Group #27196759 — subgroups

Subgroups — #27196759

Order 13184.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15693027	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₀₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₀₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13110105	C₄	1	1	yes	characteristic
6	#15693027	—	2	2	no	—
7	#15693027	—	2	2	no	—
8	C₂×#13110089	C₄	1	1	yes	characteristic
9	C₂×#13110088	C₄	1	2	yes	—
10	C₂×C₈×D₁₀₃	C₂²	1	2	yes	—
11	C₂×C₁₀₃⋊C₁₆	—	2	2	no	—
12	C₂×C₁₀₃⋊C₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₀₃	—	2	2	no	—
15	C₂×C₁₆×C₁₀₃	C₂²	1	2	yes	—
16	#13110105	—	2	2	no	—
17	#13110105	—	2	2	no	—
18	C₂×Dic₂₀₆	C₈	1	1	yes	characteristic
19	C₂×D₄₁₂	C₈	1	1	yes	characteristic
20	C₂×C₂²⋊D₁₀₃	C₈	1	2	yes	—
21	C₂×C₄×D₁₀₃	C₂×C₄	1	2	yes	—
22	#13110089	—	2	2	no	—
23	#13110088	—	2	4	no	—
24	C₂×C₁₀₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₈×D₁₀₃	—	2	2	no	—
26	C₈×D₁₀₃	—	2	2	no	—
27	C₁₀₃⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
31	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
32	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
33	C₁₆×C₁₀₃	—	2	2	no	—
34	C₂³×C₁₀₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₁₀₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	M₁₆	1	2	yes	—
39	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊D₁₀₃	—	2	4	no	—
41	C₂×Dic₁₀₃	C₂×C₈	1	2	yes	—
42	D₄₁₂	—	2	2	no	—
43	C₄×D₁₀₃	—	2	2	no	—
44	C₄×D₁₀₃	—	2	2	no	—
45	Dic₂₀₆	—	2	2	no	—
46	C₈×C₁₀₃	C₂²⋊C₄	1	2	yes	—
47	C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
49	C₁₀₃⋊C₈	—	2	2	no	—
50	C₂²×C₁₀₃	—	2	2	no	—
51	C₂²×C₁₀₃	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₁₀₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₁₀₃	—	2	2	no	—
54	C₂×D₁₀₃	—	2	4	no	—
55	C₄×C₁₀₃	C₂²⋊C₈	1	2	yes	—
56	C₄×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₀₃	C₂×M₁₆	1	1	yes	characteristic
58	Dic₁₀₃	—	2	2	no	—
59	C₂×C₁₀₃	—	2	2	no	—
60	C₂×C₁₀₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₁₀₃	#504	1	2	yes	—
62	D₁₀₃	—	4	2	no	—
63	#2113	—	103	1	no	maximal, Sylow
64	C₁₀₃	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	103	1	no	—
66	C₂×M₃₂	D₁₀₃	1	1	yes	characteristic, complemented
67	C₂²×C₁₆	—	103	1	no	—
68	C₂²⋊C₁₆	—	103	4	no	—
69	C₂×#41	—	103	1	no	—
70	#131	—	206	2	no	—
71	#131	—	206	2	no	—
72	C₂×M₁₆	—	103	1	no	—
73	C₂×M₁₆	—	103	2	no	—
74	C₂²×C₈	—	103	2	no	—
75	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
76	M₃₂	—	2	2	no	—
77	C₂×C₁₆	—	103	2	no	—
78	C₂×C₁₆	—	206	2	no	—
79	C₂×C₁₆	C₂×D₁₀₃	1	2	yes	—
80	#41	—	206	2	no	—
81	#41	—	206	2	no	—
82	C₂×Q₈	—	103	1	no	—
83	C₂×D₄	—	103	2	no	—
84	C₂×D₄	—	103	1	no	—
85	C₂²×C₄	—	103	2	no	—
86	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
87	M₁₆	—	206	2	no	—
88	M₁₆	—	206	4	no	—
89	C₂×C₈	—	103	2	no	—
90	C₂×C₈	—	206	2	no	—
91	C₂×C₈	—	206	2	no	—
92	C₂×C₈	C₄×D₁₀₃	1	1	yes	characteristic
93	C₂×C₈	D₄₁₂	1	2	yes	—
94	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
95	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	206	2	no	—
98	C₂³	—	103	2	no	—
99	C₂³	C₈×D₁₀₃	1	1	yes	characteristic
100	Q₈	—	206	2	no	—
101	D₄	—	206	4	no	—
102	D₄	—	206	2	no	—
103	C₂×C₄	—	103	2	no	—
104	C₂×C₄	—	206	2	no	—
105	C₂×C₄	—	206	2	no	—
106	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
107	C₂×C₄	C₈×D₁₀₃	1	1	yes	characteristic
108	C₂×C₄	#13110088	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
110	C₈	—	206	2	no	—
111	C₈	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
112	C₈	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
113	C₈	C₂×D₄₁₂	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	206	4	no	—
116	C₂²	—	206	2	no	—
117	C₂²	(C₂×D₁₀₃)⋊C₈	1	2	yes	—
118	C₂²	C₂×C₈×D₁₀₃	1	1	yes	characteristic
119	C₄	—	206	2	no	—
120	C₄	(C₂×D₁₀₃)⋊C₈	1	2	yes	—
121	C₄	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
122	C₄	C₂×#13110088	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	412	2	no	—
125	C₂	C₂×(C₂×D₁₀₃)⋊C₈	1	1	yes	characteristic
126	C₂	#20268209	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693027

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₀₃)⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110105

C₄

yes

—

—

C₄

yes

characteristic

C₂×#13110088

C₄

yes

—

C₂×C₈×D₁₀₃

C₂²

yes

—

C₂×C₁₀₃⋊C₁₆

—

C₂×C₁₀₃⋊C₁₆

C₂²

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

M₃₂×C₁₀₃

—

C₂×C₁₆×C₁₀₃

C₂²

yes

—

#13110105

—

#13110105

—

C₂×Dic₂₀₆

C₈

yes

characteristic

C₂×D₄₁₂

C₈

yes

characteristic

C₂×C₂²⋊D₁₀₃

C₈

yes

—

C₂×C₄×D₁₀₃

C₂×C₄

yes

—

#13110089

—

#13110088

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₈×D₁₀₃

—

C₈×D₁₀₃

—

C₁₀₃⋊C₁₆

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₁₆×C₁₀₃

—

C₂³×C₁₀₃

C₂×C₈

yes

characteristic

C₂²×D₁₀₃

C₂×C₈

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₃

M₁₆

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

C₂×C₈

yes

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

C₂²⋊C₄

yes

—

C₈×C₁₀₃

C₂×D₄

yes

characteristic

C₈×C₁₀₃

C₂×D₄

yes

characteristic

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂²⋊C₈

yes

—

C₂²×C₁₀₃

C₂²×C₈

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

C₂²⋊C₈

yes

—

C₄×C₁₀₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₃

C₂×M₁₆

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₃

#504

yes

—

D₁₀₃

—

#2113

—

103

maximal, Sylow

C₁₀₃

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

103

—

C₂×M₃₂

D₁₀₃

yes

characteristic, complemented

C₂²×C₁₆

—

103

—

C₂²⋊C₁₆

—

103

—

C₂×#41

—

103

—

#131

—

206

—

#131

—

206

—

C₂×M₁₆

—

103

—

C₂×M₁₆

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

103

—

C₂×C₁₆

—

206

—

C₂×C₁₆

C₂×D₁₀₃

yes

—

#41

—

206

—

#41

—

206

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

characteristic

M₁₆

—

206

—

M₁₆

—

206

—

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₄×D₁₀₃

yes

characteristic

C₂×C₈

D₄₁₂

yes

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

C₁₆

—

C₁₆

—

206

—

C₂³

—

103

—

C₂³

C₈×D₁₀₃

yes

characteristic

100

Q₈

—

206

—

101

D₄

—

206

—

102

D₄

—

206

—

103

C₂×C₄

—

103

—

104

C₂×C₄

—

206

—

105

C₂×C₄

—

206

—

106

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

107

C₂×C₄

C₈×D₁₀₃

yes

characteristic

108

C₂×C₄

#13110088

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

110

C₈

—

206

—

111

C₈

(C₂×D₁₀₃)⋊C₄

yes

—

112

C₈

C₂×C₂²⋊D₁₀₃

yes

characteristic

113

C₈

C₂×D₄₁₂

yes

characteristic

114

C₂²

—

115

C₂²

—

206

—

116

C₂²

—

206

—

117

C₂²

(C₂×D₁₀₃)⋊C₈

yes

—

118

C₂²

C₂×C₈×D₁₀₃

yes

characteristic

119

C₄

—

206

—

120

C₄

(C₂×D₁₀₃)⋊C₈

yes

—

121

C₄

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

122

C₄

C₂×#13110088

yes

characteristic

123

C₂

—

124

C₂

—

412

—

125

C₂

C₂×(C₂×D₁₀₃)⋊C₈

yes

characteristic

126

C₂

#20268209

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂²×C₈
Fitting	C₂×M₃₂×C₁₀₃	C₂
prime radical	C₂×#13110105	C₄
cosocle	C₂×C₈×C₁₀₃	C₂³
hypercenter	C₂×M₃₂	D₁₀₃
hypocenter	C₁₀₃	#2113
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂²×C₈

Fitting

C₂×M₃₂×C₁₀₃

C₂

prime radical

C₂×#13110105

C₄

cosocle

C₂×C₈×C₁₀₃

C₂³

hypercenter

C₂×M₃₂

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂²×C₁₀₃⋊C₁₆	C₂	semidirect
3. (C₂×D₁₀₃)⋊C₁₆	C₂	semidirect
4. C₂×M₃₂×C₁₀₃	C₂	semidirect
64. C₁₀₃	#2113	semidirect
66. C₂×M₃₂	D₁₀₃	semidirect

H ≅ G/N

type

2. C₂²×C₁₀₃⋊C₁₆

C₂

semidirect

3. (C₂×D₁₀₃)⋊C₁₆

C₂

semidirect

4. C₂×M₃₂×C₁₀₃

C₂

semidirect

64. C₁₀₃

#2113

semidirect

66. C₂×M₃₂

D₁₀₃

semidirect

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

subgroup	index
C₂×#15693027	2
C₂²×C₁₀₃⋊C₁₆	2
(C₂×D₁₀₃)⋊C₁₆	2
C₂×M₃₂×C₁₀₃	2
#2113	103

subgroup

index

C₂×#15693027

C₂²×C₁₀₃⋊C₁₆

(C₂×D₁₀₃)⋊C₁₆

C₂×M₃₂×C₁₀₃

#2113

103

Subgroups — #27196759

Conjugacy classes of subgroups

Complements

Maximal subgroups