Group #28337397 — subgroups

Subgroups — #28337397

Order 13696.

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

normal subgroups only

This group has 292 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	C₂×C₄×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20292381	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₂₁₄	—	2	2	no	—
8	C₄×Dic₂₁₄	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₇	C₄	1	1	yes	characteristic
10	C₂×Dic₁₀₇⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
12	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
13	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₇	—	2	2	no	—
16	C₂×Dic₂₁₄	—	2	2	no	—
17	C₂×Dic₂₁₄	—	2	2	no	—
18	C₂×Dic₂₁₄	C₈	1	4	yes	complemented
19	C₂²×Dic₁₀₇	C₂×C₄	1	2	yes	—
20	C₂²×Dic₁₀₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₀₇⋊C₈	—	2	2	no	—
22	C₂×C₁₀₇⋊C₈	—	2	4	no	—
23	C₄⋊Dic₁₀₇	—	2	2	no	—
24	Dic₁₀₇⋊C₄	—	2	4	no	—
25	C₄×Dic₁₀₇	—	2	2	no	—
26	C₄×Dic₁₀₇	—	2	2	no	—
27	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₂×C₈×C₁₀₇	—	2	2	no	—
32	C₂×C₈×C₁₀₇	Q₈	1	4	yes	complemented
33	C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₄²×C₁₀₇	D₄	1	2	yes	—
35	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	—	2	2	no	—
37	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₀₇	M₁₆	1	2	yes	—
41	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₀₇	#41	1	2	yes	—
45	C₂×Dic₁₀₇	—	2	2	no	—
46	C₂×Dic₁₀₇	—	2	2	no	—
47	C₂×Dic₁₀₇	—	2	2	no	—
48	C₂×Dic₁₀₇	—	2	4	no	—
49	C₂×Dic₁₀₇	C₂×C₈	1	4	yes	—
50	Dic₂₁₄	—	2	8	no	—
51	C₈×C₁₀₇	—	2	4	no	—
52	C₁₀₇⋊C₈	—	4	2	no	—
53	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
55	C₂²×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₇	#131	1	2	yes	—
58	C₄×C₁₀₇	—	2	2	no	—
59	C₄×C₁₀₇	C₂²⋊C₈	1	4	yes	—
60	C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₀₇	#124	1	1	yes	characteristic
62	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₀₇	—	2	4	no	—
64	Dic₁₀₇	—	4	2	no	—
65	C₂×C₁₀₇	#391	1	1	yes	characteristic
66	C₂×C₁₀₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₀₇	#408	1	1	yes	characteristic
68	C₂×C₁₀₇	C₈×Q₈	1	2	yes	—
69	C₂×C₁₀₇	#446	1	2	yes	—
70	#1984	—	107	1	no	maximal, Sylow
71	C₁₀₇	#1984	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×Q₈	—	107	1	no	—
73	C₂×C₄⋊C₈	—	107	1	no	—
74	C₂×C₄×C₈	D₁₀₇	1	1	yes	characteristic
75	#336	—	107	4	no	—
76	C₂²×Q₈	—	107	1	no	—
77	C₂²×C₈	—	107	2	no	—
78	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
79	C₄×Q₈	—	214	2	no	—
80	C₄×Q₈	—	214	2	no	—
81	C₂×C₄⋊C₄	—	107	2	no	—
82	C₂×C₄⋊C₄	—	107	1	no	—
83	C₂×C₄²	—	107	2	no	—
84	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
85	C₄⋊C₈	—	214	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₂×Q₈	—	107	4	no	—
88	C₂×Q₈	—	214	2	no	—
89	C₂×Q₈	—	214	2	no	—
90	C₂²×C₄	—	107	2	no	—
91	C₂²×C₄	—	107	2	no	—
92	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	214	2	no	—
98	C₂×C₈	—	214	4	no	—
99	C₂×C₈	Dic₂₁₄	1	4	yes	complemented
100	C₄⋊C₄	—	214	4	no	—
101	C₄⋊C₄	—	214	2	no	—
102	C₄²	—	214	2	no	—
103	C₄²	—	214	2	no	—
104	C₄²	D₄₂₈	1	2	yes	—
105	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
106	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
107	Q₈	—	214	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	107	4	no	—
110	C₂×C₄	—	214	2	no	—
111	C₂×C₄	—	214	4	no	—
112	C₂×C₄	—	214	2	no	—
113	C₂×C₄	—	214	2	no	—
114	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
115	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
116	C₂×C₄	C₈×D₁₀₇	1	2	yes	—
117	C₂×C₄	#13133801	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₁₄	1	2	yes	—
121	C₂×C₄	#13133818	1	2	yes	—
122	C₈	—	2	4	no	—
123	C₈	—	428	2	no	—
124	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
125	C₂²	C₄×Dic₂₁₄	1	2	yes	—
126	C₂²	C₂×C₈×D₁₀₇	1	1	yes	characteristic
127	C₂²	C₂×#13133801	1	1	yes	characteristic
128	C₂²	#16812115	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	428	2	no	—
131	C₄	—	214	4	no	—
132	C₄	(C₂×D₁₀₇)⋊C₈	1	4	yes	—
133	C₄	#16812088	1	2	yes	—
134	C₄	#16812097	1	1	yes	characteristic
135	C₄	#16812109	1	1	yes	characteristic
136	C₂	#20292533	1	1	yes	characteristic
137	C₂	C₂×(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
138	C₂	#20292593	1	1	yes	characteristic
139	C₂	C₈×Dic₂₁₄	1	2	yes	—
140	C₂	#20292757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₁₄

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

#20292381

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₄

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₄×Dic₂₁₄

—

C₄×Dic₂₁₄

—

C₂×C₄⋊Dic₁₀₇

C₄

yes

characteristic

C₂×Dic₁₀₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₈

yes

complemented

C₂²×Dic₁₀₇

C₂×C₄

yes

—

C₂²×Dic₁₀₇

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

Q₈

yes

complemented

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×C₈

yes

—

C₂×C₄×C₁₀₇

M₁₆

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂×C₈

yes

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₂²×C₁₀₇

C₂²×C₈

yes

characteristic

C₂²×C₁₀₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊C₈

yes

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#391

yes

characteristic

C₂×C₁₀₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₇

#408

yes

characteristic

C₂×C₁₀₇

C₈×Q₈

yes

—

C₂×C₁₀₇

#446

yes

—

#1984

—

107

maximal, Sylow

C₁₀₇

#1984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

107

—

C₂×C₄⋊C₈

—

107

—

C₂×C₄×C₈

D₁₀₇

yes

characteristic

#336

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₄×Q₈

—

214

—

C₄×Q₈

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

C₄×C₈

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

Dic₂₁₄

yes

complemented

100

C₄⋊C₄

—

214

—

101

C₄⋊C₄

—

214

—

102

C₄²

—

214

—

103

C₄²

—

214

—

104

C₄²

D₄₂₈

yes

—

105

C₄²

C₂²⋊D₁₀₇

yes

—

106

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

107

Q₈

—

214

—

108

C₂×C₄

—

109

C₂×C₄

—

107

—

110

C₂×C₄

—

214

—

111

C₂×C₄

—

214

—

112

C₂×C₄

—

214

—

113

C₂×C₄

—

214

—

114

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

115

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

116

C₂×C₄

C₈×D₁₀₇

yes

—

117

C₂×C₄

#13133801

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

119

C₂×C₄

C₂×D₄₂₈

yes

characteristic

120

C₂×C₄

C₂×Dic₂₁₄

yes

—

121

C₂×C₄

#13133818

yes

—

122

C₈

—

123

C₈

—

428

—

124

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

125

C₂²

C₄×Dic₂₁₄

yes

—

126

C₂²

C₂×C₈×D₁₀₇

yes

characteristic

127

C₂²

C₂×#13133801

yes

characteristic

128

C₂²

#16812115

yes

—

129

C₄

—

130

C₄

—

428

—

131

C₄

—

214

—

132

C₄

(C₂×D₁₀₇)⋊C₈

yes

—

133

C₄

#16812088

yes

—

134

C₄

#16812097

yes

characteristic

135

C₄

#16812109

yes

characteristic

136

C₂

#20292533

yes

characteristic

137

C₂

C₂×(C₂×D₁₀₇)⋊C₈

yes

characteristic

138

C₂

#20292593

yes

characteristic

139

C₂

C₈×Dic₂₁₄

yes

—

140

C₂

#20292757

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator	C₂²×C₁₀₇	C₂²×C₈
Fitting	C₂×C₄×C₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#1984
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₇

commutator

C₂²×C₁₀₇

C₂²×C₈

Fitting

C₂×C₄×C₈×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
18. C₂×Dic₂₁₄	C₈	semidirect
32. C₂×C₈×C₁₀₇	Q₈	semidirect
71. C₁₀₇	#1984	semidirect
99. C₂×C₈	Dic₂₁₄	semidirect

H ≅ G/N

type

18. C₂×Dic₂₁₄

C₈

semidirect

32. C₂×C₈×C₁₀₇

Q₈

semidirect

71. C₁₀₇

#1984

semidirect

99. C₂×C₈

Dic₂₁₄

semidirect

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

subgroup	index
C₂×C₄×Dic₂₁₄	2
C₂×C₄⋊(C₁₀₇⋊C₈)	2
#20292381	2
C₂×C₄×C₈×C₁₀₇	2
#1984	107

subgroup

index

C₂×C₄×Dic₂₁₄

C₂×C₄⋊(C₁₀₇⋊C₈)

#20292381

C₂×C₄×C₈×C₁₀₇

#1984

107

Subgroups — #28337397

Conjugacy classes of subgroups

Complements

Maximal subgroups