₇

yes

characteristic

#107

C₂×D₁₀₇

yes

characteristic

#103

—

214

—

C₄×C₈

—

107

—

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

107

—

Q₁₆

—

428

—

Q₁₆

—

214

—

Q₁₆

—

214

—

100

Q₁₆

—

214

—

101

C₂×C₈

—

107

—

102

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

428

—

105

C₄⋊C₄

—

428

—

106

C₄⋊C₄

—

428

—

107

C₄²

—

107

—

108

C₄²

—

428

—

109

C₁₆

—

110

C₁₆

—

214

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

428

—

114

Q₈

—

428

—

115

Q₈

—

428

—

116

Q₈

—

214

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

107

—

120

C₂×C₄

—

107

—

121

C₂×C₄

—

428

—

122

C₂×C₄

—

428

—

123

C₂×C₄

D₄×D₁₀₇

yes

characteristic

124

C₈

—

214

—

125

C₈

D₄×D₁₀₇

yes

characteristic

126

C₈

#13133818

yes

characteristic

127

C₂²

D₈×D₁₀₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

856

—

131

C₄

—

107

—

132

C₄

—

428

—

133

C₄

—

214

—

134

C₄

#16812082

yes

characteristic

135

C₄

#16812136

yes

characteristic

136

C₂

#20292821

yes

characteristic

137

C₂

Q₃₂×D₁₀₇

yes

characteristic

138

C₂

#20293066

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₇
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#358×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	#358	D₁₀₇
hypocenter	C₁₀₇	#2209
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₀₇

commutator, cosocle

C₂×C₈×C₁₀₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₇

Fitting

#358×C₁₀₇

C₂

socle, prime radical

C₂²×C₁₀₇

C₂×D₈

D₁₀₇

C₁₀₇

G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₁₀₇	Q₃₂	semidirect
70. C₁₀₇	#2209	semidirect

H ≅ G/N

type

65. Dic₁₀₇

Q₃₂

semidirect

70. C₁₀₇

#2209

semidirect

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

subgroup	index
C₂×Dic₈₅₆	2
C₂×#16812055	2
#20293005	2
#20292721	2
Dic₁₀₇⋊C₁₆	2
#20292430	2
#358×C₁₀₇	2
#2209	107

subgroup

index

C₂×Dic₈₅₆

Dic₁₀₇⋊C₁₆

#358×C₁₀₇

107

Subgroups — #28338510

Conjugacy classes of subgroups

Complements

Maximal subgroups