Group #34200029 — subgroups

Subgroups — #34200029

Order 20224.

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

normal subgroups only

This group has 212 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	#23761316	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	2	yes	—
42	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₇₉	#41	1	1	yes	characteristic
45	C₁₆×C₇₉	—	2	1	no	—
46	C₁₆×C₇₉	—	4	2	no	—
47	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₇₉	—	4	2	no	—
52	D₃₁₆	—	2	2	no	—
53	C₄×D₇₉	—	4	2	no	—
54	Dic₁₅₈	—	2	2	no	—
55	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₇₉	#131	1	1	yes	characteristic
57	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
58	C₈×C₇₉	#135	1	1	yes	characteristic
59	C₇₉⋊C₈	—	4	2	no	—
60	C₇₉⋊C₈	—	4	2	no	—
61	C₂²×C₇₉	#443	1	1	yes	characteristic
62	C₂×D₇₉	—	4	2	no	—
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
65	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
66	Dic₇₉	—	4	2	no	—
67	#12047	—	79	1	no	maximal, Sylow
68	C₂×C₇₉	—	2	1	no	—
69	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
70	D₇₉	—	8	2	no	—
71	#2178	—	79	1	no	—
72	#2170	—	79	2	no	—
73	#1373	D₇₉	1	1	yes	characteristic, complemented
74	#1335	—	79	2	no	—
75	#1311	—	79	1	no	—
76	C₇₉	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	—	79	1	no	—
78	#504	—	158	2	no	—
79	#443	—	79	2	no	—
80	#368	—	79	1	no	—
81	#364	C₂×D₇₉	1	2	yes	—
82	#359	—	79	2	no	—
83	#350	—	158	2	no	—
84	C₄×C₁₆	—	79	1	no	—
85	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₈²	—	79	1	no	—
87	#135	—	79	2	no	—
88	#131	—	158	2	no	—
89	Q₃₂	—	79	1	no	—
90	SD₃₂	—	79	2	no	—
91	D₁₆	—	79	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	316	2	no	—
94	C₂×C₁₆	—	316	2	no	—
95	C₂×C₁₆	—	79	1	no	—
96	C₂×C₁₆	—	79	1	no	—
97	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
99	#108	—	79	2	no	—
100	#104	—	158	2	no	—
101	C₄×C₈	—	158	1	no	—
102	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	#41	—	158	2	no	—
104	Q₁₆	—	79	2	no	—
105	SD₁₆	—	158	2	no	—
106	D₈	—	79	2	no	—
107	M₁₆	—	316	2	no	—
108	M₁₆	—	158	2	no	—
109	C₂×C₈	—	316	2	no	—
110	C₂×C₈	—	158	2	no	—
111	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
112	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
113	C₄²	#11929625	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	79	2	no	—
117	C₁₆	—	158	1	no	—
118	C₁₆	C₈×D₇₉	1	1	yes	characteristic
119	C₁₆	C₈×D₇₉	1	1	yes	characteristic
120	Q₈	—	158	2	no	—
121	D₄	—	316	2	no	—
122	D₄	—	158	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	316	2	no	—
125	C₂×C₄	#14579123	1	1	yes	characteristic
126	C₈	—	316	2	no	—
127	C₈	—	316	2	no	—
128	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
129	C₈	#14579169	1	1	yes	characteristic
130	C₈	D₈×D₇₉	1	1	yes	characteristic
131	C₈	#14579184	1	1	yes	characteristic
132	C₂²	—	316	2	no	—
133	C₂²	#19098679	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	316	2	no	—
136	C₄	#19098628	1	1	yes	characteristic
137	C₄	#19098645	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	632	2	no	—
140	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

#23761316

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

—

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

—

#12047

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₈×D₈

yes

characteristic

D₇₉

—

#2178

—

#2170

—

#1373

D₇₉

yes

characteristic, complemented

#1335

—

#1311

—

C₇₉

#12047

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

#108

—

100

#104

—

158

—

101

C₄×C₈

—

158

—

102

C₄×C₈

C₂²×D₇₉

yes

characteristic

103

#41

—

158

—

104

Q₁₆

—

105

SD₁₆

—

158

—

106

D₈

—

107

M₁₆

—

316

—

108

M₁₆

—

158

—

109

C₂×C₈

—

316

—

110

C₂×C₈

—

158

—

111

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

112

C₂×C₈

D₄×D₇₉

yes

characteristic

113

C₄²

#11929625

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

158

—

118

C₁₆

C₈×D₇₉

yes

characteristic

119

C₁₆

C₈×D₇₉

yes

characteristic

120

Q₈

—

158

—

121

D₄

—

316

—

122

D₄

—

158

—

123

C₂×C₄

—

124

C₂×C₄

—

316

—

125

C₂×C₄

#14579123

yes

characteristic

126

C₈

—

316

—

127

C₈

—

316

—

128

C₈

C₂×C₈×D₇₉

yes

characteristic

129

C₈

#14579169

yes

characteristic

130

C₈

D₈×D₇₉

yes

characteristic

131

C₈

#14579184

yes

characteristic

132

C₂²

—

316

—

133

C₂²

#19098679

yes

characteristic

134

C₄

—

135

C₄

—

316

—

136

C₄

#19098628

yes

characteristic

137

C₄

#19098645

yes

characteristic

138

C₂

—

139

C₂

—

632

—

140

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	#12047
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. #23761316	C₂	semidirect
5. C₇₉×#1373	C₂	semidirect
18. Dic₆₃₂	C₈	semidirect
19. #14579106	C₈	semidirect
20. D₁₂₆₄	C₈	semidirect
73. #1373	D₇₉	semidirect
76. C₇₉	#12047	semidirect

H ≅ G/N

type

2. #23765236

C₂

semidirect

3. #23761372

C₂

semidirect

4. #23761316

C₂

semidirect

5. C₇₉×#1373

C₂

semidirect

18. Dic₆₃₂

C₈

semidirect

19. #14579106

C₈

semidirect

20. D₁₂₆₄

C₈

semidirect

73. #1373

D₇₉

semidirect

76. C₇₉

#12047

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
#23761316	2
C₇₉×#1373	2
#12047	79

subgroup

index

C₇₉×#1373

Subgroups — #34200029

Conjugacy classes of subgroups

Complements

Maximal subgroups