Group #13051993 — subgroups

Subgroups — #89865⋊C₃ #13051993

Order 1344.

196 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 76 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	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#6432⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#6384⋊C₃	C₂	1	4	yes	maximal
4	C₄×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#89865	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂×Q₈×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
8	C₄⋊C₄×ΓL(1,𝔽₈)	—	2	1	no	—
9	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
10	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	—	2	1	no	—
12	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	—	2	1	no	—
13	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	2	yes	—
14	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
15	#6432	C₂×C₃	1	1	yes	characteristic
16	#6384	C₂×C₃	1	4	yes	—
17	C₄×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×#487	—	7	1	no	maximal, Hall
19	Q₈×ΓL(1,𝔽₈)	—	4	2	no	—
20	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
21	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
22	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
23	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
24	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
25	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
26	ΓL(1,𝔽₈)⋊C₈	—	2	2	no	—
27	C₂×C₇×Q₈	—	2	1	no	—
28	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
29	C₇×C₄⋊C₄	—	2	1	no	—
30	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
31	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
32	C₄⋊Dic₇	—	2	1	no	—
33	C₄⋊Dic₇	—	2	1	no	—
34	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	—
35	C₃×C₄⋊Q₈	—	7	1	no	—
36	C₃×C₄⋊Q₈	—	7	1	no	—
37	C₃×#103	—	7	4	no	—
38	C₃×C₄×C₈	—	7	1	no	—
39	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
40	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
41	C₄×ΓL(1,𝔽₈)	SD₁₆	1	2	yes	—
42	C₄×ΓL(1,𝔽₈)	Q₁₆	1	2	yes	—
43	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
44	ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
45	#487	—	7	1	no	Sylow
46	C₇×Q₈	—	4	2	no	—
47	C₂×C₄×C₇	—	2	2	no	—
48	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
51	C₂×Dic₇	—	2	2	no	—
52	Dic₁₄	—	2	4	no	—
53	C₇⋊C₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	7	2	no	—
55	C₂×C₃×Q₈	—	14	1	no	—
56	C₂×C₃×C₈	—	7	2	no	—
57	C₃×C₄⋊C₄	—	7	2	no	—
58	C₃×C₄⋊C₄	—	14	1	no	—
59	C₃×C₄⋊C₄	—	14	1	no	—
60	C₃×C₄⋊C₄	—	14	1	no	—
61	C₃×C₄²	—	7	1	no	—
62	C₂×ΓL(1,𝔽₈)	#124	1	1	yes	characteristic
63	C₂×ΓL(1,𝔽₈)	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×ΓL(1,𝔽₈)	C₂×Q₁₆	1	1	yes	characteristic
65	C₄⋊Q₈	—	7	1	no	—
66	C₄⋊Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic
67	#103	—	7	4	no	—
68	C₄×C₈	—	7	1	no	—
69	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
70	C₄×C₇	—	4	2	no	—
71	C₄×C₇	C₃×SD₁₆	1	2	yes	—
72	C₄×C₇	C₃×Q₁₆	1	2	yes	—
73	C₄×C₇	C₃×#41	1	2	yes	—
74	Dic₇	—	4	2	no	—
75	C₃×Q₈	—	14	4	no	—
76	C₃×Q₈	—	28	2	no	—
77	C₂×C₃×C₄	—	7	1	no	—
78	C₂×C₃×C₄	—	7	1	no	—
79	C₂×C₃×C₄	—	7	1	no	—
80	C₂×C₃×C₄	—	14	2	no	—
81	C₂×C₃×C₄	—	14	2	no	—
82	C₃×C₈	—	14	2	no	—
83	ΓL(1,𝔽₈)	#487	1	1	yes	characteristic, complemented, Hall
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	7	2	no	—
86	C₂×C₈	—	7	2	no	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	14	1	no	—
89	C₄⋊C₄	—	14	1	no	—
90	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
91	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
92	C₂×C₇	C₃×#124	1	1	yes	characteristic
93	C₂×C₇	C₂×C₃×SD₁₆	1	1	yes	characteristic
94	C₂×C₇	C₂×C₃×Q₁₆	1	1	yes	characteristic
95	C₂²×C₃	—	7	1	no	—
96	C₃×C₄	—	7	2	no	—
97	C₃×C₄	—	7	2	no	—
98	C₃×C₄	—	7	2	no	—
99	C₃×C₄	—	28	2	no	—
100	C₃×C₄	—	28	2	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	14	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	14	2	no	—
105	C₂×C₄	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
106	C₂×C₄	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
108	C₈	—	14	2	no	—
109	C₇	C₃×#487	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	7	1	no	—
111	C₂×C₃	—	7	1	no	—
112	C₂×C₃	—	7	1	no	—
113	C₂²	C₂×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
114	C₄	—	4	2	no	—
115	C₄	—	28	2	no	—
116	C₄	#1142⋊C₃	1	2	yes	—
117	C₄	#1144⋊C₃	1	2	yes	—
118	C₄	#66659	1	2	yes	—
119	C₃	—	7	1	no	Sylow
120	C₂	C₂×#1142⋊C₃	1	1	yes	characteristic
121	C₂	C₂×#1144⋊C₃	1	1	yes	characteristic
122	C₂	#6510⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

ΓL(1,𝔽₈)×C₄⋊Q₈

C₂

yes

maximal, characteristic

#6432⋊C₃

C₂

yes

maximal, characteristic

#6384⋊C₃

C₂

yes

maximal

C₄×ΓL(1,𝔽₈)⋊C₈

C₂

yes

maximal, characteristic

#89865

C₃

yes

maximal, characteristic, complemented, Hall

C₂×Q₈×ΓL(1,𝔽₈)

—

C₂×ΓL(1,𝔽₈)⋊Q₈

C₂²

yes

—

C₄⋊C₄×ΓL(1,𝔽₈)

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

—

C₂×ΓL(1,𝔽₈)⋊C₈

C₂²

yes

—

C₇×C₄⋊Q₈

C₂×C₃

yes

characteristic

#6432

C₂×C₃

yes

characteristic

#6384

C₂×C₃

yes

—

C₄×C₇⋊C₈

C₂×C₃

yes

—

—

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₄

—

ΓL(1,𝔽₈)⋊Q₈

—

ΓL(1,𝔽₈)⋊C₈

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

C₂²×C₃

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₇

C₂²×C₃

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

C₂²×C₃

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×#103

—

C₃×C₄×C₈

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

—

C₄×ΓL(1,𝔽₈)

SD₁₆

yes

—

C₄×ΓL(1,𝔽₈)

Q₁₆

yes

—

C₄×ΓL(1,𝔽₈)

#41

yes

—

ΓL(1,𝔽₈)⋊C₄

—

#487

—

Sylow

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₃×D₄

yes

characteristic

C₂×C₄×C₇

C₃×D₄

yes

characteristic

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

C₇⋊C₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

yes

C₂×SD₁₆

yes

characteristic

C₂×ΓL(1,𝔽₈)

C₂×Q₁₆

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₇)

yes

characteristic

#103

—

C₄×C₈

—

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₃×SD₁₆

yes

—

C₄×C₇

C₃×Q₁₆

yes

—

C₄×C₇

C₃×#41

yes

—

Dic₇

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

ΓL(1,𝔽₈)

#487

yes

characteristic, complemented, Hall

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

C₄²

yes

C₂×C₇

yes

C₂×C₇

C₂×C₃×SD₁₆

yes

characteristic

C₂×C₇

C₂×C₃×Q₁₆

yes

characteristic

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

(C₂²⋊D₇)⋊C₃

yes

characteristic

106

C₂×C₄

(C₂²⋊D₇)⋊C₃

yes

characteristic

107

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

108

C₈

—

109

C₇

C₃×#487

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂²

C₂×(C₂²⋊D₇)⋊C₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#1142⋊C₃

yes

—

117

C₄

#1144⋊C₃

yes

—

118

C₄

#66659

yes

—

119

C₃

—

Sylow

120

C₂

C₂×#1142⋊C₃

yes

characteristic

121

C₂

C₂×#1144⋊C₃

yes

characteristic

122

C₂

#6510⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂²⋊D₇)⋊C₃
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×C₄⋊Q₈	C₂×C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×D₄
hypercenter	C₄⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#487
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×(C₂²⋊D₇)⋊C₃

commutator, cosocle

C₂×C₄×C₇

C₂³×C₃

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₇)

Fitting

C₇×C₄⋊Q₈

C₂×C₃

socle

C₂²×C₇

C₂×C₃×D₄

prime radical

C₂²×ΓL(1,𝔽₈)

C₂×D₄

C₄⋊Q₈

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
5. #89865	C₃	semidirect
83. ΓL(1,𝔽₈)	#487	semidirect
109. C₇	C₃×#487	semidirect

H ≅ G/N

type

5. #89865

C₃

semidirect

83. ΓL(1,𝔽₈)

#487

semidirect

109. C₇

C₃×#487

semidirect

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

subgroup	index
ΓL(1,𝔽₈)×C₄⋊Q₈	2
#6432⋊C₃	2
#6384⋊C₃	2
C₄×ΓL(1,𝔽₈)⋊C₈	2
#89865	3
C₃×#487	7

subgroup

index

ΓL(1,𝔽₈)×C₄⋊Q₈

C₄×ΓL(1,𝔽₈)⋊C₈

Subgroups — #89865⋊C₃ #13051993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89865⋊C3 #13051993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89865⋊C₃ #13051993