Group #11782415 — subgroups

Subgroups — #11782415

Order 1152.

642 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₃²×#492	C₂	1	1	yes	maximal, characteristic
2	#10592779	C₂	1	2	yes	maximal, complemented
3	#10592775	C₂	1	2	yes	maximal
4	#10592379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#71254	—	3	4	no	maximal
7	C₂×C₃²×SD₁₆	—	2	2	no	—
8	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃²×#127	C₂²	1	1	yes	characteristic
10	C₂×#3754	—	2	4	no	—
11	#64078	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#63644	—	2	2	no	—
14	#63635	—	2	4	no	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#492	D₃	1	4	yes	—
17	#5080	—	3	8	no	—
18	#5076	—	3	8	no	—
19	#4680	—	3	4	no	—
20	C₈×C₃⋊C₈	—	3	4	no	—
21	C₂×C₃²×Q₈	—	2	1	no	—
22	C₂×C₃²×D₄	—	2	2	no	—
23	C₂×C₃²×D₄	—	2	1	no	—
24	C₂×#634	—	2	2	no	—
25	C₃²×SD₁₆	—	4	4	no	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄²	C₂³	1	1	yes	characteristic
29	#3754	—	4	8	no	—
30	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
31	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	D₄	1	4	yes	—
33	#1705	—	9	1	no	Sylow
34	C₂×C₃×SD₁₆	—	2	8	no	—
35	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
36	C₃×#127	C₂×D₃	1	4	yes	—
37	C₂×#214	—	6	16	no	—
38	#869	—	3	8	no	—
39	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
40	#817	—	6	8	no	—
41	#808	—	6	16	no	—
42	C₄×C₃⋊C₈	—	3	8	no	—
43	C₂³×C₃²	—	4	1	no	—
44	C₃²×Q₈	—	4	2	no	—
45	C₃²×D₄	—	4	4	no	—
46	C₃²×D₄	—	4	2	no	—
47	C₂×C₃²×C₄	—	4	1	no	—
48	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
49	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃⋊Dic₃	—	4	2	no	—
51	#634	—	4	4	no	—
52	C₃²×C₈	SD₁₆	1	4	yes	complemented
53	C₃⋊(C₃⋊C₈)	SD₁₆	1	8	yes	complemented
54	#498	—	9	2	no	—
55	#498	—	9	1	no	—
56	#492	—	9	2	no	—
57	#492	C₃⋊D₃	1	1	yes	characteristic
58	C₈²	—	9	1	no	—
59	C₂×C₃×Q₈	—	2	4	no	—
60	C₂×C₃×D₄	—	2	8	no	—
61	C₂×C₃×D₄	—	2	4	no	—
62	C₂×Dic₆	—	6	8	no	—
63	C₃×SD₁₆	—	4	16	no	—
64	C₂×C₃×C₈	#67	1	8	yes	—
65	C₃×C₄⋊C₄	—	2	8	no	—
66	C₃×C₄²	C₂²×D₃	1	4	yes	—
67	#214	—	12	32	no	—
68	C₄⋊Dic₃	—	6	8	no	—
69	C₄⋊Dic₃	—	6	8	no	—
70	C₂×C₃⋊C₈	—	3	16	no	—
71	C₂²×C₃²	—	8	2	no	—
72	C₂²×C₃²	—	8	1	no	—
73	C₂²×C₃²	#127	1	1	yes	characteristic
74	C₃²×C₄	—	8	1	no	—
75	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
76	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
77	C₃⋊Dic₃	—	8	2	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	18	4	no	—
80	C₄⋊Q₈	—	9	2	no	—
81	C₄⋊Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
82	#127	C₂×C₃⋊D₃	1	1	yes	characteristic
83	#106	—	18	4	no	—
84	#106	—	18	2	no	—
85	C₄×C₈	—	9	2	no	—
86	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
87	C₂³×C₃	—	4	4	no	—
88	C₃×Q₈	—	4	8	no	—
89	C₃×D₄	—	4	16	no	—
90	C₃×D₄	—	4	8	no	—
91	C₂×C₃×C₄	—	4	4	no	—
92	C₂×C₃×C₄	D₃×D₄	1	8	yes	—
93	C₂×C₃×C₄	C₂×#67	1	4	yes	—
94	C₂×Dic₃	—	12	8	no	—
95	Dic₆	—	12	16	no	—
96	C₃×C₈	#214	1	16	yes	complemented
97	C₃⋊C₈	—	3	32	no	—
98	C₂×C₃²	—	16	1	no	—
99	C₂×C₃²	#492	1	2	yes	—
100	C₂×C₃²	#492	1	1	yes	characteristic
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	18	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	SD₁₆	—	4	4	no	—
106	SD₁₆	—	36	8	no	—
107	C₂×C₈	—	9	4	no	—
108	C₂×C₈	#638	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	18	2	no	—
111	C₄⋊C₄	—	18	2	no	—
112	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
113	C₂²×C₃	—	8	8	no	—
114	C₂²×C₃	—	8	4	no	—
115	C₂²×C₃	#940	1	4	yes	—
116	C₃×C₄	—	8	4	no	—
117	C₃×C₄	D₃×SD₁₆	1	16	yes	—
118	C₃×C₄	C₂×#214	1	8	yes	—
119	Dic₃	—	24	8	no	—
120	C₃²	#1705	1	1	yes	characteristic, complemented, Sylow
121	C₂³	—	4	1	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	36	4	no	—
124	D₄	—	4	4	no	—
125	D₄	—	4	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	36	2	no	—
128	C₂×C₄	D₄×C₃⋊D₃	1	2	yes	—
129	C₂×C₄	C₂×#638	1	1	yes	characteristic
130	C₈	—	9	8	no	—
131	C₈	#3754	1	4	yes	complemented
132	C₂×C₃	—	16	4	no	—
133	C₂×C₃	#5076	1	4	yes	—
134	C₂×C₃	#5175	1	8	yes	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#64149	1	1	yes	characteristic
138	C₄	—	8	1	no	—
139	C₄	—	72	2	no	—
140	C₄	SD₁₆×C₃⋊D₃	1	4	yes	—
141	C₄	C₂×#3754	1	2	yes	—
142	C₃	#71254	1	4	yes	complemented
143	C₂	—	16	1	no	—
144	C₂	#10592775	1	1	yes	characteristic
145	C₂	#10592874	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#492

C₂

yes

maximal, characteristic

#10592779

C₂

yes

maximal, complemented

#10592775

C₂

yes

maximal

#10592379

C₂

yes

maximal, characteristic, complemented

C₈×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

#71254

—

maximal

C₂×C₃²×SD₁₆

—

C₃²×C₄⋊Q₈

C₂²

yes

characteristic

C₃²×#127

C₂²

yes

characteristic

C₂×#3754

—

#64078

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63644

—

#63635

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃×#492

D₃

yes

—

#5080

—

#5076

—

#4680

—

C₈×C₃⋊C₈

—

C₂×C₃²×Q₈

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

—

C₂×#634

—

C₃²×SD₁₆

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

#3754

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

#1705

—

Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×#127

C₂×D₃

yes

—

C₂×#214

—

#869

—

C₃×C₄×C₈

C₂×D₃

yes

—

#817

—

#808

—

C₄×C₃⋊C₈

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

SD₁₆

yes

complemented

C₃⋊(C₃⋊C₈)

SD₁₆

yes

—

—

—

C₃⋊D₃

yes

characteristic

C₈²

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

#214

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃⋊Dic₃

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

characteristic

#127

C₂×C₃⋊D₃

yes

characteristic

#106

—

#106

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

#214

yes

complemented

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#492

yes

—

100

C₂×C₃²

#492

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

106

SD₁₆

—

107

C₂×C₈

—

108

C₂×C₈

#638

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

C₂²×C₃⋊D₃

yes

characteristic

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

#940

yes

—

116

C₃×C₄

—

117

C₃×C₄

D₃×SD₁₆

yes

—

118

C₃×C₄

C₂×#214

yes

—

119

Dic₃

—

120

C₃²

#1705

yes

characteristic, complemented, Sylow

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₄×C₃⋊D₃

yes

—

129

C₂×C₄

C₂×#638

yes

characteristic

130

C₈

—

131

C₈

#3754

yes

complemented

132

C₂×C₃

—

133

C₂×C₃

#5076

yes

—

134

C₂×C₃

#5175

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

#64149

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

SD₁₆×C₃⋊D₃

yes

—

141

C₄

C₂×#3754

yes

—

142

C₃

#71254

yes

complemented

143

C₂

—

144

C₂

#10592775

yes

characteristic

145

C₂

#10592874

yes

—

	subgroup	G/H
center	C₂²	#64149
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#492	C₂
socle	C₂²×C₃²	#127
prime radical	C₃²×#127	C₂²
hypercenter	#492	C₃⋊D₃
hypocenter	C₃²	#1705
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#64149

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#492

C₂

socle

C₂²×C₃²

#127

prime radical

C₃²×#127

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #10592779	C₂	semidirect
4. #10592379	C₂	semidirect
5. C₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
52. C₃²×C₈	SD₁₆	semidirect
53. C₃⋊(C₃⋊C₈)	SD₁₆	semidirect
96. C₃×C₈	#214	semidirect
120. C₃²	#1705	semidirect
131. C₈	#3754	semidirect
142. C₃	#71254	semidirect

H ≅ G/N

type

2. #10592779

C₂

semidirect

4. #10592379

C₂

semidirect

5. C₈×C₃⋊(C₃⋊C₈)

C₂

semidirect

52. C₃²×C₈

SD₁₆

semidirect

53. C₃⋊(C₃⋊C₈)

SD₁₆

semidirect

96. C₃×C₈

#214

semidirect

120. C₃²

#1705

semidirect

131. C₈

#3754

semidirect

142. C₃

#71254

semidirect

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

subgroup	index
C₃²×#492	2
#10592779	2
#10592775	2
#10592379	2
C₈×C₃⋊(C₃⋊C₈)	2
#71254	3

subgroup

index

C₃²×#492

C₈×C₃⋊(C₃⋊C₈)

Subgroups — #11782415

Conjugacy classes of subgroups

Complements

Maximal subgroups