Group #27194289 — subgroups

Subgroups — #27194289

Order 13184.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20268170	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268081	C₂	1	2	yes	maximal, complemented
3	#20267540	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#555×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110095	—	2	2	no	—
7	#15693021	C₂²	1	1	yes	characteristic
8	#15692952	—	2	2	no	—
9	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
10	#15692942	—	2	2	no	—
11	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
13	#124×C₁₀₃	—	2	2	no	—
14	#121×C₁₀₃	—	2	2	no	—
15	C₄×D₄×C₁₀₃	C₂²	1	2	yes	—
16	#117×C₁₀₃	—	2	2	no	—
17	C₂×Dic₂₀₆	—	2	1	no	—
18	#13110095	—	4	4	no	—
19	C₂×C₁₀₃⋊C₈	—	2	1	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₂×Q₈×C₁₀₃	D₄	1	2	yes	—
23	C₂×D₄×C₁₀₃	—	2	2	no	—
24	C₂×D₄×C₁₀₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₃	—	2	2	no	—
26	C₄⋊C₄×C₁₀₃	—	2	2	no	—
27	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₀₃	—	2	4	no	—
29	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
30	C₄²×C₁₀₃	—	2	2	no	—
31	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂³×C₁₀₃	—	2	2	no	—
33	Q₈×C₁₀₃	—	4	2	no	—
34	D₄×C₁₀₃	—	2	2	no	—
35	D₄×C₁₀₃	—	2	2	no	—
36	D₄×C₁₀₃	—	8	2	no	—
37	C₂×C₄×C₁₀₃	—	2	2	no	—
38	C₂×C₄×C₁₀₃	—	2	2	no	—
39	C₂×C₄×C₁₀₃	—	4	2	no	—
40	C₂×C₄×C₁₀₃	—	4	2	no	—
41	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
43	C₂×Dic₁₀₃	—	4	1	no	—
44	Dic₂₀₆	—	4	2	no	—
45	C₁₀₃⋊C₈	—	2	2	no	—
46	C₁₀₃⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₃	—	4	2	no	—
48	C₂²×C₁₀₃	—	4	2	no	—
49	C₂²×C₁₀₃	—	4	2	no	—
50	C₂²×C₁₀₃	#120	1	1	yes	characteristic
51	C₄×C₁₀₃	—	2	1	no	—
52	C₄×C₁₀₃	—	4	2	no	—
53	C₄×C₁₀₃	—	4	2	no	—
54	C₄×C₁₀₃	#135	1	2	yes	—
55	C₄×C₁₀₃	#137	1	2	yes	—
56	Dic₁₀₃	—	8	1	no	—
57	C₂×C₁₀₃	—	8	2	no	—
58	C₂×C₁₀₃	#452	1	2	yes	—
59	C₂×C₁₀₃	#453	1	1	yes	characteristic
60	#1634	—	103	1	no	maximal, Sylow
61	C₁₀₃	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	D₁₀₃	1	1	yes	characteristic
63	#494	—	103	1	no	—
64	#476	—	103	2	no	—
65	#333	—	103	1	no	—
66	D₄⋊C₈	—	103	2	no	—
67	C₂×Q₁₆	—	206	2	no	—
68	C₄⋊Q₈	—	103	1	no	—
69	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₀₃	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	206	2	no	—
75	C₄⋊C₈	—	103	2	no	—
76	#102	—	206	2	no	—
77	C₄×C₈	—	103	1	no	—
78	C₂×Q₈	—	206	1	no	—
79	C₂×Q₈	C₂²⋊D₁₀₃	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₀₃	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	412	4	no	—
84	C₂×C₈	—	206	2	no	—
85	C₂×C₈	—	206	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	206	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₀₃	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	412	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	412	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
106	C₈	—	412	2	no	—
107	C₈	—	206	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#15692990	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	824	1	no	—
116	C₄	#15693041	1	2	yes	—
117	C₄	#15693051	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#20267953	1	2	yes	—
120	C₂	#20267959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268170

C₂

yes

maximal, characteristic, complemented

#20268081

C₂

yes

maximal, complemented

#20267540

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal, complemented

#555×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110095

—

#15693021

C₂²

yes

characteristic

#15692952

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

#15692942

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

#124×C₁₀₃

—

#121×C₁₀₃

—

C₄×D₄×C₁₀₃

C₂²

yes

—

#117×C₁₀₃

—

C₂×Dic₂₀₆

—

#13110095

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₂×Q₈×C₁₀₃

D₄

yes

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

D₄

yes

—

C₂²⋊C₄×C₁₀₃

—

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

—

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#135

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#452

yes

—

C₂×C₁₀₃

#453

yes

characteristic

#1634

—

103

maximal, Sylow

C₁₀₃

#1634

yes

characteristic, complemented, Sylow

#555

D₁₀₃

yes

characteristic

#494

—

103

—

#476

—

103

—

#333

—

103

—

D₄⋊C₈

—

103

—

C₂×Q₁₆

—

206

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₀₃

yes

—

#117

—

#106

—

206

—

C₄⋊C₈

—

103

—

#102

—

206

—

C₄×C₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

C₂²⋊D₁₀₃

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

—

Q₁₆

—

412

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₄⋊C₄

C₂²⋊D₁₀₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

412

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

412

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

106

C₈

—

412

—

107

C₈

—

206

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#15692990

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

824

—

116

C₄

#15693041

yes

—

117

C₄

#15693051

yes

—

118

C₂

—

119

C₂

#20267953

yes

—

120

C₂

#20267959

yes

characteristic

	subgroup	G/H
center	C₂²	#15692990
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting, prime radical	#555×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
hypercenter	#555	D₁₀₃
hypocenter	C₁₀₃	#1634
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#15692990

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting, prime radical

#555×C₁₀₃

C₂

C₂²×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #20268170	C₂	semidirect
2. #20268081	C₂	semidirect
3. #20267540	C₂	semidirect
4. D₄⋊(C₁₀₃⋊C₈)	C₂	semidirect
61. C₁₀₃	#1634	semidirect

H ≅ G/N

type

1. #20268170

C₂

semidirect

2. #20268081

C₂

semidirect

3. #20267540

C₂

semidirect

4. D₄⋊(C₁₀₃⋊C₈)

C₂

semidirect

61. C₁₀₃

#1634

semidirect

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

subgroup	index
#20268170	2
#20268081	2
#20267540	2
D₄⋊(C₁₀₃⋊C₈)	2
#555×C₁₀₃	2
#1634	103

subgroup

index

#20268170

#20268081

#20267540

D₄⋊(C₁₀₃⋊C₈)

#555×C₁₀₃

#1634

103

Subgroups — #27194289

Conjugacy classes of subgroups

Complements

Maximal subgroups