Group #34209243 — subgroups

Subgroups — #34209243

Order 20224.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23765653	C₂	1	1	yes	maximal, characteristic
2	#23761580	C₂	1	4	yes	maximal
3	C₄×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#2252	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₃₂	—	2	1	no	—
6	C₂×Dic₆₃₂	C₂²	1	2	yes	—
7	#19098943	—	2	1	no	—
8	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
9	#500×C₇₉	—	2	1	no	—
10	#366×C₇₉	—	2	1	no	—
11	#366×C₇₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₁₆	—	2	2	no	—
14	C₂×Dic₃₁₆	—	4	1	no	—
15	#14579161	—	4	1	no	—
16	Dic₆₃₂	—	2	4	no	—
17	Dic₆₃₂	—	4	2	no	—
18	C₇₉⋊C₃₂	—	2	2	no	—
19	C₄⋊Q₈×C₇₉	—	4	1	no	—
20	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
22	#107×C₇₉	—	2	2	no	—
23	#107×C₇₉	—	4	1	no	—
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	4	2	no	—
26	Dic₃₁₆	—	4	2	no	—
27	Dic₃₁₆	—	4	2	no	—
28	Dic₃₁₆	—	8	2	no	—
29	C₄⋊Dic₇₉	—	8	1	no	—
30	C₄⋊Dic₇₉	—	8	1	no	—
31	C₂×Q₈×C₇₉	—	8	1	no	—
32	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₉	—	4	2	no	—
35	C₄⋊C₄×C₇₉	—	8	1	no	—
36	C₄²×C₇₉	D₈	1	1	yes	characteristic
37	C₁₆×C₇₉	SD₁₆	1	2	yes	—
38	C₁₆×C₇₉	#41	1	2	yes	—
39	Q₈×C₇₉	—	16	2	no	—
40	C₂×C₄×C₇₉	—	8	2	no	—
41	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₇₉	—	8	2	no	—
45	Dic₁₅₈	—	8	2	no	—
46	Dic₁₅₈	—	8	2	no	—
47	C₈×C₇₉	SD₃₂	1	2	yes	—
48	C₈×C₇₉	#124	1	1	yes	characteristic
49	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₇₉	—	16	2	no	—
52	C₄×C₇₉	SD₆₄	1	2	yes	—
53	C₄×C₇₉	Q₆₄	1	2	yes	—
54	C₄×C₇₉	#486	1	1	yes	characteristic
55	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₇₉	—	16	2	no	—
57	#13720	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	#2240	1	1	yes	characteristic
59	C₂×C₇₉	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	D₇₉	1	1	yes	characteristic
62	#2247	—	79	1	no	—
63	#1416	—	79	4	no	—
64	C₄×C₃₂	—	79	1	no	—
65	C₇₉	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	79	2	no	—
67	C₂×Q₃₂	—	158	1	no	—
68	#500	—	2	1	no	—
69	#494	—	158	1	no	—
70	C₂×C₃₂	—	79	2	no	—
71	#366	—	2	1	no	—
72	#366	C₂×D₇₉	1	2	yes	—
73	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
74	C₂×Q₁₆	—	316	1	no	—
75	C₂×Q₁₆	—	158	2	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	316	1	no	—
78	Q₃₂	—	316	2	no	—
79	Q₃₂	—	158	4	no	—
80	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
82	#107	—	2	2	no	—
83	#107	—	4	1	no	—
84	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
85	C₃₂	—	158	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	316	2	no	—
88	Q₁₆	—	316	2	no	—
89	Q₁₆	—	316	2	no	—
90	Q₁₆	—	632	2	no	—
91	C₂×C₈	#11929609	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	632	1	no	—
96	C₄⋊C₄	—	632	1	no	—
97	C₄²	#11929609	1	1	yes	characteristic
98	C₁₆	#11929611	1	2	yes	—
99	C₁₆	#11929625	1	2	yes	—
100	Q₈	—	16	2	no	—
101	Q₈	—	632	2	no	—
102	Q₈	—	632	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	632	2	no	—
105	C₂×C₄	#14579102	1	1	yes	characteristic
106	C₂×C₄	#14579102	1	1	yes	characteristic
107	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
108	C₈	#14579104	1	2	yes	—
109	C₈	#14579150	1	1	yes	characteristic
110	C₈	C₂×#11929611	1	1	yes	characteristic
111	C₂²	C₂×#14579102	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	1264	2	no	—
114	C₄	#19098401	1	2	yes	—
115	C₄	#19098403	1	2	yes	—
116	C₄	#19098911	1	1	yes	characteristic
117	C₄	C₂×#14579104	1	1	yes	characteristic
118	C₂	#23765625	1	1	yes	characteristic
119	C₂	C₂×#19098401	1	1	yes	characteristic
120	C₂	C₂×#19098403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765653

C₂

yes

maximal, characteristic

#23761580

C₂

yes

maximal

C₄×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic

C₇₉×#2252

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

—

C₂×Dic₆₃₂

C₂²

yes

—

#19098943

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#500×C₇₉

—

#366×C₇₉

—

#366×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

Dic₆₃₂

—

Dic₆₃₂

—

C₇₉⋊C₃₂

—

C₄⋊Q₈×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

#41

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

SD₃₂

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

SD₆₄

yes

—

C₄×C₇₉

Q₆₄

yes

—

C₄×C₇₉

#486

yes

characteristic

C₄×C₇₉

C₂×SD₃₂

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

C₂×SD₆₄

yes

characteristic

C₂×C₇₉

C₂×Q₆₄

yes

characteristic

#2252

D₇₉

yes

characteristic

#2247

—

#1416

—

C₄×C₃₂

—

C₇₉

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

158

—

#500

—

#494

—

158

—

C₂×C₃₂

—

#366

—

#366

C₂×D₇₉

yes

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

316

—

Q₃₂

—

316

—

Q₃₂

—

158

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

#107

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

158

—

C₂×Q₈

—

C₂×Q₈

—

316

—

Q₁₆

—

316

—

Q₁₆

—

316

—

Q₁₆

—

632

—

C₂×C₈

#11929609

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

632

—

C₄⋊C₄

—

632

—

C₄²

#11929609

yes

characteristic

C₁₆

#11929611

yes

—

C₁₆

#11929625

yes

—

100

Q₈

—

101

Q₈

—

632

—

102

Q₈

—

632

—

103

C₂×C₄

—

104

C₂×C₄

—

632

—

105

C₂×C₄

#14579102

yes

characteristic

106

C₂×C₄

#14579102

yes

characteristic

107

C₂×C₄

C₂×#11929609

yes

characteristic

108

C₈

#14579104

yes

—

109

C₈

#14579150

yes

characteristic

110

C₈

C₂×#11929611

yes

characteristic

111

C₂²

C₂×#14579102

yes

characteristic

112

C₄

—

113

C₄

—

1264

—

114

C₄

#19098401

yes

—

115

C₄

#19098403

yes

—

116

C₄

#19098911

yes

characteristic

117

C₄

C₂×#14579104

yes

characteristic

118

C₂

#23765625

yes

characteristic

119

C₂

C₂×#19098401

yes

characteristic

120

C₂

C₂×#19098403

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2252	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	#2252	D₇₉
hypocenter	C₇₉	#13720
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2252

C₂

socle, prime radical

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 ◁ #2252

N	H ≅ G/N	type
65. C₇₉	#13720	semidirect

H ≅ G/N

type

65. C₇₉

#13720

semidirect

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

subgroup	index
#23765653	2
#23761580	2
C₄×C₇₉⋊C₃₂	2
C₇₉×#2252	2
#13720	79

subgroup

index

#23765653

#23761580

C₄×C₇₉⋊C₃₂

C₇₉×#2252

#13720

Subgroups — #34209243

Conjugacy classes of subgroups

Complements

Maximal subgroups