Group #27194203 — subgroups

Subgroups — #27194203

Order 13184.

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

normal subgroups only

This group has 240 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	#20268167	C₂	1	1	yes	maximal, characteristic
2	#20267991	C₂	1	2	yes	maximal, complemented
3	#20267535	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13110092	—	2	2	no	—
7	C₂×#13110090	—	2	2	no	—
8	#15693018	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
10	#15692941	—	2	2	no	—
11	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₁₀₃	C₂²	1	1	yes	characteristic
13	#124×C₁₀₃	—	2	2	no	—
14	#123×C₁₀₃	—	2	2	no	—
15	C₄×Q₈×C₁₀₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₀₃	—	2	2	no	—
17	C₂×D₄₁₂	—	2	2	no	—
18	C₂×D₄₁₂	—	2	1	no	—
19	#13110092	—	4	4	no	—
20	#13110090	—	4	4	no	—
21	C₂×C₁₀₃⋊C₈	—	2	1	no	—
22	C₂×C₁₀₃⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₀₃	D₄	1	2	yes	—
24	C₂×D₄×C₁₀₃	—	2	2	no	—
25	C₂×D₄×C₁₀₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₃	—	2	2	no	—
27	C₄⋊C₄×C₁₀₃	—	2	2	no	—
28	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₀₃	—	2	4	no	—
30	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
31	C₄²×C₁₀₃	—	2	2	no	—
32	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₃	—	2	2	no	—
34	C₂²×D₁₀₃	—	4	1	no	—
35	Q₈×C₁₀₃	—	2	2	no	—
36	Q₈×C₁₀₃	—	2	2	no	—
37	D₄×C₁₀₃	—	4	2	no	—
38	D₄×C₁₀₃	—	4	2	no	—
39	D₄×C₁₀₃	—	4	2	no	—
40	C₂×C₄×C₁₀₃	—	2	2	no	—
41	C₂×C₄×C₁₀₃	—	2	2	no	—
42	C₂×C₄×C₁₀₃	—	8	2	no	—
43	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
45	D₄₁₂	—	4	2	no	—
46	D₄₁₂	—	4	2	no	—
47	D₄₁₂	—	4	2	no	—
48	C₁₀₃⋊C₈	—	2	2	no	—
49	C₁₀₃⋊C₈	—	4	2	no	—
50	C₂²×C₁₀₃	—	4	4	no	—
51	C₂²×C₁₀₃	—	4	2	no	—
52	C₂²×C₁₀₃	#120	1	1	yes	characteristic
53	C₂×D₁₀₃	—	8	2	no	—
54	C₂×D₁₀₃	—	8	1	no	—
55	C₄×C₁₀₃	—	2	1	no	—
56	C₄×C₁₀₃	—	4	2	no	—
57	C₄×C₁₀₃	—	4	2	no	—
58	C₄×C₁₀₃	#135	1	2	yes	—
59	C₄×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₀₃	—	8	2	no	—
61	C₂×C₁₀₃	#449	1	2	yes	—
62	C₂×C₁₀₃	#453	1	1	yes	characteristic
63	D₁₀₃	—	16	1	no	—
64	#1620	—	103	1	no	maximal, Sylow
65	C₁₀₃	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	D₁₀₃	1	1	yes	characteristic, complemented
67	#493	—	103	1	no	—
68	#460	—	103	2	no	—
69	#331	—	103	1	no	—
70	Q₈⋊C₈	—	103	2	no	—
71	C₂×SD₁₆	—	206	2	no	—
72	C₂×D₈	—	206	2	no	—
73	#127	—	103	1	no	—
74	#127	C₂×D₁₀₃	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₀₃	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	103	2	no	—
80	#102	—	206	2	no	—
81	C₄×C₈	—	103	1	no	—
82	C₂×Q₈	C₂²⋊D₁₀₃	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	206	2	no	—
85	C₂×D₄	—	206	1	no	—
86	C₂×D₄	C₂²⋊D₁₀₃	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	412	4	no	—
89	D₈	—	412	4	no	—
90	C₂×C₈	—	206	2	no	—
91	C₂×C₈	—	206	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₀₃	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	412	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	412	2	no	—
106	D₄	—	412	2	no	—
107	D₄	—	412	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
113	C₈	—	412	2	no	—
114	C₈	—	206	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	824	2	no	—
118	C₂²	—	824	1	no	—
119	C₂²	#15692990	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#15693041	1	2	yes	—
124	C₄	#15693046	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	1648	1	no	—
127	C₂	#20267936	1	2	yes	—
128	C₂	#20267959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268167

C₂

yes

maximal, characteristic

#20267991

C₂

yes

maximal, complemented

#20267535

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₁₀₃⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110092

—

C₂×#13110090

—

#15693018

C₂²

yes

characteristic

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

#15692941

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

#127×C₁₀₃

C₂²

yes

characteristic

#124×C₁₀₃

—

#123×C₁₀₃

—

C₄×Q₈×C₁₀₃

C₂²

yes

—

C₄×D₄×C₁₀₃

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

#13110092

—

#13110090

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

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

—

C₂²×D₁₀₃

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#120

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#135

yes

—

C₄×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#449

yes

—

C₂×C₁₀₃

#453

yes

characteristic

D₁₀₃

—

#1620

—

103

maximal, Sylow

C₁₀₃

#1620

yes

characteristic, complemented, Sylow

#559

D₁₀₃

yes

characteristic, complemented

#493

—

103

—

#460

—

103

—

#331

—

103

—

Q₈⋊C₈

—

103

—

C₂×SD₁₆

—

206

—

C₂×D₈

—

206

—

#127

—

103

—

#127

C₂×D₁₀₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₀₃

yes

—

C₄×D₄

—

C₄⋊C₈

—

103

—

#102

—

206

—

C₄×C₈

—

103

—

C₂×Q₈

C₂²⋊D₁₀₃

yes

—

C₂×D₄

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

—

SD₁₆

—

412

—

D₈

—

412

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₀₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

C₂³

—

412

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

412

—

106

D₄

—

412

—

107

D₄

—

412

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

113

C₈

—

412

—

114

C₈

—

206

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

824

—

118

C₂²

—

824

—

119

C₂²

#15692990

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#15693041

yes

—

124

C₄

#15693046

yes

—

125

C₂

—

126

C₂

—

1648

—

127

C₂

#20267936

yes

—

128

C₂

#20267959

yes

characteristic

	subgroup	G/H
center	C₂²	#15692990
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#559×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
prime radical	#20268167	C₂
hypercenter	#559	D₁₀₃
hypocenter	C₁₀₃	#1620
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#15692990

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#559×C₁₀₃

C₂

C₂²×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #20267991	C₂	semidirect
3. #20267535	C₂	semidirect
4. Q₈⋊(C₁₀₃⋊C₈)	C₂	semidirect
5. #559×C₁₀₃	C₂	semidirect
65. C₁₀₃	#1620	semidirect
66. #559	D₁₀₃	semidirect

H ≅ G/N

type

2. #20267991

C₂

semidirect

3. #20267535

C₂

semidirect

4. Q₈⋊(C₁₀₃⋊C₈)

C₂

semidirect

5. #559×C₁₀₃

C₂

semidirect

65. C₁₀₃

#1620

semidirect

66. #559

D₁₀₃

semidirect

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

subgroup	index
#20268167	2
#20267991	2
#20267535	2
Q₈⋊(C₁₀₃⋊C₈)	2
#559×C₁₀₃	2
#1620	103

subgroup

index

#20268167

#20267991

#20267535

Q₈⋊(C₁₀₃⋊C₈)

#559×C₁₀₃

#1620

103

Subgroups — #27194203

Conjugacy classes of subgroups

Complements

Maximal subgroups