Group #27194767 — subgroups

Subgroups — #27194767

Order 13184.

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

normal subgroups only

This group has 198 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	#20268187	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268165	C₂	1	1	yes	maximal, characteristic
3	#20268164	C₂	1	1	yes	maximal, characteristic
4	#20268141	C₂	1	1	yes	maximal, characteristic, complemented
5	#20268139	C₂	1	1	yes	maximal, characteristic
6	#20267512	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13110094	—	2	2	no	—
9	C₂×#13110092	—	2	2	no	—
10	#15693021	C₂²	1	1	yes	characteristic
11	#15693018	C₂²	1	1	yes	characteristic
12	#15692952	—	2	2	no	—
13	#15692945	—	2	2	no	—
14	#15692941	—	2	2	no	—
15	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
19	#107×C₁₀₃	—	2	2	no	—
20	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₀₆	—	2	1	no	—
22	C₂×D₄₁₂	—	2	1	no	—
23	C₂×D₄₁₂	—	2	1	no	—
24	C₂×D₄₁₂	—	2	1	no	—
25	#13110094	—	4	4	no	—
26	#13110092	—	4	4	no	—
27	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₁₀₃	—	2	1	no	—
30	C₄⋊Dic₁₀₃	—	2	1	no	—
31	C₂×Q₈×C₁₀₃	—	2	1	no	—
32	C₂×Q₈×C₁₀₃	—	2	1	no	—
33	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₀₃	—	2	1	no	—
35	C₄⋊C₄×C₁₀₃	—	2	1	no	—
36	C₄⋊C₄×C₁₀₃	—	2	1	no	—
37	C₄⋊C₄×C₁₀₃	—	2	1	no	—
38	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
39	C₂²×D₁₀₃	—	4	1	no	—
40	Q₈×C₁₀₃	—	4	2	no	—
41	Q₈×C₁₀₃	—	4	2	no	—
42	C₂×C₄×C₁₀₃	—	4	1	no	—
43	C₂×C₄×C₁₀₃	—	4	1	no	—
44	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₀₃	—	4	1	no	—
48	D₄₁₂	—	4	2	no	—
49	D₄₁₂	—	4	2	no	—
50	D₄₁₂	—	4	2	no	—
51	Dic₂₀₆	—	4	2	no	—
52	C₈×C₁₀₃	SD₁₆	1	4	yes	complemented
53	C₁₀₃⋊C₈	—	2	2	no	—
54	C₁₀₃⋊C₈	—	2	2	no	—
55	C₂²×C₁₀₃	#127	1	1	yes	characteristic
56	C₂×D₁₀₃	—	8	1	no	—
57	C₂×D₁₀₃	—	8	1	no	—
58	C₂×D₁₀₃	—	8	1	no	—
59	C₄×C₁₀₃	—	8	1	no	—
60	C₄×C₁₀₃	—	8	1	no	—
61	C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
62	C₄×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₀₃	#137	1	2	yes	—
64	Dic₁₀₃	—	8	1	no	—
65	C₂×C₁₀₃	#492	1	1	yes	characteristic
66	C₂×C₁₀₃	#496	1	1	yes	characteristic
67	C₂×C₁₀₃	#497	1	1	yes	characteristic
68	D₁₀₃	—	16	1	no	—
69	#1714	—	103	1	no	maximal, Sylow
70	C₁₀₃	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	D₁₀₃	1	1	yes	characteristic, complemented
72	#498	—	103	1	no	—
73	#492	—	103	1	no	—
74	#492	—	103	1	no	—
75	#487	—	103	1	no	—
76	#486	—	103	1	no	—
77	#322	—	103	1	no	—
78	C₂×SD₁₆	—	206	2	no	—
79	C₂×SD₁₆	—	206	2	no	—
80	C₄⋊Q₈	—	103	1	no	—
81	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
83	#127	—	103	1	no	—
84	#107	—	2	2	no	—
85	#106	—	206	2	no	—
86	#103	—	206	2	no	—
87	#102	—	206	2	no	—
88	C₄×C₈	—	103	1	no	—
89	C₄×C₈	—	103	1	no	—
90	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	206	1	no	—
94	C₂×D₄	—	206	1	no	—
95	C₂×D₄	—	206	1	no	—
96	C₂×D₄	—	206	1	no	—
97	SD₁₆	—	412	4	no	—
98	SD₁₆	—	412	4	no	—
99	C₂×C₈	—	103	2	no	—
100	C₂×C₈	—	103	2	no	—
101	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	206	1	no	—
107	C₄⋊C₄	—	206	1	no	—
108	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
109	C₂³	—	412	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	412	2	no	—
113	D₄	—	412	2	no	—
114	D₄	—	412	2	no	—
115	D₄	—	412	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	412	1	no	—
119	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
122	C₈	—	206	2	no	—
123	C₈	—	206	2	no	—
124	C₈	#13110092	1	4	yes	complemented
125	C₂²	—	824	1	no	—
126	C₂²	—	824	1	no	—
127	C₂²	—	824	1	no	—
128	C₂²	#15693017	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	824	1	no	—
132	C₄	C₂×#13110092	1	2	yes	—
133	C₄	#15693045	1	2	yes	—
134	C₄	#15693053	1	2	yes	—
135	C₂	—	1648	1	no	—
136	C₂	#20268165	1	1	yes	characteristic
137	C₂	#20268179	1	1	yes	characteristic
138	C₂	#20268184	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268187

C₂

yes

maximal, characteristic, complemented

#20268165

C₂

yes

maximal, characteristic

#20268164

C₂

yes

maximal, characteristic

#20268141

C₂

yes

maximal, characteristic, complemented

#20268139

C₂

yes

maximal, characteristic

#20267512

C₂

yes

maximal, characteristic, complemented

#500×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110094

—

C₂×#13110092

—

#15693021

C₂²

yes

characteristic

#15693018

C₂²

yes

—

—

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

#107×C₁₀₃

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

#13110094

—

#13110092

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

Dic₂₀₆

—

C₈×C₁₀₃

SD₁₆

yes

complemented

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

#127

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×SD₁₆

yes

—

C₄×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

D₁₀₃

—

#1714

—

103

maximal, Sylow

C₁₀₃

#1714

yes

characteristic, complemented, Sylow

#500

D₁₀₃

yes

characteristic, complemented

#498

—

103

—

#492

—

103

—

#492

—

103

—

#487

—

103

—

#486

—

103

—

#322

—

103

—

C₂×SD₁₆

—

206

—

C₂×SD₁₆

—

206

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

#127

—

103

—

#107

—

#106

—

206

—

#103

—

206

—

#102

—

206

—

C₄×C₈

—

103

—

C₄×C₈

—

103

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

SD₁₆

—

412

—

SD₁₆

—

412

—

C₂×C₈

—

103

—

100

C₂×C₈

—

103

—

101

C₂×C₈

C₂²⋊D₁₀₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

206

—

107

C₄⋊C₄

—

206

—

108

C₄²

C₂²×D₁₀₃

yes

characteristic

109

C₂³

—

412

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

412

—

113

D₄

—

412

—

114

D₄

—

412

—

115

D₄

—

412

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

412

—

119

C₂×C₄

D₄×D₁₀₃

yes

characteristic

120

C₂×C₄

D₄×D₁₀₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

122

C₈

—

206

—

123

C₈

—

206

—

124

C₈

#13110092

yes

complemented

125

C₂²

—

824

—

126

C₂²

—

824

—

127

C₂²

—

824

—

128

C₂²

#15693017

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

824

—

132

C₄

C₂×#13110092

yes

—

133

C₄

#15693045

yes

—

134

C₄

#15693053

yes

—

135

C₂

—

1648

—

136

C₂

#20268165

yes

characteristic

137

C₂

#20268179

yes

characteristic

138

C₂

#20268184

yes

characteristic

	subgroup	G/H
center	C₂²	#15693017
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#500×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#127
prime radical	#15693018	C₂²
hypercenter	#500	D₁₀₃
hypocenter	C₁₀₃	#1714
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#15693017

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#500×C₁₀₃

C₂

C₂²×C₁₀₃

C₂²

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #20268187	C₂	semidirect
4. #20268141	C₂	semidirect
6. #20267512	C₂	semidirect
7. #500×C₁₀₃	C₂	semidirect
52. C₈×C₁₀₃	SD₁₆	semidirect
70. C₁₀₃	#1714	semidirect
71. #500	D₁₀₃	semidirect
124. C₈	#13110092	semidirect

H ≅ G/N

type

1. #20268187

C₂

semidirect

4. #20268141

C₂

semidirect

6. #20267512

C₂

semidirect

7. #500×C₁₀₃

C₂

semidirect

52. C₈×C₁₀₃

SD₁₆

semidirect

70. C₁₀₃

#1714

semidirect

71. #500

D₁₀₃

semidirect

124. C₈

#13110092

semidirect

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

subgroup	index
#20268187	2
#20268165	2
#20268164	2
#20268141	2
#20268139	2
#20267512	2
#500×C₁₀₃	2
#1714	103

subgroup

index

#500×C₁₀₃

103

Subgroups — #27194767

Conjugacy classes of subgroups

Complements

Maximal subgroups