Group #11874225 — subgroups

Subgroups — #11874225

Order 1152.

842 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 147 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×#64228	C₂	1	4	yes	maximal
2	#10593909	C₂	1	2	yes	maximal
3	C₄×#3787	C₂	1	1	yes	maximal, characteristic
4	#10593450	C₂	1	8	yes	maximal, complemented
5	C₂×#3797	C₂²	1	4	yes	—
6	C₂×#3787	C₂²	1	2	yes	—
7	C₂×#3776	C₂²	1	16	yes	—
8	#64228	—	2	16	no	—
9	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₃×#869	—	2	2	no	—
11	#63919	C₂²	1	8	yes	—
12	#63900	—	2	4	no	—
13	#63873	—	2	2	no	—
14	C₄×#622	C₂²	1	4	yes	—
15	D₃×C₄⋊Q₈	—	6	2	no	—
16	C₂×C₄×C₃⋊D₃	D₄	1	2	yes	—
17	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
18	C₂×C₃×Dic₆	—	2	4	no	—
19	C₂×#627	C₂³	1	8	yes	—
20	C₂×#624	—	2	4	no	—
21	#3797	—	2	16	no	—
22	C₂×#622	C₂³	1	4	yes	—
23	#3787	D₄	1	8	yes	—
24	#3776	—	2	32	no	—
25	C₃²×C₄²	D₄	1	1	yes	characteristic
26	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
27	C₄×C₃⋊Dic₃	C₂³	1	2	yes	—
28	C₃×C₄⋊Dic₃	—	2	8	no	—
29	#3724	—	2	8	no	—
30	C₂×#494	—	9	1	no	maximal, Sylow
31	C₂×D₃×Q₈	—	6	4	no	—
32	C₃×C₄⋊Q₈	—	6	2	no	—
33	#944	—	6	4	no	—
34	D₃×C₄⋊C₄	—	6	8	no	—
35	#888	—	6	8	no	—
36	C₄²×D₃	—	6	2	no	—
37	#869	—	2	2	no	—
38	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
41	C₂×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
42	C₂×C₃⋊Dic₃	C₂⁴	1	1	yes	characteristic
43	C₄×C₃⋊D₃	Q₁₆	1	8	yes	complemented
44	C₄×C₃⋊D₃	C₂×D₄	1	4	yes	—
45	C₂×C₃×Dic₃	—	2	8	no	—
46	C₃×Dic₆	—	4	8	no	—
47	#627	—	2	16	no	—
48	#624	—	4	8	no	—
49	#622	C₂×D₄	1	8	yes	—
50	C₂²×Q₁₆	—	9	4	no	—
51	C₂×C₄⋊Q₈	—	9	2	no	—
52	#494	—	9	8	no	—
53	C₂×C₄×C₈	—	9	1	no	—
54	C₂×C₃×Q₈	—	6	4	no	—
55	D₃×Q₈	—	12	16	no	—
56	C₂×C₄×D₃	—	6	4	no	—
57	C₂×C₄×D₃	—	6	2	no	—
58	C₂×C₄×D₃	—	6	8	no	—
59	C₂×Dic₆	—	2	4	no	—
60	C₂×Dic₆	—	6	8	no	—
61	C₃×C₄⋊C₄	—	6	8	no	—
62	C₃×C₄²	—	2	2	no	—
63	C₄⋊Dic₃	—	2	8	no	—
64	Dic₃⋊C₄	—	6	16	no	—
65	C₄×Dic₃	—	6	4	no	—
66	C₄×Dic₃	—	6	2	no	—
67	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₃⋊D₃	#127	1	2	yes	—
69	C₂×C₃⋊D₃	C₂×Q₁₆	1	4	yes	—
70	C₃²×C₄	#127	1	2	yes	—
71	C₃²×C₄	C₂×Q₁₆	1	4	yes	complemented
72	C₃⋊Dic₃	C₂×Q₁₆	1	4	yes	complemented
73	C₃⋊Dic₃	C₂²×D₄	1	2	yes	—
74	C₃×Dic₃	—	4	8	no	—
75	C₂²×Q₈	—	9	4	no	—
76	C₂×Q₁₆	—	9	16	no	—
77	C₂×Q₁₆	—	18	16	no	—
78	C₂²×C₈	—	9	2	no	—
79	C₄⋊Q₈	—	6	2	no	—
80	C₄⋊Q₈	—	9	8	no	—
81	C₄⋊Q₈	—	18	2	no	—
82	C₂×C₄⋊C₄	—	18	4	no	—
83	C₂×C₄²	—	9	1	no	—
84	C₄×C₈	—	9	4	no	—
85	C₂²×D₃	—	6	2	no	—
86	C₃×Q₈	—	12	8	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	6	8	no	—
90	C₂×Dic₃	—	2	8	no	—
91	C₂×Dic₃	—	6	4	no	—
92	C₂×Dic₃	—	6	2	no	—
93	C₄×D₃	—	6	16	no	—
94	C₄×D₃	—	6	8	no	—
95	C₄×D₃	—	12	16	no	—
96	Dic₆	—	4	8	no	—
97	Dic₆	—	12	16	no	—
98	C₂×C₃²	C₂×#127	1	1	yes	characteristic
99	C₂×C₃²	C₂²×Q₁₆	1	2	yes	—
100	C₃⋊D₃	#494	1	4	yes	complemented
101	C₂×Q₈	—	6	4	no	—
102	C₂×Q₈	—	9	8	no	—
103	C₂×Q₈	—	18	16	no	—
104	C₂²×C₄	—	9	2	no	—
105	C₂²×C₄	—	9	1	no	—
106	C₂²×C₄	—	18	4	no	—
107	Q₁₆	—	18	32	no	—
108	C₂×C₈	—	9	4	no	—
109	C₂×C₈	—	9	8	no	—
110	C₄⋊C₄	—	6	8	no	—
111	C₄⋊C₄	—	18	8	no	—
112	C₄²	—	9	2	no	—
113	C₄²	—	9	1	no	—
114	C₄²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₃	—	2	2	no	—
116	C₂×D₃	—	6	4	no	—
117	C₂×D₃	—	6	8	no	—
118	C₃×C₄	—	2	8	no	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	—	12	8	no	—
121	Dic₃	—	4	8	no	—
122	Dic₃	—	6	4	no	—
123	Dic₃	—	6	8	no	—
124	C₃²	C₂×#494	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	9	1	no	—
126	Q₈	—	12	8	no	—
127	Q₈	—	18	16	no	—
128	C₂×C₄	—	6	8	no	—
129	C₂×C₄	—	9	2	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	—	9	8	no	—
132	C₂×C₄	—	9	4	no	—
133	C₂×C₄	—	36	8	no	—
134	C₂×C₄	C₂×GO⁺(4,𝔽₂)	1	2	yes	—
135	C₂×C₄	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
136	C₈	—	9	8	no	—
137	C₂×C₃	—	2	4	no	—
138	C₂×C₃	—	2	2	no	—
139	D₃	—	6	8	no	—
140	C₂²	—	9	4	no	—
141	C₂²	—	9	2	no	—
142	C₂²	C₂²×GO⁺(4,𝔽₂)	1	1	yes	characteristic
143	C₄	—	9	4	no	—
144	C₄	—	9	2	no	—
145	C₄	—	12	8	no	—
146	C₄	#64228	1	4	yes	complemented
147	C₄	#64231	1	2	yes	—
148	C₃	—	2	2	no	—
149	C₂	—	9	4	no	—
150	C₂	C₂×#64228	1	2	yes	—
151	C₂	C₂×#64231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64228

C₂

yes

maximal

#10593909

C₂

yes

maximal

C₄×#3787

C₂

yes

maximal, characteristic

#10593450

C₂

yes

maximal, complemented

C₂×#3797

C₂²

yes

—

C₂×#3787

C₂²

yes

—

C₂×#3776

C₂²

yes

—

#64228

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

C₃×#869

—

#63919

C₂²

yes

—

#63900

—

#63873

—

C₄×#622

C₂²

yes

—

D₃×C₄⋊Q₈

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂×C₃×Dic₆

—

C₂×#627

C₂³

yes

—

C₂×#624

—

#3797

—

C₂×#622

C₂³

yes

—

#3787

D₄

yes

—

#3776

—

C₃²×C₄²

D₄

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂³

yes

—

C₃×C₄⋊Dic₃

—

#3724

—

C₂×#494

—

maximal, Sylow

C₂×D₃×Q₈

—

C₃×C₄⋊Q₈

—

#944

—

D₃×C₄⋊C₄

—

#888

—

C₄²×D₃

—

#869

—

C₂²×C₃⋊D₃

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

C₂⁴

yes

characteristic

C₄×C₃⋊D₃

Q₁₆

yes

complemented

C₄×C₃⋊D₃

C₂×D₄

yes

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

#627

—

#624

—

#622

C₂×D₄

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

C₂×C₄×C₈

—

C₂×C₃×Q₈

—

D₃×Q₈

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂²×D₄

yes

characteristic

C₂×C₃⋊D₃

#127

yes

—

C₂×C₃⋊D₃

C₂×Q₁₆

yes

—

C₃²×C₄

#127

yes

—

C₃²×C₄

C₂×Q₁₆

yes

complemented

C₃⋊Dic₃

C₂×Q₁₆

yes

complemented

C₃⋊Dic₃

C₂²×D₄

yes

—

C₃×Dic₃

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄×C₈

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

Dic₆

—

C₂×C₃²

C₂×#127

yes

characteristic

C₂×C₃²

C₂²×Q₁₆

yes

—

100

C₃⋊D₃

#494

yes

complemented

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

GO⁺(4,𝔽₂)

yes

characteristic

115

C₂²×C₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

Dic₃

—

122

Dic₃

—

123

Dic₃

—

124

C₃²

C₂×#494

yes

characteristic, complemented, Sylow

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂×GO⁺(4,𝔽₂)

yes

—

135

C₂×C₄

C₂×GO⁺(4,𝔽₂)

yes

characteristic

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

—

139

D₃

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂²×GO⁺(4,𝔽₂)

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#64228

yes

complemented

147

C₄

#64231

yes

—

148

C₃

—

149

C₂

—

150

C₂

C₂×#64228

yes

—

151

C₂

C₂×#64231

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×GO⁺(4,𝔽₂)
commutator, cosocle	C₂×C₃⋊Dic₃	C₂⁴
Fitting	C₃²×C₄²	D₄
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃⋊D₃	C₂×D₄
hypercenter	C₄²	GO⁺(4,𝔽₂)
hypocenter	C₃²	C₂×#494
LCS	G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×GO⁺(4,𝔽₂)

commutator, cosocle

C₂×C₃⋊Dic₃

C₂⁴

Fitting

C₃²×C₄²

D₄

socle

C₂²×C₃²

C₂²×D₄

prime radical

C₂²×C₃⋊D₃

C₂×D₄

hypercenter

C₄²

GO⁺(4,𝔽₂)

hypocenter

C₃²

C₂×#494

LCS

G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
4. #10593450	C₂	semidirect
43. C₄×C₃⋊D₃	Q₁₆	semidirect
71. C₃²×C₄	C₂×Q₁₆	semidirect
72. C₃⋊Dic₃	C₂×Q₁₆	semidirect
100. C₃⋊D₃	#494	semidirect
124. C₃²	C₂×#494	semidirect
146. C₄	#64228	semidirect

H ≅ G/N

type

4. #10593450

C₂

semidirect

43. C₄×C₃⋊D₃

Q₁₆

semidirect

71. C₃²×C₄

C₂×Q₁₆

semidirect

72. C₃⋊Dic₃

C₂×Q₁₆

semidirect

100. C₃⋊D₃

#494

semidirect

124. C₃²

C₂×#494

semidirect

146. C₄

#64228

semidirect

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

subgroup	index
C₂×#64228	2
#10593909	2
C₄×#3787	2
#10593450	2
C₂×#494	9

subgroup

index

Subgroups — #11874225

Conjugacy classes of subgroups

Complements

Maximal subgroups