Group #27196590 — subgroups

Subgroups — #27196590

Order 13184.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20267790	C₂	1	2	yes	maximal
2	#20267757	C₂	1	1	yes	maximal, characteristic
3	#20267560	C₂	1	2	yes	maximal, complemented
4	#20267556	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13110105	C₂²	1	1	yes	characteristic
7	C₂×#13110089	C₂²	1	2	yes	—
8	C₂×#13110089	C₂²	1	1	yes	characteristic
9	#15692944	—	2	2	no	—
10	#15692941	—	2	2	no	—
11	(C₂×D₁₀₃)⋊C₈	—	2	2	no	—
12	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
14	#106×C₁₀₃	—	2	2	no	—
15	#13110105	—	2	2	no	—
16	#13110105	—	2	2	no	—
17	C₂×Dic₂₀₆	D₄	1	1	yes	characteristic
18	C₂×D₄₁₂	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₀₃	—	2	1	no	—
20	C₂×C₄×D₁₀₃	—	2	1	no	—
21	#13110089	—	2	2	no	—
22	#13110089	—	2	2	no	—
23	#13110089	—	2	2	no	—
24	C₂×C₁₀₃⋊C₈	—	2	1	no	—
25	C₂×C₁₀₃⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₀₃	—	2	2	no	—
27	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₃	—	2	2	no	—
29	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₁₀₃	—	2	2	no	—
31	C₄⋊C₄×C₁₀₃	—	2	2	no	—
32	C₂³×C₁₀₃	#41	1	1	yes	characteristic
33	C₂²×D₁₀₃	—	2	1	no	—
34	C₂×C₄×C₁₀₃	—	4	2	no	—
35	C₂×C₄×C₁₀₃	—	8	2	no	—
36	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	#41	1	2	yes	—
39	C₂×C₄×C₁₀₃	#41	1	2	yes	—
40	C₂²⋊D₁₀₃	—	4	2	no	—
41	C₂×Dic₁₀₃	—	2	1	no	—
42	D₄₁₂	—	2	2	no	—
43	C₄×D₁₀₃	—	4	1	no	—
44	C₄×D₁₀₃	—	4	1	no	—
45	Dic₂₀₆	—	2	2	no	—
46	C₈×C₁₀₃	—	2	2	no	—
47	C₁₀₃⋊C₈	—	4	2	no	—
48	C₁₀₃⋊C₈	—	4	1	no	—
49	C₂²×C₁₀₃	—	2	2	no	—
50	C₂²×C₁₀₃	#121	1	1	yes	characteristic
51	C₂²×C₁₀₃	#126	1	2	yes	—
52	C₂×D₁₀₃	—	4	1	no	—
53	C₂×D₁₀₃	—	4	2	no	—
54	C₄×C₁₀₃	—	8	2	no	—
55	C₄×C₁₀₃	#123	1	1	yes	characteristic
56	C₄×C₁₀₃	#123	1	2	yes	—
57	C₄×C₁₀₃	#124	1	1	yes	characteristic
58	Dic₁₀₃	—	4	1	no	—
59	C₂×C₁₀₃	—	2	2	no	—
60	C₂×C₁₀₃	#397	1	1	yes	characteristic
61	C₂×C₁₀₃	#471	1	2	yes	—
62	D₁₀₃	—	8	1	no	—
63	#2079	—	103	1	no	maximal, Sylow
64	C₁₀₃	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	D₁₀₃	1	1	yes	characteristic, complemented
66	#417	—	103	2	no	—
67	#408	—	103	1	no	—
68	#343	—	103	2	no	—
69	#341	—	103	1	no	—
70	C₂×#41	—	103	1	no	—
71	C₂×M₁₆	—	103	2	no	—
72	C₂×M₁₆	—	103	1	no	—
73	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
75	#106	—	2	2	no	—
76	#103	—	206	2	no	—
77	#102	—	206	2	no	—
78	C₂²⋊C₈	—	206	2	no	—
79	#41	—	206	2	no	—
80	#41	—	206	2	no	—
81	C₂×Q₈	—	103	1	no	—
82	C₂×D₄	—	103	1	no	—
83	C₂×D₄	—	206	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	206	1	no	—
86	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
87	M₁₆	—	206	2	no	—
88	M₁₆	—	206	2	no	—
89	M₁₆	—	206	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	206	2	no	—
92	C₂×C₈	—	206	1	no	—
93	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂³	—	206	1	no	—
97	C₂³	#13110104	1	1	yes	characteristic
98	Q₈	—	206	2	no	—
99	D₄	—	412	2	no	—
100	D₄	—	206	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	412	1	no	—
104	C₂×C₄	—	412	1	no	—
105	C₂×C₄	—	206	1	no	—
106	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
108	C₂×C₄	#13110105	1	2	yes	—
109	C₂×C₄	#13110106	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	412	2	no	—
112	C₈	—	412	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	412	2	no	—
115	C₂²	—	412	1	no	—
116	C₂²	#15692992	1	1	yes	characteristic
117	C₂²	#15693016	1	2	yes	—
118	C₄	—	8	2	no	—
119	C₄	—	412	1	no	—
120	C₄	#15693004	1	1	yes	characteristic
121	C₄	#15693006	1	2	yes	—
122	C₄	#15693010	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	824	1	no	—
125	C₂	#20267717	1	1	yes	characteristic
126	C₂	#20268051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20267790

C₂

yes

maximal

#20267757

C₂

yes

maximal, characteristic

#20267560

C₂

yes

maximal, complemented

#20267556

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110105

C₂²

yes

characteristic

C₂×#13110089

C₂²

yes

—

C₂×#13110089

C₂²

yes

characteristic

#15692944

—

#15692941

—

(C₂×D₁₀₃)⋊C₈

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

#106×C₁₀₃

—

#13110105

—

#13110105

—

C₂×Dic₂₀₆

D₄

yes

characteristic

C₂×D₄₁₂

D₄

yes

characteristic

C₂×C₂²⋊D₁₀₃

—

C₂×C₄×D₁₀₃

—

#13110089

—

#13110089

—

#13110089

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₂³×C₁₀₃

#41

yes

characteristic

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#121

yes

characteristic

C₂²×C₁₀₃

#126

yes

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#123

yes

characteristic

C₄×C₁₀₃

#123

yes

—

C₄×C₁₀₃

#124

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#397

yes

characteristic

C₂×C₁₀₃

#471

yes

—

D₁₀₃

—

#2079

—

103

maximal, Sylow

C₁₀₃

#2079

yes

characteristic, complemented, Sylow

C₂×#106

D₁₀₃

yes

characteristic, complemented

#417

—

103

—

#408

—

103

—

#343

—

103

—

#341

—

103

—

C₂×#41

—

103

—

C₂×M₁₆

—

103

—

C₂×M₁₆

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

#106

—

#103

—

206

—

#102

—

206

—

C₂²⋊C₈

—

206

—

#41

—

206

—

#41

—

206

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

206

—

M₁₆

—

206

—

M₁₆

—

206

—

C₂×C₈

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

206

—

C₂³

#13110104

yes

characteristic

Q₈

—

206

—

D₄

—

412

—

100

D₄

—

206

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

412

—

104

C₂×C₄

—

412

—

105

C₂×C₄

—

206

—

106

C₂×C₄

D₄×D₁₀₃

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

108

C₂×C₄

#13110105

yes

—

109

C₂×C₄

#13110106

yes

—

110

C₈

—

111

C₈

—

412

—

112

C₈

—

412

—

113

C₂²

—

114

C₂²

—

412

—

115

C₂²

—

412

—

116

C₂²

#15692992

yes

characteristic

117

C₂²

#15693016

yes

—

118

C₄

—

119

C₄

—

412

—

120

C₄

#15693004

yes

characteristic

121

C₄

#15693006

yes

—

122

C₄

#15693010

yes

characteristic

123

C₂

—

124

C₂

—

824

—

125

C₂

#20267717

yes

characteristic

126

C₂

#20268051

yes

—

	subgroup	G/H
center	C₂²	#15692992
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×#106×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#121
prime radical	C₂×#13110105	C₂²
hypercenter	C₂×#106	D₁₀₃
hypocenter	C₁₀₃	#2079
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#15692992

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×#106×C₁₀₃

C₂

C₂²×C₁₀₃

C₂²

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #20267560	C₂	semidirect
4. #20267556	C₂	semidirect
5. C₂×#106×C₁₀₃	C₂	semidirect
64. C₁₀₃	#2079	semidirect
65. C₂×#106	D₁₀₃	semidirect

H ≅ G/N

type

3. #20267560

C₂

semidirect

4. #20267556

C₂

semidirect

5. C₂×#106×C₁₀₃

C₂

semidirect

64. C₁₀₃

#2079

semidirect

65. C₂×#106

D₁₀₃

semidirect

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

subgroup	index
#20267790	2
#20267757	2
#20267560	2
#20267556	2
C₂×#106×C₁₀₃	2
#2079	103

subgroup

index

C₂×#106×C₁₀₃

103

Subgroups — #27196590

Conjugacy classes of subgroups

Complements

Maximal subgroups