Group #27204013 — subgroups

Subgroups — #27204013

Order 13184.

456 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×Dic₄₁₂	C₂	1	1	yes	maximal, characteristic
2	#20268268	C₂	1	2	yes	maximal
3	#20267899	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₄₁₂	C₂	1	4	yes	maximal, complemented
5	#20267829	C₂	1	1	yes	maximal, characteristic
6	#20267801	C₂	1	2	yes	maximal
7	#405×C₁₀₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₀₆	C₂²	1	2	yes	—
9	C₂×Dic₄₁₂	C₄	1	8	yes	complemented
10	C₂×Dic₄₁₂	C₂²	1	2	yes	—
11	C₂×Dic₄₁₂	C₂²	1	2	yes	—
12	C₄×Dic₂₀₆	C₂²	1	8	yes	—
13	#15692986	—	2	4	no	—
14	#15692980	C₂²	1	2	yes	—
15	#15692977	—	2	4	no	—
16	#15692955	C₂²	1	2	yes	—
17	#15692953	C₂²	1	2	yes	—
18	#15692946	C₂²	1	8	yes	—
19	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
21	#117×C₁₀₃	C₂²	1	1	yes	characteristic
22	#97×C₁₀₃	C₂²	1	2	yes	—
23	C₄×C₈×C₁₀₃	C₂²	1	2	yes	—
24	C₂×Dic₂₀₆	—	2	2	no	—
25	C₂×Dic₂₀₆	—	2	4	no	—
26	C₂×Dic₂₀₆	C₂×C₄	1	8	yes	—
27	C₂×Dic₂₀₆	C₂³	1	4	yes	—
28	C₂²×Dic₁₀₃	—	2	2	no	—
29	Dic₄₁₂	C₂×C₄	1	8	yes	—
30	Dic₄₁₂	C₂×C₄	1	8	yes	—
31	C₄⋊Dic₁₀₃	C₂³	1	4	yes	—
32	Dic₁₀₃⋊C₄	—	2	8	no	—
33	C₂²⋊Dic₁₀₃	—	2	2	no	—
34	C₄×Dic₁₀₃	—	2	8	no	—
35	C₄×Dic₁₀₃	—	2	2	no	—
36	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₀₃	—	2	2	no	—
38	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
40	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
43	C₂²⋊C₄×C₁₀₃	D₄	1	2	yes	—
44	C₄²×C₁₀₃	C₂³	1	2	yes	—
45	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₀₃	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₃	#41	1	2	yes	—
50	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₃	—	2	8	no	—
52	C₂×Dic₁₀₃	—	2	4	no	—
53	C₂×Dic₁₀₃	—	2	4	no	—
54	Dic₂₀₆	—	2	8	no	—
55	Dic₂₀₆	—	2	4	no	—
56	Dic₂₀₆	C₂²×C₄	1	8	yes	—
57	C₈×C₁₀₃	—	2	2	no	—
58	C₈×C₁₀₃	C₂²×C₄	1	2	yes	—
59	C₈×C₁₀₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₁₀₃	—	2	2	no	—
61	C₂²×C₁₀₃	C₄×D₄	1	2	yes	—
62	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₃	—	2	4	no	—
64	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₀₃	C₂×#41	1	1	yes	characteristic
67	Dic₁₀₃	—	2	8	no	—
68	Dic₁₀₃	—	4	4	no	—
69	C₂×C₁₀₃	—	2	2	no	—
70	C₂×C₁₀₃	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₀₃	#578	1	2	yes	—
72	#2950	—	103	1	no	maximal, Sylow
73	C₁₀₃	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	—	103	1	no	—
75	#519	—	103	2	no	—
76	#441	—	103	4	no	—
77	C₄×Q₁₆	—	103	4	no	—
78	#427	—	103	1	no	—
79	#419	—	103	2	no	—
80	#405	D₁₀₃	1	1	yes	characteristic
81	C₂²×Q₈	—	103	2	no	—
82	C₂×Q₁₆	—	103	8	no	—
83	C₂×Q₁₆	—	103	2	no	—
84	C₂×Q₁₆	—	103	2	no	—
85	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
87	C₄×Q₈	—	103	8	no	—
88	C₄×Q₈	—	206	4	no	—
89	#117	—	103	2	no	—
90	#117	—	206	4	no	—
91	#117	C₂×D₁₀₃	1	1	yes	characteristic
92	#107	—	103	2	no	—
93	#106	—	103	2	no	—
94	#103	—	103	8	no	—
95	#97	C₂×D₁₀₃	1	2	yes	—
96	C₄×C₈	C₂×D₁₀₃	1	2	yes	—
97	C₂×Q₈	—	103	8	no	—
98	C₂×Q₈	—	103	4	no	—
99	C₂×Q₈	—	206	2	no	—
100	C₂×Q₈	—	206	4	no	—
101	C₂²×C₄	—	206	2	no	—
102	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
103	Q₁₆	—	103	8	no	—
104	Q₁₆	—	103	8	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₄×D₁₀₃	1	4	yes	—
107	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
108	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
110	C₄⋊C₄	—	103	4	no	—
111	C₄⋊C₄	—	206	8	no	—
112	C₄⋊C₄	D₄₁₂	1	2	yes	—
113	C₂²⋊C₄	—	206	2	no	—
114	C₂²⋊C₄	D₄₁₂	1	2	yes	—
115	C₄²	—	206	2	no	—
116	C₄²	—	206	8	no	—
117	C₄²	C₂²×D₁₀₃	1	2	yes	—
118	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
119	Q₈	—	103	8	no	—
120	Q₈	—	206	4	no	—
121	Q₈	—	206	8	no	—
122	C₂×C₄	—	206	4	no	—
123	C₂×C₄	—	206	4	no	—
124	C₂×C₄	—	206	8	no	—
125	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
126	C₂×C₄	C₂×D₄₁₂	1	4	yes	—
127	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
128	C₂×C₄	#13110105	1	2	yes	—
129	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂×C₄×D₁₀₃	1	2	yes	—
132	C₈	C₂×C₄×D₁₀₃	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₄₁₂	1	2	yes	—
135	C₂²	C₂²×D₄₁₂	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	412	4	no	—
138	C₄	—	206	8	no	—
139	C₄	C₄×D₄₁₂	1	2	yes	—
140	C₄	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
141	C₄	C₂×#13110105	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₄₁₂	1	1	yes	characteristic
144	C₂	#20268608	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₁₂

C₂

yes

maximal, characteristic

#20268268

C₂

yes

maximal

#20267899

C₂

yes

maximal, complemented

C₄×Dic₄₁₂

C₂

yes

maximal, complemented

#20267829

C₂

yes

maximal, characteristic

#20267801

C₂

yes

maximal

#405×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

—

C₂×Dic₄₁₂

C₄

yes

complemented

C₂×Dic₄₁₂

C₂²

yes

—

C₂×Dic₄₁₂

C₂²

yes

—

C₄×Dic₂₀₆

C₂²

yes

—

#15692986

—

#15692980

C₂²

yes

—

#15692977

—

#15692955

C₂²

yes

—

#15692953

C₂²

yes

—

#15692946

C₂²

yes

—

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

#117×C₁₀₃

C₂²

yes

characteristic

#97×C₁₀₃

C₂²

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

C₂×C₄

yes

—

C₂×Dic₂₀₆

C₂³

yes

—

C₂²×Dic₁₀₃

—

Dic₄₁₂

C₂×C₄

yes

—

Dic₄₁₂

C₂×C₄

yes

—

C₄⋊Dic₁₀₃

C₂³

yes

—

Dic₁₀₃⋊C₄

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

D₄

yes

—

C₂²⋊C₄×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

C₂²×C₄

yes

—

C₈×C₁₀₃

—

C₈×C₁₀₃

C₂²×C₄

yes

—

C₈×C₁₀₃

C₂²×C₄

yes

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₄×D₄

yes

—

C₂²×C₁₀₃

C₂²×D₄

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

C₂³×C₄

yes

characteristic

C₄×C₁₀₃

C₂×#41

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₃

#578

yes

—

#2950

—

103

maximal, Sylow

C₁₀₃

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

103

—

#519

—

103

—

#441

—

103

—

C₄×Q₁₆

—

103

—

#427

—

103

—

#419

—

103

—

#405

D₁₀₃

yes

characteristic

C₂²×Q₈

—

103

—

C₂×Q₁₆

—

103

—

C₂×Q₁₆

—

103

—

C₂×Q₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₄×Q₈

—

103

—

C₄×Q₈

—

206

—

#117

—

103

—

#117

—

206

—

#117

C₂×D₁₀₃

yes

characteristic

#107

—

103

—

#106

—

103

—

#103

—

103

—

#97

C₂×D₁₀₃

yes

—

C₄×C₈

C₂×D₁₀₃

yes

—

C₂×Q₈

—

103

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

100

C₂×Q₈

—

206

—

101

C₂²×C₄

—

206

—

102

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

103

Q₁₆

—

103

—

104

Q₁₆

—

103

—

105

M₁₆

—

106

C₂×C₈

C₄×D₁₀₃

yes

—

107

C₂×C₈

C₂²×D₁₀₃

yes

—

108

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

109

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

110

C₄⋊C₄

—

103

—

111

C₄⋊C₄

—

206

—

112

C₄⋊C₄

D₄₁₂

yes

—

113

C₂²⋊C₄

—

206

—

114

C₂²⋊C₄

D₄₁₂

yes

—

115

C₄²

—

206

—

116

C₄²

—

206

—

117

C₄²

C₂²×D₁₀₃

yes

—

118

C₂³

C₂×D₄₁₂

yes

characteristic

119

Q₈

—

103

—

120

Q₈

—

206

—

121

Q₈

—

206

—

122

C₂×C₄

—

206

—

123

C₂×C₄

—

206

—

124

C₂×C₄

—

206

—

125

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

126

C₂×C₄

C₂×D₄₁₂

yes

—

127

C₂×C₄

C₂×D₄₁₂

yes

characteristic

128

C₂×C₄

#13110105

yes

—

129

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₁₀₃

yes

—

132

C₈

C₂×C₄×D₁₀₃

yes

—

133

C₂²

—

134

C₂²

C₄×D₄₁₂

yes

—

135

C₂²

C₂²×D₄₁₂

yes

characteristic

136

C₄

—

137

C₄

—

412

—

138

C₄

—

206

—

139

C₄

C₄×D₄₁₂

yes

—

140

C₄

C₂²×C₄×D₁₀₃

yes

characteristic

141

C₄

C₂×#13110105

yes

characteristic

142

C₂

—

143

C₂

C₂×C₄×D₄₁₂

yes

characteristic

144

C₂

#20268608

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₁₂
commutator	C₄×C₁₀₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#405×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂×C₄×C₁₀₃	C₂⁴
hypercenter	#405	D₁₀₃
hypocenter	C₁₀₃	#2950
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₄₁₂

commutator

C₄×C₁₀₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₃

Fitting

#405×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂²×D₄

prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂×C₄×C₁₀₃

C₂⁴

D₁₀₃

C₁₀₃

G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
3. #20267899	C₂	semidirect
4. C₄×Dic₄₁₂	C₂	semidirect
9. C₂×Dic₄₁₂	C₄	semidirect
73. C₁₀₃	#2950	semidirect

H ≅ G/N

type

3. #20267899

C₂

semidirect

4. C₄×Dic₄₁₂

C₂

semidirect

9. C₂×Dic₄₁₂

C₄

semidirect

73. C₁₀₃

#2950

semidirect

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

subgroup	index
C₂²×Dic₄₁₂	2
#20268268	2
#20267899	2
C₄×Dic₄₁₂	2
#20267829	2
#20267801	2
#405×C₁₀₃	2
#2950	103

subgroup

index

C₂²×Dic₄₁₂

C₄×Dic₄₁₂

#405×C₁₀₃

103

Subgroups — #27204013

Conjugacy classes of subgroups

Complements

Maximal subgroups