Group #27196564 — subgroups

Subgroups — #27196564

Order 13184.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15692946	C₂	1	2	yes	maximal
2	#20267697	C₂	1	1	yes	maximal, characteristic
3	#20267657	C₂	1	1	yes	maximal, characteristic
4	#20267554	C₂	1	2	yes	maximal
5	#336×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
9	#15692946	—	2	4	no	—
10	C₁₀₃⋊#95	—	2	2	no	—
11	C₁₀₃⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₀₆	—	2	2	no	—
15	C₂×Dic₂₀₆	—	2	2	no	—
16	C₂×Dic₂₀₆	—	2	1	no	—
17	C₂×Dic₂₀₆	D₄	1	2	yes	—
18	C₂²×Dic₁₀₃	—	2	1	no	—
19	C₂²×Dic₁₀₃	—	2	2	no	—
20	C₂²×Dic₁₀₃	—	2	1	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	—	2	2	no	—
23	C₄⋊Dic₁₀₃	—	2	2	no	—
24	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₃	—	2	2	no	—
27	C₂×C₈×C₁₀₃	—	2	2	no	—
28	C₂×C₈×C₁₀₃	—	2	2	no	—
29	C₄²×C₁₀₃	—	2	2	no	—
30	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	—	2	2	no	—
32	C₂×C₄×C₁₀₃	—	2	2	no	—
33	C₂×C₄×C₁₀₃	—	2	2	no	—
34	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₃	#41	1	2	yes	—
36	C₂×C₄×C₁₀₃	#41	1	2	yes	—
37	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
38	C₂×Dic₁₀₃	—	2	2	no	—
39	C₂×Dic₁₀₃	—	4	1	no	—
40	C₂×Dic₁₀₃	—	4	1	no	—
41	C₂×Dic₁₀₃	—	4	2	no	—
42	C₂×Dic₁₀₃	—	4	1	no	—
43	C₂×Dic₁₀₃	—	4	2	no	—
44	C₂×Dic₁₀₃	—	4	2	no	—
45	C₂×Dic₁₀₃	—	4	2	no	—
46	Dic₂₀₆	—	2	4	no	—
47	Dic₂₀₆	—	4	2	no	—
48	C₈×C₁₀₃	—	4	2	no	—
49	C₂²×C₁₀₃	#121	1	1	yes	characteristic
50	C₂²×C₁₀₃	#123	1	1	yes	characteristic
51	C₂²×C₁₀₃	#124	1	1	yes	characteristic
52	C₂²×C₁₀₃	#135	1	2	yes	—
53	C₂²×C₁₀₃	#137	1	2	yes	—
54	C₄×C₁₀₃	—	4	2	no	—
55	C₄×C₁₀₃	#123	1	2	yes	—
56	C₄×C₁₀₃	#126	1	2	yes	—
57	Dic₁₀₃	—	4	2	no	—
58	Dic₁₀₃	—	8	2	no	—
59	Dic₁₀₃	—	8	1	no	—
60	C₂×C₁₀₃	#397	1	1	yes	characteristic
61	C₂×C₁₀₃	#464	1	2	yes	—
62	C₂×C₁₀₃	#485	1	2	yes	—
63	C₂×C₁₀₃	#488	1	1	yes	characteristic
64	C₂×C₁₀₃	#491	1	1	yes	characteristic
65	#2074	—	103	1	no	maximal, Sylow
66	C₁₀₃	#2074	1	1	yes	characteristic, complemented, Sylow
67	C₂×#103	—	103	2	no	—
68	#391	—	103	1	no	—
69	#382	—	103	1	no	—
70	#340	—	103	2	no	—
71	#336	D₁₀₃	1	1	yes	characteristic
72	C₂²×Q₈	—	103	1	no	—
73	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	103	1	no	—
75	C₂×C₄⋊C₄	—	103	2	no	—
76	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
77	#103	—	206	4	no	—
78	#95	—	206	2	no	—
79	#95	—	206	2	no	—
80	C₂×Q₈	—	103	2	no	—
81	C₂×Q₈	—	206	2	no	—
82	C₂×Q₈	—	206	2	no	—
83	C₂×Q₈	—	206	1	no	—
84	C₂²×C₄	—	206	2	no	—
85	C₂²×C₄	—	206	1	no	—
86	C₂²×C₄	—	206	1	no	—
87	C₂²×C₄	D₄₁₂	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	206	2	no	—
93	C₄⋊C₄	—	206	2	no	—
94	C₄⋊C₄	—	206	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
97	Q₈	—	412	2	no	—
98	Q₈	—	206	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	412	2	no	—
103	C₂×C₄	—	412	1	no	—
104	C₂×C₄	—	412	2	no	—
105	C₂×C₄	—	412	2	no	—
106	C₂×C₄	—	412	2	no	—
107	C₂×C₄	—	412	1	no	—
108	C₂×C₄	—	412	1	no	—
109	C₂×C₄	—	206	2	no	—
110	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
111	C₂×C₄	#13110104	1	2	yes	—
112	C₂×C₄	#13110105	1	2	yes	—
113	C₂×C₄	#13110106	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#15692996	1	1	yes	characteristic
116	C₂²	#15693005	1	1	yes	characteristic
117	C₂²	#15693009	1	1	yes	characteristic
118	C₂²	#15693043	1	2	yes	—
119	C₂²	#15693054	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	824	1	no	—
122	C₄	—	824	2	no	—
123	C₄	—	412	2	no	—
124	C₄	#15693003	1	2	yes	—
125	C₄	#15693016	1	2	yes	—
126	C₂	#20267718	1	1	yes	characteristic
127	C₂	#20268014	1	2	yes	—
128	C₂	#20268133	1	2	yes	—
129	C₂	#20268146	1	1	yes	characteristic
130	C₂	#20268160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692946

C₂

yes

maximal

#20267697

C₂

yes

maximal, characteristic

#20267657

C₂

yes

maximal, characteristic

#20267554

C₂

yes

maximal

#336×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

#15692946

—

C₁₀₃⋊#95

—

C₁₀₃⋊#95

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

D₄

yes

—

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₂²×C₁₀₃

#137

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#123

yes

—

C₄×C₁₀₃

#126

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#397

yes

characteristic

C₂×C₁₀₃

#464

yes

—

C₂×C₁₀₃

#485

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#2074

yes

characteristic, complemented, Sylow

C₂×#103

—

103

—

#391

—

103

—

#382

—

103

—

#340

—

103

—

#336

D₁₀₃

yes

characteristic

C₂²×Q₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#103

—

206

—

#95

—

206

—

#95

—

206

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

—

C₂³

C₂×D₄₁₂

yes

characteristic

Q₈

—

412

—

Q₈

—

206

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

412

—

103

C₂×C₄

—

412

—

104

C₂×C₄

—

412

—

105

C₂×C₄

—

412

—

106

C₂×C₄

—

412

—

107

C₂×C₄

—

412

—

108

C₂×C₄

—

412

—

109

C₂×C₄

—

206

—

110

C₂×C₄

D₄×D₁₀₃

yes

characteristic

111

C₂×C₄

#13110104

yes

—

112

C₂×C₄

#13110105

yes

—

113

C₂×C₄

#13110106

yes

characteristic

114

C₈

—

115

C₂²

#15692996

yes

characteristic

116

C₂²

#15693005

yes

characteristic

117

C₂²

#15693009

yes

characteristic

118

C₂²

#15693043

yes

—

119

C₂²

#15693054

yes

—

120

C₄

—

121

C₄

—

824

—

122

C₄

—

824

—

123

C₄

—

412

—

124

C₄

#15693003

yes

—

125

C₄

#15693016

yes

—

126

C₂

#20267718

yes

characteristic

127

C₂

#20268014

yes

—

128

C₂

#20268133

yes

—

129

C₂

#20268146

yes

characteristic

130

C₂

#20268160

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#336×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	#336	D₁₀₃
hypocenter	C₁₀₃	#2074
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#336×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
66. C₁₀₃	#2074	semidirect

H ≅ G/N

type

66. C₁₀₃

#2074

semidirect

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

subgroup	index
C₂×#15692946	2
#20267697	2
#20267657	2
#20267554	2
#336×C₁₀₃	2
#2074	103

subgroup

index

#336×C₁₀₃

103

Subgroups — #27196564

Conjugacy classes of subgroups

Complements

Maximal subgroups