Group #10698041 — subgroups

Subgroups — #10698041

Order 768.

538 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 145 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×#4681	C₂	1	3	yes	maximal
2	#69777	C₂	1	3	yes	maximal
3	C₃×#1273	C₂	1	1	yes	maximal, characteristic
4	#10319	—	3	1	no	maximal, Sylow
5	C₂×#869	C₂²	1	3	yes	—
6	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
7	C₂×#816	—	2	6	no	—
8	#4935	C₂²	1	1	yes	characteristic
9	#4681	—	2	6	no	—
10	#4681	—	2	6	no	—
11	#4551	—	2	6	no	—
12	C₂×#487	—	3	3	no	—
13	#1990	—	3	3	no	—
14	#1273	D₃	1	1	yes	characteristic
15	C₂²×Dic₆	—	2	3	no	—
16	C₂²×C₃×C₈	D₄	1	6	yes	—
17	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₃	—	2	3	no	—
19	C₂×C₄⋊Dic₃	—	2	3	no	—
20	C₂×C₄⋊Dic₃	—	2	3	no	—
21	#869	—	2	3	no	—
22	#869	—	2	3	no	—
23	#869	—	2	3	no	—
24	#869	D₄	1	6	yes	—
25	C₃×C₄×C₈	—	2	3	no	—
26	C₃×C₄×C₈	—	2	3	no	—
27	#816	—	2	24	no	—
28	C₂×C₄⋊Q₈	—	3	3	no	—
29	#487	—	6	6	no	—
30	#487	—	6	6	no	—
31	C₂×#103	—	6	6	no	—
32	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
33	#384	—	3	1	no	—
34	#340	—	6	6	no	—
35	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
36	C₂²×Dic₃	—	4	3	no	—
37	C₂²×Dic₃	—	4	1	no	—
38	C₂×Dic₆	—	2	6	no	—
39	C₂×Dic₆	—	4	6	no	—
40	C₂×Dic₆	—	4	6	no	—
41	C₂×Dic₆	—	4	3	no	—
42	C₂×C₃×C₈	—	2	6	no	—
43	C₂×C₃×C₈	—	2	6	no	—
44	C₂×C₃×C₈	—	2	6	no	—
45	C₃×C₄²	C₂×D₄	1	3	yes	—
46	C₃×C₄²	#41	1	1	yes	characteristic
47	C₄⋊Dic₃	—	2	6	no	—
48	C₄⋊Dic₃	—	2	6	no	—
49	C₄⋊Dic₃	—	4	3	no	—
50	C₄⋊Dic₃	—	4	3	no	—
51	C₄⋊Dic₃	—	4	6	no	—
52	C₂²×Q₈	—	6	3	no	—
53	C₂²×C₈	D₁₂	1	6	yes	—
54	C₄⋊Q₈	—	3	6	no	—
55	C₄⋊Q₈	—	6	3	no	—
56	C₄⋊Q₈	—	6	3	no	—
57	C₄⋊Q₈	—	6	3	no	—
58	C₂×C₄⋊C₄	—	6	3	no	—
59	C₂×C₄⋊C₄	—	6	3	no	—
60	C₂×C₄⋊C₄	—	6	3	no	—
61	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
62	#103	—	6	24	no	—
63	C₄×C₈	—	2	3	no	—
64	C₄×C₈	—	2	3	no	—
65	C₂³×C₃	#127	1	1	yes	characteristic
66	C₂×C₃×C₄	#120	1	3	yes	—
67	C₂×C₃×C₄	#121	1	3	yes	—
68	C₂×C₃×C₄	#135	1	6	yes	—
69	C₂×C₃×C₄	#137	1	6	yes	—
70	C₂×Dic₃	—	4	6	no	—
71	C₂×Dic₃	—	8	3	no	—
72	C₂×Dic₃	—	8	3	no	—
73	C₂×Dic₃	—	8	3	no	—
74	Dic₆	—	4	12	no	—
75	Dic₆	—	8	6	no	—
76	C₃×C₈	—	4	6	no	—
77	C₂×Q₈	—	6	6	no	—
78	C₂×Q₈	—	12	3	no	—
79	C₂×Q₈	—	12	6	no	—
80	C₂×Q₈	—	12	6	no	—
81	C₂²×C₄	—	12	3	no	—
82	C₂²×C₄	—	12	1	no	—
83	C₂²×C₄	C₂×D₁₂	1	3	yes	—
84	C₂×C₈	—	2	6	no	—
85	C₂×C₈	—	2	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	6	6	no	—
88	C₄⋊C₄	—	6	6	no	—
89	C₄⋊C₄	—	12	3	no	—
90	C₄⋊C₄	—	12	3	no	—
91	C₄⋊C₄	—	12	6	no	—
92	C₄²	D₃×D₄	1	3	yes	—
93	C₄²	#239	1	1	yes	characteristic
94	C₂²×C₃	#392	1	1	yes	characteristic
95	C₂²×C₃	#495	1	3	yes	—
96	C₂²×C₃	#497	1	3	yes	—
97	C₃×C₄	#452	1	6	yes	—
98	C₃×C₄	#464	1	6	yes	—
99	Dic₃	—	8	6	no	—
100	Dic₃	—	16	1	no	—
101	C₂³	#874	1	1	yes	characteristic
102	Q₈	—	12	12	no	—
103	Q₈	—	24	6	no	—
104	C₂×C₄	—	12	6	no	—
105	C₂×C₄	—	24	3	no	—
106	C₂×C₄	—	24	3	no	—
107	C₂×C₄	—	24	3	no	—
108	C₂×C₄	#882	1	3	yes	—
109	C₂×C₄	#895	1	3	yes	—
110	C₂×C₄	#904	1	6	yes	—
111	C₂×C₄	#909	1	6	yes	—
112	C₈	—	4	6	no	—
113	C₂×C₃	#1711	1	1	yes	characteristic
114	C₂×C₃	#1718	1	3	yes	—
115	C₂×C₃	#2015	1	3	yes	—
116	C₂²	#4671	1	1	yes	characteristic
117	C₂²	#4696	1	3	yes	—
118	C₂²	#4715	1	3	yes	—
119	C₄	—	24	6	no	—
120	C₄	—	48	1	no	—
121	C₄	#4733	1	6	yes	—
122	C₄	#4845	1	6	yes	—
123	C₃	#10319	1	1	yes	characteristic, complemented, Sylow
124	C₂	#68061	1	1	yes	characteristic
125	C₂	#68096	1	3	yes	—
126	C₂	#68677	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4681

C₂

yes

maximal

#69777

C₂

yes

maximal

C₃×#1273

C₂

yes

maximal, characteristic

#10319

—

maximal, Sylow

C₂×#869

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×#816

—

#4935

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

C₂²×Dic₆

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

—

—

—

D₄

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

#816

—

C₂×C₄⋊Q₈

—

#487

—

#487

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₃

yes

—

#384

—

#340

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

#127

yes

characteristic

C₂×C₃×C₄

#120

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

#135

yes

—

C₂×C₃×C₄

#137

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₃×D₄

yes

—

C₄²

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#497

yes

—

C₃×C₄

#452

yes

—

C₃×C₄

#464

yes

—

Dic₃

—

100

Dic₃

—

101

C₂³

#874

yes

characteristic

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#882

yes

—

109

C₂×C₄

#895

yes

—

110

C₂×C₄

#904

yes

—

111

C₂×C₄

#909

yes

—

112

C₈

—

113

C₂×C₃

#1711

yes

characteristic

114

C₂×C₃

#1718

yes

—

115

C₂×C₃

#2015

yes

—

116

C₂²

#4671

yes

characteristic

117

C₂²

#4696

yes

—

118

C₂²

#4715

yes

—

119

C₄

—

120

C₄

—

121

C₄

#4733

yes

—

122

C₄

#4845

yes

—

123

C₃

#10319

yes

characteristic, complemented, Sylow

124

C₂

#68061

yes

characteristic

125

C₂

#68096

yes

—

126

C₂

#68677

yes

—

	subgroup	G/H
center	C₂³	#874
commutator, cosocle	C₂×C₃×C₄²	C₂³
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1273	C₂
socle, prime radical	C₂³×C₃	#127
hypercenter	#1273	D₃
hypocenter	C₃	#10319
LCS	G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#874

commutator, cosocle

C₂×C₃×C₄²

C₂³

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
123. C₃	#10319	semidirect

H ≅ G/N

type

123. C₃

#10319

semidirect

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

subgroup	index
C₂×#4681	2
#69777	2
C₃×#1273	2
#10319	3

subgroup

index

Subgroups — #10698041

Conjugacy classes of subgroups

Complements

Maximal subgroups