Group #34200024 — subgroups

Subgroups — #34200024

Order 20224.

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

normal subgroups only

This group has 216 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	#23765282	C₂	1	1	yes	maximal, characteristic
2	#23765236	C₂	1	2	yes	maximal, complemented
3	#23761372	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098995	C₄	1	1	yes	characteristic
7	#19098981	—	2	2	no	—
8	#19098681	C₂²	1	2	yes	—
9	#19098393	C₄	1	1	yes	characteristic
10	#19098371	C₄	1	2	yes	—
11	#19098348	—	2	2	no	—
12	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	#364×C₇₉	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#14579183	C₂×C₄	1	2	yes	—
17	#14579167	—	2	2	no	—
18	Dic₆₃₂	C₈	1	1	yes	characteristic, complemented
19	#14579106	C₈	1	2	yes	complemented
20	D₁₂₆₄	C₈	1	1	yes	characteristic, complemented
21	#14579100	—	4	2	no	—
22	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₇₉	—	4	2	no	—
25	#14579097	C₂×C₄	1	2	yes	—
26	#14579089	—	2	2	no	—
27	C₄×C₇₉⋊C₈	—	2	1	no	—
28	M₃₂×C₇₉	—	2	2	no	—
29	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	#11929624	—	2	2	no	—
33	Dic₃₁₆	C₂×C₈	1	2	yes	—
34	#11929613	—	2	2	no	—
35	D₆₃₂	C₂×C₈	1	2	yes	—
36	#11929608	—	2	2	no	—
37	#11929607	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₈×D₇₉	—	4	2	no	—
40	C₇₉⋊C₁₆	—	2	1	no	—
41	C₇₉⋊C₁₆	D₈	1	1	yes	characteristic
42	C₇₉⋊C₁₆	D₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₇₉	#41	1	1	yes	characteristic
46	C₁₆×C₇₉	—	2	1	no	—
47	C₁₆×C₇₉	—	4	2	no	—
48	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₇₉	—	4	2	no	—
53	D₃₁₆	—	2	2	no	—
54	C₄×D₇₉	—	4	2	no	—
55	Dic₁₅₈	—	2	2	no	—
56	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₇₉	#131	1	1	yes	characteristic
58	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
59	C₈×C₇₉	#135	1	1	yes	characteristic
60	C₇₉⋊C₈	—	2	4	no	—
61	C₇₉⋊C₈	—	4	2	no	—
62	C₂²×C₇₉	#443	1	1	yes	characteristic
63	C₂×D₇₉	—	4	2	no	—
64	C₄×C₇₉	—	2	2	no	—
65	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
66	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
67	Dic₇₉	—	4	2	no	—
68	#12046	—	79	1	no	maximal, Sylow
69	C₂×C₇₉	—	2	1	no	—
70	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
71	D₇₉	—	8	2	no	—
72	#2178	—	79	1	no	—
73	#2170	—	79	2	no	—
74	#1373	D₇₉	1	1	yes	characteristic, complemented
75	#1335	—	79	2	no	—
76	C₈×C₁₆	—	79	1	no	—
77	C₇₉	#12046	1	1	yes	characteristic, complemented, Sylow
78	#508	—	79	1	no	—
79	#504	—	158	2	no	—
80	#443	—	79	2	no	—
81	#368	—	79	1	no	—
82	#364	C₂×D₇₉	1	2	yes	—
83	#359	—	79	2	no	—
84	#350	—	158	2	no	—
85	C₄×C₁₆	—	79	1	no	—
86	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
87	C₈²	—	79	1	no	—
88	#135	—	79	2	no	—
89	#131	—	158	2	no	—
90	Q₃₂	—	79	1	no	—
91	SD₃₂	—	79	2	no	—
92	D₁₆	—	79	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	316	2	no	—
95	C₂×C₁₆	—	316	2	no	—
96	C₂×C₁₆	—	79	1	no	—
97	C₂×C₁₆	—	79	1	no	—
98	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
100	#108	—	79	2	no	—
101	#104	—	158	2	no	—
102	C₄×C₈	—	158	1	no	—
103	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	#41	—	158	2	no	—
105	Q₁₆	—	79	2	no	—
106	SD₁₆	—	158	2	no	—
107	D₈	—	79	2	no	—
108	M₁₆	—	316	2	no	—
109	M₁₆	—	158	2	no	—
110	C₂×C₈	—	316	2	no	—
111	C₂×C₈	—	158	2	no	—
112	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
113	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
114	C₄²	#11929625	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	79	1	no	—
118	C₁₆	—	79	1	no	—
119	C₁₆	—	158	1	no	—
120	C₁₆	C₈×D₇₉	1	1	yes	characteristic
121	C₁₆	C₈×D₇₉	1	1	yes	characteristic
122	Q₈	—	158	2	no	—
123	D₄	—	316	2	no	—
124	D₄	—	158	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	316	2	no	—
127	C₂×C₄	#14579123	1	1	yes	characteristic
128	C₈	—	316	2	no	—
129	C₈	—	158	4	no	—
130	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
131	C₈	#14579169	1	1	yes	characteristic
132	C₈	D₈×D₇₉	1	1	yes	characteristic
133	C₈	#14579184	1	1	yes	characteristic
134	C₂²	—	316	2	no	—
135	C₂²	#19098679	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	316	2	no	—
138	C₄	#19098628	1	1	yes	characteristic
139	C₄	#19098645	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	632	2	no	—
142	C₂	#23762194	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765282

C₂

yes

maximal, characteristic

#23765236

C₂

yes

maximal, complemented

#23761372

C₂

yes

maximal, complemented

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#1373

C₂

yes

maximal, characteristic, complemented

#19098995

C₄

yes

characteristic

#19098981

—

#19098681

C₂²

yes

—

#19098393

C₄

yes

characteristic

#19098371

C₄

yes

—

#19098348

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#364×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#14579183

C₂×C₄

yes

—

#14579167

—

Dic₆₃₂

C₈

yes

characteristic, complemented

#14579106

C₈

yes

complemented

D₁₂₆₄

C₈

yes

characteristic, complemented

#14579100

—

C₂×C₇₉⋊C₁₆

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

D₄

yes

characteristic

C₁₆×D₇₉

—

#14579097

C₂×C₄

yes

—

#14579089

—

C₄×C₇₉⋊C₈

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

#11929624

—

Dic₃₁₆

C₂×C₈

yes

—

#11929613

—

D₆₃₂

C₂×C₈

yes

—

#11929608

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

D₈

yes

characteristic

C₇₉⋊C₁₆

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂²⋊D₇₉

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

#135

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#443

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

C₄×D₈

yes

characteristic

Dic₇₉

—

#12046

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₈×D₈

yes

characteristic

D₇₉

—

#2178

—

#2170

—

#1373

D₇₉

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₇₉

#12046

yes

characteristic, complemented, Sylow

#508

—

#504

—

158

—

#443

—

#368

—

#364

C₂×D₇₉

yes

—

#359

—

#350

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

#135

—

#131

—

158

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

316

—

C₂×C₁₆

—

316

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

characteristic

100

#108

—

101

#104

—

158

—

102

C₄×C₈

—

158

—

103

C₄×C₈

C₂²×D₇₉

yes

characteristic

104

#41

—

158

—

105

Q₁₆

—

106

SD₁₆

—

158

—

107

D₈

—

108

M₁₆

—

316

—

109

M₁₆

—

158

—

110

C₂×C₈

—

316

—

111

C₂×C₈

—

158

—

112

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

113

C₂×C₈

D₄×D₇₉

yes

characteristic

114

C₄²

#11929625

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

158

—

120

C₁₆

C₈×D₇₉

yes

characteristic

121

C₁₆

C₈×D₇₉

yes

characteristic

122

Q₈

—

158

—

123

D₄

—

316

—

124

D₄

—

158

—

125

C₂×C₄

—

126

C₂×C₄

—

316

—

127

C₂×C₄

#14579123

yes

characteristic

128

C₈

—

316

—

129

C₈

—

158

—

130

C₈

C₂×C₈×D₇₉

yes

characteristic

131

C₈

#14579169

yes

characteristic

132

C₈

D₈×D₇₉

yes

characteristic

133

C₈

#14579184

yes

characteristic

134

C₂²

—

316

—

135

C₂²

#19098679

yes

characteristic

136

C₄

—

137

C₄

—

316

—

138

C₄

#19098628

yes

characteristic

139

C₄

#19098645

yes

characteristic

140

C₂

—

141

C₂

—

632

—

142

C₂

#23762194

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₇₉
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1373	C₂
socle	C₂×C₇₉	C₈×D₈
prime radical	#19098995	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1373	D₇₉
hypocenter	C₇₉	#12046
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₇₉

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1373

C₂

socle

C₂×C₇₉

C₈×D₈

prime radical

#19098995

C₄

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #23765236	C₂	semidirect
3. #23761372	C₂	semidirect
4. C₁₆×C₇₉⋊C₈	C₂	semidirect
5. C₇₉×#1373	C₂	semidirect
18. Dic₆₃₂	C₈	semidirect
19. #14579106	C₈	semidirect
20. D₁₂₆₄	C₈	semidirect
74. #1373	D₇₉	semidirect
77. C₇₉	#12046	semidirect

H ≅ G/N

type

2. #23765236

C₂

semidirect

3. #23761372

C₂

semidirect

4. C₁₆×C₇₉⋊C₈

C₂

semidirect

5. C₇₉×#1373

C₂

semidirect

18. Dic₆₃₂

C₈

semidirect

19. #14579106

C₈

semidirect

20. D₁₂₆₄

C₈

semidirect

74. #1373

D₇₉

semidirect

77. C₇₉

#12046

semidirect

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

subgroup	index
#23765282	2
#23765236	2
#23761372	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#1373	2
#12046	79

subgroup

index

#23765282

#23765236

#23761372

C₁₆×C₇₉⋊C₈

C₇₉×#1373

#12046

Subgroups — #34200024

Conjugacy classes of subgroups

Complements

Maximal subgroups