Group #13052021 — subgroups

Subgroups — #89891⋊C₃ #13052021

Order 1344.

228 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 104 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×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	#6489⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	(ΓL(1,𝔽₈)⋊C₄)⋊C₈	C₂	1	4	yes	maximal, complemented
5	#89891	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	M₁₆×ΓL(1,𝔽₈)	—	2	2	no	—
8	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
9	(C₂²⋊Dic₇)⋊C₃	C₄	1	2	yes	—
10	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₄	1	2	yes	—
11	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
12	C₂×ΓL(1,𝔽₈)⋊C₈	—	2	2	no	—
13	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic
15	#6489	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
17	Dic₇⋊C₈	C₂×C₃	1	4	yes	complemented
18	C₃×#424	—	7	1	no	maximal, Hall
19	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
22	C₂×C₄×ΓL(1,𝔽₈)	Q₈	1	2	yes	—
23	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
24	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	4	yes	—
25	C₈×ΓL(1,𝔽₈)	—	2	2	no	—
26	ΓL(1,𝔽₈)⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
28	C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
30	C₂²⋊Dic₇	C₃×C₄	1	2	yes	—
31	C₄⋊Dic₇	C₃×C₄	1	2	yes	—
32	C₄×Dic₇	C₂²×C₃	1	2	yes	—
33	C₂×C₇⋊C₈	—	2	2	no	—
34	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	—
35	C₂×C₃×M₁₆	—	7	1	no	—
36	C₂²×C₃×C₈	—	7	1	no	—
37	C₃×#117	—	7	1	no	—
38	C₃×C₄⋊C₈	—	7	4	no	—
39	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
40	C₂²×ΓL(1,𝔽₈)	C₄⋊C₄	1	2	yes	—
41	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
42	C₄×ΓL(1,𝔽₈)	C₄⋊C₄	1	2	yes	—
43	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
44	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
45	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
46	#424	—	7	1	no	Sylow
47	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
51	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
52	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
53	C₇×C₈	—	2	2	no	—
54	C₇⋊C₈	—	2	2	no	—
55	C₂²×C₃×C₄	—	7	1	no	—
56	C₃×M₁₆	—	14	2	no	—
57	C₂×C₃×C₈	—	7	2	no	—
58	C₂×C₃×C₈	—	7	2	no	—
59	C₂×C₃×C₈	—	14	2	no	—
60	C₃×C₄⋊C₄	—	7	2	no	—
61	C₃×C₂²⋊C₄	—	7	2	no	—
62	C₃×C₄²	—	7	2	no	—
63	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
64	C₂×ΓL(1,𝔽₈)	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×ΓL(1,𝔽₈)	#131	1	2	yes	—
66	C₂×M₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic
67	C₂²×C₈	—	7	1	no	—
68	#117	—	7	1	no	—
69	C₄⋊C₈	—	7	4	no	—
70	C₂²×C₇	—	2	2	no	—
71	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
72	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
74	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
75	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
76	Dic₇	—	2	4	no	—
77	C₂³×C₃	—	7	1	no	—
78	C₂×C₃×C₄	—	7	4	no	—
79	C₂×C₃×C₄	—	7	2	no	—
80	C₂×C₃×C₄	—	7	1	no	—
81	C₂×C₃×C₄	—	7	2	no	—
82	C₂×C₃×C₄	—	7	1	no	—
83	C₃×C₈	—	14	2	no	—
84	C₃×C₈	—	14	2	no	—
85	ΓL(1,𝔽₈)	#424	1	1	yes	characteristic, complemented, Hall
86	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	7	2	no	—
89	C₂×C₈	—	14	2	no	—
90	C₂×C₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
91	C₄⋊C₄	—	7	2	no	—
92	C₂²⋊C₄	—	7	2	no	—
93	C₄²	—	7	2	no	—
94	C₂×C₇	—	2	2	no	—
95	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇	C₃×#131	1	2	yes	—
97	C₂²×C₃	—	7	1	no	—
98	C₂²×C₃	—	7	2	no	—
99	C₂²×C₃	—	14	2	no	—
100	C₃×C₄	—	7	1	no	—
101	C₃×C₄	—	7	1	no	—
102	C₃×C₄	—	7	2	no	—
103	C₃×C₄	—	14	4	no	—
104	C₂³	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
105	C₂×C₄	—	7	4	no	—
106	C₂×C₄	ΓL(1,𝔽₈)⋊Q₈	1	2	yes	—
107	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
108	C₂×C₄	(C₂²⋊D₇)⋊C₃	1	2	yes	—
109	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	14	2	no	—
112	C₇	C₃×#424	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₃	—	7	2	no	—
114	C₂×C₃	—	7	1	no	—
115	C₂×C₃	—	14	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	1	2	yes	—
118	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
119	C₄	—	14	4	no	—
120	C₄	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	1	2	yes	—
121	C₄	C₂×ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
122	C₄	C₂×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
123	C₃	—	7	1	no	Sylow
124	C₂	—	2	2	no	—
125	C₂	#6470⋊C₃	1	2	yes	—
126	C₂	C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic

#6489⋊C₃

C₂

yes

maximal, characteristic

C₂²×ΓL(1,𝔽₈)⋊C₈

C₂

yes

maximal, characteristic

(ΓL(1,𝔽₈)⋊C₄)⋊C₈

C₂

yes

maximal, complemented

#89891

C₃

yes

maximal, characteristic, complemented, Hall

C₂²×C₄×ΓL(1,𝔽₈)

C₂²

yes

characteristic

M₁₆×ΓL(1,𝔽₈)

—

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

—

(C₂²⋊Dic₇)⋊C₃

C₄

yes

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₄

yes

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

—

C₂×ΓL(1,𝔽₈)⋊C₈

—

C₂×ΓL(1,𝔽₈)⋊C₈

C₂²

yes

—

C₂×C₇×M₁₆

C₂×C₃

yes

characteristic

#6489

C₂×C₃

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₃

yes

characteristic

Dic₇⋊C₈

C₂×C₃

yes

complemented

C₃×#424

—

maximal, Hall

C₂³×ΓL(1,𝔽₈)

C₂×C₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

C₂×C₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

—

C₂×C₄×ΓL(1,𝔽₈)

Q₈

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₄

C₂×C₄

yes

—

C₈×ΓL(1,𝔽₈)

—

ΓL(1,𝔽₈)⋊C₈

—

C₂²×C₄×C₇

C₂²×C₃

yes

characteristic

C₇×M₁₆

—

C₂×C₇×C₈

C₂²×C₃

yes

—

C₂²⋊Dic₇

C₃×C₄

yes

—

C₄⋊Dic₇

C₃×C₄

yes

—

C₄×Dic₇

C₂²×C₃

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂²×C₃

yes

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

—

C₃×#117

—

C₃×C₄⋊C₈

—

C₂²×ΓL(1,𝔽₈)

—

C₂²×ΓL(1,𝔽₈)

C₄⋊C₄

yes

—

C₂²×ΓL(1,𝔽₈)

C₂²×C₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

C₄⋊C₄

yes

—

C₄×ΓL(1,𝔽₈)

C₂×D₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

C₂×Q₈

yes

—

—

C₂³×C₇

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₇

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₇

C₃×D₄

yes

—

C₂×C₄×C₇

C₃×Q₈

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×Dic₇

C₂×C₃×C₄

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₃×C₄

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

—

C₂×ΓL(1,𝔽₈)

C₂×C₄⋊C₄

yes

characteristic

C₂×ΓL(1,𝔽₈)

#131

yes

—

C₂×M₁₆

AGL(1,𝔽₇)

yes

characteristic

C₂²×C₈

—

#117

—

C₄⋊C₈

—

C₂²×C₇

—

C₂²×C₇

C₃×C₄⋊C₄

yes

—

C₂²×C₇

C₂²×C₃×C₄

yes

characteristic

C₄×C₇

C₃×C₄⋊C₄

yes

—

C₄×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

C₂×C₃×Q₈

yes

characteristic

Dic₇

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#424

yes

characteristic, complemented, Hall

C₂²×C₄

C₂×AGL(1,𝔽₇)

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₇)

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

C₂×C₃×C₄⋊C₄

yes

characteristic

C₂×C₇

C₃×#131

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₂³

C₄×AGL(1,𝔽₇)

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

ΓL(1,𝔽₈)⋊Q₈

yes

—

107

C₂×C₄

C₄×AGL(1,𝔽₇)

yes

characteristic

108

C₂×C₄

(C₂²⋊D₇)⋊C₃

yes

—

109

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

110

C₈

—

111

C₈

—

112

C₇

C₃×#424

yes

characteristic, complemented, Sylow

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

C₂²

—

117

C₂²

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

yes

—

118

C₂²

C₂×C₄×AGL(1,𝔽₇)

yes

characteristic

119

C₄

—

120

C₄

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

yes

—

121

C₄

C₂×ΓL(1,𝔽₈)⋊Q₈

yes

characteristic

122

C₄

C₂×(C₂²⋊D₇)⋊C₃

yes

characteristic

123

C₃

—

Sylow

124

C₂

—

125

C₂

#6470⋊C₃

yes

—

126

C₂

C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator, socle	C₂²×C₇	C₂²×C₃×C₄
Fitting	C₂×C₇×M₁₆	C₂×C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂×M₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#424
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×AGL(1,𝔽₇)

commutator, socle

C₂²×C₇

C₂²×C₃×C₄

Fitting

C₂×C₇×M₁₆

C₂×C₃

prime radical

C₂³×ΓL(1,𝔽₈)

C₂×C₄

cosocle

C₂×C₄×C₇

C₂³×C₃

C₂×M₁₆

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
4. (ΓL(1,𝔽₈)⋊C₄)⋊C₈	C₂	semidirect
5. #89891	C₃	semidirect
17. Dic₇⋊C₈	C₂×C₃	semidirect
85. ΓL(1,𝔽₈)	#424	semidirect
112. C₇	C₃×#424	semidirect

H ≅ G/N

type

4. (ΓL(1,𝔽₈)⋊C₄)⋊C₈

C₂

semidirect

5. #89891

C₃

semidirect

17. Dic₇⋊C₈

C₂×C₃

semidirect

85. ΓL(1,𝔽₈)

#424

semidirect

112. C₇

C₃×#424

semidirect

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

subgroup	index
C₂×M₁₆×ΓL(1,𝔽₈)	2
#6489⋊C₃	2
C₂²×ΓL(1,𝔽₈)⋊C₈	2
(ΓL(1,𝔽₈)⋊C₄)⋊C₈	2
#89891	3
C₃×#424	7

subgroup

index

C₂×M₁₆×ΓL(1,𝔽₈)

C₂²×ΓL(1,𝔽₈)⋊C₈

Subgroups — #89891⋊C₃ #13052021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89891⋊C3 #13052021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89891⋊C₃ #13052021