Group #11800070 — subgroups

Subgroups — #11800070

Order 1152.

420 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 161 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×#500	C₂	1	1	yes	maximal, characteristic
2	#10592398	C₂	1	1	yes	maximal, characteristic
3	#10589358	C₂	1	4	yes	maximal
4	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	#71382	—	3	4	no	maximal
6	C₃²×C₄⋊Q₈	—	2	1	no	—
7	C₂×#3746	—	2	1	no	—
8	C₂×#3746	C₂²	1	2	yes	—
9	#64078	—	2	1	no	—
10	C₃²×#107	—	2	1	no	—
11	C₃²×#107	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
14	C₃×#500	D₃	1	4	yes	—
15	#4699	—	3	4	no	—
16	#4492	—	3	16	no	—
17	C₄×C₃⋊C₁₆	—	3	4	no	—
18	C₂×C₃²×Q₈	—	4	1	no	—
19	C₂×#634	—	2	2	no	—
20	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	—	2	2	no	—
23	C₃²×C₄⋊C₄	—	4	1	no	—
24	C₃²×C₄²	D₄	1	1	yes	characteristic
25	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
26	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
27	#3746	—	2	4	no	—
28	#3746	—	4	2	no	—
29	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
30	#2241	—	9	1	no	Sylow
31	C₃×C₄⋊Q₈	—	2	4	no	—
32	C₂×Dic₁₂	—	3	8	no	—
33	C₂×Dic₁₂	—	6	4	no	—
34	#869	—	6	4	no	—
35	C₃×#107	—	2	4	no	—
36	C₃×#107	C₂×D₃	1	8	yes	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	C₂×C₃⋊C₁₆	—	3	8	no	—
39	C₃²×Q₈	—	8	2	no	—
40	C₂×C₃²×C₄	—	4	2	no	—
41	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
42	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
43	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃⋊Dic₃	—	4	2	no	—
45	#634	—	4	2	no	—
46	#634	—	4	2	no	—
47	C₃²×C₈	SD₁₆	1	2	yes	—
48	C₃²×C₈	#41	1	2	yes	—
49	#500	C₃⋊D₃	1	1	yes	characteristic
50	#494	—	9	1	no	—
51	#358	—	9	4	no	—
52	C₄×C₁₆	—	9	1	no	—
53	C₂×C₃×Q₈	—	4	4	no	—
54	C₂×Dic₆	—	6	8	no	—
55	C₂×C₃×C₈	#67	1	4	yes	—
56	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
57	C₃×C₄⋊C₄	—	2	8	no	—
58	C₃×C₄⋊C₄	—	4	4	no	—
59	C₃×C₄²	#67	1	4	yes	—
60	C₄⋊Dic₃	—	12	4	no	—
61	C₄⋊Dic₃	—	12	4	no	—
62	Dic₁₂	—	6	16	no	—
63	Dic₁₂	—	12	8	no	—
64	C₃⋊C₁₆	—	6	8	no	—
65	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
66	C₃²×C₄	—	8	2	no	—
67	C₃²×C₄	SD₃₂	1	2	yes	—
68	C₃²×C₄	Q₃₂	1	2	yes	—
69	C₃²×C₄	#124	1	1	yes	characteristic
70	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
71	C₃⋊Dic₃	—	8	2	no	—
72	C₂×Q₁₆	—	9	2	no	—
73	C₂×Q₁₆	—	18	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	18	1	no	—
76	C₂×C₁₆	—	9	2	no	—
77	#107	—	2	1	no	—
78	#107	C₂×C₃⋊D₃	1	2	yes	—
79	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
80	C₃×Q₈	—	8	8	no	—
81	C₂×C₃×C₄	—	4	8	no	—
82	C₂×C₃×C₄	#213	1	4	yes	—
83	C₂×C₃×C₄	#213	1	4	yes	—
84	C₂×C₃×C₄	C₂×#67	1	4	yes	—
85	C₂×Dic₃	—	12	8	no	—
86	Dic₆	—	12	8	no	—
87	Dic₆	—	12	8	no	—
88	C₃×C₈	#215	1	8	yes	—
89	C₃×C₈	#239	1	8	yes	—
90	C₂×C₃²	#486	1	1	yes	characteristic
91	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
92	C₂×C₃²	C₂×Q₃₂	1	1	yes	characteristic
93	C₂×Q₈	—	4	1	no	—
94	C₂×Q₈	—	18	2	no	—
95	Q₁₆	—	18	4	no	—
96	Q₁₆	—	36	2	no	—
97	C₂×C₈	#638	1	1	yes	characteristic
98	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	36	1	no	—
102	C₄⋊C₄	—	36	1	no	—
103	C₄²	#638	1	1	yes	characteristic
104	C₁₆	—	18	2	no	—
105	C₂²×C₃	C₂×#213	1	4	yes	—
106	C₃×C₄	—	8	8	no	—
107	C₃×C₄	#827	1	8	yes	—
108	C₃×C₄	#829	1	8	yes	—
109	C₃×C₄	C₂×#215	1	4	yes	—
110	C₃×C₄	#947	1	4	yes	—
111	Dic₃	—	24	8	no	—
112	C₃²	#2241	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	8	2	no	—
114	Q₈	—	36	2	no	—
115	Q₈	—	36	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	36	2	no	—
118	C₂×C₄	#3753	1	1	yes	characteristic
119	C₂×C₄	#3753	1	1	yes	characteristic
120	C₂×C₄	C₂×#638	1	1	yes	characteristic
121	C₈	#3755	1	2	yes	—
122	C₈	#3832	1	2	yes	—
123	C₂×C₃	#5088	1	4	yes	—
124	C₂×C₃	C₂×#827	1	4	yes	—
125	C₂×C₃	C₂×#829	1	4	yes	—
126	C₂²	C₂×#3753	1	1	yes	characteristic
127	C₄	—	8	2	no	—
128	C₄	—	72	2	no	—
129	C₄	#63654	1	2	yes	—
130	C₄	#63656	1	2	yes	—
131	C₄	C₂×#3755	1	1	yes	characteristic
132	C₄	#64156	1	1	yes	characteristic
133	C₃	#71382	1	4	yes	complemented
134	C₂	#10592787	1	1	yes	characteristic
135	C₂	C₂×#63654	1	1	yes	characteristic
136	C₂	C₂×#63656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#500

C₂

yes

maximal, characteristic

#10592398

C₂

yes

maximal, characteristic

#10589358

C₂

yes

maximal

C₄×C₃⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic

#71382

—

maximal

C₃²×C₄⋊Q₈

—

C₂×#3746

—

C₂×#3746

C₂²

yes

—

#64078

—

C₃²×#107

—

C₃²×#107

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

C₃×#500

D₃

yes

—

#4699

—

#4492

—

C₄×C₃⋊C₁₆

—

C₂×C₃²×Q₈

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

D₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3746

—

#3746

—

C₃⋊(C₃⋊C₁₆)

—

#2241

—

Sylow

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₃×#107

—

C₃×#107

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₂×C₃⋊C₁₆

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

SD₁₆

yes

—

C₃²×C₈

#41

yes

—

#500

C₃⋊D₃

yes

characteristic

#494

—

#358

—

C₄×C₁₆

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

#67

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

SD₃₂

yes

—

C₃²×C₄

Q₃₂

yes

—

C₃²×C₄

#124

yes

characteristic

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

#107

—

#107

C₂×C₃⋊D₃

yes

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#213

yes

—

C₂×C₃×C₄

#213

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

#215

yes

—

C₃×C₈

#239

yes

—

C₂×C₃²

#486

yes

characteristic

C₂×C₃²

C₂×SD₃₂

yes

characteristic

C₂×C₃²

C₂×Q₃₂

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#638

yes

characteristic

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

#638

yes

characteristic

104

C₁₆

—

105

C₂²×C₃

C₂×#213

yes

—

106

C₃×C₄

—

107

C₃×C₄

#827

yes

—

108

C₃×C₄

#829

yes

—

109

C₃×C₄

C₂×#215

yes

—

110

C₃×C₄

#947

yes

—

111

Dic₃

—

112

C₃²

#2241

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#3753

yes

characteristic

119

C₂×C₄

#3753

yes

characteristic

120

C₂×C₄

C₂×#638

yes

characteristic

121

C₈

#3755

yes

—

122

C₈

#3832

yes

—

123

C₂×C₃

#5088

yes

—

124

C₂×C₃

C₂×#827

yes

—

125

C₂×C₃

C₂×#829

yes

—

126

C₂²

C₂×#3753

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#63654

yes

—

130

C₄

#63656

yes

—

131

C₄

C₂×#3755

yes

characteristic

132

C₄

#64156

yes

characteristic

133

C₃

#71382

yes

complemented

134

C₂

#10592787

yes

characteristic

135

C₂

C₂×#63654

yes

characteristic

136

C₂

C₂×#63656

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#3753
commutator, cosocle	C₂×C₃²×C₈	C₂³
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#500	C₂
socle, prime radical	C₂²×C₃²	C₂×D₈
hypercenter	#500	C₃⋊D₃
hypocenter	C₃²	#2241
LCS	G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#3753

commutator, cosocle

C₂×C₃²×C₈

C₂³

Frattini

C₂×C₈

C₂²×C₃⋊D₃

Fitting

C₃²×#500

C₂

socle, prime radical

C₂²×C₃²

C₂×D₈

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
112. C₃²	#2241	semidirect
133. C₃	#71382	semidirect

H ≅ G/N

type

112. C₃²

#2241

semidirect

133. C₃

#71382

semidirect

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

subgroup	index
C₃²×#500	2
#10592398	2
#10589358	2
C₄×C₃⋊(C₃⋊C₁₆)	2
#71382	3

subgroup

index

C₃²×#500

#10592398

#10589358

C₄×C₃⋊(C₃⋊C₁₆)

#71382

Subgroups — #11800070

Conjugacy classes of subgroups

Complements

Maximal subgroups