Group #27205693 — subgroups

Subgroups — #27205693

Order 13184.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20268348	C₂	1	1	yes	maximal, characteristic
2	#20268201	C₂	1	4	yes	maximal, complemented
3	#20268190	C₂	1	4	yes	maximal, complemented
4	#20267831	C₂	1	4	yes	maximal
5	C₄×#13110089	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₁₀₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13110089	C₂²	1	2	yes	—
8	#15693021	C₂²	1	2	yes	—
9	#15693013	C₂²	1	2	yes	—
10	#15693000	—	2	2	no	—
11	C₄×Dic₂₀₆	—	2	2	no	—
12	#15692980	C₂²	1	2	yes	—
13	#15692954	C₂²	1	8	yes	—
14	#15692952	C₂²	1	8	yes	—
15	#15692932	C₂²	1	2	yes	—
16	C₄×C₁₀₃⋊C₈	C₂²	1	2	yes	—
17	#125×C₁₀₃	—	2	2	no	—
18	#125×C₁₀₃	C₂²	1	2	yes	—
19	#125×C₁₀₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₀₆	—	2	2	no	—
24	#13110089	Q₈	1	8	yes	complemented
25	C₂×C₁₀₃⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₃	—	2	2	no	—
27	C₄⋊Dic₁₀₃	C₂³	1	4	yes	—
28	Dic₁₀₃⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₀₃	—	2	2	no	—
30	C₄×Dic₁₀₃	—	2	2	no	—
31	C₂²×C₄×C₁₀₃	—	2	2	no	—
32	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₃	—	2	4	no	—
35	C₄⋊C₄×C₁₀₃	—	2	4	no	—
36	C₄⋊C₄×C₁₀₃	—	2	2	no	—
37	C₄⋊C₄×C₁₀₃	C₂³	1	4	yes	—
38	C₄²×C₁₀₃	D₄	1	2	yes	—
39	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₃	—	2	4	no	—
43	C₂×C₄×C₁₀₃	—	4	2	no	—
44	C₂×C₄×C₁₀₃	—	4	4	no	—
45	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₃	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₃	—	2	4	no	—
51	Dic₂₀₆	—	4	4	no	—
52	C₁₀₃⋊C₈	C₂×Q₈	1	8	yes	—
53	C₂²×C₁₀₃	—	2	2	no	—
54	C₂²×C₁₀₃	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₀₃	—	2	2	no	—
57	C₄×C₁₀₃	—	2	2	no	—
58	C₄×C₁₀₃	—	4	4	no	—
59	C₄×C₁₀₃	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₀₃	C₂²×Q₈	1	2	yes	—
61	Dic₁₀₃	—	4	4	no	—
62	C₂×C₁₀₃	—	2	2	no	—
63	C₂×C₁₀₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₀₃	#575	1	2	yes	—
65	#3166	—	103	1	no	maximal, Sylow
66	C₁₀₃	#3166	1	1	yes	characteristic, complemented, Sylow
67	#533	—	103	1	no	—
68	C₂×#125	D₁₀₃	1	1	yes	characteristic
69	#501	—	103	4	no	—
70	#499	—	103	4	no	—
71	#428	—	103	4	no	—
72	C₄×M₁₆	—	103	1	no	—
73	C₂×M₁₆	—	103	2	no	—
74	C₄⋊Q₈	—	103	2	no	—
75	#125	—	2	2	no	—
76	#125	—	103	2	no	—
77	#125	C₂×D₁₀₃	1	2	yes	—
78	#125	C₂×D₁₀₃	1	2	yes	—
79	C₂²⋊Q₈	—	206	2	no	—
80	C₄×Q₈	—	206	2	no	—
81	#117	—	103	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
85	#107	—	103	8	no	—
86	#106	—	103	8	no	—
87	#97	—	103	2	no	—
88	C₄×C₈	—	103	2	no	—
89	C₂×Q₈	—	206	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
93	M₁₆	—	103	8	no	—
94	C₂×C₈	—	103	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	103	4	no	—
99	C₄⋊C₄	—	206	4	no	—
100	C₄⋊C₄	—	206	2	no	—
101	C₄⋊C₄	C₂²×D₁₀₃	1	4	yes	—
102	C₂²⋊C₄	—	206	2	no	—
103	C₄²	—	206	2	no	—
104	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
105	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
106	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
108	Q₈	—	412	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	206	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₀₃	1	4	yes	—
117	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
118	C₈	—	103	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#15693019	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	412	4	no	—
126	C₄	#15693019	1	2	yes	—
127	C₄	C₂×Q₈×D₁₀₃	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#15693019	1	1	yes	characteristic
130	C₂	#20268589	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20268348

C₂

yes

maximal, characteristic

#20268201

C₂

yes

maximal, complemented

#20268190

C₂

yes

maximal, complemented

#20267831

C₂

yes

maximal

C₄×#13110089

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

#15693021

C₂²

yes

—

#15693013

C₂²

yes

—

#15693000

—

C₄×Dic₂₀₆

—

#15692980

C₂²

yes

—

#15692954

C₂²

yes

—

#15692952

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

C₂×Dic₂₀₆

—

#13110089

Q₈

yes

complemented

C₂×C₁₀₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

C₂³

yes

—

Dic₁₀₃⋊C₄

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

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

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

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

characteristic

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

C₂×Q₈

yes

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₄⋊Q₈

yes

—

C₂²×C₁₀₃

C₂²×D₄

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄⋊Q₈

yes

—

C₄×C₁₀₃

C₂²×Q₈

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₃

#575

yes

—

#3166

—

103

maximal, Sylow

C₁₀₃

#3166

yes

characteristic, complemented, Sylow

#533

—

103

—

C₂×#125

D₁₀₃

yes

characteristic

#501

—

103

—

#499

—

103

—

#428

—

103

—

C₄×M₁₆

—

103

—

C₂×M₁₆

—

103

—

C₄⋊Q₈

—

103

—

#125

—

#125

—

103

—

#125

C₂×D₁₀₃

yes

—

#125

C₂×D₁₀₃

yes

—

C₂²⋊Q₈

—

206

—

C₄×Q₈

—

206

—

#117

—

103

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#107

—

103

—

#106

—

103

—

#97

—

103

—

C₄×C₈

—

103

—

C₂×Q₈

—

206

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

103

—

C₂×C₈

—

103

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

100

C₄⋊C₄

—

206

—

101

C₄⋊C₄

C₂²×D₁₀₃

yes

—

102

C₂²⋊C₄

—

206

—

103

C₄²

—

206

—

104

C₄²

C₂²⋊D₁₀₃

yes

—

105

C₄²

C₂²×D₁₀₃

yes

characteristic

106

C₄²

C₂²×D₁₀₃

yes

characteristic

107

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

108

Q₈

—

412

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

206

—

113

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

116

C₂×C₄

Q₈×D₁₀₃

yes

—

117

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

118

C₈

—

103

—

119

C₂²

—

120

C₂²

#15693019

yes

—

121

C₂²

C₂²×C₂²⋊D₁₀₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

412

—

126

C₄

#15693019

yes

—

127

C₄

C₂×Q₈×D₁₀₃

yes

—

128

C₂

—

129

C₂

C₂×#15693019

yes

characteristic

130

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	C₂³×C₁₀₃	C₂×D₄
hypercenter	C₂×#125	D₁₀₃
hypocenter	C₁₀₃	#3166
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

prime radical

C₂³×C₁₀₃

C₂×D₄

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20268201	C₂	semidirect
3. #20268190	C₂	semidirect
24. #13110089	Q₈	semidirect
66. C₁₀₃	#3166	semidirect

H ≅ G/N

type

2. #20268201

C₂

semidirect

3. #20268190

C₂

semidirect

24. #13110089

Q₈

semidirect

66. C₁₀₃

#3166

semidirect

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

subgroup	index
#20268348	2
#20268201	2
#20268190	2
#20267831	2
C₄×#13110089	2
C₂×#125×C₁₀₃	2
#3166	103

subgroup

index

C₂×#125×C₁₀₃

103

Subgroups — #27205693

Conjugacy classes of subgroups

Complements

Maximal subgroups