Group #27196676 — subgroups

Subgroups — #27196676

Order 13184.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15692953	C₂	1	2	yes	maximal, complemented
2	C₂×#15692943	C₂	1	2	yes	maximal
3	#20267693	C₂	1	1	yes	maximal, characteristic
4	#20267661	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₃	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
11	#15692953	—	2	4	no	—
12	#15692943	—	2	4	no	—
13	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂³×D₁₀₃	—	2	1	no	—
16	C₂×D₄₁₂	—	2	2	no	—
17	C₂×D₄₁₂	—	2	2	no	—
18	C₂×D₄₁₂	—	2	1	no	—
19	C₂×D₄₁₂	Q₈	1	2	yes	—
20	C₂²×Dic₁₀₃	—	2	1	no	—
21	C₂²×Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	—	2	2	no	—
23	C₄⋊Dic₁₀₃	—	2	2	no	—
24	C₄⋊Dic₁₀₃	—	2	2	no	—
25	C₄⋊Dic₁₀₃	—	4	4	no	—
26	(C₂×D₁₀₃)⋊C₄	—	4	4	no	—
27	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₀₃	—	2	2	no	—
30	C₂×C₈×C₁₀₃	—	2	2	no	—
31	C₂×C₈×C₁₀₃	D₄	1	4	yes	—
32	C₄²×C₁₀₃	—	2	2	no	—
33	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₁₀₃	—	2	2	no	—
35	C₂²×D₁₀₃	—	4	1	no	—
36	C₂²×D₁₀₃	—	4	1	no	—
37	C₂²×D₁₀₃	—	4	2	no	—
38	C₂²×D₁₀₃	—	4	2	no	—
39	C₂×C₄×C₁₀₃	—	2	2	no	—
40	C₂×C₄×C₁₀₃	—	2	2	no	—
41	C₂×C₄×C₁₀₃	SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	#41	1	2	yes	—
45	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
46	C₂×Dic₁₀₃	—	4	1	no	—
47	C₂×Dic₁₀₃	—	4	2	no	—
48	C₂×Dic₁₀₃	—	4	2	no	—
49	C₂×Dic₁₀₃	—	4	4	no	—
50	D₄₁₂	—	2	4	no	—
51	D₄₁₂	—	4	2	no	—
52	C₈×C₁₀₃	—	2	4	no	—
53	C₂²×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₃	#123	1	1	yes	characteristic
55	C₂²×C₁₀₃	#127	1	1	yes	characteristic
56	C₂²×C₁₀₃	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×D₁₀₃	—	4	2	no	—
59	C₂×D₁₀₃	—	4	4	no	—
60	C₂×D₁₀₃	—	8	2	no	—
61	C₂×D₁₀₃	—	8	1	no	—
62	C₂×D₁₀₃	—	8	1	no	—
63	C₄×C₁₀₃	—	2	4	no	—
64	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₀₃	#123	1	2	yes	—
66	Dic₁₀₃	—	8	1	no	—
67	Dic₁₀₃	—	8	2	no	—
68	C₂×C₁₀₃	#399	1	1	yes	characteristic
69	C₂×C₁₀₃	#476	1	2	yes	—
70	C₂×C₁₀₃	#481	1	2	yes	—
71	C₂×C₁₀₃	#492	1	1	yes	characteristic
72	C₂×C₁₀₃	#496	1	1	yes	characteristic
73	D₁₀₃	—	8	2	no	—
74	#2094	—	103	1	no	maximal, Sylow
75	C₁₀₃	#2094	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	103	2	no	—
77	C₂×#102	—	103	2	no	—
78	#390	—	103	1	no	—
79	#384	—	103	1	no	—
80	#336	D₁₀₃	1	1	yes	characteristic, complemented
81	C₂²×D₄	—	103	1	no	—
82	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
83	C₂×C₄⋊C₄	—	103	2	no	—
84	C₂×C₄⋊C₄	—	103	1	no	—
85	C₂×C₄⋊C₄	—	206	2	no	—
86	C₂×C₂²⋊C₄	—	206	2	no	—
87	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
88	#106	—	206	4	no	—
89	#102	—	206	4	no	—
90	C₂⁴	—	206	1	no	—
91	C₂×D₄	—	103	2	no	—
92	C₂×D₄	—	206	2	no	—
93	C₂×D₄	—	206	1	no	—
94	C₂×D₄	—	206	2	no	—
95	C₂²×C₄	—	206	1	no	—
96	C₂²×C₄	—	206	2	no	—
97	C₂²×C₄	D₄₁₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₄₁₂	1	4	yes	—
102	C₄⋊C₄	—	412	4	no	—
103	C₄⋊C₄	—	206	2	no	—
104	C₄⋊C₄	—	206	2	no	—
105	C₄⋊C₄	—	206	2	no	—
106	C₂²⋊C₄	—	412	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	412	1	no	—
109	C₂³	—	412	1	no	—
110	C₂³	—	412	2	no	—
111	C₂³	—	412	2	no	—
112	C₂³	—	206	2	no	—
113	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
114	D₄	—	412	2	no	—
115	D₄	—	206	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	412	1	no	—
119	C₂×C₄	—	412	2	no	—
120	C₂×C₄	—	412	2	no	—
121	C₂×C₄	—	412	4	no	—
122	C₂×C₄	#13110094	1	4	yes	—
123	C₂×C₄	C₂×D₄₁₂	1	2	yes	—
124	C₂×C₄	Q₈×D₁₀₃	1	1	yes	characteristic
125	C₂×C₄	#13110104	1	2	yes	—
126	C₂×C₄	#13110104	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂²	—	824	2	no	—
129	C₂²	—	824	1	no	—
130	C₂²	—	824	1	no	—
131	C₂²	—	412	4	no	—
132	C₂²	—	412	2	no	—
133	C₂²	#15693001	1	1	yes	characteristic
134	C₂²	#15693005	1	1	yes	characteristic
135	C₂²	#15693018	1	1	yes	characteristic
136	C₂²	C₂×#13110094	1	2	yes	—
137	C₂²	#15693049	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	824	1	no	—
140	C₄	—	824	2	no	—
141	C₄	#15693001	1	2	yes	—
142	C₄	#15693005	1	2	yes	—
143	C₂	—	824	2	no	—
144	C₂	#20267728	1	1	yes	characteristic
145	C₂	#20268082	1	2	yes	—
146	C₂	#20268110	1	2	yes	—
147	C₂	#20268164	1	1	yes	characteristic
148	C₂	#20268180	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692953

C₂

yes

maximal, complemented

C₂×#15692943

C₂

yes

maximal

#20267693

C₂

yes

maximal, characteristic

#20267661

C₂

yes

maximal, characteristic, complemented

#336×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₁₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₂×(C₂×D₁₀₃)⋊C₄

—

#15692953

—

#15692943

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂³×D₁₀₃

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

Q₈

yes

—

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

(C₂×D₁₀₃)⋊C₄

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

SD₁₆

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

C₂²⋊Q₈

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

C₂×SD₁₆

yes

—

C₂²×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

#123

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#399

yes

characteristic

C₂×C₁₀₃

#476

yes

—

C₂×C₁₀₃

#481

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

D₁₀₃

—

#2094

—

103

maximal, Sylow

C₁₀₃

#2094

yes

characteristic, complemented, Sylow

C₂×#106

—

103

—

C₂×#102

—

103

—

#390

—

103

—

#384

—

103

—

#336

D₁₀₃

yes

characteristic, complemented

C₂²×D₄

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₂²⋊C₄

—

206

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#106

—

206

—

#102

—

206

—

C₂⁴

—

206

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₄₁₂

yes

—

102

C₄⋊C₄

—

412

—

103

C₄⋊C₄

—

206

—

104

C₄⋊C₄

—

206

—

105

C₄⋊C₄

—

206

—

106

C₂²⋊C₄

—

412

—

107

C₄²

—

108

C₂³

—

412

—

109

C₂³

—

412

—

110

C₂³

—

412

—

111

C₂³

—

412

—

112

C₂³

—

206

—

113

C₂³

C₂×D₄₁₂

yes

characteristic

114

D₄

—

412

—

115

D₄

—

206

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

412

—

119

C₂×C₄

—

412

—

120

C₂×C₄

—

412

—

121

C₂×C₄

—

412

—

122

C₂×C₄

#13110094

yes

—

123

C₂×C₄

C₂×D₄₁₂

yes

—

124

C₂×C₄

Q₈×D₁₀₃

yes

characteristic

125

C₂×C₄

#13110104

yes

—

126

C₂×C₄

#13110104

yes

characteristic

127

C₈

—

128

C₂²

—

824

—

129

C₂²

—

824

—

130

C₂²

—

824

—

131

C₂²

—

412

—

132

C₂²

—

412

—

133

C₂²

#15693001

yes

characteristic

134

C₂²

#15693005

yes

characteristic

135

C₂²

#15693018

yes

characteristic

136

C₂²

C₂×#13110094

yes

—

137

C₂²

#15693049

yes

—

138

C₄

—

139

C₄

—

824

—

140

C₄

—

824

—

141

C₄

#15693001

yes

—

142

C₄

#15693005

yes

—

143

C₂

—

824

—

144

C₂

#20267728

yes

characteristic

145

C₂

#20268082

yes

—

146

C₂

#20268110

yes

—

147

C₂

#20268164

yes

characteristic

148

C₂

#20268180

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#336×C₁₀₃	C₂
socle	C₂³×C₁₀₃	C₂×D₄
prime radical	C₂²×D₄₁₂	C₂²
hypercenter	#336	D₁₀₃
hypocenter	C₁₀₃	#2094
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#336×C₁₀₃

C₂

socle

C₂³×C₁₀₃

C₂×D₄

prime radical

C₂²×D₄₁₂

C₂²

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#15692953	C₂	semidirect
4. #20267661	C₂	semidirect
5. #336×C₁₀₃	C₂	semidirect
75. C₁₀₃	#2094	semidirect
80. #336	D₁₀₃	semidirect

H ≅ G/N

type

1. C₂×#15692953

C₂

semidirect

4. #20267661

C₂

semidirect

5. #336×C₁₀₃

C₂

semidirect

75. C₁₀₃

#2094

semidirect

80. #336

D₁₀₃

semidirect

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

subgroup	index
C₂×#15692953	2
C₂×#15692943	2
#20267693	2
#20267661	2
#336×C₁₀₃	2
#2094	103

subgroup

index

#336×C₁₀₃

103

Subgroups — #27196676

Conjugacy classes of subgroups

Complements

Maximal subgroups