Group #34199824 — subgroups

Subgroups — #34199824

Order 20224.

234 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 147 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	#23762148	C₂	1	2	yes	maximal
2	#23761466	C₂	1	4	yes	maximal, complemented
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098616	—	2	2	no	—
5	#19098316	—	2	2	no	—
6	#19098314	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	#346×C₇₉	C₄	1	2	yes	—
11	#346×C₇₉	C₂²	1	2	yes	—
12	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
13	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
14	C₄×C₇₉⋊C₈	—	4	2	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	—	2	2	no	—
19	C₂×C₁₆×C₇₉	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₇₉⋊C₈	—	4	4	no	—
27	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
29	C₂×C₈×C₇₉	—	2	2	no	—
30	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
34	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₉	SD₁₆	1	2	yes	—
38	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₇₉	—	2	2	no	—
40	C₁₆×C₇₉	C₂×C₈	1	4	yes	—
41	C₂³×C₇₉	#106	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	4	2	no	—
43	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₇₉	#106	1	2	yes	—
45	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#108	1	2	yes	—
47	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
49	C₈×C₇₉	—	2	1	no	—
50	C₈×C₇₉	—	2	1	no	—
51	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
52	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
54	C₇₉⋊C₈	—	8	4	no	—
55	C₂²×C₇₉	—	2	2	no	—
56	C₂²×C₇₉	#334	1	2	yes	—
57	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
58	C₄×C₇₉	—	2	2	no	—
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	#334	1	2	yes	—
61	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
63	#11999	—	79	1	no	maximal, Sylow
64	C₂×C₇₉	—	2	2	no	—
65	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
66	C₂×C₇₉	#2155	1	2	yes	—
67	C₂×#346	D₇₉	1	1	yes	characteristic
68	#1564	—	79	2	no	—
69	#1368	—	79	4	no	—
70	C₇₉	#11999	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
72	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
73	#431	—	158	2	no	—
74	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
75	#346	Dic₇₉	1	2	yes	—
76	#346	C₂×D₇₉	1	2	yes	—
77	#335	—	158	2	no	—
78	#334	—	79	4	no	—
79	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
80	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
81	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₇₉⋊C₈	1	4	yes	complemented
84	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
86	C₄⋊C₈	—	158	4	no	—
87	C₂²⋊C₈	—	316	2	no	—
88	C₄×C₈	—	316	2	no	—
89	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
90	C₄×C₈	D₃₁₆	1	1	yes	characteristic
91	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
92	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
93	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
94	C₂²×C₄	#11929613	1	2	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	316	4	no	—
97	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
98	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
99	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
100	C₂×C₈	#11929608	1	2	yes	—
101	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
102	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
103	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
104	C₄²	#11929613	1	2	yes	—
105	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
108	C₂³	#14579093	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
111	C₂×C₄	#14579093	1	2	yes	—
112	C₂×C₄	#14579093	1	1	yes	characteristic
113	C₂×C₄	#14579097	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
115	C₂×C₄	C₂×#11929613	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	632	4	no	—
119	C₈	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
120	C₈	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
121	C₈	C₂×#11929608	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#19098314	1	2	yes	—
124	C₂²	C₂×#14579093	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	#19098314	1	2	yes	—
128	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
129	C₄	C₂×#14579097	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#19098314	1	1	yes	characteristic
132	C₂	#23765173	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762148

C₂

yes

maximal

#23761466

C₂

yes

maximal, complemented

C₂×#346×C₇₉

C₂

yes

maximal, characteristic

#19098616

—

#19098316

—

#19098314

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₄

yes

characteristic

C₂²×C₁₆×C₇₉

C₄

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₄

yes

—

#346×C₇₉

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₈

yes

complemented

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

M₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

—

C₂³×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

#106

yes

—

C₂×C₄×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

#108

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₄⋊C₈

yes

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#334

yes

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#334

yes

—

C₄×C₇₉

C₂×C₄⋊C₈

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#334

yes

characteristic

C₂×C₇₉

#2155

yes

—

C₂×#346

D₇₉

yes

characteristic

#1564

—

#1368

—

C₇₉

#11999

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₇₉

yes

characteristic

C₂²×C₁₆

Dic₇₉

yes

characteristic

#431

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#346

Dic₇₉

yes

—

#346

C₂×D₇₉

yes

—

#335

—

158

—

#334

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₇₉⋊C₈

yes

complemented

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

316

—

C₄×C₈

—

316

—

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

#11929613

yes

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₂×C₇₉⋊C₈

yes

—

100

C₂×C₈

#11929608

yes

—

101

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

102

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

103

C₄²

C₄⋊Dic₇₉

yes

characteristic

104

C₄²

#11929613

yes

—

105

C₄²

C₂×D₃₁₆

yes

characteristic

106

C₁₆

—

107

C₁₆

C₂×C₇₉⋊C₈

yes

—

108

C₂³

#14579093

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

—

111

C₂×C₄

#14579093

yes

—

112

C₂×C₄

#14579093

yes

characteristic

113

C₂×C₄

#14579097

yes

—

114

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

115

C₂×C₄

C₂×#11929613

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

632

—

119

C₈

C₄⋊(C₇₉⋊C₈)

yes

—

120

C₈

C₂²×C₇₉⋊C₈

yes

characteristic

121

C₈

C₂×#11929608

yes

characteristic

122

C₂²

—

123

C₂²

#19098314

yes

—

124

C₂²

C₂×#14579093

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#19098314

yes

—

128

C₄

C₂×C₄⋊(C₇₉⋊C₈)

yes

characteristic

129

C₄

C₂×#14579097

yes

characteristic

130

C₂

—

131

C₂

C₂×#19098314

yes

characteristic

132

C₂

#23765173

yes

—

	subgroup	G/H
center	C₂×C₄	#14579093
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#11999
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#14579093

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₂×#346×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #23761466	C₂	semidirect
19. C₂×C₁₆×C₇₉	C₈	semidirect
70. C₇₉	#11999	semidirect
83. C₂×C₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

2. #23761466

C₂

semidirect

19. C₂×C₁₆×C₇₉

C₈

semidirect

70. C₇₉

#11999

semidirect

83. C₂×C₁₆

C₇₉⋊C₈

semidirect

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

subgroup	index
#23762148	2
#23761466	2
C₂×#346×C₇₉	2
#11999	79

subgroup

index

#23762148

#23761466

C₂×#346×C₇₉

#11999

Subgroups — #34199824

Conjugacy classes of subgroups

Complements

Maximal subgroups