Group #27194821 — subgroups

Subgroups — #27194821

Order 13184.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20268196	C₂	1	1	yes	maximal, characteristic
2	#20268196	C₂	1	1	yes	maximal, characteristic
3	#20268189	C₂	1	1	yes	maximal, characteristic
4	#20268171	C₂	1	1	yes	maximal, characteristic
5	#20268142	C₂	1	1	yes	maximal, characteristic
6	#20268142	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₀₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₁₂	—	2	2	no	—
9	#15693021	C₂²	1	1	yes	characteristic
10	#15693021	C₂²	1	1	yes	characteristic
11	#15693021	C₂²	1	1	yes	characteristic
12	#15693021	C₂²	1	1	yes	characteristic
13	#15692955	—	2	2	no	—
14	#15692955	—	2	2	no	—
15	#15692953	—	2	2	no	—
16	#15692946	—	2	2	no	—
17	#15692946	—	2	2	no	—
18	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₀₆	—	2	1	no	—
22	C₂×Dic₂₀₆	—	2	1	no	—
23	C₂×Dic₂₀₆	—	2	1	no	—
24	C₂×Dic₂₀₆	—	2	1	no	—
25	Dic₄₁₂	—	4	4	no	—
26	C₄⋊Dic₁₀₃	—	2	1	no	—
27	C₄⋊Dic₁₀₃	—	2	1	no	—
28	C₄⋊Dic₁₀₃	—	2	1	no	—
29	C₄⋊Dic₁₀₃	—	2	1	no	—
30	C₄⋊Dic₁₀₃	—	2	1	no	—
31	C₄⋊Dic₁₀₃	—	2	1	no	—
32	C₄⋊Dic₁₀₃	—	2	1	no	—
33	C₄⋊Dic₁₀₃	—	2	1	no	—
34	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
35	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
36	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
37	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₃	—	4	1	no	—
42	C₂×Dic₁₀₃	—	4	1	no	—
43	C₂×Dic₁₀₃	—	4	1	no	—
44	C₂×Dic₁₀₃	—	4	1	no	—
45	Dic₂₀₆	—	4	2	no	—
46	Dic₂₀₆	—	4	2	no	—
47	Dic₂₀₆	—	4	2	no	—
48	Dic₂₀₆	—	4	2	no	—
49	C₈×C₁₀₃	—	2	2	no	—
50	C₈×C₁₀₃	—	2	2	no	—
51	C₈×C₁₀₃	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₀₃	#127	1	1	yes	characteristic
53	C₄×C₁₀₃	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₀₃	#137	1	2	yes	—
56	Dic₁₀₃	—	8	1	no	—
57	Dic₁₀₃	—	8	1	no	—
58	Dic₁₀₃	—	8	1	no	—
59	Dic₁₀₃	—	8	1	no	—
60	C₂×C₁₀₃	#494	1	1	yes	characteristic
61	C₂×C₁₀₃	#496	1	1	yes	characteristic
62	C₂×C₁₀₃	#497	1	1	yes	characteristic
63	#1723	—	103	1	no	maximal, Sylow
64	C₁₀₃	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	103	1	no	—
66	#500	—	103	1	no	—
67	#498	—	103	1	no	—
68	#494	—	103	1	no	—
69	#487	—	103	1	no	—
70	#487	—	103	1	no	—
71	#322	D₁₀₃	1	1	yes	characteristic
72	C₂×Q₁₆	—	206	2	no	—
73	C₄⋊Q₈	—	103	1	no	—
74	C₄⋊Q₈	—	103	1	no	—
75	C₄⋊Q₈	—	103	1	no	—
76	C₄⋊Q₈	—	103	1	no	—
77	#107	—	206	2	no	—
78	#107	—	206	2	no	—
79	#106	—	206	2	no	—
80	#103	—	206	2	no	—
81	#103	—	206	2	no	—
82	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
85	C₂×Q₈	—	206	1	no	—
86	C₂×Q₈	—	206	1	no	—
87	C₂×Q₈	—	206	1	no	—
88	C₂×Q₈	—	206	1	no	—
89	Q₁₆	—	412	4	no	—
90	C₂×C₈	D₄₁₂	1	2	yes	—
91	C₂×C₈	D₄₁₂	1	2	yes	—
92	C₂×C₈	D₄₁₂	1	2	yes	—
93	C₄⋊C₄	—	206	1	no	—
94	C₄⋊C₄	—	206	1	no	—
95	C₄⋊C₄	—	206	1	no	—
96	C₄⋊C₄	—	206	1	no	—
97	C₄⋊C₄	—	206	1	no	—
98	C₄⋊C₄	—	206	1	no	—
99	C₄⋊C₄	—	206	1	no	—
100	C₄⋊C₄	—	206	1	no	—
101	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
102	Q₈	—	412	2	no	—
103	Q₈	—	412	2	no	—
104	Q₈	—	412	2	no	—
105	Q₈	—	412	2	no	—
106	C₂×C₄	—	412	1	no	—
107	C₂×C₄	—	412	1	no	—
108	C₂×C₄	—	412	1	no	—
109	C₂×C₄	—	412	1	no	—
110	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₄₁₂	1	4	yes	complemented
116	C₂²	#15693018	1	1	yes	characteristic
117	C₄	—	824	1	no	—
118	C₄	—	824	1	no	—
119	C₄	—	824	1	no	—
120	C₄	—	824	1	no	—
121	C₄	C₂×Dic₄₁₂	1	2	yes	—
122	C₄	#15693049	1	2	yes	—
123	C₄	#15693054	1	2	yes	—
124	C₂	#20268171	1	1	yes	characteristic
125	C₂	#20268180	1	1	yes	characteristic
126	C₂	#20268185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268196

C₂

yes

maximal, characteristic

#20268196

C₂

yes

maximal, characteristic

#20268189

C₂

yes

maximal, characteristic

#20268171

C₂

yes

maximal, characteristic

#20268142

C₂

yes

maximal, characteristic

#20268142

C₂

yes

maximal, characteristic

#322×C₁₀₃

C₂

yes

maximal, characteristic

C₂×Dic₄₁₂

—

#15693021

C₂²

yes

characteristic

#15693021

C₂²

yes

characteristic

#15693021

C₂²

yes

characteristic

#15693021

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

Dic₄₁₂

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₈×C₁₀₃

Q₁₆

yes

complemented

C₂²×C₁₀₃

#127

yes

characteristic

C₄×C₁₀₃

C₂×Q₁₆

yes

—

C₄×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1723

yes

characteristic, complemented, Sylow

#500

—

103

—

#500

—

103

—

#498

—

103

—

#494

—

103

—

#487

—

103

—

#487

—

103

—

#322

D₁₀₃

yes

characteristic

C₂×Q₁₆

—

206

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

#107

—

206

—

#107

—

206

—

#106

—

206

—

#103

—

206

—

#103

—

206

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

Q₁₆

—

412

—

C₂×C₈

D₄₁₂

yes

—

C₂×C₈

D₄₁₂

yes

—

C₂×C₈

D₄₁₂

yes

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

100

C₄⋊C₄

—

206

—

101

C₄²

C₂²×D₁₀₃

yes

characteristic

102

Q₈

—

412

—

103

Q₈

—

412

—

104

Q₈

—

412

—

105

Q₈

—

412

—

106

C₂×C₄

—

412

—

107

C₂×C₄

—

412

—

108

C₂×C₄

—

412

—

109

C₂×C₄

—

412

—

110

C₂×C₄

C₂×D₄₁₂

yes

characteristic

111

C₂×C₄

C₂×D₄₁₂

yes

characteristic

112

C₂×C₄

C₂×D₄₁₂

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₄₁₂

yes

complemented

116

C₂²

#15693018

yes

characteristic

117

C₄

—

824

—

118

C₄

—

824

—

119

C₄

—

824

—

120

C₄

—

824

—

121

C₄

C₂×Dic₄₁₂

yes

—

122

C₄

#15693049

yes

—

123

C₄

#15693054

yes

—

124

C₂

#20268171

yes

characteristic

125

C₂

#20268180

yes

characteristic

126

C₂

#20268185

yes

characteristic

	subgroup	G/H
center	C₂²	#15693018
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#322×C₁₀₃	C₂
socle, prime radical	C₂²×C₁₀₃	#127
hypercenter	#322	D₁₀₃
hypocenter	C₁₀₃	#1723
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#15693018

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#322×C₁₀₃

C₂

C₂²×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₀₃	Q₁₆	semidirect
64. C₁₀₃	#1723	semidirect
115. C₈	Dic₄₁₂	semidirect

H ≅ G/N

type

51. C₈×C₁₀₃

Q₁₆

semidirect

64. C₁₀₃

#1723

semidirect

115. C₈

Dic₄₁₂

semidirect

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

subgroup	index
#20268196	2
#20268189	2
#20268171	2
#20268142	2
#322×C₁₀₃	2
#1723	103

subgroup

index

#322×C₁₀₃

103

Subgroups — #27194821

Conjugacy classes of subgroups

Complements

Maximal subgroups