Group #27196696 — subgroups

Subgroups — #27196696

Order 13184.

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

normal subgroups only

This group has 210 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	#20267819	C₂	1	2	yes	maximal, complemented
2	#20267807	C₂	1	2	yes	maximal
3	#20267759	C₂	1	1	yes	maximal, characteristic
4	#20267548	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13110105	C₂²	1	1	yes	characteristic
7	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
8	#15692980	C₂²	1	2	yes	—
9	Dic₁₀₃⋊C₈	—	2	4	no	—
10	#15692947	—	2	4	no	—
11	(C₂×D₁₀₃)⋊C₈	—	2	2	no	—
12	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
13	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
14	#108×C₁₀₃	—	2	2	no	—
15	#13110105	—	2	2	no	—
16	#13110105	—	2	2	no	—
17	C₂×Dic₂₀₆	Q₈	1	1	yes	characteristic
18	C₂×D₄₁₂	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₀₃	—	2	1	no	—
20	C₂×C₄×D₁₀₃	—	2	1	no	—
21	C₂×C₁₀₃⋊C₈	—	2	1	no	—
22	C₂×C₁₀₃⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₀₃	—	2	2	no	—
24	C₂²⋊Dic₁₀₃	—	2	2	no	—
25	C₄×Dic₁₀₃	D₄	1	4	yes	—
26	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₀₃	—	2	2	no	—
28	M₁₆×C₁₀₃	—	2	2	no	—
29	M₁₆×C₁₀₃	—	2	2	no	—
30	C₂×C₈×C₁₀₃	—	2	2	no	—
31	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
32	C₂³×C₁₀₃	#41	1	1	yes	characteristic
33	C₂²×D₁₀₃	—	2	1	no	—
34	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₀₃	#41	1	2	yes	—
38	C₂²⋊D₁₀₃	—	4	2	no	—
39	C₂×Dic₁₀₃	—	2	1	no	—
40	C₂×Dic₁₀₃	—	2	4	no	—
41	D₄₁₂	—	2	2	no	—
42	C₄×D₁₀₃	—	4	1	no	—
43	C₄×D₁₀₃	—	4	1	no	—
44	Dic₂₀₆	—	2	2	no	—
45	C₈×C₁₀₃	—	2	2	no	—
46	C₈×C₁₀₃	—	4	2	no	—
47	C₁₀₃⋊C₈	—	4	1	no	—
48	C₂²×C₁₀₃	—	4	1	no	—
49	C₂²×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₃	#123	1	2	yes	—
51	C₂×D₁₀₃	—	4	2	no	—
52	C₂×D₁₀₃	—	4	1	no	—
53	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
54	C₄×C₁₀₃	#123	1	1	yes	characteristic
55	C₄×C₁₀₃	#127	1	1	yes	characteristic
56	Dic₁₀₃	—	4	4	no	—
57	Dic₁₀₃	—	4	1	no	—
58	C₂×C₁₀₃	—	2	2	no	—
59	C₂×C₁₀₃	—	2	1	no	—
60	C₂×C₁₀₃	#399	1	1	yes	characteristic
61	D₁₀₃	—	8	1	no	—
62	#2098	—	103	1	no	maximal, Sylow
63	C₁₀₃	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	D₁₀₃	1	1	yes	characteristic, complemented
65	#424	—	103	2	no	—
66	#421	—	103	2	no	—
67	#408	—	103	1	no	—
68	H(3,ℤ/4)	—	103	1	no	—
69	C₂×#41	—	103	1	no	—
70	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
71	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
72	C₂²×C₈	—	103	1	no	—
73	#117	—	103	2	no	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	206	4	no	—
76	#104	—	206	4	no	—
77	C₂²⋊C₈	—	206	2	no	—
78	#41	—	206	2	no	—
79	#41	—	206	2	no	—
80	C₂×Q₈	—	103	1	no	—
81	C₂×D₄	—	103	1	no	—
82	C₂×D₄	—	206	1	no	—
83	C₂²×C₄	—	206	1	no	—
84	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	206	2	no	—
90	C₂×C₈	—	206	1	no	—
91	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
92	C₄⋊C₄	—	206	2	no	—
93	C₂²⋊C₄	—	206	2	no	—
94	C₄²	—	103	4	no	—
95	C₂³	—	206	1	no	—
96	C₂³	#13110106	1	1	yes	characteristic
97	Q₈	—	206	2	no	—
98	D₄	—	412	2	no	—
99	D₄	—	206	2	no	—
100	C₂×C₄	—	412	1	no	—
101	C₂×C₄	—	412	1	no	—
102	C₂×C₄	—	206	1	no	—
103	C₂×C₄	—	206	4	no	—
104	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₃	1	1	yes	characteristic
107	C₂×C₄	#13110105	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	412	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	412	2	no	—
113	C₂²	—	412	1	no	—
114	C₂²	#15692997	1	1	yes	characteristic
115	C₂²	#15693006	1	2	yes	—
116	C₄	—	412	4	no	—
117	C₄	—	412	1	no	—
118	C₄	#15692999	1	2	yes	—
119	C₄	#15693004	1	1	yes	characteristic
120	C₄	#15693017	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	824	1	no	—
124	C₂	#20267726	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267819

C₂

yes

maximal, complemented

#20267807

C₂

yes

maximal

#20267759

C₂

yes

maximal, characteristic

#20267548

C₂

yes

maximal, characteristic, complemented

#430×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110105

C₂²

yes

characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

#15692980

C₂²

yes

—

Dic₁₀₃⋊C₈

—

#15692947

—

(C₂×D₁₀₃)⋊C₈

—

C₂×M₁₆×C₁₀₃

C₂²

yes

—

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

#108×C₁₀₃

—

#13110105

—

#13110105

—

C₂×Dic₂₀₆

Q₈

yes

characteristic

C₂×D₄₁₂

Q₈

yes

characteristic

C₂×C₂²⋊D₁₀₃

—

C₂×C₄×D₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

#41

yes

characteristic

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₃

#123

yes

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

yes

C₄×C₁₀₃

yes

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#399

yes

characteristic

D₁₀₃

—

#2098

—

103

maximal, Sylow

C₁₀₃

#2098

yes

characteristic, complemented, Sylow

#430

D₁₀₃

yes

characteristic, complemented

#424

—

103

—

#421

—

103

—

#408

—

103

—

H(3,ℤ/4)

—

103

—

C₂×#41

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

—

103

—

#117

—

103

—

#108

—

C₄⋊C₈

—

206

—

#104

—

206

—

C₂²⋊C₈

—

206

—

#41

—

206

—

#41

—

206

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₄⋊C₄

—

206

—

C₂²⋊C₄

—

206

—

C₄²

—

103

—

C₂³

—

206

—

C₂³

#13110106

yes

characteristic

Q₈

—

206

—

D₄

—

412

—

D₄

—

206

—

100

C₂×C₄

—

412

—

101

C₂×C₄

—

412

—

102

C₂×C₄

—

206

—

103

C₂×C₄

—

206

—

104

C₂×C₄

D₄×D₁₀₃

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₃

yes

characteristic

107

C₂×C₄

#13110105

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

412

—

111

C₂²

—

112

C₂²

—

412

—

113

C₂²

—

412

—

114

C₂²

#15692997

yes

characteristic

115

C₂²

#15693006

yes

—

116

C₄

—

412

—

117

C₄

—

412

—

118

C₄

#15692999

yes

—

119

C₄

#15693004

yes

characteristic

120

C₄

#15693017

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

824

—

124

C₂

#20267726

yes

characteristic

	subgroup	G/H
center	C₄	#15693017
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#430×C₁₀₃	C₂
socle	C₂×C₁₀₃	#399
prime radical	C₂×#13110105	C₂²
hypercenter	#430	D₁₀₃
hypocenter	C₁₀₃	#2098
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#15693017

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#430×C₁₀₃

C₂

C₂×C₁₀₃

C₂²

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #20267819	C₂	semidirect
4. #20267548	C₂	semidirect
5. #430×C₁₀₃	C₂	semidirect
63. C₁₀₃	#2098	semidirect
64. #430	D₁₀₃	semidirect

H ≅ G/N

type

1. #20267819

C₂

semidirect

4. #20267548

C₂

semidirect

5. #430×C₁₀₃

C₂

semidirect

63. C₁₀₃

#2098

semidirect

64. #430

D₁₀₃

semidirect

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

subgroup	index
#20267819	2
#20267807	2
#20267759	2
#20267548	2
#430×C₁₀₃	2
#2098	103

subgroup

index

#430×C₁₀₃

103

Subgroups — #27196696

Conjugacy classes of subgroups

Complements

Maximal subgroups