Group #34209237 — subgroups

Subgroups — #34209237

Order 20224.

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

normal subgroups only

This group has 202 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	#23765675	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761578	C₂	1	4	yes	maximal, complemented
3	C₄×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#2246	C₂	1	1	yes	maximal, characteristic
5	#19098968	—	2	1	no	—
6	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
7	#19098389	—	2	1	no	—
8	#19098389	C₂²	1	2	yes	—
9	C₂×D₁₆×C₇₉	—	2	1	no	—
10	C₂×D₁₆×C₇₉	C₂²	1	2	yes	—
11	#493×C₇₉	—	2	1	no	—
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#14579161	—	4	1	no	—
14	#14579095	—	2	2	no	—
15	#14579095	—	4	1	no	—
16	C₇₉⋊C₃₂	—	2	2	no	—
17	C₂×D₈×C₇₉	—	2	2	no	—
18	C₂×D₈×C₇₉	—	4	1	no	—
19	#127×C₇₉	—	4	1	no	—
20	D₁₆×C₇₉	—	2	4	no	—
21	D₁₆×C₇₉	—	4	2	no	—
22	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	8	1	no	—
26	C₄⋊Dic₇₉	—	4	2	no	—
27	C₄⋊Dic₇₉	—	8	1	no	—
28	C₂×D₄×C₇₉	—	4	2	no	—
29	C₂×D₄×C₇₉	—	8	1	no	—
30	C₂×D₄×C₇₉	—	8	1	no	—
31	D₈×C₇₉	—	4	2	no	—
32	D₈×C₇₉	—	4	2	no	—
33	D₈×C₇₉	—	8	2	no	—
34	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₇₉	D₈	1	1	yes	characteristic
37	C₁₆×C₇₉	SD₁₆	1	2	yes	—
38	C₁₆×C₇₉	#41	1	2	yes	—
39	C₂³×C₇₉	—	8	2	no	—
40	D₄×C₇₉	—	8	2	no	—
41	D₄×C₇₉	—	8	2	no	—
42	D₄×C₇₉	—	16	2	no	—
43	D₄×C₇₉	—	16	2	no	—
44	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₉	—	8	2	no	—
48	Dic₁₅₈	—	16	2	no	—
49	C₈×C₇₉	SD₃₂	1	2	yes	—
50	C₈×C₇₉	#124	1	1	yes	characteristic
51	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
52	C₂²×C₇₉	—	16	2	no	—
53	C₂²×C₇₉	—	16	2	no	—
54	C₂²×C₇₉	—	16	2	no	—
55	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₉	D₃₂	1	2	yes	—
57	C₄×C₇₉	SD₆₄	1	2	yes	—
58	C₄×C₇₉	#486	1	1	yes	characteristic
59	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
60	Dic₇₉	—	16	2	no	—
61	#13719	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	32	2	no	—
63	C₂×C₇₉	#2240	1	1	yes	characteristic
64	C₂×C₇₉	C₂×D₃₂	1	1	yes	characteristic
65	C₂×C₇₉	C₂×SD₆₄	1	1	yes	characteristic
66	#2252	—	79	1	no	—
67	#2246	D₇₉	1	1	yes	characteristic
68	#1415	—	79	4	no	—
69	C₄×C₃₂	—	79	1	no	—
70	C₇₉	#13719	1	1	yes	characteristic, complemented, Sylow
71	C₂×D₁₆	—	2	1	no	—
72	C₂×D₁₆	C₂×D₇₉	1	2	yes	—
73	#500	—	158	1	no	—
74	#493	—	2	1	no	—
75	C₂×C₃₂	—	79	2	no	—
76	#366	—	79	2	no	—
77	#366	—	158	1	no	—
78	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
79	C₂×D₈	—	2	2	no	—
80	C₂×D₈	—	4	1	no	—
81	C₄⋊Q₈	—	316	1	no	—
82	#127	—	4	1	no	—
83	D₁₆	—	2	4	no	—
84	D₁₆	—	4	2	no	—
85	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
87	#107	—	316	1	no	—
88	#107	—	158	2	no	—
89	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
90	C₃₂	—	158	2	no	—
91	C₂×Q₈	—	632	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	8	1	no	—
94	C₂×D₄	—	8	1	no	—
95	D₈	—	4	2	no	—
96	D₈	—	4	2	no	—
97	D₈	—	8	2	no	—
98	C₂×C₈	#11929609	1	1	yes	characteristic
99	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
100	C₄⋊C₄	—	316	2	no	—
101	C₄⋊C₄	—	632	1	no	—
102	C₄²	#11929609	1	1	yes	characteristic
103	C₁₆	#11929612	1	2	yes	—
104	C₁₆	#11929623	1	2	yes	—
105	C₂³	—	8	2	no	—
106	Q₈	—	1264	2	no	—
107	D₄	—	8	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	16	2	no	—
110	D₄	—	16	2	no	—
111	C₂×C₄	—	632	2	no	—
112	C₂×C₄	#14579102	1	1	yes	characteristic
113	C₂×C₄	#14579102	1	1	yes	characteristic
114	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
115	C₈	#14579105	1	2	yes	—
116	C₈	#14579148	1	1	yes	characteristic
117	C₈	C₂×#11929612	1	1	yes	characteristic
118	C₂²	—	16	2	no	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	C₂×#14579102	1	1	yes	characteristic
122	C₄	—	1264	2	no	—
123	C₄	#19098398	1	2	yes	—
124	C₄	#19098401	1	2	yes	—
125	C₄	#19098909	1	1	yes	characteristic
126	C₄	C₂×#14579105	1	1	yes	characteristic
127	C₂	—	32	2	no	—
128	C₂	#23765623	1	1	yes	characteristic
129	C₂	C₂×#19098398	1	1	yes	characteristic
130	C₂	C₂×#19098401	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic, complemented

#23761578

C₂

yes

maximal, complemented

C₄×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₇₉×#2246

C₂

yes

maximal, characteristic

#19098968

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#19098389

—

#19098389

C₂²

yes

—

C₂×D₁₆×C₇₉

—

C₂×D₁₆×C₇₉

C₂²

yes

—

#493×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

C₇₉⋊C₃₂

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#127×C₇₉

—

D₁₆×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

#41

yes

—

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

SD₃₂

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

D₃₂

yes

—

C₄×C₇₉

SD₆₄

yes

—

C₄×C₇₉

#486

yes

characteristic

C₄×C₇₉

C₂×SD₃₂

yes

characteristic

Dic₇₉

—

#13719

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2240

yes

characteristic

C₂×C₇₉

C₂×D₃₂

yes

characteristic

C₂×C₇₉

C₂×SD₆₄

yes

characteristic

#2252

—

#2246

D₇₉

yes

characteristic

#1415

—

C₄×C₃₂

—

C₇₉

#13719

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

C₂×D₇₉

yes

—

#500

—

158

—

#493

—

C₂×C₃₂

—

#366

—

#366

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

316

—

#127

—

D₁₆

—

D₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

316

—

#107

—

158

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

158

—

C₂×Q₈

—

632

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

#11929609

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

100

C₄⋊C₄

—

316

—

101

C₄⋊C₄

—

632

—

102

C₄²

#11929609

yes

characteristic

103

C₁₆

#11929612

yes

—

104

C₁₆

#11929623

yes

—

105

C₂³

—

106

Q₈

—

1264

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

632

—

112

C₂×C₄

#14579102

yes

characteristic

113

C₂×C₄

#14579102

yes

characteristic

114

C₂×C₄

C₂×#11929609

yes

characteristic

115

C₈

#14579105

yes

—

116

C₈

#14579148

yes

characteristic

117

C₈

C₂×#11929612

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#14579102

yes

characteristic

122

C₄

—

1264

—

123

C₄

#19098398

yes

—

124

C₄

#19098401

yes

—

125

C₄

#19098909

yes

characteristic

126

C₄

C₂×#14579105

yes

characteristic

127

C₂

—

128

C₂

#23765623

yes

characteristic

129

C₂

C₂×#19098398

yes

characteristic

130

C₂

C₂×#19098401

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting, prime radical	C₇₉×#2246	C₂
socle	C₂²×C₇₉	C₂×D₁₆
hypercenter	#2246	D₇₉
hypocenter	C₇₉	#13719
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting, prime radical

C₇₉×#2246

C₂

socle

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 ◁ #2246

N	H ≅ G/N	type
1. #23765675	C₂	semidirect
2. #23761578	C₂	semidirect
3. C₄×C₇₉⋊C₃₂	C₂	semidirect
70. C₇₉	#13719	semidirect

H ≅ G/N

type

1. #23765675

C₂

semidirect

2. #23761578

C₂

semidirect

3. C₄×C₇₉⋊C₃₂

C₂

semidirect

70. C₇₉

#13719

semidirect

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

subgroup	index
#23765675	2
#23761578	2
C₄×C₇₉⋊C₃₂	2
C₇₉×#2246	2
#13719	79

subgroup

index

#23765675

#23761578

C₄×C₇₉⋊C₃₂

C₇₉×#2246

#13719

Subgroups — #34209237

Conjugacy classes of subgroups

Complements

Maximal subgroups