Group #34200160 — subgroups

Subgroups — #34200160

Order 20224.

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

normal subgroups only

This group has 204 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	#23765287	C₂	1	1	yes	maximal, characteristic
2	#23765243	C₂	1	2	yes	maximal, complemented
3	#23761468	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761377	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098995	C₄	1	1	yes	characteristic
7	#19098979	—	2	2	no	—
8	#19098681	C₂²	1	2	yes	—
9	#19098393	C₄	1	1	yes	characteristic
10	#19098384	C₂²	1	2	yes	—
11	#19098371	C₄	1	2	yes	—
12	#19098346	—	2	2	no	—
13	#346×C₇₉	C₂²	1	1	yes	characteristic
14	#346×C₇₉	C₂²	1	1	yes	characteristic
15	#322×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	#14579101	—	2	2	no	—
22	#14579100	—	4	2	no	—
23	C₁₆×D₇₉	—	4	2	no	—
24	#14579097	C₂×C₄	1	2	yes	—
25	#14579089	—	2	2	no	—
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
29	C₄×C₈×C₇₉	—	2	1	no	—
30	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	#11929624	—	2	2	no	—
32	Dic₃₁₆	C₂×C₈	1	2	yes	—
33	#11929613	—	2	2	no	—
34	D₆₃₂	C₂×C₈	1	2	yes	—
35	#11929608	—	2	2	no	—
36	#11929607	—	4	2	no	—
37	C₈×D₇₉	—	4	2	no	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂×C₈×C₇₉	—	4	1	no	—
40	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₉	#41	1	1	yes	characteristic
43	C₁₆×C₇₉	—	2	1	no	—
44	C₁₆×C₇₉	—	2	2	no	—
45	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₇₉	—	4	2	no	—
50	D₃₁₆	—	2	2	no	—
51	C₄×D₇₉	—	4	2	no	—
52	Dic₁₅₈	—	2	2	no	—
53	C₈×C₇₉	—	8	1	no	—
54	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₇₉	#131	1	1	yes	characteristic
56	C₈×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₇₉⋊C₈	—	4	2	no	—
58	C₂²×C₇₉	#444	1	1	yes	characteristic
59	C₂×D₇₉	—	4	2	no	—
60	C₄×C₇₉	—	4	1	no	—
61	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
62	C₄×C₇₉	#442	1	1	yes	characteristic
63	Dic₇₉	—	4	2	no	—
64	#12068	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	—	2	1	no	—
66	C₂×C₇₉	#1580	1	1	yes	characteristic
67	D₇₉	—	8	2	no	—
68	#2179	—	79	1	no	—
69	#2171	—	79	2	no	—
70	#1369	—	79	1	no	—
71	#1336	—	79	2	no	—
72	#1313	D₇₉	1	1	yes	characteristic, complemented
73	C₇₉	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	79	1	no	—
75	#504	—	158	2	no	—
76	#443	—	79	2	no	—
77	#368	—	79	1	no	—
78	#364	—	79	2	no	—
79	#359	—	79	2	no	—
80	#350	—	158	2	no	—
81	#346	C₂×D₇₉	1	1	yes	characteristic
82	#346	C₂×D₇₉	1	1	yes	characteristic
83	#322	C₂×D₇₉	1	1	yes	characteristic
84	#135	—	79	2	no	—
85	#131	—	158	2	no	—
86	Q₃₂	—	79	1	no	—
87	SD₃₂	—	79	2	no	—
88	D₁₆	—	79	1	no	—
89	M₃₂	—	316	2	no	—
90	M₃₂	—	158	2	no	—
91	C₂×C₁₆	—	316	2	no	—
92	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
94	C₂×C₁₆	D₃₁₆	1	2	yes	—
95	#108	—	79	2	no	—
96	#104	—	158	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
99	#41	—	158	2	no	—
100	Q₁₆	—	79	2	no	—
101	SD₁₆	—	158	2	no	—
102	D₈	—	79	2	no	—
103	M₁₆	—	316	2	no	—
104	M₁₆	—	158	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	316	2	no	—
107	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
108	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
109	C₄²	#11929624	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	316	2	no	—
113	C₁₆	C₈×D₇₉	1	1	yes	characteristic
114	C₁₆	C₈×D₇₉	1	1	yes	characteristic
115	Q₈	—	158	2	no	—
116	D₄	—	316	2	no	—
117	D₄	—	158	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	316	2	no	—
120	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
121	C₈	—	8	1	no	—
122	C₈	—	316	2	no	—
123	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
124	C₈	#14579167	1	1	yes	characteristic
125	C₈	#14579189	1	2	yes	—
126	C₂²	—	316	2	no	—
127	C₂²	#19098686	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	316	2	no	—
130	C₄	C₈×D₃₁₆	1	1	yes	characteristic
131	C₄	#19098676	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	632	2	no	—
134	C₂	#23762225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765287

C₂

yes

maximal, characteristic

#23765243

C₂

yes

maximal, complemented

#23761468

C₂

yes

maximal, characteristic, complemented

#23761377

C₂

yes

maximal, complemented

C₇₉×#1313

C₂

yes

maximal, characteristic, complemented

#19098995

C₄

yes

characteristic

#19098979

—

#19098681

C₂²

yes

—

#19098393

C₄

yes

characteristic

#19098384

C₂²

yes

—

#19098371

C₄

yes

—

#19098346

—

#346×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₂²

yes

characteristic

#14579183

C₂×C₄

yes

—

#14579167

—

Dic₆₃₂

C₈

yes

characteristic, complemented

#14579106

C₈

yes

complemented

D₁₂₆₄

C₈

yes

characteristic, complemented

#14579101

—

#14579100

—

C₁₆×D₇₉

—

#14579097

C₂×C₄

yes

—

#14579089

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

#11929624

—

Dic₃₁₆

C₂×C₈

yes

—

#11929613

—

D₆₃₂

C₂×C₈

yes

—

#11929608

—

#11929607

—

C₈×D₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

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

—

C₈×C₇₉

C₂²×C₈

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

#444

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

#442

yes

characteristic

Dic₇₉

—

#12068

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1580

yes

characteristic

D₇₉

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

158

—

—

—

—

—

—

158

—

#346

C₂×D₇₉

yes

characteristic

#346

C₂×D₇₉

yes

characteristic

#322

C₂×D₇₉

yes

characteristic

#135

—

#131

—

158

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

316

—

M₃₂

—

158

—

C₂×C₁₆

—

316

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

D₃₁₆

yes

—

#108

—

#104

—

158

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

#41

—

158

—

100

Q₁₆

—

101

SD₁₆

—

158

—

102

D₈

—

103

M₁₆

—

316

—

104

M₁₆

—

158

—

105

C₂×C₈

—

106

C₂×C₈

—

316

—

107

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

108

C₂×C₈

C₂×D₃₁₆

yes

characteristic

109

C₄²

#11929624

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

316

—

113

C₁₆

C₈×D₇₉

yes

characteristic

114

C₁₆

C₈×D₇₉

yes

characteristic

115

Q₈

—

158

—

116

D₄

—

316

—

117

D₄

—

158

—

118

C₂×C₄

—

119

C₂×C₄

—

316

—

120

C₂×C₄

C₄×D₃₁₆

yes

characteristic

121

C₈

—

122

C₈

—

316

—

123

C₈

C₂×C₈×D₇₉

yes

characteristic

124

C₈

#14579167

yes

characteristic

125

C₈

#14579189

yes

—

126

C₂²

—

316

—

127

C₂²

#19098686

yes

characteristic

128

C₄

—

129

C₄

—

316

—

130

C₄

C₈×D₃₁₆

yes

characteristic

131

C₄

#19098676

yes

characteristic

132

C₂

—

133

C₂

—

632

—

134

C₂

#23762225

yes

characteristic

	subgroup	G/H
center	C₄	#19098676
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1313	C₂
socle	C₂×C₇₉	#1580
prime radical	#19098995	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1313	D₇₉
hypocenter	C₇₉	#12068
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

subgroup

G/H

center

C₄

#19098676

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1313

C₂

C₂×C₇₉

C₄

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

N	H ≅ G/N	type
2. #23765243	C₂	semidirect
3. #23761468	C₂	semidirect
4. #23761377	C₂	semidirect
5. C₇₉×#1313	C₂	semidirect
18. Dic₆₃₂	C₈	semidirect
19. #14579106	C₈	semidirect
20. D₁₂₆₄	C₈	semidirect
72. #1313	D₇₉	semidirect
73. C₇₉	#12068	semidirect

H ≅ G/N

type

2. #23765243

C₂

semidirect

3. #23761468

C₂

semidirect

4. #23761377

C₂

semidirect

5. C₇₉×#1313

C₂

semidirect

18. Dic₆₃₂

C₈

semidirect

19. #14579106

C₈

semidirect

20. D₁₂₆₄

C₈

semidirect

72. #1313

D₇₉

semidirect

73. C₇₉

#12068

semidirect

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

subgroup	index
#23765287	2
#23765243	2
#23761468	2
#23761377	2
C₇₉×#1313	2
#12068	79

subgroup

index

C₇₉×#1313

Subgroups — #34200160

Conjugacy classes of subgroups

Complements

Maximal subgroups