Group #34175749 — subgroups

Subgroups — #34175749

Order 20224.

110 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 59 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⋊(C₇₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#23761534	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2183	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₉⋊C₃₂	—	2	1	no	—
5	C₂×C₇₉⋊C₃₂	—	2	1	no	—
6	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₇₉	—	2	1	no	—
8	C₄⋊C₁₆×C₇₉	C₄	1	1	yes	characteristic
9	#346×C₇₉	—	2	1	no	—
10	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	C₇₉⋊C₃₂	—	2	2	no	—
12	C₇₉⋊C₃₂	—	4	1	no	—
13	C₄×Q₈×C₇₉	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₇₉	—	2	1	no	—
15	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
17	C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₄⋊C₈×C₇₉	C₈	1	1	yes	characteristic
19	C₄×C₈×C₇₉	—	2	1	no	—
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₇₉	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₉	—	2	1	no	—
23	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
25	C₄⋊C₄×C₇₉	—	2	1	no	—
26	C₄⋊C₄×C₇₉	C₁₆	1	1	yes	characteristic
27	C₄²×C₇₉	—	2	1	no	—
28	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
29	C₁₆×C₇₉	—	2	1	no	—
30	C₁₆×C₇₉	—	4	1	no	—
31	C₁₆×C₇₉	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₇₉	Q₁₆	1	1	yes	characteristic
33	Q₈×C₇₉	—	4	1	no	—
34	C₂×C₄×C₇₉	—	2	1	no	—
35	C₂×C₄×C₇₉	—	2	1	no	—
36	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₉	M₃₂	1	1	yes	characteristic
39	C₈×C₇₉	—	2	1	no	—
40	C₈×C₇₉	—	4	1	no	—
41	C₈×C₇₉	#103	1	1	yes	characteristic
42	C₈×C₇₉	#104	1	1	yes	characteristic
43	C₂²×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₇₉	—	2	1	no	—
45	C₄×C₇₉	—	2	1	no	—
46	C₄×C₇₉	—	4	1	no	—
47	C₄×C₇₉	—	4	1	no	—
48	C₄×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
49	C₄×C₇₉	#350	1	1	yes	characteristic
50	#7393	—	79	1	no	maximal, Sylow
51	C₂×C₇₉	Q₈⋊C₁₆	1	1	yes	characteristic
52	C₂×C₇₉	#1400	1	1	yes	characteristic
53	C₂×C₇₉	#1401	1	1	yes	characteristic
54	#2183	D₇₉	1	1	yes	characteristic
55	C₄⋊C₃₂	—	79	1	no	—
56	#1397	—	79	1	no	—
57	C₇₉	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	Dic₇₉	1	1	yes	characteristic
59	C₂×C₃₂	—	158	1	no	—
60	C₂×C₃₂	—	158	1	no	—
61	C₄⋊C₁₆	—	2	1	no	—
62	C₄⋊C₁₆	Dic₇₉	1	1	yes	characteristic
63	#346	—	2	1	no	—
64	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
65	C₄×Q₈	C₇₉⋊C₈	1	1	yes	characteristic
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₇₉⋊C₈	1	1	yes	characteristic
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
73	C₃₂	—	316	1	no	—
74	C₃₂	—	158	2	no	—
75	C₂×Q₈	C₇₉⋊C₁₆	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
78	C₂×C₈	#11929608	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	C₇₉⋊C₁₆	1	1	yes	characteristic
81	C₄²	—	2	1	no	—
82	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	4	1	no	—
85	C₁₆	#11929611	1	1	yes	characteristic
86	C₁₆	#11929614	1	1	yes	characteristic
87	Q₈	—	4	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
91	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
92	C₂×C₄	#14579101	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	1	no	—
95	C₈	#14579085	1	1	yes	characteristic
96	C₈	#14579088	1	1	yes	characteristic
97	C₂²	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
103	C₄	#19098347	1	1	yes	characteristic
104	C₂	Q₈⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
105	C₂	#23761540	1	1	yes	characteristic
106	C₂	#23761542	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₇₉⋊C₃₂)

C₂

yes

maximal, characteristic

#23761534

C₂

yes

maximal, characteristic

C₇₉×#2183

C₂

yes

maximal, characteristic

C₂×C₇₉⋊C₃₂

—

C₂×C₇₉⋊C₃₂

—

C₈×Q₈×C₇₉

C₄

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊C₁₆×C₇₉

C₄

yes

characteristic

#346×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₇₉⋊C₃₂

—

C₇₉⋊C₃₂

—

C₄×Q₈×C₇₉

C₈

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₈

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₉

C₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₁₆

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

SD₁₆

yes

characteristic

C₁₆×C₇₉

Q₁₆

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₉

M₃₂

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₂²×C₇₉

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

Q₈⋊C₈

yes

C₄×C₇₉

yes

—

C₂×C₇₉

Q₈⋊C₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

C₄⋊C₃₂

—

#1397

—

C₇₉

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

Dic₇₉

yes

characteristic

C₂×C₃₂

—

158

—

C₂×C₃₂

—

158

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Dic₇₉

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×Q₈

C₇₉⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₇₉⋊C₈

yes

characteristic

C₄×C₈

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₃₂

—

316

—

C₃₂

—

158

—

C₂×Q₈

C₇₉⋊C₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

#11929608

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₇₉⋊C₁₆

yes

characteristic

C₄²

—

C₄²

C₂×C₇₉⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

C₂×C₄

C₂×C₇₉⋊C₁₆

yes

characteristic

C₂×C₄

#14579101

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

C₂²⋊(C₇₉⋊C₁₆)

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

Q₈⋊(C₇₉⋊C₈)

yes

characteristic

103

C₄

#19098347

yes

characteristic

104

C₂

Q₈⋊(C₇₉⋊C₁₆)

yes

characteristic

105

C₂

#23761540

yes

characteristic

106

C₂

#23761542

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₉
Fitting	C₇₉×#2183	C₂
socle, prime radical	C₂²×C₇₉	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	#2183	D₇₉
hypocenter	C₇₉	#7393
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₇₉

Fitting

C₇₉×#2183

C₂

socle, prime radical

C₂²×C₇₉

C₂²⋊C₁₆

cosocle

C₄×C₁₆×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

N	H ≅ G/N	type
57. C₇₉	#7393	semidirect

H ≅ G/N

type

57. C₇₉

#7393

semidirect

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

subgroup	index
C₄⋊(C₇₉⋊C₃₂)	2
#23761534	2
C₇₉×#2183	2
#7393	79

subgroup

index

C₄⋊(C₇₉⋊C₃₂)

#23761534

C₇₉×#2183

#7393

Subgroups — #34175749

Conjugacy classes of subgroups

Complements

Maximal subgroups