Group #23765666 — subgroups

Subgroups — #23765666

Order 10112.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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×#14579106	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₂₆₄	C₂	1	2	yes	maximal, complemented
3	#19098947	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098940	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₂×#11929613	—	2	1	no	—
8	C₂×D₆₃₂	—	2	1	no	—
9	C₂×D₆₃₂	C₂²	1	2	yes	—
10	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
11	#14579158	—	2	1	no	—
12	#14579149	—	2	1	no	—
13	#14579106	—	2	4	no	—
14	D₁₂₆₄	—	2	4	no	—
15	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
16	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	2	1	no	—
18	C₂×D₃₁₆	—	2	1	no	—
19	C₂×D₃₁₆	—	2	2	no	—
20	C₂×D₃₁₆	—	4	2	no	—
21	Dic₃₁₆	—	2	2	no	—
22	#11929613	—	4	2	no	—
23	D₆₃₂	—	2	2	no	—
24	D₆₃₂	—	2	2	no	—
25	D₆₃₂	—	2	2	no	—
26	D₆₃₂	—	4	2	no	—
27	(C₂×D₇₉)⋊C₄	—	4	2	no	—
28	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₄²×C₇₉	D₄	1	1	yes	characteristic
31	C₁₆×C₇₉	D₄	1	4	yes	complemented
32	C₂²×D₇₉	—	4	1	no	—
33	C₂²×D₇₉	—	4	2	no	—
34	C₂×C₄×C₇₉	D₈	1	2	yes	—
35	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₇₉	—	4	1	no	—
37	D₃₁₆	—	4	1	no	—
38	D₃₁₆	—	4	2	no	—
39	D₃₁₆	—	4	2	no	—
40	D₃₁₆	—	4	1	no	—
41	D₃₁₆	—	8	4	no	—
42	Dic₁₅₈	—	4	1	no	—
43	Dic₁₅₈	—	4	1	no	—
44	C₈×C₇₉	D₈	1	2	yes	—
45	C₈×C₇₉	C₂×D₄	1	2	yes	—
46	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
47	C₂×D₇₉	—	8	1	no	—
48	C₂×D₇₉	—	8	2	no	—
49	C₂×D₇₉	—	8	2	no	—
50	C₂×D₇₉	—	8	4	no	—
51	C₄×C₇₉	—	2	2	no	—
52	C₄×C₇₉	#127	1	1	yes	characteristic
53	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
54	Dic₇₉	—	8	1	no	—
55	C₂×C₇₉	#493	1	1	yes	characteristic
56	C₂×C₇₉	#509	1	2	yes	—
57	D₇₉	—	16	1	no	—
58	D₇₉	—	16	2	no	—
59	#2250	—	79	1	no	maximal, Sylow
60	C₇₉	#2250	1	1	yes	characteristic, complemented, Sylow
61	C₂×SD₃₂	—	79	2	no	—
62	C₂×D₁₆	—	79	2	no	—
63	#495	—	79	1	no	—
64	#493	—	79	1	no	—
65	#346	D₇₉	1	1	yes	characteristic, complemented
66	C₂×Q₁₆	—	79	1	no	—
67	C₂×SD₁₆	—	158	1	no	—
68	C₂×D₈	—	79	2	no	—
69	C₂×D₈	—	79	1	no	—
70	C₂×D₈	—	158	1	no	—
71	#127	—	158	1	no	—
72	#124	—	158	1	no	—
73	SD₃₂	—	158	4	no	—
74	D₁₆	—	158	4	no	—
75	C₂×C₁₆	C₂×D₇₉	1	2	yes	—
76	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
77	C₂×Q₈	—	158	1	no	—
78	C₂×D₄	—	316	2	no	—
79	C₂×D₄	—	158	1	no	—
80	C₂×D₄	—	158	2	no	—
81	Q₁₆	—	158	2	no	—
82	SD₁₆	—	316	2	no	—
83	D₈	—	316	2	no	—
84	D₈	—	158	2	no	—
85	D₈	—	158	2	no	—
86	D₈	—	158	2	no	—
87	C₂×C₈	D₃₁₆	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
89	C₂²⋊C₄	—	316	2	no	—
90	C₄²	D₃₁₆	1	1	yes	characteristic
91	C₁₆	D₃₁₆	1	4	yes	complemented
92	C₂³	—	316	1	no	—
93	C₂³	—	316	2	no	—
94	Q₈	—	316	1	no	—
95	Q₈	—	316	1	no	—
96	D₄	—	316	1	no	—
97	D₄	—	316	2	no	—
98	D₄	—	316	2	no	—
99	D₄	—	316	1	no	—
100	D₄	—	632	4	no	—
101	C₂×C₄	—	316	1	no	—
102	C₂×C₄	D₆₃₂	1	2	yes	—
103	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
104	C₈	D₆₃₂	1	2	yes	—
105	C₈	C₂×D₃₁₆	1	2	yes	—
106	C₂²	—	632	2	no	—
107	C₂²	—	632	1	no	—
108	C₂²	—	632	4	no	—
109	C₂²	—	632	2	no	—
110	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	632	1	no	—
113	C₄	#14579158	1	1	yes	characteristic
114	C₄	C₂×D₆₃₂	1	1	yes	characteristic
115	C₂	—	1264	2	no	—
116	C₂	—	1264	1	no	—
117	C₂	#19098940	1	1	yes	characteristic
118	C₂	#19099001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579106

C₂

yes

maximal, complemented

C₂×D₁₂₆₄

C₂

yes

maximal, complemented

#19098947

C₂

yes

maximal, characteristic, complemented

#19098940

C₂

yes

maximal, characteristic, complemented

#346×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929613

—

C₂×D₆₃₂

—

C₂×D₆₃₂

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

—

—

—

D₁₂₆₄

—

C₂×C₁₆×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₁₆×C₇₉

D₄

yes

complemented

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

D₈

yes

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#127

yes

characteristic

C₄×C₇₉

C₂×D₈

yes

characteristic

Dic₇₉

—

C₂×C₇₉

#493

yes

characteristic

C₂×C₇₉

#509

yes

—

D₇₉

—

D₇₉

—

#2250

—

maximal, Sylow

C₇₉

#2250

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#493

—

#346

D₇₉

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×SD₁₆

—

158

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

158

—

#127

—

158

—

#124

—

158

—

SD₃₂

—

158

—

D₁₆

—

158

—

C₂×C₁₆

C₂×D₇₉

yes

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

158

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

Q₁₆

—

158

—

SD₁₆

—

316

—

D₈

—

316

—

D₈

—

158

—

D₈

—

158

—

D₈

—

158

—

C₂×C₈

D₃₁₆

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₂²⋊C₄

—

316

—

C₄²

D₃₁₆

yes

characteristic

C₁₆

D₃₁₆

yes

complemented

C₂³

—

316

—

C₂³

—

316

—

Q₈

—

316

—

Q₈

—

316

—

D₄

—

316

—

D₄

—

316

—

D₄

—

316

—

D₄

—

316

—

100

D₄

—

632

—

101

C₂×C₄

—

316

—

102

C₂×C₄

D₆₃₂

yes

—

103

C₂×C₄

C₂×D₃₁₆

yes

characteristic

104

C₈

D₆₃₂

yes

—

105

C₈

C₂×D₃₁₆

yes

—

106

C₂²

—

632

—

107

C₂²

—

632

—

108

C₂²

—

632

—

109

C₂²

—

632

—

110

C₂²

C₂×D₆₃₂

yes

characteristic

111

C₄

—

112

C₄

—

632

—

113

C₄

#14579158

yes

characteristic

114

C₄

C₂×D₆₃₂

yes

characteristic

115

C₂

—

1264

—

116

C₂

—

1264

—

117

C₂

#19098940

yes

characteristic

118

C₂

#19099001

yes

—

	subgroup	G/H
center	C₂²	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#346	D₇₉
hypocenter	C₇₉	#2250
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×D₆₃₂

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#346×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
1. C₂×#14579106	C₂	semidirect
2. C₂×D₁₂₆₄	C₂	semidirect
3. #19098947	C₂	semidirect
4. #19098940	C₂	semidirect
5. #346×C₇₉	C₂	semidirect
31. C₁₆×C₇₉	D₄	semidirect
60. C₇₉	#2250	semidirect
65. #346	D₇₉	semidirect
91. C₁₆	D₃₁₆	semidirect

H ≅ G/N

type

1. C₂×#14579106

C₂

semidirect

2. C₂×D₁₂₆₄

C₂

semidirect

3. #19098947

C₂

semidirect

4. #19098940

C₂

semidirect

5. #346×C₇₉

C₂

semidirect

31. C₁₆×C₇₉

D₄

semidirect

60. C₇₉

#2250

semidirect

65. #346

D₇₉

semidirect

91. C₁₆

D₃₁₆

semidirect

Subgroups — #23765666

Conjugacy classes of subgroups

Complements