Group #10707207 — subgroups

Subgroups — #10707207

Order 768.

198 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₃×#2176	C₂	1	1	yes	maximal, characteristic
2	#70906	C₂	1	2	yes	maximal
3	#67764	C₂	1	2	yes	maximal
4	#67513	C₂	1	1	yes	maximal, characteristic
5	#67511	C₂	1	1	yes	maximal, characteristic
6	#12088	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₃₂	C₄	1	1	yes	characteristic
8	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
9	#5024	—	2	2	no	—
10	C₃×#367	C₄	1	1	yes	characteristic
11	C₃×#358	C₄	1	2	yes	—
12	C₃×#346	C₂²	1	1	yes	characteristic
13	#4535	—	2	2	no	—
14	#4461	C₂²	1	1	yes	characteristic
15	#4457	C₂²	1	2	yes	—
16	#4455	C₂²	1	1	yes	characteristic
17	#2176	D₃	1	1	yes	characteristic
18	#1593	—	3	2	no	—
19	#1368	—	3	1	no	—
20	#1334	—	3	2	no	—
21	#1313	—	3	1	no	—
22	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
23	C₃×C₄×Q₈	—	2	2	no	—
24	C₃×Q₃₂	—	2	2	no	—
25	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₃×#107	C₂×C₄	1	2	yes	—
28	C₃×#103	—	2	2	no	—
29	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊C₁₆	—	2	1	no	—
31	C₄⋊(C₃⋊C₈)	—	2	2	no	—
32	C₄⋊(C₃⋊C₈)	—	4	2	no	—
33	#803	—	4	2	no	—
34	C₄×C₃⋊C₈	D₄	1	2	yes	—
35	C₂×Q₃₂	Dic₃	1	1	yes	characteristic
36	#446	—	6	2	no	—
37	C₄×Q₁₆	C₂×D₃	1	2	yes	—
38	#367	Dic₃	1	1	yes	characteristic
39	#358	Dic₃	1	2	yes	—
40	#346	—	3	1	no	—
41	#346	C₂×D₃	1	1	yes	characteristic
42	#334	—	3	2	no	—
43	Q₈⋊C₈	—	6	2	no	—
44	#322	—	3	1	no	—
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₃×Q₁₆	—	2	2	no	—
47	C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₈	#41	1	1	yes	characteristic
50	C₃×C₄⋊C₄	—	2	2	no	—
51	C₃×C₄⋊C₄	—	4	2	no	—
52	C₃×C₄²	—	4	2	no	—
53	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃⋊C₈	—	2	2	no	—
55	C₂×C₃⋊C₈	—	4	2	no	—
56	C₃×C₁₆	—	2	1	no	—
57	C₃×C₁₆	M₁₆	1	2	yes	—
58	C₃⋊C₁₆	—	4	1	no	—
59	C₂×Q₁₆	C₂×Dic₃	1	2	yes	—
60	C₄×Q₈	—	2	2	no	—
61	Q₃₂	—	2	2	no	—
62	C₂×C₁₆	—	6	1	no	—
63	C₂×C₁₆	C₂×Dic₃	1	1	yes	characteristic
64	C₂×C₁₆	C₂×Dic₃	1	1	yes	characteristic
65	#107	C₂×Dic₃	1	2	yes	—
66	C₄⋊C₈	—	6	2	no	—
67	C₄⋊C₈	—	12	2	no	—
68	#103	—	2	2	no	—
69	#97	—	12	2	no	—
70	C₄×C₈	—	3	2	no	—
71	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₃×Q₈	—	4	2	no	—
73	C₃×Q₈	—	4	2	no	—
74	C₂×C₃×C₄	—	4	2	no	—
75	C₂×C₃×C₄	—	4	2	no	—
76	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
78	C₃×C₈	—	2	1	no	—
79	C₃×C₈	C₂×M₁₆	1	1	yes	characteristic
80	C₃×C₈	#131	1	1	yes	characteristic
81	C₃⋊C₈	—	4	2	no	—
82	C₃⋊C₈	—	8	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	Q₁₆	—	2	2	no	—
85	Q₁₆	—	2	2	no	—
86	C₂×C₈	—	6	2	no	—
87	C₂×C₈	—	12	2	no	—
88	C₂×C₈	#237	1	1	yes	characteristic
89	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₃×D₄	1	1	yes	characteristic
94	C₁₆	—	2	1	no	—
95	C₁₆	—	12	1	no	—
96	C₁₆	#208	1	2	yes	—
97	C₂²×C₃	#442	1	1	yes	characteristic
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	—	8	2	no	—
100	C₃×C₄	—	8	2	no	—
101	C₃×C₄	#436	1	1	yes	characteristic
102	C₃×C₄	#444	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	Q₈	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	#911	1	2	yes	—
108	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	12	2	no	—
111	C₈	—	24	2	no	—
112	C₈	C₂×#208	1	1	yes	characteristic
113	C₈	#948	1	1	yes	characteristic
114	C₂×C₃	#1594	1	1	yes	characteristic
115	C₂×C₃	#2180	1	2	yes	—
116	C₂²	#5152	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	8	2	no	—
119	C₄	—	8	2	no	—
120	C₄	#5012	1	1	yes	characteristic
121	C₄	#5197	1	1	yes	characteristic
122	C₃	#12088	1	1	yes	characteristic, complemented, Sylow
123	C₂	#70882	1	1	yes	characteristic
124	C₂	#71521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2176

C₂

yes

maximal, characteristic

#70906

C₂

yes

maximal

#67764

C₂

yes

maximal

#67513

C₂

yes

maximal, characteristic

#67511

C₂

yes

maximal, characteristic

#12088

—

maximal, Sylow

C₂×C₃×Q₃₂

C₄

yes

characteristic

C₃×C₄×Q₁₆

C₂²

yes

—

#5024

—

C₃×#367

C₄

yes

characteristic

C₃×#358

C₄

yes

—

C₃×#346

C₂²

yes

characteristic

#4535

—

#4461

C₂²

yes

characteristic

#4457

C₂²

yes

—

#4455

C₂²

yes

characteristic

#2176

D₃

yes

—

—

—

—

C₂×C₃×Q₁₆

C₂×C₄

yes

—

C₃×C₄×Q₈

—

C₃×Q₃₂

—

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₃×#107

C₂×C₄

yes

—

C₃×#103

—

C₃×C₄×C₈

C₂³

yes

characteristic

C₂×C₃⋊C₁₆

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

D₄

yes

—

C₂×Q₃₂

Dic₃

yes

characteristic

#446

—

C₄×Q₁₆

C₂×D₃

yes

—

#367

Dic₃

yes

characteristic

#358

Dic₃

yes

—

#346

—

#346

C₂×D₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

M₁₆

yes

—

C₃⋊C₁₆

—

C₂×Q₁₆

C₂×Dic₃

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₃

yes

characteristic

C₂×C₁₆

C₂×Dic₃

yes

characteristic

#107

C₂×Dic₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×D₄

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₈

—

C₃×C₈

C₂×M₁₆

yes

characteristic

C₃×C₈

#131

yes

characteristic

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#237

yes

characteristic

C₂×C₈

C₂²×Dic₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₃×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#208

yes

—

C₂²×C₃

#442

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

#436

yes

characteristic

102

C₃×C₄

#444

yes

characteristic

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#911

yes

—

108

C₂×C₄

D₄×Dic₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×#208

yes

characteristic

113

C₈

#948

yes

characteristic

114

C₂×C₃

#1594

yes

characteristic

115

C₂×C₃

#2180

yes

—

116

C₂²

#5152

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#5012

yes

characteristic

121

C₄

#5197

yes

characteristic

122

C₃

#12088

yes

characteristic, complemented, Sylow

123

C₂

#70882

yes

characteristic

124

C₂

#71521

yes

—

	subgroup	G/H
center	C₂²	#5152
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2176	C₂
socle, prime radical	C₂²×C₃	#442
cosocle	C₃×C₄×C₈	C₂³
hypercenter	#2176	D₃
hypocenter	C₃	#12088
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#5152

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 ◁ #2176

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
C₃×#2176	2
#70906	2
#67764	2
#67513	2
#67511	2
#12088	3

subgroup

index

Subgroups — #10707207

Conjugacy classes of subgroups

Complements

Maximal subgroups