Group #34203004 — subgroups

Subgroups — #34203004

Order 20224.

262 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23765673	C₂	1	4	yes	maximal
2	#23763018	C₂	1	1	yes	maximal, characteristic
3	C₈×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1713	C₂	1	1	yes	maximal, characteristic
5	#19098968	—	2	1	no	—
6	#19098968	C₂²	1	2	yes	—
7	#19098943	—	2	1	no	—
8	#19098388	—	2	8	no	—
9	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	#500×C₇₉	—	2	1	no	—
11	#500×C₇₉	C₂²	1	2	yes	—
12	#494×C₇₉	—	2	1	no	—
13	C₈²×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₁₆	—	4	2	no	—
15	#14579161	—	2	2	no	—
16	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
17	#14579095	—	2	4	no	—
18	#14579095	—	4	2	no	—
19	C₂×Q₁₆×C₇₉	—	4	2	no	—
20	C₄⋊Q₈×C₇₉	—	2	2	no	—
21	#107×C₇₉	—	2	4	no	—
22	#107×C₇₉	—	4	2	no	—
23	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₅₈	—	4	2	no	—
27	Dic₃₁₆	—	8	4	no	—
28	C₄⋊Dic₇₉	—	4	2	no	—
29	C₄⋊Dic₇₉	—	4	2	no	—
30	C₇₉⋊C₁₆	Q₁₆	1	8	yes	complemented
31	C₂×Q₈×C₇₉	—	4	2	no	—
32	Q₁₆×C₇₉	—	8	4	no	—
33	C₂×C₈×C₇₉	D₈	1	2	yes	—
34	C₂×C₈×C₇₉	D₈	1	2	yes	—
35	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
36	C₄⋊C₄×C₇₉	—	4	2	no	—
37	C₄⋊C₄×C₇₉	—	4	2	no	—
38	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₇₉	—	8	4	no	—
40	C₂×C₄×C₇₉	—	8	2	no	—
41	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₇₉	—	8	2	no	—
45	Dic₁₅₈	—	8	4	no	—
46	C₈×C₇₉	D₁₆	1	4	yes	—
47	C₈×C₇₉	Q₃₂	1	4	yes	—
48	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
49	C₂²×C₇₉	#493	1	1	yes	characteristic
50	C₄×C₇₉	—	16	2	no	—
51	C₄×C₇₉	#494	1	2	yes	—
52	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
53	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
54	Dic₇₉	—	16	2	no	—
55	#12521	—	79	1	no	maximal, Sylow
56	C₂×C₇₉	#1713	1	1	yes	characteristic
57	C₂×C₇₉	#2246	1	1	yes	characteristic
58	C₂×C₇₉	#2247	1	1	yes	characteristic
59	#2252	—	79	4	no	—
60	#1713	—	79	1	no	—
61	#1713	D₇₉	1	1	yes	characteristic
62	C₈×C₁₆	—	79	1	no	—
63	C₇₉	#12521	1	1	yes	characteristic, complemented, Sylow
64	#500	—	2	1	no	—
65	#500	—	79	2	no	—
66	#500	—	158	1	no	—
67	#500	C₂×D₇₉	1	2	yes	—
68	#494	—	2	1	no	—
69	#494	—	158	1	no	—
70	#366	—	158	8	no	—
71	C₄×C₁₆	—	79	2	no	—
72	C₈²	C₂×D₇₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	4	2	no	—
74	C₂×Q₁₆	—	316	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	158	2	no	—
77	C₂×C₁₆	—	79	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	316	2	no	—
81	#107	—	158	4	no	—
82	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	316	2	no	—
87	Q₁₆	—	8	4	no	—
88	Q₁₆	—	632	4	no	—
89	C₂×C₈	#11929609	1	2	yes	—
90	C₂×C₈	#11929609	1	2	yes	—
91	C₂×C₈	D₄×D₇₉	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	316	2	no	—
95	C₄⋊C₄	—	316	2	no	—
96	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
97	C₁₆	—	79	8	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	632	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	632	2	no	—
102	C₂×C₄	#14579157	1	1	yes	characteristic
103	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
104	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
105	C₈	#14579102	1	4	yes	—
106	C₈	#14579107	1	4	yes	—
107	C₈	Q₁₆×D₇₉	1	4	yes	—
108	C₂²	#19098939	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	1264	2	no	—
111	C₄	#19098941	1	2	yes	—
112	C₄	C₂×#14579102	1	2	yes	—
113	C₄	C₂×#14579107	1	2	yes	—
114	C₂	#23763016	1	1	yes	characteristic
115	C₂	#23765649	1	1	yes	characteristic
116	C₂	#23765652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765673

C₂

yes

maximal

#23763018

C₂

yes

maximal, characteristic

C₈×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic

C₇₉×#1713

C₂

yes

maximal, characteristic

#19098968

—

#19098968

C₂²

yes

—

#19098943

—

#19098388

—

C₄×C₇₉⋊C₁₆

C₂²

yes

—

#500×C₇₉

—

#500×C₇₉

C₂²

yes

—

#494×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

#14579161

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579095

—

#14579095

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

Q₃₂

yes

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#494

yes

—

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

C₂×Q₃₂

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₈×C₁₆

—

C₇₉

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

#500

—

158

—

#500

C₂×D₇₉

yes

—

#494

—

#494

—

158

—

#366

—

158

—

C₄×C₁₆

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

316

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

316

—

#107

—

158

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

Q₁₆

—

Q₁₆

—

632

—

C₂×C₈

#11929609

yes

—

C₂×C₈

#11929609

yes

—

C₂×C₈

D₄×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

632

—

100

C₂×C₄

—

101

C₂×C₄

—

632

—

102

C₂×C₄

#14579157

yes

characteristic

103

C₂×C₄

C₂×#11929609

yes

characteristic

104

C₂×C₄

C₂×#11929609

yes

characteristic

105

C₈

#14579102

yes

—

106

C₈

#14579107

yes

—

107

C₈

Q₁₆×D₇₉

yes

—

108

C₂²

#19098939

yes

characteristic

109

C₄

—

110

C₄

—

1264

—

111

C₄

#19098941

yes

—

112

C₄

C₂×#14579102

yes

—

113

C₄

C₂×#14579107

yes

—

114

C₂

#23763016

yes

characteristic

115

C₂

#23765649

yes

characteristic

116

C₂

#23765652

yes

characteristic

	subgroup	G/H
center	C₂²	#19098939
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1713	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	#1713	D₇₉
hypocenter	C₇₉	#12521
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19098939

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1713

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
30. C₇₉⋊C₁₆	Q₁₆	semidirect
63. C₇₉	#12521	semidirect

H ≅ G/N

type

30. C₇₉⋊C₁₆

Q₁₆

semidirect

63. C₇₉

#12521

semidirect

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

subgroup	index
#23765673	2
#23763018	2
C₈×C₇₉⋊C₁₆	2
C₇₉×#1713	2
#12521	79

subgroup

index

#23765673

#23763018

C₈×C₇₉⋊C₁₆

C₇₉×#1713

#12521

Subgroups — #34203004

Conjugacy classes of subgroups

Complements

Maximal subgroups