Group #34209255 — subgroups

Subgroups — #34209255

Order 20224.

194 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23765679	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761578	C₂	1	4	yes	maximal, complemented
3	#23761534	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#2246	C₂	1	1	yes	maximal, characteristic
5	#19098965	—	2	1	no	—
6	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
7	#19098391	—	2	1	no	—
8	#19098389	C₂²	1	2	yes	—
9	C₂×D₁₆×C₇₉	—	2	1	no	—
10	C₂×D₁₆×C₇₉	C₂²	1	2	yes	—
11	#493×C₇₉	—	2	1	no	—
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#14579153	—	4	1	no	—
14	#14579095	—	2	2	no	—
15	#14579093	—	4	1	no	—
16	C₇₉⋊C₃₂	—	2	2	no	—
17	C₂×D₈×C₇₉	—	2	2	no	—
18	C₂×D₈×C₇₉	—	4	1	no	—
19	#127×C₇₉	—	4	1	no	—
20	D₁₆×C₇₉	—	2	4	no	—
21	D₁₆×C₇₉	—	4	2	no	—
22	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₄⋊Dic₇₉	—	4	2	no	—
26	Dic₇₉⋊C₄	—	8	2	no	—
27	C₂×D₄×C₇₉	—	4	2	no	—
28	C₂×D₄×C₇₉	—	8	2	no	—
29	D₈×C₇₉	—	4	2	no	—
30	D₈×C₇₉	—	4	2	no	—
31	D₈×C₇₉	—	8	2	no	—
32	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₇₉	D₈	1	1	yes	characteristic
35	C₁₆×C₇₉	SD₁₆	1	2	yes	—
36	C₁₆×C₇₉	#41	1	2	yes	—
37	C₂³×C₇₉	—	8	2	no	—
38	D₄×C₇₉	—	8	2	no	—
39	D₄×C₇₉	—	8	2	no	—
40	D₄×C₇₉	—	16	4	no	—
41	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
42	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
43	C₂×Dic₇₉	—	8	2	no	—
44	C₈×C₇₉	SD₃₂	1	2	yes	—
45	C₈×C₇₉	#124	1	1	yes	characteristic
46	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
47	C₂²×C₇₉	—	16	2	no	—
48	C₂²×C₇₉	—	16	4	no	—
49	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₇₉	—	2	2	no	—
51	C₄×C₇₉	#486	1	1	yes	characteristic
52	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
53	Dic₇₉	—	16	2	no	—
54	#13723	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	—	32	2	no	—
56	C₂×C₇₉	#2240	1	1	yes	characteristic
57	C₂×C₇₉	#2263	1	2	yes	—
58	#2253	—	79	1	no	—
59	#2246	D₇₉	1	1	yes	characteristic
60	#1415	—	79	4	no	—
61	#1397	—	79	1	no	—
62	C₇₉	#13723	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₁₆	—	2	1	no	—
64	C₂×D₁₆	C₂×D₇₉	1	2	yes	—
65	#499	—	158	1	no	—
66	#493	—	2	1	no	—
67	C₂×C₃₂	—	79	2	no	—
68	#367	—	158	1	no	—
69	#366	—	79	2	no	—
70	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
71	C₂×D₈	—	2	2	no	—
72	C₂×D₈	—	4	1	no	—
73	#127	—	4	1	no	—
74	#125	—	316	1	no	—
75	D₁₆	—	2	4	no	—
76	D₁₆	—	4	2	no	—
77	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
79	#107	—	158	2	no	—
80	#106	—	316	1	no	—
81	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
82	C₃₂	—	158	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	8	2	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	D₈	—	8	2	no	—
88	C₂×C₈	#11929609	1	1	yes	characteristic
89	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
90	C₄⋊C₄	—	316	2	no	—
91	C₄⋊C₄	—	632	2	no	—
92	C₄²	#11929609	1	1	yes	characteristic
93	C₁₆	#11929612	1	2	yes	—
94	C₁₆	#11929623	1	2	yes	—
95	C₂³	—	8	2	no	—
96	D₄	—	8	2	no	—
97	D₄	—	8	2	no	—
98	D₄	—	16	4	no	—
99	C₂×C₄	—	632	2	no	—
100	C₂×C₄	#14579102	1	2	yes	—
101	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
102	C₈	#14579105	1	2	yes	—
103	C₈	#14579148	1	1	yes	characteristic
104	C₈	C₂×#11929612	1	1	yes	characteristic
105	C₂²	—	16	2	no	—
106	C₂²	—	16	4	no	—
107	C₂²	C₂×#14579102	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	1264	2	no	—
110	C₄	#19098909	1	1	yes	characteristic
111	C₄	C₂×#14579105	1	1	yes	characteristic
112	C₂	—	32	2	no	—
113	C₂	#23765623	1	1	yes	characteristic
114	C₂	#23765713	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765679

C₂

yes

maximal, characteristic, complemented

#23761578

C₂

yes

maximal, complemented

#23761534

C₂

yes

maximal, characteristic, complemented

C₇₉×#2246

C₂

yes

maximal, characteristic

#19098965

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#19098391

—

#19098389

C₂²

yes

—

C₂×D₁₆×C₇₉

—

C₂×D₁₆×C₇₉

C₂²

yes

—

#493×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

C₇₉⋊C₃₂

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#127×C₇₉

—

D₁₆×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

#41

yes

—

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

SD₃₂

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#486

yes

characteristic

C₄×C₇₉

C₂×SD₃₂

yes

characteristic

Dic₇₉

—

#13723

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2240

yes

characteristic

C₂×C₇₉

#2263

yes

—

#2253

—

#2246

D₇₉

yes

characteristic

#1415

—

#1397

—

C₇₉

#13723

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

C₂×D₇₉

yes

—

#499

—

158

—

#493

—

C₂×C₃₂

—

#367

—

158

—

#366

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×D₈

—

C₂×D₈

—

#127

—

#125

—

316

—

D₁₆

—

D₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

158

—

#106

—

316

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

158

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

#11929609

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

316

—

C₄⋊C₄

—

632

—

C₄²

#11929609

yes

characteristic

C₁₆

#11929612

yes

—

C₁₆

#11929623

yes

—

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

632

—

100

C₂×C₄

#14579102

yes

—

101

C₂×C₄

C₂×#11929609

yes

characteristic

102

C₈

#14579105

yes

—

103

C₈

#14579148

yes

characteristic

104

C₈

C₂×#11929612

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#14579102

yes

characteristic

108

C₄

—

109

C₄

—

1264

—

110

C₄

#19098909

yes

characteristic

111

C₄

C₂×#14579105

yes

characteristic

112

C₂

—

113

C₂

#23765623

yes

characteristic

114

C₂

#23765713

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting, prime radical	C₇₉×#2246	C₂
socle	C₂²×C₇₉	C₂×D₁₆
hypercenter	#2246	D₇₉
hypocenter	C₇₉	#13723
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting, prime radical

C₇₉×#2246

C₂

socle

C₂²×C₇₉

C₂×D₁₆

D₇₉

C₇₉

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

N	H ≅ G/N	type
1. #23765679	C₂	semidirect
2. #23761578	C₂	semidirect
3. #23761534	C₂	semidirect
62. C₇₉	#13723	semidirect

H ≅ G/N

type

1. #23765679

C₂

semidirect

2. #23761578

C₂

semidirect

3. #23761534

C₂

semidirect

62. C₇₉

#13723

semidirect

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

subgroup	index
#23765679	2
#23761578	2
#23761534	2
C₇₉×#2246	2
#13723	79

subgroup

index

C₇₉×#2246

Subgroups — #34209255

Conjugacy classes of subgroups

Complements

Maximal subgroups