Group #34208772 — subgroups

Subgroups — #34208772

Order 20224.

190 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23765184	C₂	1	2	yes	maximal
2	#23761602	C₂	1	2	yes	maximal, complemented
3	#23761600	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098601	—	2	2	no	—
6	#19098391	—	2	2	no	—
7	#19098389	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₉	C₄	1	4	yes	complemented
10	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	#14579120	—	4	2	no	—
13	#14579095	—	2	4	no	—
14	#14579093	—	4	2	no	—
15	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
19	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
20	C₃₂×C₇₉	C₂×C₄	1	2	yes	—
21	C₃₂×C₇₉	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₇₉	—	4	4	no	—
23	C₂²⋊Dic₇₉	—	8	2	no	—
24	C₄×Dic₇₉	—	8	2	no	—
25	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
26	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
28	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
29	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
31	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
32	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₇₉	D₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₉	#107	1	2	yes	—
35	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
36	C₂×C₄×C₇₉	Q₃₂	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
38	C₂×Dic₇₉	—	8	4	no	—
39	C₈×C₇₉	#107	1	2	yes	—
40	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂²×C₇₉	—	2	2	no	—
43	C₂²×C₇₉	#366	1	2	yes	—
44	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
45	C₄×C₇₉	#366	1	2	yes	—
46	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
47	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
48	Dic₇₉	—	16	4	no	—
49	#13633	—	79	1	no	maximal, Sylow
50	C₂×C₇₉	—	2	2	no	—
51	C₂×C₇₉	C₂×#366	1	1	yes	characteristic
52	C₂×C₇₉	#2262	1	2	yes	—
53	C₂²×C₃₂	D₇₉	1	1	yes	characteristic
54	#2158	—	79	2	no	—
55	#1424	—	79	2	no	—
56	#1423	—	79	2	no	—
57	C₇₉	#13633	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
59	#427	—	158	2	no	—
60	C₂×C₃₂	Dic₇₉	1	4	yes	complemented
61	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
62	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
63	#367	—	158	2	no	—
64	#366	—	79	4	no	—
65	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
66	#117	—	316	2	no	—
67	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
68	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
69	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
70	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
71	#107	—	158	4	no	—
72	#106	—	316	2	no	—
73	C₃₂	C₂×Dic₇₉	1	2	yes	—
74	C₃₂	C₂×Dic₇₉	1	2	yes	—
75	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
77	C₂×C₈	D₆₃₂	1	1	yes	characteristic
78	C₂×C₈	Dic₃₁₆	1	2	yes	—
79	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
80	C₄⋊C₄	—	316	4	no	—
81	C₂²⋊C₄	—	632	2	no	—
82	C₄²	—	632	2	no	—
83	C₁₆	C₄⋊Dic₇₉	1	2	yes	—
84	C₁₆	C₂²×Dic₇₉	1	1	yes	characteristic
85	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
86	C₂³	D₁₂₆₄	1	1	yes	characteristic
87	C₂×C₄	—	632	4	no	—
88	C₂×C₄	#14579095	1	2	yes	—
89	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
90	C₂×C₄	Dic₆₃₂	1	2	yes	—
91	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
92	C₈	#14579095	1	2	yes	—
93	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
94	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#19098389	1	2	yes	—
97	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
98	C₄	—	1264	4	no	—
99	C₄	#19098389	1	2	yes	—
100	C₄	C₂×#14579095	1	1	yes	characteristic
101	C₄	C₂×Dic₆₃₂	1	1	yes	characteristic
102	C₂	—	2	2	no	—
103	C₂	C₂×#19098389	1	1	yes	characteristic
104	C₂	#23765709	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765184

C₂

yes

maximal

#23761602

C₂

yes

maximal, complemented

#23761600

C₂

yes

maximal, complemented

C₂²×C₃₂×C₇₉

C₂

yes

maximal, characteristic

#19098601

—

#19098391

—

#19098389

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×C₃₂×C₇₉

C₄

yes

complemented

C₂×C₃₂×C₇₉

C₂²

yes

characteristic

C₂×C₃₂×C₇₉

C₂²

yes

—

—

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₃₂×C₇₉

C₂×C₄

yes

—

C₃₂×C₇₉

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

Q₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

Q₃₂

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

#107

yes

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₂×Q₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#366

yes

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

#366

yes

—

C₄×C₇₉

C₂×#107

yes

characteristic

C₄×C₇₉

C₂×Q₃₂

yes

characteristic

Dic₇₉

—

#13633

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#366

yes

characteristic

C₂×C₇₉

#2262

yes

—

C₂²×C₃₂

D₇₉

yes

—

—

—

C₇₉

#13633

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#427

—

158

—

C₂×C₃₂

Dic₇₉

yes

complemented

C₂×C₃₂

C₂×D₇₉

yes

characteristic

C₂×C₃₂

C₂×D₇₉

yes

characteristic

#367

—

158

—

#366

—

C₂²×C₈

D₃₁₆

yes

characteristic

#117

—

316

—

C₂×C₁₆

Dic₁₅₈

yes

—

C₂×C₁₆

D₃₁₆

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

158

—

#106

—

316

—

C₃₂

C₂×Dic₇₉

yes

—

C₃₂

C₂×Dic₇₉

yes

—

C₂²×C₄

D₆₃₂

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

D₆₃₂

yes

characteristic

C₂×C₈

Dic₃₁₆

yes

—

C₂×C₈

C₂×D₃₁₆

yes

characteristic

C₄⋊C₄

—

316

—

C₂²⋊C₄

—

632

—

C₄²

—

632

—

C₁₆

C₄⋊Dic₇₉

yes

—

C₁₆

C₂²×Dic₇₉

yes

characteristic

C₁₆

C₂×Dic₁₅₈

yes

characteristic

C₂³

D₁₂₆₄

yes

characteristic

C₂×C₄

—

632

—

C₂×C₄

#14579095

yes

—

C₂×C₄

D₁₂₆₄

yes

characteristic

C₂×C₄

Dic₆₃₂

yes

—

C₂×C₄

C₂×D₆₃₂

yes

characteristic

C₈

#14579095

yes

—

C₈

C₂×C₄⋊Dic₇₉

yes

characteristic

C₈

C₂×Dic₃₁₆

yes

characteristic

C₂²

—

C₂²

#19098389

yes

—

C₂²

C₂×D₁₂₆₄

yes

characteristic

C₄

—

1264

—

C₄

#19098389

yes

—

100

C₄

C₂×#14579095

yes

characteristic

101

C₄

C₂×Dic₆₃₂

yes

characteristic

102

C₂

—

103

C₂

C₂×#19098389

yes

characteristic

104

C₂

#23765709

yes

—

	subgroup	G/H
center	C₂×C₄	D₁₂₆₄
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂³×C₇₉	D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂²×C₃₂	D₇₉
hypocenter	C₇₉	#13633
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₄

D₁₂₆₄

commutator

C₁₆×C₇₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₂²×C₃₂×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

C₂³×C₇₉

D₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

hypercenter

C₂²×C₃₂

D₇₉

hypocenter

C₇₉

#13633

LCS

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. #23761602	C₂	semidirect
3. #23761600	C₂	semidirect
9. C₂×C₃₂×C₇₉	C₄	semidirect
57. C₇₉	#13633	semidirect
60. C₂×C₃₂	Dic₇₉	semidirect

H ≅ G/N

type

2. #23761602

C₂

semidirect

3. #23761600

C₂

semidirect

9. C₂×C₃₂×C₇₉

C₄

semidirect

57. C₇₉

#13633

semidirect

60. C₂×C₃₂

Dic₇₉

semidirect

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

subgroup	index
#23765184	2
#23761602	2
#23761600	2
C₂²×C₃₂×C₇₉	2
#13633	79

subgroup

index

C₂²×C₃₂×C₇₉

Subgroups — #34208772

Conjugacy classes of subgroups

Complements

Maximal subgroups