Group #27194260 — subgroups

Subgroups — #27194260

Order 13184.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20268163	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268081	C₂	1	2	yes	maximal, complemented
3	#20267535	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	D₄²×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110093	—	2	2	no	—
7	#15693021	C₂²	1	1	yes	characteristic, complemented
8	#15692952	—	2	2	no	—
9	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	complemented
10	#15692942	—	2	2	no	—
11	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₁₀₃	—	2	2	no	—
13	#127×C₁₀₃	C₂²	1	1	yes	characteristic
14	#121×C₁₀₃	—	2	2	no	—
15	#120×C₁₀₃	—	2	2	no	—
16	C₄×D₄×C₁₀₃	C₂²	1	2	yes	—
17	C₂×Dic₂₀₆	—	2	1	no	—
18	#13110093	—	4	4	no	—
19	C₂×C₁₀₃⋊C₈	—	2	1	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₂⁴×C₁₀₃	—	2	2	no	—
23	C₂×D₄×C₁₀₃	—	2	4	no	—
24	C₂×D₄×C₁₀₃	—	2	2	no	—
25	C₂×D₄×C₁₀₃	—	4	4	no	—
26	C₂×D₄×C₁₀₃	—	4	4	no	—
27	C₂×D₄×C₁₀₃	D₄	1	2	yes	—
28	C₂×D₄×C₁₀₃	D₄	1	2	yes	—
29	C₂²×C₄×C₁₀₃	—	2	2	no	—
30	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
31	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
32	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
33	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
34	C₂³×C₁₀₃	—	2	2	no	—
35	C₂³×C₁₀₃	—	2	2	no	—
36	C₂³×C₁₀₃	—	4	8	no	—
37	C₂³×C₁₀₃	—	8	2	no	—
38	D₄×C₁₀₃	—	2	2	no	—
39	D₄×C₁₀₃	—	4	2	no	—
40	D₄×C₁₀₃	—	4	8	no	—
41	D₄×C₁₀₃	—	4	2	no	—
42	D₄×C₁₀₃	—	4	2	no	—
43	D₄×C₁₀₃	SD₁₆	1	4	yes	complemented
44	C₂×C₄×C₁₀₃	—	2	2	no	—
45	C₂×C₄×C₁₀₃	—	4	2	no	—
46	C₂×C₄×C₁₀₃	—	4	2	no	—
47	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
49	C₂×Dic₁₀₃	—	4	1	no	—
50	Dic₂₀₆	—	4	2	no	—
51	C₁₀₃⋊C₈	—	2	2	no	—
52	C₁₀₃⋊C₈	—	4	2	no	—
53	C₂²×C₁₀₃	—	2	4	no	—
54	C₂²×C₁₀₃	—	4	2	no	—
55	C₂²×C₁₀₃	—	4	2	no	—
56	C₂²×C₁₀₃	—	4	4	no	—
57	C₂²×C₁₀₃	—	4	2	no	—
58	C₂²×C₁₀₃	—	8	8	no	—
59	C₂²×C₁₀₃	#120	1	1	yes	characteristic
60	C₄×C₁₀₃	—	2	1	no	—
61	C₄×C₁₀₃	—	4	2	no	—
62	C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
63	C₄×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
64	Dic₁₀₃	—	8	1	no	—
65	C₂×C₁₀₃	—	4	4	no	—
66	C₂×C₁₀₃	—	8	2	no	—
67	C₂×C₁₀₃	#450	1	2	yes	—
68	C₂×C₁₀₃	#453	1	1	yes	characteristic
69	#1629	—	103	1	no	maximal, Sylow
70	C₁₀₃	#1629	1	1	yes	characteristic, complemented, Sylow
71	D₄²	D₁₀₃	1	1	yes	characteristic
72	#492	—	103	1	no	—
73	#476	—	103	2	no	—
74	#331	—	103	1	no	—
75	D₄⋊C₈	—	103	2	no	—
76	C₂²×D₄	—	2	2	no	—
77	C₂×SD₁₆	—	206	2	no	—
78	C₄⋊Q₈	—	103	1	no	—
79	#127	C₂×D₁₀₃	1	1	yes	characteristic
80	#121	—	2	2	no	—
81	#120	—	2	2	no	—
82	C₄×D₄	C₂×D₁₀₃	1	2	yes	—
83	#106	—	206	2	no	—
84	C₄⋊C₈	—	103	2	no	—
85	#102	—	206	2	no	—
86	C₄×C₈	—	103	1	no	—
87	C₂⁴	—	2	2	no	—
88	C₂×Q₈	—	206	1	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	4	4	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	C₂²⋊D₁₀₃	1	2	yes	—
94	C₂×D₄	C₂²⋊D₁₀₃	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	SD₁₆	—	412	4	no	—
97	C₂×C₈	—	206	2	no	—
98	C₂×C₈	—	206	1	no	—
99	C₄⋊C₄	—	206	2	no	—
100	C₄⋊C₄	C₂²⋊D₁₀₃	1	2	yes	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	8	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	412	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	8	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	#13110093	1	4	yes	complemented
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	412	1	no	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
121	C₈	—	412	2	no	—
122	C₈	—	206	2	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	8	8	no	—
129	C₂²	#15692990	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	824	1	no	—
133	C₄	C₂×#13110093	1	2	yes	—
134	C₄	#15693047	1	2	yes	—
135	C₂	—	4	4	no	—
136	C₂	—	8	2	no	—
137	C₂	#20267942	1	2	yes	—
138	C₂	#20267959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268163

C₂

yes

maximal, characteristic, complemented

#20268081

C₂

yes

maximal, complemented

#20267535

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal, complemented

D₄²×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110093

—

#15693021

C₂²

yes

characteristic, complemented

#15692952

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

complemented

#15692942

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic, complemented

C₂²×D₄×C₁₀₃

—

#127×C₁₀₃

C₂²

yes

characteristic

#121×C₁₀₃

—

#120×C₁₀₃

—

C₄×D₄×C₁₀₃

C₂²

yes

—

C₂×Dic₂₀₆

—

#13110093

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₂⁴×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

D₄

yes

—

C₂×D₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

D₄

yes

—

C₂²⋊C₄×C₁₀₃

—

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

—

C₂³×C₁₀₃

—

C₂³×C₁₀₃

—

C₂³×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

SD₁₆

yes

complemented

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×SD₁₆

yes

—

C₄×C₁₀₃

AGL(1,ℤ/8)

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#450

yes

—

C₂×C₁₀₃

#453

yes

characteristic

#1629

—

103

maximal, Sylow

C₁₀₃

#1629

yes

characteristic, complemented, Sylow

D₄²

D₁₀₃

yes

characteristic

#492

—

103

—

#476

—

103

—

#331

—

103

—

D₄⋊C₈

—

103

—

C₂²×D₄

—

C₂×SD₁₆

—

206

—

C₄⋊Q₈

—

103

—

#127

C₂×D₁₀₃

yes

characteristic

#121

—

#120

—

C₄×D₄

C₂×D₁₀₃

yes

—

#106

—

206

—

C₄⋊C₈

—

103

—

#102

—

206

—

C₄×C₈

—

103

—

C₂⁴

—

C₂×Q₈

—

206

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₃

yes

—

C₂×D₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

—

SD₁₆

—

412

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

206

—

100

C₄⋊C₄

C₂²⋊D₁₀₃

yes

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₁₀₃

yes

characteristic

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

Q₈

—

412

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

#13110093

yes

complemented

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

412

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

121

C₈

—

412

—

122

C₈

—

206

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#15692990

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

824

—

133

C₄

C₂×#13110093

yes

—

134

C₄

#15693047

yes

—

135

C₂

—

136

C₂

—

137

C₂

#20267942

yes

—

138

C₂

#20267959

yes

characteristic

	subgroup	G/H
center	C₂²	#15692990
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting, prime radical	D₄²×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
hypercenter	D₄²	D₁₀₃
hypocenter	C₁₀₃	#1629
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄²

subgroup

G/H

center

C₂²

#15692990

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting, prime radical

D₄²×C₁₀₃

C₂

socle

C₂²×C₁₀₃

#120

hypercenter

D₄²

D₁₀₃

hypocenter

C₁₀₃

#1629

LCS

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄²

N	H ≅ G/N	type
1. #20268163	C₂	semidirect
2. #20268081	C₂	semidirect
3. #20267535	C₂	semidirect
4. D₄⋊(C₁₀₃⋊C₈)	C₂	semidirect
7. #15693021	C₂²	semidirect
9. C₄⋊(C₁₀₃⋊C₈)	C₂²	semidirect
11. C₄×C₁₀₃⋊C₈	C₂²	semidirect
43. D₄×C₁₀₃	SD₁₆	semidirect
70. C₁₀₃	#1629	semidirect
114. D₄	#13110093	semidirect

H ≅ G/N

type

1. #20268163

C₂

semidirect

2. #20268081

C₂

semidirect

3. #20267535

C₂

semidirect

4. D₄⋊(C₁₀₃⋊C₈)

C₂

semidirect

7. #15693021

C₂²

semidirect

9. C₄⋊(C₁₀₃⋊C₈)

C₂²

semidirect

11. C₄×C₁₀₃⋊C₈

C₂²

semidirect

43. D₄×C₁₀₃

SD₁₆

semidirect

70. C₁₀₃

#1629

semidirect

114. D₄

#13110093

semidirect

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

subgroup	index
#20268163	2
#20268081	2
#20267535	2
D₄⋊(C₁₀₃⋊C₈)	2
D₄²×C₁₀₃	2
#1629	103

subgroup

index

#20268163

#20268081

#20267535

D₄⋊(C₁₀₃⋊C₈)

D₄²×C₁₀₃

#1629

103

Subgroups — #27194260

Conjugacy classes of subgroups

Complements

Maximal subgroups