Group #27194739 — subgroups

Subgroups — #27194739

Order 13184.

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

normal subgroups only

This group has 180 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	#20268194	C₂	1	1	yes	maximal, characteristic
2	#20268189	C₂	1	1	yes	maximal, characteristic
3	#20268170	C₂	1	1	yes	maximal, characteristic
4	#20268143	C₂	1	1	yes	maximal, characteristic
5	#20268141	C₂	1	1	yes	maximal, characteristic
6	C₈×C₁₀₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13110095	—	2	2	no	—
9	#15693021	C₂²	1	1	yes	characteristic
10	#15693021	C₂²	1	1	yes	characteristic
11	#15692954	—	2	2	no	—
12	#15692953	—	2	2	no	—
13	#15692945	—	2	2	no	—
14	#15692944	—	2	2	no	—
15	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
19	#106×C₁₀₃	—	2	2	no	—
20	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₀₆	—	2	1	no	—
22	C₂×Dic₂₀₆	—	2	1	no	—
23	#13110095	—	4	4	no	—
24	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
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	C₂×Q₈×C₁₀₃	—	2	1	no	—
31	C₂×Q₈×C₁₀₃	—	2	1	no	—
32	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₀₃	—	2	1	no	—
34	C₄⋊C₄×C₁₀₃	—	2	1	no	—
35	C₄⋊C₄×C₁₀₃	—	2	1	no	—
36	C₄⋊C₄×C₁₀₃	—	2	1	no	—
37	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
38	Q₈×C₁₀₃	—	4	2	no	—
39	Q₈×C₁₀₃	—	4	2	no	—
40	C₂×C₄×C₁₀₃	—	4	1	no	—
41	C₂×C₄×C₁₀₃	—	4	1	no	—
42	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₃	—	4	1	no	—
46	C₂×Dic₁₀₃	—	4	1	no	—
47	Dic₂₀₆	—	4	2	no	—
48	Dic₂₀₆	—	4	2	no	—
49	C₈×C₁₀₃	Q₁₆	1	4	yes	complemented
50	C₁₀₃⋊C₈	—	2	2	no	—
51	C₁₀₃⋊C₈	SD₁₆	1	4	yes	—
52	C₂²×C₁₀₃	#127	1	1	yes	characteristic
53	C₄×C₁₀₃	—	8	1	no	—
54	C₄×C₁₀₃	—	8	1	no	—
55	C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₀₃	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₀₃	#135	1	2	yes	—
58	Dic₁₀₃	—	8	1	no	—
59	Dic₁₀₃	—	8	1	no	—
60	C₂×C₁₀₃	#492	1	1	yes	characteristic
61	C₂×C₁₀₃	#494	1	1	yes	characteristic
62	C₂×C₁₀₃	#495	1	1	yes	characteristic
63	#1707	—	103	1	no	maximal, Sylow
64	C₁₀₃	#1707	1	1	yes	characteristic, complemented, Sylow
65	#500	—	103	1	no	—
66	#498	—	103	1	no	—
67	#498	D₁₀₃	1	1	yes	characteristic
68	#494	—	103	1	no	—
69	#487	—	103	1	no	—
70	#487	—	103	1	no	—
71	C₈²	—	103	1	no	—
72	C₂×Q₁₆	—	206	2	no	—
73	C₄⋊Q₈	—	103	1	no	—
74	C₄⋊Q₈	—	103	1	no	—
75	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
77	#107	—	206	2	no	—
78	#106	—	2	2	no	—
79	#106	—	206	2	no	—
80	#103	—	206	2	no	—
81	#103	—	206	2	no	—
82	C₄×C₈	—	103	1	no	—
83	C₄×C₈	—	103	1	no	—
84	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	206	1	no	—
88	C₂×Q₈	—	206	1	no	—
89	Q₁₆	—	412	4	no	—
90	C₂×C₈	—	103	2	no	—
91	C₂×C₈	—	103	2	no	—
92	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	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	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	412	2	no	—
105	Q₈	—	412	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	412	1	no	—
109	C₂×C₄	—	412	1	no	—
110	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
113	C₈	—	103	4	no	—
114	C₈	—	206	2	no	—
115	C₈	#13110095	1	4	yes	complemented
116	C₂²	#15693017	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	824	1	no	—
120	C₄	—	824	1	no	—
121	C₄	SD₁₆×D₁₀₃	1	2	yes	—
122	C₄	C₂×#13110095	1	2	yes	—
123	C₄	#15693042	1	2	yes	—
124	C₂	#20268162	1	1	yes	characteristic
125	C₂	#20268170	1	1	yes	characteristic
126	C₂	#20268172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268194

C₂

yes

maximal, characteristic

#20268189

C₂

yes

maximal, characteristic

#20268170

C₂

yes

maximal, characteristic

#20268143

C₂

yes

maximal, characteristic

#20268141

C₂

yes

maximal, characteristic

C₈×C₁₀₃⋊C₈

C₂

yes

maximal, characteristic

#498×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110095

—

#15693021

C₂²

yes

characteristic

#15693021

C₂²

yes

—

—

—

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

#106×C₁₀₃

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

#13110095

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

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

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

Q₁₆

yes

complemented

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

SD₁₆

yes

—

C₂²×C₁₀₃

#127

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×SD₁₆

yes

—

C₄×C₁₀₃

C₂×Q₁₆

yes

—

C₄×C₁₀₃

#135

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1707

yes

characteristic, complemented, Sylow

#500

—

103

—

#498

—

103

—

#498

D₁₀₃

yes

characteristic

#494

—

103

—

#487

—

103

—

#487

—

103

—

C₈²

—

103

—

C₂×Q₁₆

—

206

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

#107

—

206

—

#106

—

#106

—

206

—

#103

—

206

—

#103

—

206

—

C₄×C₈

—

103

—

C₄×C₈

—

103

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

Q₁₆

—

412

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

100

C₄⋊C₄

—

206

—

101

C₄²

C₂²×D₁₀₃

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

412

—

105

Q₈

—

412

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

412

—

109

C₂×C₄

—

412

—

110

C₂×C₄

D₄×D₁₀₃

yes

characteristic

111

C₂×C₄

D₄×D₁₀₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

113

C₈

—

103

—

114

C₈

—

206

—

115

C₈

#13110095

yes

complemented

116

C₂²

#15693017

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

824

—

120

C₄

—

824

—

121

C₄

SD₁₆×D₁₀₃

yes

—

122

C₄

C₂×#13110095

yes

—

123

C₄

#15693042

yes

—

124

C₂

#20268162

yes

characteristic

125

C₂

#20268170

yes

characteristic

126

C₂

#20268172

yes

characteristic

	subgroup	G/H
center	C₂²	#15693017
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#498×C₁₀₃	C₂
socle, prime radical	C₂²×C₁₀₃	#127
hypercenter	#498	D₁₀₃
hypocenter	C₁₀₃	#1707
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#15693017

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#498×C₁₀₃

C₂

C₂²×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
49. C₈×C₁₀₃	Q₁₆	semidirect
64. C₁₀₃	#1707	semidirect
115. C₈	#13110095	semidirect

H ≅ G/N

type

49. C₈×C₁₀₃

Q₁₆

semidirect

64. C₁₀₃

#1707

semidirect

115. C₈

#13110095

semidirect

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

subgroup	index
#20268194	2
#20268189	2
#20268170	2
#20268143	2
#20268141	2
C₈×C₁₀₃⋊C₈	2
#498×C₁₀₃	2
#1707	103

subgroup

index

C₈×C₁₀₃⋊C₈

#498×C₁₀₃

#1707

103

Subgroups — #27194739

Conjugacy classes of subgroups

Complements

Maximal subgroups