Group #11711541 — subgroups

Subgroups — #11711541

Order 768.

171 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 39 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×#504⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#73126	C₂	1	2	yes	maximal
3	#12843	—	3	1	no	maximal, Sylow
4	C₂×#131⋊C₃	C₂²	1	1	yes	characteristic
5	#5426	—	2	2	no	—
6	#504⋊C₃	C₄	1	2	yes	complemented
7	#4505	—	4	1	no	maximal
8	C₂×#504	D₃	1	1	yes	characteristic
9	C₂×#368	—	3	1	no	—
10	C₂×#367	—	3	1	no	—
11	#1816	—	3	2	no	—
12	#1387	—	3	2	no	—
13	C₂×#41⋊C₃	D₄	1	1	yes	characteristic
14	#131⋊C₃	C₂×C₄	1	1	yes	characteristic
15	#131⋊C₃	Q₈	1	1	yes	characteristic
16	#859	—	4	2	no	—
17	C₂×C₃×C₁₆	—	4	1	no	—
18	#818	—	4	2	no	—
19	C₂×#131	C₂×D₃	1	1	yes	characteristic
20	#504	—	3	2	no	—
21	#504	Dic₃	1	2	yes	complemented
22	C₂×M₃₂	—	3	1	no	—
23	C₂²×C₁₆	—	3	1	no	—
24	C₂×#108	—	3	2	no	—
25	C₂×#107	—	3	2	no	—
26	#417	—	6	2	no	—
27	#368	—	6	2	no	—
28	#367	—	6	1	no	—
29	#367	—	6	1	no	—
30	#343	—	6	2	no	—
31	#41⋊C₃	C₄⋊C₄	1	1	yes	characteristic
32	#41⋊C₃	Q₁₆	1	1	yes	characteristic
33	C₂×SL(2,𝔽₃)	D₈	1	1	yes	characteristic
34	C₂×C₃×C₈	—	4	1	no	—
35	C₄⋊Dic₃	—	8	2	no	—
36	C₃×C₁₆	—	4	2	no	—
37	C₂×#41	D₁₂	1	1	yes	characteristic
38	#131	—	3	1	no	—
39	#131	—	3	1	no	—
40	#131	Dic₆	1	1	yes	characteristic
41	#131	C₂×Dic₃	1	1	yes	characteristic
42	C₂×M₁₆	—	3	1	no	—
43	C₂×M₁₆	—	6	2	no	—
44	C₂²×C₈	—	3	1	no	—
45	C₂×C₄⋊C₄	—	6	2	no	—
46	M₃₂	—	3	2	no	—
47	M₃₂	—	6	1	no	—
48	C₂×C₁₆	—	3	1	no	—
49	C₂×C₁₆	—	3	2	no	—
50	C₂×C₁₆	—	3	2	no	—
51	C₂×C₁₆	S₄	1	1	yes	characteristic
52	#108	—	6	2	no	—
53	#108	—	6	2	no	—
54	#107	—	6	2	no	—
55	#107	—	6	2	no	—
56	#103	—	12	2	no	—
57	#102	—	12	2	no	—
58	C₂×C₃×C₄	—	4	1	no	—
59	C₂×Dic₃	—	16	2	no	—
60	SL(2,𝔽₃)	#107	1	1	yes	characteristic
61	C₃×C₈	—	4	1	no	—
62	C₃×C₈	—	4	1	no	—
63	#41	—	3	1	no	—
64	#41	—	3	1	no	—
65	#41	Dic₁₂	1	1	yes	characteristic
66	#41	C₄⋊Dic₃	1	1	yes	characteristic
67	C₂×Q₈	D₂₄	1	1	yes	characteristic
68	C₂×D₄	—	3	1	no	—
69	C₂²×C₄	—	3	1	no	—
70	C₂²×C₄	—	12	2	no	—
71	M₁₆	—	3	1	no	—
72	M₁₆	—	3	1	no	—
73	M₁₆	—	6	1	no	—
74	M₁₆	—	12	2	no	—
75	M₁₆	—	12	2	no	—
76	C₂×C₈	—	3	1	no	—
77	C₂×C₈	—	3	1	no	—
78	C₂×C₈	—	3	1	no	—
79	C₂×C₈	—	3	1	no	—
80	C₂×C₈	—	3	1	no	—
81	C₂×C₈	—	12	2	no	—
82	C₂×C₈	C₂×S₄	1	1	yes	characteristic
83	C₄⋊C₄	—	12	2	no	—
84	C₄⋊C₄	—	12	2	no	—
85	C₁₆	—	3	2	no	—
86	C₁₆	SL(2,ℤ/4)	1	2	yes	—
87	C₂²×C₃	—	4	1	no	—
88	C₃×C₄	—	4	1	no	—
89	C₃×C₄	—	4	1	no	—
90	Dic₃	—	32	2	no	—
91	C₂³	—	3	1	no	—
92	Q₈	—	3	1	no	—
93	Q₈	#818	1	1	yes	characteristic
94	D₄	—	3	1	no	—
95	D₄	—	3	1	no	—
96	D₄	—	6	1	no	—
97	C₂×C₄	—	3	1	no	—
98	C₂×C₄	—	3	1	no	—
99	C₂×C₄	—	3	1	no	—
100	C₂×C₄	—	3	1	no	—
101	C₂×C₄	—	3	1	no	—
102	C₂×C₄	—	24	2	no	—
103	C₂×C₄	—	48	2	no	—
104	C₂×C₄	C₄⋊S₄	1	1	yes	characteristic
105	C₈	—	3	1	no	—
106	C₈	—	3	1	no	—
107	C₈	—	24	2	no	—
108	C₈	A₄⋊Q₈	1	1	yes	characteristic
109	C₈	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
110	C₂×C₃	—	4	2	no	—
111	C₂×C₃	—	4	1	no	—
112	C₂²	—	3	1	no	—
113	C₂²	—	3	1	no	—
114	C₂²	—	6	2	no	—
115	C₂²	#5402	1	1	yes	characteristic
116	C₄	—	3	1	no	—
117	C₄	—	3	1	no	—
118	C₄	—	48	2	no	—
119	C₄	#5398	1	1	yes	characteristic
120	C₄	C₄⋊SL(2,ℤ/4)	1	1	yes	characteristic
121	C₃	—	4	1	no	Sylow
122	C₂	—	6	1	no	—
123	C₂	—	6	1	no	—
124	C₂	#73041	1	2	yes	—
125	C₂	#73111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504⋊C₃

C₂

yes

maximal, characteristic

#73126

C₂

yes

maximal

#12843

—

maximal, Sylow

C₂×#131⋊C₃

C₂²

yes

characteristic

#5426

—

#504⋊C₃

C₄

yes

—

D₃

yes

—

—

—

—

C₂×#41⋊C₃

D₄

yes

characteristic

#131⋊C₃

C₂×C₄

yes

characteristic

#131⋊C₃

Q₈

yes

characteristic

#859

—

C₂×C₃×C₁₆

—

#818

—

C₂×#131

C₂×D₃

yes

characteristic

#504

—

#504

Dic₃

yes

complemented

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

characteristic

#41⋊C₃

Q₁₆

yes

characteristic

C₂×SL(2,𝔽₃)

D₈

yes

characteristic

C₂×C₃×C₈

—

C₄⋊Dic₃

—

C₃×C₁₆

—

C₂×#41

D₁₂

yes

—

—

Dic₆

yes

characteristic

#131

C₂×Dic₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

S₄

yes

—

—

—

—

—

—

C₂×C₃×C₄

—

C₂×Dic₃

—

SL(2,𝔽₃)

#107

yes

characteristic

C₃×C₈

—

C₃×C₈

—

#41

—

#41

—

#41

Dic₁₂

yes

characteristic

#41

C₄⋊Dic₃

yes

characteristic

C₂×Q₈

D₂₄

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×S₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₁₆

—

C₁₆

SL(2,ℤ/4)

yes

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

C₂³

—

Q₈

—

Q₈

#818

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₄⋊S₄

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

A₄⋊Q₈

yes

characteristic

109

C₈

C₂×SL(2,ℤ/4)

yes

characteristic

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#5402

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#5398

yes

characteristic

120

C₄

C₄⋊SL(2,ℤ/4)

yes

characteristic

121

C₃

—

Sylow

122

C₂

—

123

C₂

—

124

C₂

#73041

yes

—

125

C₂

#73111

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#5402
commutator	#131⋊C₃	C₂×C₄
Frattini	C₂×C₈	C₂×S₄
Fitting	C₂×#504	D₃
prime radical	C₂×#41⋊C₃	D₄
cosocle	C₂×#131⋊C₃	C₂²
hypercenter	C₂×C₁₆	S₄
hypocenter	SL(2,𝔽₃)	#107
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

C₂²

C₂×C₄

C₂×C₈

C₂×S₄

Fitting

C₂×#504

D₃

prime radical

C₂×#41⋊C₃

D₄

cosocle

C₂×#131⋊C₃

C₂²

hypercenter

C₂×C₁₆

S₄

G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
6. #504⋊C₃	C₄	semidirect
21. #504	Dic₃	semidirect

H ≅ G/N

type

6. #504⋊C₃

C₄

semidirect

21. #504

Dic₃

semidirect

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

subgroup	index
C₂×#504⋊C₃	2
#73126	2
#12843	3
#4505	4

subgroup

index

C₂×#504⋊C₃

#73126

#12843

#4505

Subgroups — #11711541

Conjugacy classes of subgroups

Complements

Maximal subgroups