Group #27205547 — subgroups

Subgroups — #27205547

Order 13184.

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

normal subgroups only

This group has 392 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	#20268341	C₂	1	1	yes	maximal, characteristic
2	#20268161	C₂	1	2	yes	maximal, complemented
3	#20268157	C₂	1	2	yes	maximal, complemented
4	#20268148	C₂	1	4	yes	maximal, complemented
5	#20267790	C₂	1	4	yes	maximal, complemented
6	C₄×#13110089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13110105	C₂²	1	2	yes	—
9	C₂×#13110089	C₂²	1	2	yes	—
10	#15693021	C₂²	1	1	yes	characteristic
11	#15693018	C₂²	1	1	yes	characteristic
12	#15693009	C₂²	1	2	yes	—
13	#15692995	—	2	2	no	—
14	C₄×D₄₁₂	—	2	2	no	—
15	#15692944	C₂²	1	8	yes	—
16	#15692941	C₂²	1	8	yes	—
17	#15692932	C₂²	1	2	yes	—
18	C₄×C₁₀₃⋊C₈	C₂²	1	2	yes	—
19	#125×C₁₀₃	—	2	2	no	—
20	#125×C₁₀₃	C₂²	1	2	yes	—
21	#125×C₁₀₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
25	#13110105	—	2	8	no	—
26	C₂×Dic₂₀₆	C₂³	1	2	yes	—
27	C₂×D₄₁₂	—	2	2	no	—
28	C₂×D₄₁₂	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₀₃	—	2	2	no	—
30	C₂×C₄×D₁₀₃	—	2	2	no	—
31	#13110089	—	2	4	no	—
32	C₂×C₁₀₃⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₀₃	—	2	2	no	—
34	(C₂×D₁₀₃)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₁₀₃	—	2	2	no	—
36	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
37	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₃	—	2	4	no	—
39	C₄⋊C₄×C₁₀₃	—	2	4	no	—
40	C₄⋊C₄×C₁₀₃	—	2	2	no	—
41	C₄⋊C₄×C₁₀₃	C₂³	1	4	yes	—
42	C₄²×C₁₀₃	D₄	1	2	yes	—
43	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
44	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
45	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₁₀₃	—	2	2	no	—
47	C₂×C₄×C₁₀₃	—	2	4	no	—
48	C₂×C₄×C₁₀₃	—	4	2	no	—
49	C₂×C₄×C₁₀₃	—	4	4	no	—
50	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₃	#41	1	4	yes	—
54	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₁₀₃	—	4	4	no	—
56	C₂×Dic₁₀₃	—	2	2	no	—
57	D₄₁₂	—	2	4	no	—
58	D₄₁₂	—	4	4	no	—
59	C₄×D₁₀₃	—	4	4	no	—
60	Dic₂₀₆	—	2	4	no	—
61	C₁₀₃⋊C₈	—	2	4	no	—
62	C₂²×C₁₀₃	—	2	2	no	—
63	C₂²×C₁₀₃	#124	1	2	yes	—
64	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₀₃	—	4	2	no	—
66	C₂×D₁₀₃	—	4	4	no	—
67	C₄×C₁₀₃	—	2	2	no	—
68	C₄×C₁₀₃	—	2	2	no	—
69	C₄×C₁₀₃	—	4	4	no	—
70	C₄×C₁₀₃	#124	1	2	yes	—
71	C₄×C₁₀₃	C₂×#41	1	2	yes	—
72	Dic₁₀₃	—	4	2	no	—
73	C₂×C₁₀₃	—	2	2	no	—
74	C₂×C₁₀₃	C₂×#124	1	1	yes	characteristic
75	C₂×C₁₀₃	#575	1	2	yes	—
76	D₁₀₃	—	8	2	no	—
77	#3142	—	103	1	no	maximal, Sylow
78	C₁₀₃	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	—	103	1	no	—
80	C₂×#125	D₁₀₃	1	1	yes	characteristic, complemented
81	#491	—	103	2	no	—
82	#490	—	103	2	no	—
83	#488	—	103	4	no	—
84	#417	—	103	4	no	—
85	C₄×M₁₆	—	103	1	no	—
86	C₂×#41	—	103	2	no	—
87	C₂×M₁₆	—	103	2	no	—
88	C₄⋊Q₈	—	103	1	no	—
89	#127	—	103	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₁₀₃	1	2	yes	—
92	#125	C₂×D₁₀₃	1	2	yes	—
93	#124	—	103	2	no	—
94	#121	—	206	2	no	—
95	C₄×D₄	—	206	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
98	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
99	#103	—	103	8	no	—
100	#102	—	103	8	no	—
101	#97	—	103	2	no	—
102	C₄×C₈	—	103	2	no	—
103	#41	—	206	8	no	—
104	C₂×Q₈	—	103	2	no	—
105	C₂×D₄	—	103	2	no	—
106	C₂×D₄	—	206	2	no	—
107	C₂×D₄	—	206	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	206	2	no	—
110	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
112	M₁₆	—	206	4	no	—
113	C₂×C₈	—	103	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	206	2	no	—
118	C₄⋊C₄	C₂²×D₁₀₃	1	4	yes	—
119	C₂²⋊C₄	—	206	4	no	—
120	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
121	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
122	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
123	C₂³	—	206	2	no	—
124	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
125	Q₈	—	206	4	no	—
126	D₄	—	412	4	no	—
127	D₄	—	412	4	no	—
128	D₄	—	206	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	412	4	no	—
133	C₂×C₄	—	206	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
137	C₂×C₄	#13110106	1	4	yes	—
138	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
139	C₈	—	206	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	412	2	no	—
142	C₂²	—	412	4	no	—
143	C₂²	#15693010	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	412	2	no	—
149	C₄	#15693010	1	2	yes	—
150	C₄	C₂×#13110106	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	824	2	no	—
153	C₂	C₂×#15693010	1	1	yes	characteristic
154	C₂	#20268589	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20268341

C₂

yes

maximal, characteristic

#20268161

C₂

yes

maximal, complemented

#20268157

C₂

yes

maximal, complemented

#20268148

C₂

yes

maximal, complemented

#20267790

C₂

yes

maximal, complemented

C₄×#13110089

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110105

C₂²

yes

—

C₂×#13110089

C₂²

yes

—

#15693021

C₂²

yes

characteristic

#15693018

C₂²

yes

characteristic

#15693009

C₂²

yes

—

#15692995

—

C₄×D₄₁₂

—

#15692944

C₂²

yes

—

#15692941

C₂²

yes

—

#15692932

C₂²

yes

—

C₄×C₁₀₃⋊C₈

C₂²

yes

—

#125×C₁₀₃

—

#125×C₁₀₃

C₂²

yes

—

#125×C₁₀₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#13110105

—

C₂×Dic₂₀₆

C₂³

yes

—

C₂×D₄₁₂

—

C₂×D₄₁₂

C₂³

yes

—

C₂×C₂²⋊D₁₀₃

—

C₂×C₄×D₁₀₃

—

#13110089

—

C₂×C₁₀₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₃

—

(C₂×D₁₀₃)⋊C₄

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

C₂³

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

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

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#124

yes

—

C₂²×C₁₀₃

C₂²×D₄

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#124

yes

—

C₄×C₁₀₃

C₂×#41

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×#124

yes

characteristic

C₂×C₁₀₃

#575

yes

—

D₁₀₃

—

#3142

—

103

maximal, Sylow

C₁₀₃

#3142

yes

characteristic, complemented, Sylow

#532

—

103

—

C₂×#125

D₁₀₃

yes

characteristic, complemented

#491

—

103

—

#490

—

103

—

#488

—

103

—

#417

—

103

—

C₄×M₁₆

—

103

—

C₂×#41

—

103

—

C₂×M₁₆

—

103

—

C₄⋊Q₈

—

103

—

#127

—

103

—

#125

—

#125

C₂×D₁₀₃

yes

—

#125

C₂×D₁₀₃

yes

—

#124

—

103

—

#121

—

206

—

C₄×D₄

—

206

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#103

—

103

—

100

#102

—

103

—

101

#97

—

103

—

102

C₄×C₈

—

103

—

103

#41

—

206

—

104

C₂×Q₈

—

103

—

105

C₂×D₄

—

103

—

106

C₂×D₄

—

206

—

107

C₂×D₄

—

206

—

108

C₂²×C₄

—

109

C₂²×C₄

—

206

—

110

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

111

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

112

M₁₆

—

206

—

113

C₂×C₈

—

103

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

206

—

118

C₄⋊C₄

C₂²×D₁₀₃

yes

—

119

C₂²⋊C₄

—

206

—

120

C₄²

C₂²⋊D₁₀₃

yes

—

121

C₄²

C₂²×D₁₀₃

yes

characteristic

122

C₄²

C₂²×D₁₀₃

yes

characteristic

123

C₂³

—

206

—

124

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

125

Q₈

—

206

—

126

D₄

—

412

—

127

D₄

—

412

—

128

D₄

—

206

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

412

—

133

C₂×C₄

—

206

—

134

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

137

C₂×C₄

#13110106

yes

—

138

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

139

C₈

—

206

—

140

C₂²

—

141

C₂²

—

412

—

142

C₂²

—

412

—

143

C₂²

#15693010

yes

—

144

C₂²

C₂²×C₂²⋊D₁₀₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

412

—

149

C₄

#15693010

yes

—

150

C₄

C₂×#13110106

yes

—

151

C₂

—

152

C₂

—

824

—

153

C₂

C₂×#15693010

yes

characteristic

154

C₂

#20268589

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₃
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	C₂×#125×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	#20268341	C₂
hypercenter	C₂×#125	D₁₀₃
hypocenter	C₁₀₃	#3142
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₃

commutator, cosocle

C₂×C₄×C₁₀₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₃

Fitting

C₂×#125×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂²×D₄

C₂

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20268161	C₂	semidirect
3. #20268157	C₂	semidirect
4. #20268148	C₂	semidirect
5. #20267790	C₂	semidirect
6. C₄×#13110089	C₂	semidirect
7. C₂×#125×C₁₀₃	C₂	semidirect
78. C₁₀₃	#3142	semidirect
80. C₂×#125	D₁₀₃	semidirect

H ≅ G/N

type

2. #20268161

C₂

semidirect

3. #20268157

C₂

semidirect

4. #20268148

C₂

semidirect

5. #20267790

C₂

semidirect

6. C₄×#13110089

C₂

semidirect

7. C₂×#125×C₁₀₃

C₂

semidirect

78. C₁₀₃

#3142

semidirect

80. C₂×#125

D₁₀₃

semidirect

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

subgroup	index
#20268341	2
#20268161	2
#20268157	2
#20268148	2
#20267790	2
C₄×#13110089	2
C₂×#125×C₁₀₃	2
#3142	103

subgroup

index

C₂×#125×C₁₀₃

103

Subgroups — #27205547

Conjugacy classes of subgroups

Complements

Maximal subgroups