Group #34204652 — subgroups

Subgroups — #34204652

Order 20224.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#23763564	C₂	1	2	yes	maximal
2	#23761492	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579095	C₂²	1	4	yes	—
5	C₂×#14579093	—	2	2	no	—
6	#19098429	—	2	2	no	—
7	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₉	—	2	4	no	—
11	#14579095	—	2	8	no	—
12	#14579093	—	4	4	no	—
13	C₇₉⋊#95	—	4	4	no	—
14	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₇₉	—	2	4	no	—
18	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₇₉	Q₈	1	2	yes	—
21	C₂²×Dic₇₉	—	4	4	no	—
22	C₄⋊Dic₇₉	—	4	4	no	—
23	C₄⋊Dic₇₉	—	4	4	no	—
24	C₂²×C₄×C₇₉	D₈	1	2	yes	—
25	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
28	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₇₉	#41	1	4	yes	—
31	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₇₉	Q₁₆	1	2	yes	—
33	C₁₆×C₇₉	—	2	4	no	—
34	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₇₉	—	2	4	no	—
36	C₂×C₄×C₇₉	#107	1	4	yes	—
37	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	#124	1	2	yes	—
39	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
40	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₇₉	—	8	4	no	—
42	C₂×Dic₇₉	—	8	8	no	—
43	C₈×C₇₉	#106	1	4	yes	—
44	C₈×C₇₉	#117	1	2	yes	—
45	C₈×C₇₉	#125	1	2	yes	—
46	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
47	C₂²×C₇₉	#486	1	2	yes	—
48	C₂²×C₇₉	#508	1	4	yes	—
49	C₄×C₇₉	—	2	4	no	—
50	C₄×C₇₉	#382	1	1	yes	characteristic
51	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
52	C₄×C₇₉	#487	1	2	yes	—
53	Dic₇₉	—	16	4	no	—
54	#12889	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	#1834	1	1	yes	characteristic
56	C₂×C₇₉	#2158	1	2	yes	—
57	C₂×C₇₉	#2242	1	4	yes	—
58	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
59	#1846	—	79	2	no	—
60	#1380	—	79	4	no	—
61	C₇₉	#12889	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
63	C₂×#107	—	79	4	no	—
64	C₂×#106	—	158	2	no	—
65	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
66	#382	—	158	2	no	—
67	C₄×C₁₆	Dic₇₉	1	4	yes	complemented
68	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	158	4	no	—
71	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
74	#107	—	158	8	no	—
75	#106	—	316	4	no	—
76	C₄×C₈	Dic₁₅₈	1	2	yes	—
77	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
78	#95	—	316	4	no	—
79	C₂²×C₄	—	316	4	no	—
80	C₂²×C₄	D₆₃₂	1	2	yes	—
81	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
83	C₂×C₈	#11929613	1	4	yes	—
84	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
86	C₂×C₈	#11929624	1	4	yes	—
87	C₄⋊C₄	—	316	4	no	—
88	C₄⋊C₄	—	316	4	no	—
89	C₄²	C₄⋊Dic₇₉	1	2	yes	—
90	C₄²	Dic₃₁₆	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	632	4	no	—
95	C₂×C₄	—	632	8	no	—
96	C₂×C₄	#14579095	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
98	C₂×C₄	#14579149	1	2	yes	—
99	C₂×C₄	C₂×#11929613	1	2	yes	—
100	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
101	C₈	#14579093	1	4	yes	—
102	C₈	#14579120	1	2	yes	—
103	C₈	#14579153	1	2	yes	—
104	C₂²	C₂×#14579095	1	1	yes	characteristic
105	C₂²	#19098910	1	2	yes	—
106	C₂²	#19098995	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	1264	4	no	—
109	C₄	#19098429	1	1	yes	characteristic
110	C₄	C₂×#14579093	1	1	yes	characteristic
111	C₄	#19098914	1	2	yes	—
112	C₂	#23763528	1	1	yes	characteristic
113	C₂	#23765184	1	2	yes	—
114	C₂	#23765632	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763564

C₂

yes

maximal

#23761492

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

#19098429

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

—

#14579095

—

#14579093

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#106

yes

—

C₈×C₇₉

#117

yes

—

C₈×C₇₉

#125

yes

—

C₂²×C₇₉

C₂×#107

yes

characteristic

C₂²×C₇₉

#486

yes

—

C₂²×C₇₉

#508

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2242

yes

—

C₂×C₄×C₁₆

D₇₉

yes

characteristic

#1846

—

#1380

—

C₇₉

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₉

yes

—

C₂×#107

—

C₂×#106

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#382

—

158

—

C₄×C₁₆

Dic₇₉

yes

complemented

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

#107

—

158

—

#106

—

316

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

#95

—

316

—

C₂²×C₄

—

316

—

C₂²×C₄

D₆₃₂

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

#11929613

yes

—

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

C₂×C₈

#11929624

yes

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

C₄⋊Dic₇₉

yes

—

C₄²

Dic₃₁₆

yes

—

C₁₆

—

C₂³

C₂×D₆₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

632

—

C₂×C₄

—

632

—

C₂×C₄

#14579095

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

#14579149

yes

—

C₂×C₄

C₂×#11929613

yes

—

100

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

101

C₈

#14579093

yes

—

102

C₈

#14579120

yes

—

103

C₈

#14579153

yes

—

104

C₂²

C₂×#14579095

yes

characteristic

105

C₂²

#19098910

yes

—

106

C₂²

#19098995

yes

—

107

C₄

—

108

C₄

—

1264

—

109

C₄

#19098429

yes

characteristic

110

C₄

C₂×#14579093

yes

characteristic

111

C₄

#19098914

yes

—

112

C₂

#23763528

yes

characteristic

113

C₂

#23765184

yes

—

114

C₂

#23765632

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12889
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×C₄×C₁₆×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₉

hypocenter

C₇₉

#12889

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₇₉	C₄	semidirect
61. C₇₉	#12889	semidirect
67. C₄×C₁₆	Dic₇₉	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₇₉

C₄

semidirect

61. C₇₉

#12889

semidirect

67. C₄×C₁₆

Dic₇₉

semidirect

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

subgroup	index
#23763564	2
#23761492	2
C₂×C₄×C₁₆×C₇₉	2
#12889	79

subgroup

index

#23763564

#23761492

C₂×C₄×C₁₆×C₇₉

#12889

Subgroups — #34204652

Conjugacy classes of subgroups

Complements

Maximal subgroups