Group #10698026 — subgroups

Subgroups — #10698026

Order 768.

586 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 169 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×#4680	C₂	1	3	yes	maximal
2	#69771	C₂	1	3	yes	maximal
3	C₃×#1273	C₂	1	1	yes	maximal, characteristic
4	#10316	—	3	1	no	maximal, Sylow
5	C₂×#869	C₂²	1	3	yes	—
6	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
7	C₂×#817	—	2	6	no	—
8	C₂×#816	—	2	6	no	—
9	#4935	C₂²	1	1	yes	characteristic
10	#4680	—	2	12	no	—
11	C₂×#498	—	3	3	no	—
12	#1956	—	3	3	no	—
13	#1273	D₃	1	1	yes	characteristic
14	C₂²×Dic₆	—	2	3	no	—
15	C₂²×C₃×C₈	D₄	1	6	yes	—
16	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊Dic₃	—	2	6	no	—
18	C₂×C₄⋊Dic₃	—	2	3	no	—
19	#869	—	2	6	no	—
20	#869	—	2	3	no	—
21	#869	D₄	1	6	yes	—
22	C₃×C₄×C₈	—	2	6	no	—
23	#817	—	2	24	no	—
24	#816	—	4	12	no	—
25	C₂×C₄⋊Q₈	—	3	3	no	—
26	#498	—	6	12	no	—
27	C₂×#106	—	6	6	no	—
28	C₂×#103	—	6	6	no	—
29	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
30	#384	—	3	1	no	—
31	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
32	C₂²×Dic₃	—	4	3	no	—
33	C₂²×Dic₃	—	4	1	no	—
34	C₂×Dic₆	—	2	6	no	—
35	C₂×Dic₆	—	4	6	no	—
36	C₂×Dic₆	—	4	6	no	—
37	C₂×Dic₆	—	4	3	no	—
38	C₂×C₃×C₈	—	2	6	no	—
39	C₂×C₃×C₈	—	2	6	no	—
40	C₂×C₃×C₈	SD₁₆	1	12	yes	—
41	C₃×C₄²	C₂×D₄	1	3	yes	—
42	C₃×C₄²	#41	1	1	yes	characteristic
43	C₄⋊Dic₃	—	2	12	no	—
44	C₄⋊Dic₃	—	4	6	no	—
45	C₄⋊Dic₃	—	4	6	no	—
46	C₂²×Q₈	—	6	3	no	—
47	C₂²×C₈	D₁₂	1	6	yes	—
48	C₄⋊Q₈	—	3	6	no	—
49	C₄⋊Q₈	—	6	6	no	—
50	C₄⋊Q₈	—	6	3	no	—
51	C₂×C₄⋊C₄	—	6	6	no	—
52	C₂×C₄⋊C₄	—	6	3	no	—
53	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
54	#106	—	6	24	no	—
55	#103	—	12	12	no	—
56	C₄×C₈	—	2	6	no	—
57	C₂³×C₃	#127	1	1	yes	characteristic
58	C₂×C₃×C₄	#120	1	3	yes	—
59	C₂×C₃×C₄	#121	1	3	yes	—
60	C₂×C₃×C₄	C₂×SD₁₆	1	6	yes	—
61	C₂×C₃×C₄	#137	1	6	yes	—
62	C₂×Dic₃	—	4	6	no	—
63	C₂×Dic₃	—	8	3	no	—
64	C₂×Dic₃	—	8	3	no	—
65	C₂×Dic₃	—	8	3	no	—
66	Dic₆	—	4	12	no	—
67	Dic₆	—	8	6	no	—
68	C₃×C₈	—	2	12	no	—
69	C₂×Q₈	—	6	6	no	—
70	C₂×Q₈	—	12	3	no	—
71	C₂×Q₈	—	12	6	no	—
72	C₂×Q₈	—	12	6	no	—
73	C₂²×C₄	—	12	3	no	—
74	C₂²×C₄	—	12	1	no	—
75	C₂²×C₄	C₂×D₁₂	1	3	yes	—
76	C₂×C₈	—	2	6	no	—
77	C₂×C₈	—	2	6	no	—
78	C₂×C₈	#204	1	12	yes	—
79	C₄⋊C₄	—	6	12	no	—
80	C₄⋊C₄	—	12	6	no	—
81	C₄⋊C₄	—	12	6	no	—
82	C₄²	D₃×D₄	1	3	yes	—
83	C₄²	#239	1	1	yes	characteristic
84	C₂²×C₃	#392	1	1	yes	characteristic
85	C₂²×C₃	#492	1	3	yes	—
86	C₂²×C₃	#497	1	3	yes	—
87	C₃×C₄	#448	1	6	yes	—
88	C₃×C₄	#461	1	6	yes	—
89	Dic₃	—	8	6	no	—
90	Dic₃	—	16	1	no	—
91	C₂³	#874	1	1	yes	characteristic
92	Q₈	—	12	12	no	—
93	Q₈	—	24	6	no	—
94	C₂×C₄	—	12	6	no	—
95	C₂×C₄	—	24	3	no	—
96	C₂×C₄	—	24	3	no	—
97	C₂×C₄	—	24	3	no	—
98	C₂×C₄	#882	1	3	yes	—
99	C₂×C₄	#895	1	3	yes	—
100	C₂×C₄	C₂×#204	1	6	yes	—
101	C₂×C₄	#909	1	6	yes	—
102	C₈	—	2	12	no	—
103	C₂×C₃	#1705	1	1	yes	characteristic
104	C₂×C₃	#1716	1	3	yes	—
105	C₂×C₃	#2013	1	3	yes	—
106	C₂²	#4671	1	1	yes	characteristic
107	C₂²	#4693	1	3	yes	—
108	C₂²	#4715	1	3	yes	—
109	C₄	—	24	6	no	—
110	C₄	—	48	1	no	—
111	C₄	#4738	1	6	yes	—
112	C₄	#4841	1	6	yes	—
113	C₃	#10316	1	1	yes	characteristic, complemented, Sylow
114	C₂	#68057	1	1	yes	characteristic
115	C₂	#68095	1	3	yes	—
116	C₂	#68687	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4680

C₂

yes

maximal

#69771

C₂

yes

maximal

C₃×#1273

C₂

yes

maximal, characteristic

#10316

—

maximal, Sylow

C₂×#869

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×#817

—

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

—

#869

—

#869

—

#869

D₄

yes

—

C₃×C₄×C₈

—

#817

—

#816

—

C₂×C₄⋊Q₈

—

#498

—

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₃

yes

—

#384

—

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

SD₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#106

—

#103

—

C₄×C₈

—

C₂³×C₃

#127

yes

characteristic

C₂×C₃×C₄

#120

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

C₂×SD₁₆

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

#204

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₃×D₄

yes

—

C₄²

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#497

yes

—

C₃×C₄

#448

yes

—

C₃×C₄

#461

yes

—

Dic₃

—

Dic₃

—

C₂³

#874

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#882

yes

—

C₂×C₄

#895

yes

—

100

C₂×C₄

C₂×#204

yes

—

101

C₂×C₄

#909

yes

—

102

C₈

—

103

C₂×C₃

#1705

yes

characteristic

104

C₂×C₃

#1716

yes

—

105

C₂×C₃

#2013

yes

—

106

C₂²

#4671

yes

characteristic

107

C₂²

#4693

yes

—

108

C₂²

#4715

yes

—

109

C₄

—

110

C₄

—

111

C₄

#4738

yes

—

112

C₄

#4841

yes

—

113

C₃

#10316

yes

characteristic, complemented, Sylow

114

C₂

#68057

yes

characteristic

115

C₂

#68095

yes

—

116

C₂

#68687

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	#10316
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
113. C₃	#10316	semidirect

H ≅ G/N

type

113. C₃

#10316

semidirect

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

subgroup	index
C₂×#4680	2
#69771	2
C₃×#1273	2
#10316	3

subgroup

index

Subgroups — #10698026

Conjugacy classes of subgroups

Complements

Maximal subgroups