Group #11772081 — subgroups

Subgroups — #11772081

Order 1152.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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	#10592398	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#10589324	C₂	1	1	yes	maximal, characteristic
4	#67561	—	3	4	no	maximal
5	C₂×#3746	—	2	1	no	—
6	C₂×#3746	C₄	1	2	yes	—
7	#64078	—	2	1	no	—
8	C₂×C₃²×C₁₆	—	2	1	no	—
9	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	#4699	—	3	4	no	—
12	C₃×C₄⋊C₁₆	D₃	1	4	yes	—
13	#4458	—	3	4	no	—
14	C₂×#634	—	2	2	no	—
15	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
17	C₃²×C₄²	D₄	1	1	yes	characteristic
18	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
19	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
20	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
21	#3746	—	4	2	no	—
22	#3746	—	4	1	no	—
23	#3746	—	4	1	no	—
24	C₃²×C₁₆	—	4	1	no	—
25	#1362	—	9	1	no	Sylow
26	C₂×Dic₁₂	—	3	8	no	—
27	C₂×Dic₁₂	—	6	4	no	—
28	#869	—	6	4	no	—
29	C₂×C₃×C₁₆	—	2	4	no	—
30	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
31	C₄⋊(C₃⋊C₈)	—	6	4	no	—
32	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
34	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	—	4	2	no	—
36	#634	—	4	2	no	—
37	#634	—	4	2	no	—
38	C₃²×C₈	—	2	1	no	—
39	C₃²×C₈	D₈	1	1	yes	characteristic
40	C₃²×C₈	SD₁₆	1	1	yes	characteristic
41	C₃⋊(C₃⋊C₈)	—	8	1	no	—
42	#494	—	9	1	no	—
43	C₄⋊C₁₆	C₃⋊D₃	1	1	yes	characteristic
44	#335	—	9	1	no	—
45	C₂×Dic₆	—	6	8	no	—
46	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
47	C₂×C₃×C₈	#67	1	4	yes	—
48	C₃×C₄²	D₁₂	1	4	yes	—
49	C₄⋊Dic₃	—	12	4	no	—
50	C₄⋊Dic₃	—	12	4	no	—
51	C₂×C₃⋊C₈	—	12	4	no	—
52	Dic₁₂	—	12	8	no	—
53	Dic₁₂	—	12	4	no	—
54	Dic₁₂	—	12	4	no	—
55	C₃×C₁₆	—	4	4	no	—
56	C₂²×C₃²	#102	1	1	yes	characteristic
57	C₃²×C₄	—	2	1	no	—
58	C₃²×C₄	#100	1	1	yes	characteristic
59	C₃²×C₄	#102	1	1	yes	characteristic
60	C₃²×C₄	SD₃₂	1	1	yes	characteristic
61	C₃²×C₄	Q₃₂	1	1	yes	characteristic
62	C₃⋊Dic₃	—	8	2	no	—
63	C₂×Q₁₆	—	9	2	no	—
64	C₂×Q₁₆	—	18	1	no	—
65	C₄⋊Q₈	—	18	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₄⋊C₈	—	18	1	no	—
68	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₂×C₃×C₄	#204	1	4	yes	—
70	C₂×C₃×C₄	D₂₄	1	4	yes	—
71	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
72	C₂×Dic₃	—	12	8	no	—
73	Dic₆	—	12	8	no	—
74	Dic₆	—	12	8	no	—
75	C₃×C₈	—	2	4	no	—
76	C₃×C₈	#213	1	4	yes	—
77	C₃×C₈	#215	1	4	yes	—
78	C₃⋊C₈	—	24	4	no	—
79	C₂×C₃²	#331	1	1	yes	characteristic
80	C₂×C₃²	#358	1	1	yes	characteristic
81	C₂×C₃²	#362	1	1	yes	characteristic
82	C₂×Q₈	—	18	2	no	—
83	Q₁₆	—	36	2	no	—
84	Q₁₆	—	36	1	no	—
85	Q₁₆	—	36	1	no	—
86	C₂×C₈	—	36	1	no	—
87	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
88	C₂×C₈	#638	1	1	yes	characteristic
89	C₄⋊C₄	—	36	1	no	—
90	C₄⋊C₄	—	36	1	no	—
91	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
92	C₁₆	—	4	1	no	—
93	C₂²×C₃	#821	1	4	yes	—
94	C₃×C₄	—	2	4	no	—
95	C₃×C₄	#800	1	4	yes	—
96	C₃×C₄	Dic₂₄	1	4	yes	—
97	C₃×C₄	#809	1	4	yes	—
98	C₃×C₄	#823	1	4	yes	—
99	Dic₃	—	24	8	no	—
100	C₃²	#1362	1	1	yes	characteristic, complemented, Sylow
101	Q₈	—	36	2	no	—
102	Q₈	—	36	2	no	—
103	C₂×C₄	—	36	2	no	—
104	C₂×C₄	#3744	1	1	yes	characteristic
105	C₂×C₄	#3745	1	1	yes	characteristic
106	C₂×C₄	#3752	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	72	1	no	—
109	C₈	#3753	1	1	yes	characteristic
110	C₈	#3755	1	1	yes	characteristic
111	C₂×C₃	#4485	1	4	yes	—
112	C₂×C₃	#4503	1	4	yes	—
113	C₂×C₃	#4517	1	4	yes	—
114	C₂²	#63648	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	72	2	no	—
117	C₄	#63627	1	1	yes	characteristic
118	C₄	#63628	1	1	yes	characteristic
119	C₄	#63636	1	1	yes	characteristic
120	C₄	#63650	1	1	yes	characteristic
121	C₃	#67561	1	4	yes	complemented
122	C₂	#10589351	1	1	yes	characteristic
123	C₂	#10589369	1	1	yes	characteristic
124	C₂	#10589383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592398

C₂

yes

maximal, characteristic

C₃²×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#10589324

C₂

yes

maximal, characteristic

—

—

C₄

yes

—

#64078

—

C₂×C₃²×C₁₆

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊C₈))

—

#4699

—

C₃×C₄⋊C₁₆

D₃

yes

—

#4458

—

C₂×#634

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄²

D₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

#3746

—

#3746

—

#3746

—

C₃²×C₁₆

—

#1362

—

Sylow

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₄⋊(C₃⋊C₈)

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

D₈

yes

characteristic

C₃²×C₈

SD₁₆

yes

characteristic

C₃⋊(C₃⋊C₈)

—

#494

—

C₄⋊C₁₆

C₃⋊D₃

yes

characteristic

#335

—

C₂×Dic₆

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

D₁₂

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₂²×C₃²

#102

yes

characteristic

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃²×C₄

SD₃₂

yes

characteristic

C₃²×C₄

Q₃₂

yes

characteristic

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

#204

yes

—

C₂×C₃×C₄

D₂₄

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#213

yes

—

C₃×C₈

#215

yes

—

C₃⋊C₈

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

C₂×C₈

#638

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄⋊(C₃⋊D₃)

yes

characteristic

C₁₆

—

C₂²×C₃

#821

yes

—

C₃×C₄

—

C₃×C₄

#800

yes

—

C₃×C₄

Dic₂₄

yes

—

C₃×C₄

#809

yes

—

C₃×C₄

#823

yes

—

Dic₃

—

100

C₃²

#1362

yes

characteristic, complemented, Sylow

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

#3744

yes

characteristic

105

C₂×C₄

#3745

yes

characteristic

106

C₂×C₄

#3752

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

#3753

yes

characteristic

110

C₈

#3755

yes

characteristic

111

C₂×C₃

#4485

yes

—

112

C₂×C₃

#4503

yes

—

113

C₂×C₃

#4517

yes

—

114

C₂²

#63648

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

#63627

yes

characteristic

118

C₄

#63628

yes

characteristic

119

C₄

#63636

yes

characteristic

120

C₄

#63650

yes

characteristic

121

C₃

#67561

yes

complemented

122

C₂

#10589351

yes

characteristic

123

C₂

#10589369

yes

characteristic

124

C₂

#10589383

yes

characteristic

	subgroup	G/H
center	C₂²	#63648
commutator	C₂×C₃²×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×C₃⋊D₃
Fitting	C₃²×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃²	#102
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	C₄⋊C₁₆	C₃⋊D₃
hypocenter	C₃²	#1362
LCS	G ▷ C₂×C₃²×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#63648

commutator

C₂×C₃²×C₈

C₂×C₄

Frattini

C₄×C₈

C₂×C₃⋊D₃

Fitting

C₃²×C₄⋊C₁₆

C₂

socle, prime radical

C₂²×C₃²

#102

cosocle

C₃²×C₄×C₈

C₂²

hypercenter

C₄⋊C₁₆

C₃⋊D₃

hypocenter

C₃²

#1362

LCS

G ▷ C₂×C₃²×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
100. C₃²	#1362	semidirect
121. C₃	#67561	semidirect

H ≅ G/N

type

100. C₃²

#1362

semidirect

121. C₃

#67561

semidirect

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

subgroup	index
#10592398	2
C₃²×C₄⋊C₁₆	2
#10589324	2
#67561	3

subgroup

index

#10592398

C₃²×C₄⋊C₁₆

#10589324

#67561

Subgroups — #11772081

Conjugacy classes of subgroups

Complements

Maximal subgroups