Group #27196096 — subgroups

Subgroups — #27196096

Order 13184.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#15693021	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₀₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20267553	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₆	—	2	1	no	—
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	#15693021	—	2	2	no	—
8	#15693021	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
11	C₄⋊(C₁₀₃⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₃	—	2	1	no	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	#95×C₁₀₃	—	2	1	no	—
16	C₂×Dic₂₀₆	—	2	4	no	—
17	C₂×Dic₂₀₆	—	4	2	no	—
18	C₂×Dic₂₀₆	—	4	2	no	—
19	C₂²×Dic₁₀₃	—	2	2	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₂×C₁₀₃⋊C₈	—	2	4	no	—
22	C₄⋊Dic₁₀₃	—	2	4	no	—
23	C₄⋊Dic₁₀₃	—	2	2	no	—
24	C₄⋊Dic₁₀₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₃	—	2	2	no	—
26	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₃	—	2	2	no	—
30	C₄⋊C₄×C₁₀₃	Q₈	1	4	yes	complemented
31	C₄²×C₁₀₃	—	4	2	no	—
32	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₃	D₄	1	2	yes	—
34	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₃	—	2	2	no	—
36	C₂×C₄×C₁₀₃	—	2	4	no	—
37	C₂×C₄×C₁₀₃	—	4	2	no	—
38	C₂×C₄×C₁₀₃	—	4	2	no	—
39	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₃	D₈	1	2	yes	—
42	C₂×C₄×C₁₀₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₀₃	#41	1	2	yes	—
47	C₂×Dic₁₀₃	—	2	4	no	—
48	C₂×Dic₁₀₃	—	4	2	no	—
49	C₂×Dic₁₀₃	—	4	2	no	—
50	Dic₂₀₆	—	4	8	no	—
51	C₁₀₃⋊C₈	—	4	2	no	—
52	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₀₃	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₀₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₀₃	#137	1	2	yes	—
57	C₄×C₁₀₃	—	2	2	no	—
58	C₄×C₁₀₃	—	4	4	no	—
59	C₄×C₁₀₃	#102	1	4	yes	—
60	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
61	C₄×C₁₀₃	#124	1	1	yes	characteristic
62	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₀₃	—	4	4	no	—
64	C₂×C₁₀₃	#391	1	1	yes	characteristic
65	C₂×C₁₀₃	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₀₃	#419	1	1	yes	characteristic
67	C₂×C₁₀₃	#474	1	2	yes	—
68	C₂×C₁₀₃	#475	1	2	yes	—
69	#1988	—	103	1	no	maximal, Sylow
70	C₁₀₃	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	103	1	no	—
72	C₂×C₄⋊C₈	—	103	1	no	—
73	C₄×C₄⋊C₄	D₁₀₃	1	1	yes	characteristic
74	#340	—	103	4	no	—
75	C₂²×Q₈	—	206	1	no	—
76	C₂²×C₈	—	103	2	no	—
77	C₄⋊Q₈	—	103	4	no	—
78	C₄⋊Q₈	—	206	2	no	—
79	C₂×C₄⋊C₄	—	103	2	no	—
80	C₂×C₄⋊C₄	—	206	1	no	—
81	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
84	C₄⋊C₈	—	206	2	no	—
85	#95	—	2	1	no	—
86	C₂×Q₈	—	412	2	no	—
87	C₂×Q₈	—	412	2	no	—
88	C₂×Q₈	—	206	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	206	2	no	—
91	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
94	C₂×C₈	—	206	2	no	—
95	C₂×C₈	—	206	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	103	4	no	—
98	C₄⋊C₄	—	206	4	no	—
99	C₄⋊C₄	—	206	2	no	—
100	C₄⋊C₄	Dic₂₀₆	1	4	yes	complemented
101	C₄²	—	4	2	no	—
102	C₄²	C₄×D₁₀₃	1	2	yes	—
103	C₄²	D₄₁₂	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
105	Q₈	—	412	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	412	2	no	—
111	C₂×C₄	—	412	2	no	—
112	C₂×C₄	—	206	4	no	—
113	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
114	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
115	C₂×C₄	#13110090	1	2	yes	—
116	C₂×C₄	#13110092	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₀₆	1	2	yes	—
120	C₂×C₄	#13110105	1	2	yes	—
121	C₈	—	412	2	no	—
122	C₂²	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
123	C₂²	#15693000	1	2	yes	—
124	C₂²	C₂×#13110090	1	1	yes	characteristic
125	C₂²	C₂×#13110092	1	1	yes	characteristic
126	C₂²	#15693051	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	412	4	no	—
130	C₄	#15692941	1	4	yes	—
131	C₄	C₄×Dic₂₀₆	1	2	yes	—
132	C₄	#15693009	1	1	yes	characteristic
133	C₄	#15693021	1	1	yes	characteristic
134	C₂	#20267697	1	1	yes	characteristic
135	C₂	C₂×#15692941	1	1	yes	characteristic
136	C₂	#20267799	1	1	yes	characteristic
137	C₂	#20268071	1	2	yes	—
138	C₂	#20268076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693021

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal, characteristic

#20267553

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

#15693021

—

#15693021

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊(C₁₀₃⋊C₈)

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#95×C₁₀₃

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

Q₈

yes

complemented

C₄²×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

D₈

yes

—

C₂×C₄×C₁₀₃

SD₁₆

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₂²⋊Q₈

yes

—

C₂²×C₁₀₃

C₂×D₈

yes

characteristic

C₂²×C₁₀₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₃

#137

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#102

yes

—

C₄×C₁₀₃

C₄×Q₈

yes

—

C₄×C₁₀₃

#124

yes

characteristic

C₄×C₁₀₃

C₄⋊Q₈

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

C₂×C₁₀₃

#475

yes

—

#1988

—

103

maximal, Sylow

C₁₀₃

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

103

—

C₂×C₄⋊C₈

—

103

—

C₄×C₄⋊C₄

D₁₀₃

yes

characteristic

#340

—

103

—

C₂²×Q₈

—

206

—

C₂²×C₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

206

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

206

—

#95

—

C₂×Q₈

—

412

—

C₂×Q₈

—

412

—

C₂×Q₈

—

206

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

100

C₄⋊C₄

Dic₂₀₆

yes

complemented

101

C₄²

—

102

C₄²

C₄×D₁₀₃

yes

—

103

C₄²

D₄₁₂

yes

—

104

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

105

Q₈

—

412

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

412

—

111

C₂×C₄

—

412

—

112

C₂×C₄

—

206

—

113

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

114

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

115

C₂×C₄

#13110090

yes

—

116

C₂×C₄

#13110092

yes

—

117

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

118

C₂×C₄

C₂×D₄₁₂

yes

characteristic

119

C₂×C₄

C₂×Dic₂₀₆

yes

—

120

C₂×C₄

#13110105

yes

—

121

C₈

—

412

—

122

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

123

C₂²

#15693000

yes

—

124

C₂²

C₂×#13110090

yes

characteristic

125

C₂²

C₂×#13110092

yes

characteristic

126

C₂²

#15693051

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

412

—

130

C₄

#15692941

yes

—

131

C₄

C₄×Dic₂₀₆

yes

—

132

C₄

#15693009

yes

characteristic

133

C₄

#15693021

yes

characteristic

134

C₂

#20267697

yes

characteristic

135

C₂

C₂×#15692941

yes

characteristic

136

C₂

#20267799

yes

characteristic

137

C₂

#20268071

yes

—

138

C₂

#20268076

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₄×C₄⋊C₄×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₃
hypocenter	C₁₀₃	#1988
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₄×C₄⋊C₄×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₃

hypocenter

C₁₀₃

#1988

LCS

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #15693021	C₄	semidirect
30. C₄⋊C₄×C₁₀₃	Q₈	semidirect
70. C₁₀₃	#1988	semidirect
100. C₄⋊C₄	Dic₂₀₆	semidirect

H ≅ G/N

type

8. #15693021

C₄

semidirect

30. C₄⋊C₄×C₁₀₃

Q₈

semidirect

70. C₁₀₃

#1988

semidirect

100. C₄⋊C₄

Dic₂₀₆

semidirect

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

subgroup	index
C₂×#15693021	2
C₂×C₄⋊(C₁₀₃⋊C₈)	2
#20267553	2
C₄×C₄⋊C₄×C₁₀₃	2
#1988	103

subgroup

index

C₂×#15693021

C₂×C₄⋊(C₁₀₃⋊C₈)

#20267553

C₄×C₄⋊C₄×C₁₀₃

#1988

103

Subgroups — #27196096

Conjugacy classes of subgroups

Complements

Maximal subgroups