Group #23765121 — subgroups

Subgroups — #23765121

Order 10112.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#19098601	C₂	1	1	yes	maximal, characteristic
3	#19098368	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₅₈	—	2	1	no	—
6	C₂×Dic₃₁₆	—	2	1	no	—
7	C₂×Dic₃₁₆	—	2	2	no	—
8	C₂×Dic₃₁₆	C₄	1	4	yes	complemented
9	C₂×Dic₃₁₆	C₂²	1	2	yes	—
10	#14579120	—	2	1	no	—
11	#14579095	C₂²	1	2	yes	—
12	#14579093	—	2	1	no	—
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	M₃₂×C₇₉	—	2	2	no	—
15	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂×Dic₁₅₈	—	2	4	no	—
17	C₂×Dic₁₅₈	—	2	2	no	—
18	C₂×Dic₁₅₈	—	4	1	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂²×Dic₇₉	—	4	1	no	—
21	Dic₃₁₆	—	2	4	no	—
22	Dic₃₁₆	—	2	2	no	—
23	Dic₃₁₆	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₂²⋊Dic₇₉	—	4	1	no	—
26	C₄×Dic₇₉	—	4	1	no	—
27	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₉	D₄	1	2	yes	—
30	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₁₆×C₇₉	—	2	2	no	—
33	C₂³×C₇₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
36	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₇₉	—	4	4	no	—
39	C₂×Dic₇₉	—	4	2	no	—
40	C₂×Dic₇₉	—	4	2	no	—
41	Dic₁₅₈	—	2	4	no	—
42	Dic₁₅₈	—	4	4	no	—
43	Dic₁₅₈	—	4	2	no	—
44	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₇₉	—	2	2	no	—
48	C₂²×C₇₉	#102	1	2	yes	—
49	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₇₉	#102	1	2	yes	—
51	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
53	Dic₇₉	—	4	4	no	—
54	Dic₇₉	—	8	2	no	—
55	C₂×C₇₉	—	2	2	no	—
56	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
57	C₂×C₇₉	#510	1	2	yes	—
58	#2141	—	79	1	no	maximal, Sylow
59	C₇₉	#2141	1	1	yes	characteristic, complemented, Sylow
60	C₂²×Q₁₆	—	79	1	no	—
61	C₂×M₃₂	D₇₉	1	1	yes	characteristic
62	#427	—	79	1	no	—
63	#358	—	79	4	no	—
64	C₂²×Q₈	—	158	1	no	—
65	C₂×Q₁₆	—	79	2	no	—
66	C₂×Q₁₆	—	79	4	no	—
67	C₂×Q₁₆	—	158	2	no	—
68	C₂×Q₁₆	—	158	1	no	—
69	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
70	#117	—	158	1	no	—
71	M₃₂	—	2	2	no	—
72	C₂×C₁₆	C₂×D₇₉	1	2	yes	—
73	#107	—	79	2	no	—
74	#106	—	158	1	no	—
75	C₂×Q₈	—	316	1	no	—
76	C₂×Q₈	—	316	2	no	—
77	C₂×Q₈	—	158	4	no	—
78	C₂×Q₈	—	158	2	no	—
79	C₂²×C₄	—	316	1	no	—
80	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
81	Q₁₆	—	79	4	no	—
82	Q₁₆	—	158	4	no	—
83	Q₁₆	—	158	2	no	—
84	C₂×C₈	C₄×D₇₉	1	2	yes	—
85	C₂×C₈	D₃₁₆	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
87	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
88	C₄⋊C₄	—	158	2	no	—
89	C₂²⋊C₄	—	316	1	no	—
90	C₄²	—	316	1	no	—
91	C₁₆	—	2	2	no	—
92	C₂³	D₆₃₂	1	1	yes	characteristic
93	Q₈	—	316	4	no	—
94	Q₈	—	316	2	no	—
95	Q₈	—	158	4	no	—
96	C₂×C₄	—	316	4	no	—
97	C₂×C₄	—	316	2	no	—
98	C₂×C₄	—	316	2	no	—
99	C₂×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
100	C₂×C₄	D₆₃₂	1	1	yes	characteristic
101	C₂×C₄	#11929613	1	2	yes	—
102	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
103	C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#14579083	1	2	yes	—
108	C₂²	C₂×D₆₃₂	1	1	yes	characteristic
109	C₄	—	316	4	no	—
110	C₄	—	632	2	no	—
111	C₄	#14579083	1	2	yes	—
112	C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#11929613	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#14579083	1	1	yes	characteristic
116	C₂	#19099006	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic

#19098601

C₂

yes

maximal, characteristic

#19098368

C₂

yes

maximal, complemented

C₂×M₃₂×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₄

yes

complemented

C₂×Dic₃₁₆

C₂²

yes

—

#14579120

—

#14579095

C₂²

yes

—

#14579093

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#102

yes

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

#102

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#102

yes

characteristic

C₂×C₇₉

#510

yes

—

#2141

—

maximal, Sylow

C₇₉

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×M₃₂

D₇₉

yes

characteristic

#427

—

#358

—

C₂²×Q₈

—

158

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

#117

—

158

—

M₃₂

—

C₂×C₁₆

C₂×D₇₉

yes

—

#107

—

#106

—

158

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

316

—

C₂²×C₄

D₃₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

158

—

Q₁₆

—

158

—

C₂×C₈

C₄×D₇₉

yes

—

C₂×C₈

D₃₁₆

yes

characteristic

C₂×C₈

C₂²⋊D₇₉

yes

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

158

—

C₂²⋊C₄

—

316

—

C₄²

—

316

—

C₁₆

—

C₂³

D₆₃₂

yes

characteristic

Q₈

—

316

—

Q₈

—

316

—

Q₈

—

158

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

(C₂×D₇₉)⋊C₄

yes

—

100

C₂×C₄

D₆₃₂

yes

characteristic

101

C₂×C₄

#11929613

yes

—

102

C₂×C₄

C₂×D₃₁₆

yes

characteristic

103

C₈

(C₂×D₇₉)⋊C₄

yes

—

104

C₈

C₂×C₄×D₇₉

yes

characteristic

105

C₈

C₂×C₂²⋊D₇₉

yes

characteristic

106

C₂²

—

107

C₂²

#14579083

yes

—

108

C₂²

C₂×D₆₃₂

yes

characteristic

109

C₄

—

316

—

110

C₄

—

632

—

111

C₄

#14579083

yes

—

112

C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

113

C₄

C₂×#11929613

yes

characteristic

114

C₂

—

115

C₂

C₂×#14579083

yes

characteristic

116

C₂

#19099006

yes

—

	subgroup	G/H
center	C₂²	C₂×D₆₃₂
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	C₂×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂³×C₇₉	D₈
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂×M₃₂	D₇₉
hypocenter	C₇₉	#2141
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₆₃₂

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

C₂×M₃₂×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂³×C₇₉

D₈

cosocle

C₂×C₈×C₇₉

C₂³

hypercenter

C₂×M₃₂

D₇₉

hypocenter

C₇₉

#2141

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #19098368	C₂	semidirect
8. C₂×Dic₃₁₆	C₄	semidirect
59. C₇₉	#2141	semidirect

H ≅ G/N

type

3. #19098368

C₂

semidirect

8. C₂×Dic₃₁₆

C₄

semidirect

59. C₇₉

#2141

semidirect

Subgroups — #23765121

Conjugacy classes of subgroups

Complements