Group #26203915 — subgroups

Subgroups — C₂×#20242975 #26203915

Order 12032.

994 conjugacy classes of subgroups; only the 331 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 331 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×#15689521	C₂	1	3	yes	maximal, complemented
2	C₂×#15689496	C₂	1	3	yes	maximal
3	#20242975	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#10625325	—	2	6	no	—
6	C₂×#13094024	C₂²	1	3	yes	—
7	C₂×#13094021	C₂²	1	1	yes	characteristic
8	#15689521	—	2	6	no	—
9	#15689521	C₂²	1	12	yes	complemented
10	#15689496	—	2	6	no	—
11	#15689496	C₂²	1	12	yes	—
12	C₂×#13093956	—	2	6	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
14	C₈²×C₄₇	C₂²	1	4	yes	complemented
15	C₂²×Dic₉₄	—	2	3	no	—
16	C₂²×D₁₈₈	—	2	3	no	—
17	C₂×#10625325	—	2	24	no	—
18	C₂×#10625325	—	4	24	no	—
19	#13094024	—	2	6	no	—
20	#13094024	—	2	3	no	—
21	#13094024	C₂³	1	6	yes	—
22	#13094021	—	2	3	no	—
23	#13094021	C₂³	1	2	yes	—
24	C₂×C₄⋊Dic₄₇	—	2	6	no	—
25	#13093956	—	2	24	no	—
26	C₂²×C₈×C₄₇	D₄	1	6	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	6	yes	—
29	C₄×C₈×C₄₇	C₂³	1	6	yes	—
30	C₂³×D₄₇	—	4	1	no	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
32	C₂²×Dic₄₇	—	4	3	no	—
33	C₂×D₁₈₈	—	2	6	no	—
34	C₂×D₁₈₈	—	4	12	no	—
35	C₂×D₁₈₈	—	4	3	no	—
36	C₂×Dic₉₄	—	2	6	no	—
37	C₂×Dic₉₄	—	4	12	no	—
38	C₂×Dic₉₄	—	4	3	no	—
39	C₂×C₈×C₄₇	SD₁₆	1	24	yes	complemented
40	C₂×C₈×C₄₇	C₂×D₄	1	12	yes	—
41	C₄²×C₄₇	C₂×D₄	1	3	yes	—
42	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₄⋊Dic₄₇	—	2	12	no	—
44	C₄⋊Dic₄₇	—	4	6	no	—
45	#10625325	—	4	48	no	—
46	C₂³×C₄₇	#127	1	1	yes	characteristic
47	C₂²×D₄₇	—	4	2	no	—
48	C₂²×D₄₇	—	8	6	no	—
49	C₂×C₄×C₄₇	#127	1	3	yes	—
50	C₂×C₄×C₄₇	C₂×SD₁₆	1	12	yes	—
51	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
52	C₂×Dic₄₇	—	4	6	no	—
53	C₂×Dic₄₇	—	8	6	no	—
54	D₁₈₈	—	4	12	no	—
55	D₁₈₈	—	8	6	no	—
56	Dic₉₄	—	4	12	no	—
57	Dic₉₄	—	8	6	no	—
58	C₈×C₄₇	C₂×SD₁₆	1	24	yes	complemented
59	C₂×#1705	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	#492	1	6	yes	—
61	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
62	C₂×D₄₇	—	8	6	no	—
63	C₂×D₄₇	—	16	4	no	—
64	C₄×C₄₇	#492	1	6	yes	—
65	C₄×C₄₇	C₂²×SD₁₆	1	6	yes	—
66	Dic₄₇	—	8	6	no	—
67	C₂×#498	—	47	3	no	—
68	C₂×#492	—	47	3	no	—
69	#1705	—	47	8	no	—
70	C₂×C₈²	D₄₇	1	1	yes	characteristic, complemented
71	C₂×C₄₇	#1705	1	4	yes	complemented
72	C₂×C₄₇	C₂×#492	1	3	yes	—
73	D₄₇	—	16	2	no	—
74	C₂²×SD₁₆	—	94	6	no	—
75	C₂×C₄⋊Q₈	—	47	3	no	—
76	C₂×#127	—	47	1	no	—
77	#498	—	47	12	no	—
78	#498	—	94	6	no	—
79	#492	—	47	12	no	—
80	#492	—	94	6	no	—
81	C₂×#106	—	94	6	no	—
82	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
83	C₈²	C₂×D₄₇	1	4	yes	complemented
84	C₄₇	C₂×#1705	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	94	3	no	—
86	C₂²×D₄	—	94	3	no	—
87	C₂×SD₁₆	—	94	24	no	—
88	C₂×SD₁₆	—	188	24	no	—
89	C₂²×C₈	D₁₈₈	1	6	yes	—
90	C₄⋊Q₈	—	47	6	no	—
91	C₄⋊Q₈	—	94	6	no	—
92	C₄⋊Q₈	—	94	3	no	—
93	#127	—	47	2	no	—
94	#127	—	94	3	no	—
95	C₂×C₄⋊C₄	—	94	6	no	—
96	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
97	#106	—	94	24	no	—
98	C₄×C₈	D₁₈₈	1	6	yes	—
99	C₄×C₈	C₂²×D₄₇	1	6	yes	—
100	C₂⁴	—	188	1	no	—
101	C₂×Q₈	—	94	6	no	—
102	C₂×Q₈	—	188	12	no	—
103	C₂×Q₈	—	188	3	no	—
104	C₂×D₄	—	94	6	no	—
105	C₂×D₄	—	188	3	no	—
106	C₂×D₄	—	188	12	no	—
107	C₂²×C₄	—	188	3	no	—
108	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
109	SD₁₆	—	188	48	no	—
110	C₂×C₈	#10625325	1	24	yes	complemented
111	C₂×C₈	C₂×D₁₈₈	1	12	yes	—
112	C₄⋊C₄	—	94	12	no	—
113	C₄⋊C₄	—	188	6	no	—
114	C₄²	C₂×D₁₈₈	1	3	yes	—
115	C₄²	C₂³×D₄₇	1	1	yes	characteristic
116	C₂³	—	376	6	no	—
117	C₂³	—	188	2	no	—
118	C₂³	#13094021	1	1	yes	characteristic
119	Q₈	—	376	6	no	—
120	Q₈	—	188	12	no	—
121	D₄	—	376	6	no	—
122	D₄	—	188	12	no	—
123	C₂×C₄	—	376	6	no	—
124	C₂×C₄	—	188	6	no	—
125	C₂×C₄	#13094021	1	3	yes	—
126	C₂×C₄	C₂×#10625325	1	12	yes	—
127	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
128	C₈	C₂×#10625325	1	24	yes	complemented
129	C₂²	—	376	6	no	—
130	C₂²	—	752	4	no	—
131	C₂²	#15689496	1	6	yes	—
132	C₂²	C₂×#13094021	1	1	yes	characteristic
133	C₄	—	376	6	no	—
134	C₄	#15689496	1	6	yes	—
135	C₄	C₂²×#10625325	1	6	yes	—
136	C₂	—	752	2	no	—
137	C₂	#20242975	1	4	yes	direct product factor
138	C₂	C₂×#15689496	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689521

C₂

yes

maximal, complemented

C₂×#15689496

C₂

yes

maximal

#20242975

C₂

yes

maximal, direct product factor

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×#10625325

—

C₂×#13094024

C₂²

yes

—

C₂×#13094021

C₂²

yes

characteristic

#15689521

—

#15689521

C₂²

yes

complemented

#15689496

—

#15689496

C₂²

yes

—

C₂×#13093956

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

complemented

C₂²×Dic₉₄

—

C₂²×D₁₈₈

—

—

—

—

—

C₂³

yes

—

#13094021

—

#13094021

C₂³

yes

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

SD₁₆

yes

complemented

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂⁴

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625325

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

#127

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#127

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#492

yes

—

C₄×C₄₇

C₂²×SD₁₆

yes

—

Dic₄₇

—

C₂×#498

—

C₂×#492

—

#1705

—

C₂×C₈²

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1705

yes

complemented

C₂×C₄₇

C₂×#492

yes

—

D₄₇

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

complemented

C₄₇

C₂×#1705

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

188

—

C₂²×C₈

D₁₈₈

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#106

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂²×D₄₇

yes

—

100

C₂⁴

—

188

—

101

C₂×Q₈

—

102

C₂×Q₈

—

188

—

103

C₂×Q₈

—

188

—

104

C₂×D₄

—

105

C₂×D₄

—

188

—

106

C₂×D₄

—

188

—

107

C₂²×C₄

—

188

—

108

C₂²×C₄

C₂×D₁₈₈

yes

—

109

SD₁₆

—

188

—

110

C₂×C₈

#10625325

yes

complemented

111

C₂×C₈

C₂×D₁₈₈

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

188

—

114

C₄²

C₂×D₁₈₈

yes

—

115

C₄²

C₂³×D₄₇

yes

characteristic

116

C₂³

—

376

—

117

C₂³

—

188

—

118

C₂³

#13094021

yes

characteristic

119

Q₈

—

376

—

120

Q₈

—

188

—

121

D₄

—

376

—

122

D₄

—

188

—

123

C₂×C₄

—

376

—

124

C₂×C₄

—

188

—

125

C₂×C₄

#13094021

yes

—

126

C₂×C₄

C₂×#10625325

yes

—

127

C₂×C₄

C₂²×D₁₈₈

yes

—

128

C₈

C₂×#10625325

yes

complemented

129

C₂²

—

376

—

130

C₂²

—

752

—

131

C₂²

#15689496

yes

—

132

C₂²

C₂×#13094021

yes

characteristic

133

C₄

—

376

—

134

C₄

#15689496

yes

—

135

C₄

C₂²×#10625325

yes

—

136

C₂

—

752

—

137

C₂

#20242975

yes

direct product factor

138

C₂

C₂×#15689496

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	C₂×#1705
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13094021

commutator, cosocle

C₄²×C₄₇

C₂⁴

Frattini

C₄²

C₂³×D₄₇

Fitting

C₂×C₈²×C₄₇

C₂

C₂³×C₄₇

C₂²

C₂×C₈²

D₄₇

hypocenter

C₄₇

C₂×#1705

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
1. C₂×#15689521	C₂	semidirect
3. #20242975	C₂	direct (N × H)
4. C₂×C₈²×C₄₇	C₂	semidirect
9. #15689521	C₂²	semidirect
14. C₈²×C₄₇	C₂²	semidirect
39. C₂×C₈×C₄₇	SD₁₆	semidirect
58. C₈×C₄₇	C₂×SD₁₆	semidirect
70. C₂×C₈²	D₄₇	semidirect
71. C₂×C₄₇	#1705	semidirect
83. C₈²	C₂×D₄₇	semidirect
84. C₄₇	C₂×#1705	semidirect
110. C₂×C₈	#10625325	semidirect
128. C₈	C₂×#10625325	semidirect
137. C₂	#20242975	direct (N × H)

H ≅ G/N

type

1. C₂×#15689521

C₂

semidirect

3. #20242975

C₂

direct (N × H)

4. C₂×C₈²×C₄₇

C₂

semidirect

9. #15689521

C₂²

semidirect

14. C₈²×C₄₇

C₂²

semidirect

39. C₂×C₈×C₄₇

SD₁₆

semidirect

58. C₈×C₄₇

C₂×SD₁₆

semidirect

70. C₂×C₈²

D₄₇

semidirect

71. C₂×C₄₇

#1705

semidirect

83. C₈²

C₂×D₄₇

semidirect

84. C₄₇

C₂×#1705

semidirect

110. C₂×C₈

#10625325

semidirect

128. C₈

C₂×#10625325

semidirect

137. C₂

#20242975

direct (N × H)

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

subgroup	index
C₂×#15689521	2
C₂×#15689496	2
#20242975	2
C₂×C₈²×C₄₇	2
C₂×#1705	47

subgroup

index

C₂×C₈²×C₄₇

Subgroups — C₂×#20242975 #26203915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20242975 #26203915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242975 #26203915