Group #27194992 — subgroups

Subgroups — #27194992

Order 13184.

316 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 161 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×Dic₁₀₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₃	C₂	1	2	yes	maximal
3	C₄²×Dic₁₀₃	C₂	1	1	yes	maximal, characteristic
4	#20267546	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
11	Dic₁₀₃⋊C₈	C₄	1	8	yes	complemented
12	C₈×Dic₁₀₃	—	2	4	no	—
13	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₀₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	—	2	2	no	—
20	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₃	—	2	4	no	—
22	C₄×Dic₁₀₃	C₈	1	8	yes	complemented
23	C₄×Dic₁₀₃	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₀₃	D₄	1	2	yes	—
25	C₄×Dic₁₀₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₀₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₃	—	2	2	no	—
29	C₂×C₈×C₁₀₃	—	2	2	no	—
30	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
31	C₄²×C₁₀₃	—	2	2	no	—
32	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	—	2	2	no	—
34	C₂×C₄×C₁₀₃	—	2	2	no	—
35	C₂×C₄×C₁₀₃	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	#41	1	2	yes	—
40	C₂×Dic₁₀₃	—	2	2	no	—
41	C₂×Dic₁₀₃	C₄²	1	2	yes	—
42	C₂×Dic₁₀₃	C₄²	1	2	yes	—
43	C₂×Dic₁₀₃	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₁₀₃	C₂×C₈	1	4	yes	—
45	C₂×Dic₁₀₃	C₂×C₈	1	4	yes	—
46	C₂×Dic₁₀₃	M₁₆	1	4	yes	—
47	C₈×C₁₀₃	—	2	4	no	—
48	C₁₀₃⋊C₈	—	2	4	no	—
49	C₂²×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₃	#117	1	1	yes	characteristic
52	C₂²×C₁₀₃	C₂²×C₈	1	2	yes	—
53	C₂²×C₁₀₃	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₀₃	—	2	4	no	—
55	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
56	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
57	Dic₁₀₃	—	2	2	no	—
58	Dic₁₀₃	C₄×C₈	1	4	yes	complemented
59	Dic₁₀₃	C₄⋊C₈	1	8	yes	complemented
60	C₂×C₁₀₃	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₀₃	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂×C₁₀₃	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₀₃	C₂×C₄⋊C₈	1	2	yes	—
64	C₂×C₁₀₃	#431	1	2	yes	—
65	C₄×C₄⋊C₈	—	103	1	no	maximal, Sylow
66	C₁₀₃	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	103	2	no	—
68	C₂×C₄×C₈	—	103	2	no	—
69	C₄³	—	103	1	no	—
70	#336	—	103	1	no	—
71	#336	D₁₀₃	1	1	yes	characteristic
72	C₂²×C₈	—	103	2	no	—
73	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
74	C₂×C₄²	—	103	2	no	—
75	C₂×C₄²	—	103	1	no	—
76	C₂×C₄²	—	103	1	no	—
77	C₂×C₄²	—	103	2	no	—
78	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
79	C₄⋊C₈	—	103	8	no	—
80	C₄×C₈	—	206	4	no	—
81	C₂²×C₄	—	103	1	no	—
82	C₂²×C₄	—	103	2	no	—
83	C₂²×C₄	—	103	1	no	—
84	C₂²×C₄	C₄×D₁₀₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	103	4	no	—
89	C₂×C₈	—	206	2	no	—
90	C₂×C₈	—	206	2	no	—
91	C₂×C₈	C₄×D₁₀₃	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	103	8	no	—
94	C₄²	—	103	4	no	—
95	C₄²	—	103	2	no	—
96	C₄²	—	103	2	no	—
97	C₄²	—	206	4	no	—
98	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	103	2	no	—
102	C₂×C₄	—	103	4	no	—
103	C₂×C₄	—	103	4	no	—
104	C₂×C₄	—	103	4	no	—
105	C₂×C₄	—	103	4	no	—
106	C₂×C₄	—	103	2	no	—
107	C₂×C₄	—	206	2	no	—
108	C₂×C₄	C₈×D₁₀₃	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
110	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	#13110104	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	206	4	no	—
115	C₂²	C₄²×D₁₀₃	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₀₃	1	1	yes	characteristic
117	C₂²	#15692977	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₁₀₃	1	2	yes	—
119	C₂²	M₁₆×D₁₀₃	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	103	8	no	—
122	C₄	—	103	4	no	—
123	C₄	—	206	2	no	—
124	C₄	(C₂²⋊D₁₀₃)⋊C₄	1	2	yes	—
125	C₄	#15692986	1	2	yes	—
126	C₂	#20267642	1	1	yes	characteristic
127	C₂	C₄×C₈×D₁₀₃	1	1	yes	characteristic
128	C₂	#20267743	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₁₀₃	1	2	yes	—
130	C₂	#20267841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₀₃⋊C₈

C₂

yes

maximal

C₂×C₈×Dic₁₀₃

C₂

yes

maximal

C₄²×Dic₁₀₃

C₂

yes

maximal, characteristic

#20267546

C₂

yes

maximal, characteristic

#336×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₀₃

C₄

yes

—

C₂×C₄×Dic₁₀₃

C₄

yes

—

C₂×C₄×Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₃

C₂²

yes

characteristic

Dic₁₀₃⋊C₈

C₄

yes

complemented

C₈×Dic₁₀₃

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

C₈

yes

complemented

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₄×Dic₁₀₃

D₄

yes

—

C₄×Dic₁₀₃

Q₈

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×C₈

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₄²

yes

—

C₂×Dic₁₀₃

C₄²

yes

—

C₂×Dic₁₀₃

C₄⋊C₄

yes

—

C₂×Dic₁₀₃

C₂×C₈

yes

—

C₂×Dic₁₀₃

C₂×C₈

yes

—

C₂×Dic₁₀₃

M₁₆

yes

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₄²

yes

characteristic

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

#117

yes

characteristic

C₂²×C₁₀₃

C₂²×C₈

yes

—

C₂²×C₁₀₃

C₂×M₁₆

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

C₄×Q₈

yes

—

Dic₁₀₃

—

Dic₁₀₃

C₄×C₈

yes

complemented

Dic₁₀₃

C₄⋊C₈

yes

complemented

C₂×C₁₀₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₃

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₃

C₄×M₁₆

yes

characteristic

C₂×C₁₀₃

C₂×C₄⋊C₈

yes

—

C₂×C₁₀₃

#431

yes

—

C₄×C₄⋊C₈

—

103

maximal, Sylow

C₁₀₃

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

103

—

C₂×C₄×C₈

—

103

—

C₄³

—

103

—

#336

—

103

—

#336

D₁₀₃

yes

characteristic

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄²

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

103

—

C₄×C₈

—

206

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₄²

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

206

—

C₂³

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

103

—

102

C₂×C₄

—

103

—

103

C₂×C₄

—

103

—

104

C₂×C₄

—

103

—

105

C₂×C₄

—

103

—

106

C₂×C₄

—

103

—

107

C₂×C₄

—

206

—

108

C₂×C₄

C₈×D₁₀₃

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

110

C₂×C₄

D₄×D₁₀₃

yes

characteristic

111

C₂×C₄

Q₈×D₁₀₃

yes

characteristic

112

C₂×C₄

#13110104

yes

—

113

C₈

—

114

C₈

—

206

—

115

C₂²

C₄²×D₁₀₃

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₀₃

yes

characteristic

117

C₂²

#15692977

yes

characteristic

118

C₂²

C₂×C₈×D₁₀₃

yes

—

119

C₂²

M₁₆×D₁₀₃

yes

—

120

C₄

—

121

C₄

—

103

—

122

C₄

—

103

—

123

C₄

—

206

—

124

C₄

(C₂²⋊D₁₀₃)⋊C₄

yes

—

125

C₄

#15692986

yes

—

126

C₂

#20267642

yes

characteristic

127

C₂

C₄×C₈×D₁₀₃

yes

characteristic

128

C₂

#20267743

yes

characteristic

129

C₂

C₄⋊C₈×D₁₀₃

yes

—

130

C₂

#20267841

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₃
commutator	C₂×C₁₀₃	C₂×C₄×C₈
Fitting	#336×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#336	D₁₀₃
hypocenter	C₁₀₃	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₃

commutator

C₂×C₁₀₃

C₂×C₄×C₈

Fitting

#336×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

#336

D₁₀₃

hypocenter

C₁₀₃

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. Dic₁₀₃⋊C₈	C₄	semidirect
22. C₄×Dic₁₀₃	C₈	semidirect
58. Dic₁₀₃	C₄×C₈	semidirect
59. Dic₁₀₃	C₄⋊C₈	semidirect
66. C₁₀₃	C₄×C₄⋊C₈	semidirect

H ≅ G/N

type

11. Dic₁₀₃⋊C₈

C₄

semidirect

22. C₄×Dic₁₀₃

C₈

semidirect

58. Dic₁₀₃

C₄×C₈

semidirect

59. Dic₁₀₃

C₄⋊C₈

semidirect

66. C₁₀₃

C₄×C₄⋊C₈

semidirect

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

subgroup	index
C₂×Dic₁₀₃⋊C₈	2
C₂×C₈×Dic₁₀₃	2
C₄²×Dic₁₀₃	2
#20267546	2
#336×C₁₀₃	2
C₄×C₄⋊C₈	103

subgroup

index

C₂×Dic₁₀₃⋊C₈

C₂×C₈×Dic₁₀₃

C₄²×Dic₁₀₃

#20267546

#336×C₁₀₃

C₄×C₄⋊C₈

103

Subgroups — #27194992

Conjugacy classes of subgroups

Complements

Maximal subgroups