Group #10707206 — subgroups

Subgroups — #10707206

Order 768.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#69082	C₂	1	1	yes	maximal, characteristic
2	#68524	C₂	1	2	yes	maximal
3	C₃×#1368	C₂	1	1	yes	maximal, characteristic
4	#67511	C₂	1	1	yes	maximal, characteristic
5	#67466	C₂	1	2	yes	maximal
6	#12088	—	3	1	no	maximal, Sylow
7	C₂×Dic₂₄	C₄	1	1	yes	characteristic
8	#4831	—	2	2	no	—
9	C₄×Dic₁₂	C₂²	1	2	yes	—
10	C₃×#346	C₂²	1	1	yes	characteristic
11	C₃×#334	C₂²	1	2	yes	—
12	#4505	C₄	1	1	yes	characteristic
13	#4503	C₄	1	2	yes	—
14	#4484	—	2	2	no	—
15	#4461	C₂²	1	1	yes	characteristic
16	#4455	C₂²	1	1	yes	characteristic
17	#2176	—	3	1	no	—
18	#1593	—	3	2	no	—
19	#1368	D₃	1	1	yes	characteristic
20	#1334	—	3	2	no	—
21	#1313	—	3	1	no	—
22	C₂×Dic₁₂	C₂×C₄	1	2	yes	—
23	C₄×Dic₆	—	2	2	no	—
24	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄⋊C₈	—	2	2	no	—
27	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
28	#818	C₂×C₄	1	2	yes	—
29	#816	—	2	2	no	—
30	#815	—	4	2	no	—
31	Dic₃⋊C₈	—	4	2	no	—
32	C₂×C₃⋊C₁₆	—	2	1	no	—
33	C₄×C₃⋊C₈	D₄	1	2	yes	—
34	Dic₂₄	—	2	2	no	—
35	C₂×Q₃₂	—	3	1	no	—
36	#446	—	6	2	no	—
37	C₄×Q₁₆	—	3	2	no	—
38	#367	—	3	1	no	—
39	#358	—	3	2	no	—
40	#346	—	3	1	no	—
41	#346	C₂×D₃	1	1	yes	characteristic
42	#334	C₂×D₃	1	2	yes	—
43	Q₈⋊C₈	—	6	2	no	—
44	#322	—	3	1	no	—
45	C₂×Dic₆	—	2	2	no	—
46	C₂×C₃×C₈	—	4	2	no	—
47	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₃×C₈	#41	1	1	yes	characteristic
49	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₃	—	2	2	no	—
51	Dic₃⋊C₄	—	4	2	no	—
52	C₄×Dic₃	—	4	2	no	—
53	C₂×C₃⋊C₈	—	2	2	no	—
54	Dic₁₂	—	2	2	no	—
55	Dic₁₂	—	2	2	no	—
56	C₃×C₁₆	—	2	1	no	—
57	C₃×C₁₆	M₁₆	1	2	yes	—
58	C₃⋊C₁₆	—	4	1	no	—
59	C₂×Q₁₆	—	3	2	no	—
60	C₄×Q₈	—	6	2	no	—
61	Q₃₂	—	6	2	no	—
62	C₂×C₁₆	—	6	1	no	—
63	C₂×C₁₆	C₄×D₃	1	1	yes	characteristic
64	C₂×C₁₆	C₄×D₃	1	1	yes	characteristic
65	#107	—	3	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	12	2	no	—
68	#103	—	6	2	no	—
69	#97	—	12	2	no	—
70	C₄×C₈	—	3	2	no	—
71	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₃	—	4	2	no	—
75	C₂×Dic₃	—	4	2	no	—
76	Dic₆	—	4	2	no	—
77	Dic₆	—	4	2	no	—
78	C₃×C₈	—	2	1	no	—
79	C₃×C₈	—	8	2	no	—
80	C₃×C₈	C₂×M₁₆	1	1	yes	characteristic
81	C₃×C₈	#131	1	1	yes	characteristic
82	C₃⋊C₈	—	4	2	no	—
83	C₂×Q₈	—	6	2	no	—
84	Q₁₆	—	6	2	no	—
85	Q₁₆	—	6	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	6	2	no	—
88	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
89	C₂×C₈	#239	1	1	yes	characteristic
90	C₄⋊C₄	—	6	2	no	—
91	C₄⋊C₄	—	12	2	no	—
92	C₄²	—	12	2	no	—
93	C₄²	D₃×D₄	1	1	yes	characteristic
94	C₁₆	—	2	1	no	—
95	C₁₆	—	12	1	no	—
96	C₁₆	ΓL(1,𝔽₂₅)	1	2	yes	—
97	C₂²×C₃	#442	1	1	yes	characteristic
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	#436	1	1	yes	characteristic
100	C₃×C₄	#444	1	1	yes	characteristic
101	Dic₃	—	8	2	no	—
102	Dic₃	—	8	2	no	—
103	Q₈	—	12	2	no	—
104	Q₈	—	12	2	no	—
105	C₂×C₄	—	12	2	no	—
106	C₂×C₄	—	12	2	no	—
107	C₂×C₄	#893	1	1	yes	characteristic
108	C₂×C₄	#914	1	2	yes	—
109	C₈	—	2	1	no	—
110	C₈	—	8	2	no	—
111	C₈	—	12	2	no	—
112	C₈	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
113	C₈	#907	1	1	yes	characteristic
114	C₂×C₃	#1594	1	1	yes	characteristic
115	C₂×C₃	#2180	1	2	yes	—
116	C₂²	#4869	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	24	2	no	—
119	C₄	—	24	2	no	—
120	C₄	#4838	1	1	yes	characteristic
121	C₄	#4894	1	1	yes	characteristic
122	C₃	#12088	1	1	yes	characteristic, complemented, Sylow
123	C₂	#68525	1	1	yes	characteristic
124	C₂	#69309	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#69082

C₂

yes

maximal, characteristic

#68524

C₂

yes

maximal

C₃×#1368

C₂

yes

maximal, characteristic

#67511

C₂

yes

maximal, characteristic

#67466

C₂

yes

maximal

#12088

—

maximal, Sylow

C₂×Dic₂₄

C₄

yes

characteristic

#4831

—

C₄×Dic₁₂

C₂²

yes

—

C₃×#346

C₂²

yes

characteristic

C₃×#334

C₂²

yes

—

#4505

C₄

yes

characteristic

#4503

C₄

yes

—

#4484

—

#4461

C₂²

yes

characteristic

#4455

C₂²

yes

—

—

D₃

yes

characteristic

#1334

—

#1313

—

C₂×Dic₁₂

C₂×C₄

yes

—

C₄×Dic₆

—

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄×C₈

C₂³

yes

characteristic

#818

C₂×C₄

yes

—

#816

—

#815

—

Dic₃⋊C₈

—

C₂×C₃⋊C₁₆

—

C₄×C₃⋊C₈

D₄

yes

—

Dic₂₄

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

—

—

—

C₂×D₃

yes

characteristic

#334

C₂×D₃

yes

—

Q₈⋊C₈

—

#322

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃×C₁₆

M₁₆

yes

—

C₃⋊C₁₆

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₃

yes

characteristic

C₂×C₁₆

C₄×D₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×M₁₆

yes

characteristic

C₃×C₈

#131

yes

characteristic

C₃⋊C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×D₃

yes

characteristic

C₂×C₈

#239

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₃×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

ΓL(1,𝔽₂₅)

yes

—

C₂²×C₃

#442

yes

characteristic

C₃×C₄

—

C₃×C₄

#436

yes

characteristic

100

C₃×C₄

#444

yes

characteristic

101

Dic₃

—

102

Dic₃

—

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#893

yes

characteristic

108

C₂×C₄

#914

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

113

C₈

#907

yes

characteristic

114

C₂×C₃

#1594

yes

characteristic

115

C₂×C₃

#2180

yes

—

116

C₂²

#4869

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#4838

yes

characteristic

121

C₄

#4894

yes

characteristic

122

C₃

#12088

yes

characteristic, complemented, Sylow

123

C₂

#68525

yes

characteristic

124

C₂

#69309

yes

—

	subgroup	G/H
center	C₂²	#4869
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1368	C₂
socle, prime radical	C₂²×C₃	#442
cosocle	C₃×C₄×C₈	C₂³
hypercenter	#1368	D₃
hypocenter	C₃	#12088
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#4869

commutator

C₂×C₃×C₈

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₃×C₄×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
122. C₃	#12088	semidirect

H ≅ G/N

type

122. C₃

#12088

semidirect

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

subgroup	index
#69082	2
#68524	2
C₃×#1368	2
#67511	2
#67466	2
#12088	3

subgroup

index

Subgroups — #10707206

Conjugacy classes of subgroups

Complements

Maximal subgroups