Group #34175723 — subgroups

Subgroups — #34175723

Order 20224.

126 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
2	#23761472	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₃₂	C₄	1	1	yes	characteristic
5	C₄×Dic₃₁₆	—	2	1	no	—
6	#19098389	C₄	1	1	yes	characteristic
7	#19098368	—	2	1	no	—
8	#19098316	—	2	1	no	—
9	C₂×C₃₂×C₇₉	—	2	1	no	—
10	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁₆	—	2	1	no	—
12	C₄×Dic₁₅₈	—	4	1	no	—
13	Dic₆₃₂	—	2	1	no	—
14	Dic₆₃₂	C₈	1	2	yes	complemented
15	#14579095	—	2	1	no	—
16	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
17	#14579086	—	4	1	no	—
18	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₃₂×C₇₉	—	2	2	no	—
22	C₂×Dic₁₅₈	—	4	1	no	—
23	Dic₃₁₆	—	2	2	no	—
24	Dic₃₁₆	—	4	1	no	—
25	C₂×C₇₉⋊C₈	—	8	1	no	—
26	C₄⋊Dic₇₉	—	4	1	no	—
27	Dic₇₉⋊C₄	—	8	1	no	—
28	C₄×Dic₇₉	—	8	1	no	—
29	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
31	C₄²×C₇₉	D₈	1	1	yes	characteristic
32	C₁₆×C₇₉	—	2	1	no	—
33	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
35	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
36	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
37	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
38	C₂×Dic₇₉	—	8	1	no	—
39	C₂×Dic₇₉	—	8	1	no	—
40	Dic₁₅₈	—	4	2	no	—
41	Dic₁₅₈	—	8	1	no	—
42	C₈×C₇₉	—	2	1	no	—
43	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₈×C₇₉	#104	1	1	yes	characteristic
45	C₇₉⋊C₈	—	16	1	no	—
46	C₂²×C₇₉	#357	1	1	yes	characteristic
47	C₄×C₇₉	—	2	1	no	—
48	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
49	C₄×C₇₉	#359	1	1	yes	characteristic
50	C₄×C₇₉	SD₆₄	1	1	yes	characteristic
51	C₄×C₇₉	Q₆₄	1	1	yes	characteristic
52	Dic₇₉	—	8	2	no	—
53	Dic₇₉	—	16	1	no	—
54	#7384	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	#1331	1	1	yes	characteristic
56	C₂×C₇₉	#1416	1	1	yes	characteristic
57	C₂×C₇₉	#1417	1	1	yes	characteristic
58	C₄×Q₃₂	—	79	1	no	—
59	C₄×C₃₂	D₇₉	1	1	yes	characteristic
60	#1371	—	79	1	no	—
61	C₇₉	#7384	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₃₂	—	79	1	no	—
63	C₄×Q₁₆	—	158	1	no	—
64	C₂×C₃₂	—	2	1	no	—
65	#366	—	79	1	no	—
66	#358	—	158	1	no	—
67	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
68	#335	—	158	1	no	—
69	C₂×Q₁₆	—	158	1	no	—
70	C₄×Q₈	—	316	1	no	—
71	Q₃₂	—	79	2	no	—
72	Q₃₂	—	158	1	no	—
73	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
74	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
75	#107	—	158	1	no	—
76	C₄⋊C₈	—	316	1	no	—
77	#103	—	316	1	no	—
78	C₄×C₈	D₃₁₆	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₂×Q₈	—	316	1	no	—
81	Q₁₆	—	316	1	no	—
82	Q₁₆	—	158	2	no	—
83	C₂×C₈	—	632	1	no	—
84	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
85	C₂×C₈	#11929613	1	1	yes	characteristic
86	C₄⋊C₄	—	316	1	no	—
87	C₄⋊C₄	—	632	1	no	—
88	C₄²	—	632	1	no	—
89	C₄²	D₆₃₂	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	C₈×D₇₉	1	1	yes	characteristic
92	C₁₆	#11929607	1	1	yes	characteristic
93	Q₈	—	316	2	no	—
94	Q₈	—	632	1	no	—
95	C₂×C₄	—	632	1	no	—
96	C₂×C₄	—	632	1	no	—
97	C₂×C₄	#14579083	1	1	yes	characteristic
98	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
99	C₂×C₄	#14579106	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	1264	1	no	—
102	C₈	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
103	C₈	#14579089	1	1	yes	characteristic
104	C₂²	#19098365	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	632	2	no	—
107	C₄	—	1264	1	no	—
108	C₄	#19098292	1	1	yes	characteristic
109	C₄	#19098371	1	1	yes	characteristic
110	C₄	#19098402	1	1	yes	characteristic
111	C₄	Dic₁₂₆₄	1	1	yes	characteristic
112	C₂	#23761362	1	1	yes	characteristic
113	C₂	#23761582	1	1	yes	characteristic
114	C₂	#23761585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₆₃₂

C₂

yes

maximal, characteristic

#23761472

C₂

yes

maximal, characteristic

C₄×C₃₂×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₄

yes

characteristic

C₄×Dic₃₁₆

—

#19098389

C₄

yes

characteristic

#19098368

—

#19098316

—

C₂×C₃₂×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₄×Dic₁₅₈

—

Dic₆₃₂

—

Dic₆₃₂

C₈

yes

complemented

#14579095

—

C₄⋊(C₇₉⋊C₈)

—

#14579086

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

M₁₆

yes

characteristic

C₂×C₄×C₇₉

#102

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊C₈

yes

characteristic

C₈×C₇₉

#104

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#357

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

D₄⋊C₈

yes

characteristic

C₄×C₇₉

#359

yes

characteristic

C₄×C₇₉

SD₆₄

yes

characteristic

C₄×C₇₉

Q₆₄

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₄×Q₃₂

—

C₄×C₃₂

D₇₉

yes

characteristic

#1371

—

C₇₉

#7384

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₄×Q₁₆

—

158

—

C₂×C₃₂

—

#366

—

#358

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

158

—

C₂×Q₁₆

—

158

—

C₄×Q₈

—

316

—

Q₃₂

—

Q₃₂

—

158

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

#107

—

158

—

C₄⋊C₈

—

316

—

#103

—

316

—

C₄×C₈

D₃₁₆

yes

characteristic

C₃₂

—

C₂×Q₈

—

316

—

Q₁₆

—

316

—

Q₁₆

—

158

—

C₂×C₈

—

632

—

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂×C₈

#11929613

yes

characteristic

C₄⋊C₄

—

316

—

C₄⋊C₄

—

632

—

C₄²

—

632

—

C₄²

D₆₃₂

yes

characteristic

C₁₆

—

C₁₆

C₈×D₇₉

yes

characteristic

C₁₆

#11929607

yes

characteristic

Q₈

—

316

—

Q₈

—

632

—

C₂×C₄

—

632

—

C₂×C₄

—

632

—

C₂×C₄

#14579083

yes

characteristic

C₂×C₄

D₁₂₆₄

yes

characteristic

C₂×C₄

#14579106

yes

characteristic

100

C₈

—

101

C₈

—

1264

—

102

C₈

(C₂×D₇₉)⋊C₈

yes

characteristic

103

C₈

#14579089

yes

characteristic

104

C₂²

#19098365

yes

characteristic

105

C₄

—

106

C₄

—

632

—

107

C₄

—

1264

—

108

C₄

#19098292

yes

characteristic

109

C₄

#19098371

yes

characteristic

110

C₄

#19098402

yes

characteristic

111

C₄

Dic₁₂₆₄

yes

characteristic

112

C₂

#23761362

yes

characteristic

113

C₂

#23761582

yes

characteristic

114

C₂

#23761585

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₆₄
commutator	C₁₆×C₇₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#357
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#7384
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂×C₄

D₁₂₆₄

commutator

C₁₆×C₇₉

C₂×C₈

Frattini

C₄×C₁₆

C₂×D₇₉

Fitting

C₄×C₃₂×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

#357

cosocle

C₄×C₁₆×C₇₉

C₂²

hypercenter

C₄×C₃₂

D₇₉

hypocenter

C₇₉

#7384

LCS

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

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
14. Dic₆₃₂	C₈	semidirect
61. C₇₉	#7384	semidirect

H ≅ G/N

type

14. Dic₆₃₂

C₈

semidirect

61. C₇₉

#7384

semidirect

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

subgroup	index
C₄×Dic₆₃₂	2
#23761472	2
C₄×C₃₂×C₇₉	2
#7384	79

subgroup

index

C₄×Dic₆₃₂

#23761472

C₄×C₃₂×C₇₉

#7384

Subgroups — #34175723

Conjugacy classes of subgroups

Complements

Maximal subgroups