Group #10709921 — subgroups

Subgroups — #10709921

Order 768.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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×#1709	C₂	1	1	yes	maximal, characteristic
2	#71312	C₂	1	2	yes	maximal, complemented
3	#71291	C₂	1	2	yes	maximal
4	#68065	C₂	1	1	yes	maximal, characteristic, complemented
5	#67437	C₂	1	1	yes	maximal, characteristic, complemented
6	#12520	—	3	1	no	maximal, Sylow
7	C₃×#494	C₂²	1	1	yes	characteristic
8	C₃×#493	C₂²	1	1	yes	characteristic
9	C₃×#486	—	2	2	no	—
10	C₂×#829	—	2	4	no	—
11	C₂×#827	—	2	4	no	—
12	#4699	C₂²	1	2	yes	—
13	#4682	—	2	2	no	—
14	C₃×C₈²	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
16	#2248	—	3	2	no	—
17	#2247	—	3	2	no	—
18	#1713	—	3	1	no	—
19	#1709	D₃	1	1	yes	characteristic
20	#1311	—	3	1	no	—
21	C₂×C₃×Q₁₆	—	2	2	no	—
22	C₂×C₃×D₈	—	2	2	no	—
23	C₃×C₄⋊Q₈	—	2	1	no	—
24	C₃×#127	—	2	1	no	—
25	C₂×Dic₁₂	—	2	2	no	—
26	C₂×Dic₁₂	—	2	2	no	—
27	#869	—	2	2	no	—
28	C₃×#102	—	4	4	no	—
29	C₃×C₄×C₈	D₄	1	2	yes	—
30	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
31	#829	—	4	8	no	—
32	#827	—	4	8	no	—
33	#818	—	4	4	no	—
34	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
35	C₂×Q₃₂	—	6	4	no	—
36	C₂×SD₃₂	—	6	4	no	—
37	#500	—	6	2	no	—
38	#494	—	3	2	no	—
39	#494	C₂×D₃	1	1	yes	characteristic
40	#493	C₂×D₃	1	1	yes	characteristic
41	#486	—	2	2	no	—
42	C₄×C₁₆	—	3	2	no	—
43	C₈²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×D₄	—	4	1	no	—
46	C₂×C₃×D₄	—	4	2	no	—
47	C₂×Dic₆	—	4	2	no	—
48	C₃×Q₁₆	—	4	4	no	—
49	C₃×D₈	—	4	4	no	—
50	C₂×C₃×C₈	D₈	1	4	yes	—
51	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
52	C₃×C₄⋊C₄	—	4	2	no	—
53	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Dic₃	—	4	4	no	—
55	Dic₁₂	—	4	4	no	—
56	Dic₁₂	—	4	4	no	—
57	C₃⋊C₁₆	D₈	1	8	yes	—
58	C₂×Q₁₆	—	2	2	no	—
59	C₂×Q₁₆	—	6	2	no	—
60	C₂×Q₁₆	—	6	2	no	—
61	C₂×D₈	—	2	2	no	—
62	C₄⋊Q₈	—	2	1	no	—
63	C₄⋊Q₈	—	6	2	no	—
64	#127	—	2	1	no	—
65	Q₃₂	—	12	8	no	—
66	SD₃₂	—	12	8	no	—
67	C₂×C₁₆	—	3	4	no	—
68	#107	—	12	4	no	—
69	#102	—	4	4	no	—
70	C₄×C₈	#67	1	2	yes	—
71	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₂³×C₃	—	8	1	no	—
73	C₃×Q₈	—	8	2	no	—
74	C₃×D₄	—	8	2	no	—
75	C₃×D₄	—	8	4	no	—
76	C₂×C₃×C₄	—	8	1	no	—
77	C₂×C₃×C₄	#127	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
79	C₂×Dic₃	—	8	2	no	—
80	Dic₆	—	8	2	no	—
81	Dic₆	—	8	2	no	—
82	C₃×C₈	—	2	4	no	—
83	C₃×C₈	C₂×D₈	1	4	yes	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	12	2	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	4	1	no	—
88	Q₁₆	—	4	4	no	—
89	Q₁₆	—	12	4	no	—
90	Q₁₆	—	12	4	no	—
91	D₈	—	4	4	no	—
92	C₂×C₈	#213	1	4	yes	—
93	C₂×C₈	D₃×D₄	1	2	yes	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	12	4	no	—
96	C₄²	C₂×#67	1	1	yes	characteristic
97	C₁₆	—	3	8	no	—
98	C₂²×C₃	—	16	1	no	—
99	C₂²×C₃	—	16	2	no	—
100	C₂²×C₃	#493	1	1	yes	characteristic
101	C₃×C₄	—	16	1	no	—
102	C₃×C₄	#493	1	2	yes	—
103	C₃×C₄	#510	1	4	yes	—
104	Dic₃	—	16	2	no	—
105	C₂³	—	8	1	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	24	2	no	—
108	Q₈	—	24	2	no	—
109	D₄	—	8	2	no	—
110	D₄	—	8	4	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	24	2	no	—
113	C₂×C₄	C₂×#213	1	2	yes	—
114	C₂×C₄	#940	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	D₃×D₈	1	4	yes	—
117	C₂×C₃	—	32	1	no	—
118	C₂×C₃	#1712	1	1	yes	characteristic
119	C₂×C₃	#2251	1	2	yes	—
120	C₂²	—	16	1	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#5073	1	1	yes	characteristic
123	C₄	—	16	1	no	—
124	C₄	—	48	2	no	—
125	C₄	#5151	1	2	yes	—
126	C₄	#5195	1	4	yes	—
127	C₃	#12520	1	1	yes	characteristic, complemented, Sylow
128	C₂	—	32	1	no	—
129	C₂	#71276	1	1	yes	characteristic
130	C₂	#71322	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1709

C₂

yes

maximal, characteristic

#71312

C₂

yes

maximal, complemented

#71291

C₂

yes

maximal

#68065

C₂

yes

maximal, characteristic, complemented

#67437

C₂

yes

maximal, characteristic, complemented

#12520

—

maximal, Sylow

C₃×#494

C₂²

yes

characteristic

C₃×#493

C₂²

yes

—

—

—

C₂²

yes

—

#4682

—

C₃×C₈²

C₂²

yes

characteristic

C₄×C₃⋊C₁₆

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#1311

—

C₂×C₃×Q₁₆

—

C₂×C₃×D₈

—

C₃×C₄⋊Q₈

—

C₃×#127

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₃×#102

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

—

—

—

C₂×C₃⋊C₁₆

D₄

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₃

yes

characteristic

#493

C₂×D₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×D₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

C₃⋊C₁₆

D₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

#67

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

C₂×D₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

C₂×C₈

#213

yes

—

C₂×C₈

D₃×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×#67

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

#493

yes

characteristic

101

C₃×C₄

—

102

C₃×C₄

#493

yes

—

103

C₃×C₄

#510

yes

—

104

Dic₃

—

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×#213

yes

—

114

C₂×C₄

#940

yes

characteristic

115

C₈

—

116

C₈

D₃×D₈

yes

—

117

C₂×C₃

—

118

C₂×C₃

#1712

yes

characteristic

119

C₂×C₃

#2251

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#5073

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

#5151

yes

—

126

C₄

#5195

yes

—

127

C₃

#12520

yes

characteristic, complemented, Sylow

128

C₂

—

129

C₂

#71276

yes

characteristic

130

C₂

#71322

yes

—

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1709	C₂
socle	C₂²×C₃	#493
prime radical	C₃×#493	C₂²
hypercenter	#1709	D₃
hypocenter	C₃	#12520
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#5073

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #71312	C₂	semidirect
4. #68065	C₂	semidirect
5. #67437	C₂	semidirect
127. C₃	#12520	semidirect

H ≅ G/N

type

2. #71312

C₂

semidirect

4. #68065

C₂

semidirect

5. #67437

C₂

semidirect

127. C₃

#12520

semidirect

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

subgroup	index
C₃×#1709	2
#71312	2
#71291	2
#68065	2
#67437	2
#12520	3

subgroup

index

Subgroups — #10709921

Conjugacy classes of subgroups

Complements

Maximal subgroups