Group #27195366 — subgroups

Subgroups — #27195366

Order 13184.

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

normal subgroups only

This group has 252 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	#20268558	C₂	1	1	yes	maximal, characteristic
2	#20267840	C₂	1	2	yes	maximal, complemented
3	#20267758	C₂	1	1	yes	maximal, characteristic
4	#20267555	C₂	1	2	yes	maximal, complemented
5	#413×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	#15693028	—	2	4	no	—
7	C₂×#13110089	—	2	2	no	—
8	C₂×#13110089	C₂²	1	1	yes	characteristic
9	C₂×#13110089	C₂²	1	2	yes	—
10	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
11	#15692957	—	2	4	no	—
12	C₂²⋊(C₁₀₃⋊C₈)	—	2	2	no	—
13	C₂×#41×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
15	#101×C₁₀₃	C₄	1	2	yes	—
16	#100×C₁₀₃	C₄	1	2	yes	—
17	#13110089	—	2	4	no	—
18	#13110089	—	2	2	no	—
19	#13110089	—	2	2	no	—
20	#13110089	D₄	1	4	yes	—
21	C₂×C₁₀₃⋊C₈	—	2	4	no	—
22	C₂×C₁₀₃⋊C₈	—	2	2	no	—
23	C₂×C₁₀₃⋊C₈	—	2	1	no	—
24	C₂×C₁₀₃⋊C₈	—	2	1	no	—
25	C₂×C₁₀₃⋊C₈	—	2	2	no	—
26	#41×C₁₀₃	—	2	2	no	—
27	#41×C₁₀₃	—	2	2	no	—
28	C₂×Q₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₀₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₃	—	2	2	no	—
34	M₁₆×C₁₀₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₁₀₃	—	2	2	no	—
36	C₂³×C₁₀₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₀₃	—	2	2	no	—
39	D₄×C₁₀₃	—	2	4	no	—
40	D₄×C₁₀₃	—	2	2	no	—
41	C₂×C₄×C₁₀₃	—	2	2	no	—
42	C₂×C₄×C₁₀₃	—	2	2	no	—
43	C₂×C₄×C₁₀₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₃	#41	1	2	yes	—
48	C₈×C₁₀₃	—	2	4	no	—
49	C₁₀₃⋊C₈	—	2	4	no	—
50	C₁₀₃⋊C₈	—	4	1	no	—
51	C₁₀₃⋊C₈	—	4	1	no	—
52	C₂²×C₁₀₃	—	2	2	no	—
53	C₂²×C₁₀₃	—	4	1	no	—
54	C₂²×C₁₀₃	—	4	2	no	—
55	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₀₃	—	2	2	no	—
58	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₃	#120	1	1	yes	characteristic
60	C₄×C₁₀₃	#123	1	1	yes	characteristic
61	C₂×C₁₀₃	—	2	1	no	—
62	C₂×C₁₀₃	—	2	2	no	—
63	C₂×C₁₀₃	—	4	2	no	—
64	C₂×C₁₀₃	#385	1	1	yes	characteristic
65	#1858	—	103	1	no	maximal, Sylow
66	C₁₀₃	#1858	1	1	yes	characteristic, complemented, Sylow
67	#568	—	103	1	no	—
68	#430	—	103	2	no	—
69	#413	D₁₀₃	1	1	yes	characteristic
70	#408	—	103	1	no	—
71	#341	—	103	2	no	—
72	C₂×#41	C₂×D₁₀₃	1	1	yes	characteristic
73	#131	—	206	4	no	—
74	C₂×M₁₆	—	103	2	no	—
75	C₂×M₁₆	—	103	1	no	—
76	C₂×M₁₆	—	206	2	no	—
77	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
78	C₂²×C₈	—	103	1	no	—
79	#108	—	206	4	no	—
80	#101	Dic₁₀₃	1	2	yes	—
81	#100	Dic₁₀₃	1	2	yes	—
82	C₂²⋊C₈	—	206	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₂×Dic₁₀₃	1	1	yes	characteristic
86	C₂×D₄	Dic₂₀₆	1	2	yes	—
87	C₂×D₄	C₂×Dic₁₀₃	1	1	yes	characteristic
88	C₂²×C₄	D₄₁₂	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	103	4	no	—
92	M₁₆	—	206	2	no	—
93	M₁₆	—	206	4	no	—
94	M₁₆	—	206	2	no	—
95	M₁₆	C₂×Dic₁₀₃	1	4	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	206	1	no	—
98	C₂×C₈	—	206	1	no	—
99	C₂×C₈	—	206	2	no	—
100	C₂×C₈	—	206	4	no	—
101	C₂×C₈	—	206	2	no	—
102	C₂³	C₄⋊Dic₁₀₃	1	2	yes	—
103	C₂³	C₂×Dic₂₀₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₄⋊Dic₁₀₃	1	2	yes	—
110	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
112	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
113	C₂×C₄	#13110104	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	412	1	no	—
116	C₈	—	412	1	no	—
117	C₈	—	206	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₄⋊Dic₁₀₃	1	1	yes	characteristic
122	C₂²	#15692998	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	D₄×Dic₁₀₃	1	2	yes	—
125	C₄	#15692991	1	1	yes	characteristic
126	C₄	#15693005	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#20267666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268558

C₂

yes

maximal, characteristic

#20267840

C₂

yes

maximal, complemented

#20267758

C₂

yes

maximal, characteristic

#20267555

C₂

yes

maximal, complemented

#413×C₁₀₃

C₂

yes

maximal, characteristic

#15693028

—

C₂×#13110089

—

C₂×#13110089

C₂²

yes

characteristic

C₂×#13110089

C₂²

yes

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

#15692957

—

C₂²⋊(C₁₀₃⋊C₈)

—

C₂×#41×C₁₀₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₃

C₂²

yes

—

#101×C₁₀₃

C₄

yes

—

#100×C₁₀₃

C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

#41×C₁₀₃

—

#41×C₁₀₃

—

C₂×Q₈×C₁₀₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₃

Q₈

yes

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

—

C₂³×C₁₀₃

C₄⋊C₄

yes

—

C₂³×C₁₀₃

C₂×Q₈

yes

characteristic

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

yes

C₄×C₁₀₃

yes

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#385

yes

characteristic

#1858

—

103

maximal, Sylow

C₁₀₃

#1858

yes

characteristic, complemented, Sylow

#568

—

103

—

#430

—

103

—

#413

D₁₀₃

yes

characteristic

#408

—

103

—

#341

—

103

—

C₂×#41

C₂×D₁₀₃

yes

characteristic

#131

—

206

—

C₂×M₁₆

—

103

—

C₂×M₁₆

—

103

—

C₂×M₁₆

—

206

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

C₂²×C₈

—

103

—

#108

—

206

—

#101

Dic₁₀₃

yes

—

#100

Dic₁₀₃

yes

—

C₂²⋊C₈

—

206

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₀₃

yes

characteristic

C₂×D₄

Dic₂₀₆

yes

—

C₂×D₄

C₂×Dic₁₀₃

yes

characteristic

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

103

—

M₁₆

—

206

—

M₁₆

—

206

—

M₁₆

—

206

—

M₁₆

C₂×Dic₁₀₃

yes

—

C₂×C₈

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

100

C₂×C₈

—

206

—

101

C₂×C₈

—

206

—

102

C₂³

C₄⋊Dic₁₀₃

yes

—

103

C₂³

C₂×Dic₂₀₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊Dic₁₀₃

yes

—

110

C₂×C₄

C₂²×Dic₁₀₃

yes

characteristic

111

C₂×C₄

D₄×D₁₀₃

yes

—

112

C₂×C₄

C₂×D₄₁₂

yes

characteristic

113

C₂×C₄

#13110104

yes

—

114

C₈

—

115

C₈

—

412

—

116

C₈

—

412

—

117

C₈

—

206

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₄⋊Dic₁₀₃

yes

characteristic

122

C₂²

#15692998

yes

—

123

C₄

—

124

C₄

D₄×Dic₁₀₃

yes

—

125

C₄

#15692991

yes

characteristic

126

C₄

#15693005

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20267666

yes

characteristic

	subgroup	G/H
center	C₄	#15692991
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#413×C₁₀₃	C₂
socle	C₂×C₁₀₃	#385
prime radical	C₂×#41×C₁₀₃	C₂²
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#413	D₁₀₃
hypocenter	C₁₀₃	#1858
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#15692991

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#413×C₁₀₃

C₂

socle

C₂×C₁₀₃

#385

prime radical

C₂×#41×C₁₀₃

C₂²

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #20267840	C₂	semidirect
4. #20267555	C₂	semidirect
66. C₁₀₃	#1858	semidirect

H ≅ G/N

type

2. #20267840

C₂

semidirect

4. #20267555

C₂

semidirect

66. C₁₀₃

#1858

semidirect

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

subgroup	index
#20268558	2
#20267840	2
#20267758	2
#20267555	2
#413×C₁₀₃	2
#1858	103

subgroup

index

#413×C₁₀₃

103

Subgroups — #27195366

Conjugacy classes of subgroups

Complements

Maximal subgroups