Group #34200294 — subgroups

Subgroups — #34200294

Order 20224.

226 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 77 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	#23762314	C₂	1	2	yes	maximal, complemented
2	#23761466	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761367	C₂	1	2	yes	maximal, complemented
4	#23761320	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2177	C₂	1	1	yes	maximal, characteristic
6	#19098641	—	2	2	no	—
7	#19098339	C₂²	1	1	yes	characteristic
8	#19098314	C₂²	1	2	yes	—
9	D₄⋊(C₇₉⋊C₈)	—	2	2	no	—
10	#19098284	C₂²	1	1	yes	characteristic
11	C₂×D₁₆×C₇₉	C₄	1	1	yes	characteristic
12	C₄×D₈×C₇₉	C₂²	1	2	yes	—
13	#367×C₇₉	C₄	1	1	yes	characteristic
14	#357×C₇₉	C₄	1	2	yes	—
15	#346×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929608	—	4	2	no	—
17	C₂×C₇₉⋊C₁₆	—	2	1	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
20	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
21	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×D₄×C₇₉	—	2	2	no	—
23	D₁₆×C₇₉	—	2	2	no	—
24	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	#107×C₇₉	C₂×C₄	1	2	yes	—
27	#102×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	#11929608	—	8	4	no	—
30	C₂×C₇₉⋊C₈	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₇₉⋊C₁₆	—	4	1	no	—
33	C₂×D₄×C₇₉	—	2	2	no	—
34	C₂²×C₄×C₇₉	—	4	2	no	—
35	D₈×C₇₉	—	2	2	no	—
36	D₈×C₇₉	—	2	2	no	—
37	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	2	2	no	—
40	C₂²⋊C₄×C₇₉	—	4	2	no	—
41	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	1	no	—
43	C₁₆×C₇₉	M₁₆	1	2	yes	complemented
44	C₂³×C₇₉	—	4	2	no	—
45	D₄×C₇₉	—	4	2	no	—
46	D₄×C₇₉	—	4	2	no	—
47	C₂×C₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	—	8	2	no	—
49	C₂×C₄×C₇₉	—	8	2	no	—
50	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₇₉	#131	1	1	yes	characteristic
55	C₇₉⋊C₈	—	4	2	no	—
56	C₇₉⋊C₈	—	8	2	no	—
57	C₂²×C₇₉	—	8	2	no	—
58	C₂²×C₇₉	—	8	2	no	—
59	C₂²×C₇₉	—	8	2	no	—
60	C₂²×C₇₉	#442	1	1	yes	characteristic
61	C₄×C₇₉	—	2	2	no	—
62	C₄×C₇₉	—	8	2	no	—
63	C₄×C₇₉	#436	1	1	yes	characteristic
64	C₄×C₇₉	#444	1	1	yes	characteristic
65	#12089	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	—	16	2	no	—
67	C₂×C₇₉	#1594	1	1	yes	characteristic
68	C₂×C₇₉	AGL(1,ℤ/16)	1	2	yes	—
69	#2177	D₇₉	1	1	yes	characteristic
70	#1594	—	79	2	no	—
71	#1368	—	79	1	no	—
72	#1333	—	79	2	no	—
73	#1313	—	79	1	no	—
74	C₇₉	#12089	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	Dic₇₉	1	1	yes	characteristic
76	C₄×D₈	C₂×D₇₉	1	2	yes	—
77	#436	—	158	2	no	—
78	#367	Dic₇₉	1	1	yes	characteristic
79	#357	Dic₇₉	1	2	yes	—
80	#346	—	79	1	no	—
81	#346	C₂×D₇₉	1	1	yes	characteristic
82	#334	—	79	2	no	—
83	D₄⋊C₈	—	158	2	no	—
84	#322	—	79	1	no	—
85	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
86	C₂×M₁₆	—	316	2	no	—
87	C₄×D₄	—	2	2	no	—
88	D₁₆	—	2	2	no	—
89	C₂×C₁₆	—	158	1	no	—
90	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
92	#107	C₂×Dic₇₉	1	2	yes	—
93	C₄⋊C₈	—	158	2	no	—
94	#102	—	2	2	no	—
95	C₂²⋊C₈	—	316	2	no	—
96	C₄×C₈	—	79	2	no	—
97	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×D₄	—	2	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	2	no	—
102	M₁₆	—	632	4	no	—
103	C₂×C₈	—	316	2	no	—
104	C₂×C₈	—	158	2	no	—
105	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
106	C₂×C₈	#11929623	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₇₉	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	316	1	no	—
112	C₁₆	#11929608	1	2	yes	complemented
113	C₂³	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
120	C₂×C₄	#14579185	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	316	2	no	—
123	C₈	—	632	2	no	—
124	C₈	C₂×#11929608	1	1	yes	characteristic
125	C₈	#14579168	1	1	yes	characteristic
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	#19098675	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	8	2	no	—
132	C₄	#19098641	1	1	yes	characteristic
133	C₄	#19098685	1	1	yes	characteristic
134	C₂	—	16	2	no	—
135	C₂	#23762314	1	1	yes	characteristic
136	C₂	#23765299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762314

C₂

yes

maximal, complemented

#23761466

C₂

yes

maximal, characteristic, complemented

#23761367

C₂

yes

maximal, complemented

#23761320

C₂

yes

maximal, characteristic, complemented

C₇₉×#2177

C₂

yes

maximal, characteristic

#19098641

—

#19098339

C₂²

yes

characteristic

#19098314

C₂²

yes

—

D₄⋊(C₇₉⋊C₈)

—

#19098284

C₂²

yes

characteristic

C₂×D₁₆×C₇₉

C₄

yes

characteristic

C₄×D₈×C₇₉

C₂²

yes

—

#367×C₇₉

C₄

yes

characteristic

#357×C₇₉

C₄

yes

—

#346×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×D₈×C₇₉

C₂×C₄

yes

—

C₄×D₄×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

C₂×C₄

yes

—

#102×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

M₁₆

yes

complemented

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#442

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1594

yes

characteristic

C₂×C₇₉

AGL(1,ℤ/16)

yes

—

#2177

D₇₉

yes

—

—

—

—

C₇₉

#12089

yes

characteristic, complemented, Sylow

C₂×D₁₆

Dic₇₉

yes

characteristic

C₄×D₈

C₂×D₇₉

yes

—

#436

—

158

—

#367

Dic₇₉

yes

characteristic

#357

Dic₇₉

yes

—

#346

—

#346

C₂×D₇₉

yes

characteristic

#334

—

D₄⋊C₈

—

158

—

#322

—

C₂×D₈

C₂×Dic₇₉

yes

—

C₂×M₁₆

—

316

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

#107

C₂×Dic₇₉

yes

—

C₄⋊C₈

—

158

—

#102

—

C₂²⋊C₈

—

316

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

—

102

M₁₆

—

632

—

103

C₂×C₈

—

316

—

104

C₂×C₈

—

158

—

105

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

106

C₂×C₈

#11929623

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

D₄×D₇₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

316

—

112

C₁₆

#11929608

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×Dic₇₉

yes

characteristic

120

C₂×C₄

#14579185

yes

—

121

C₈

—

122

C₈

—

316

—

123

C₈

—

632

—

124

C₈

C₂×#11929608

yes

characteristic

125

C₈

#14579168

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#19098675

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#19098641

yes

characteristic

133

C₄

#19098685

yes

characteristic

134

C₂

—

135

C₂

#23762314

yes

characteristic

136

C₂

#23765299

yes

—

	subgroup	G/H
center	C₂²	#19098675
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2177	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂×D₁₆×C₇₉	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#2177	D₇₉
hypocenter	C₇₉	#12089
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

subgroup

G/H

center

C₂²

#19098675

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2177

C₂

socle

C₂²×C₇₉

#442

prime radical

C₂×D₁₆×C₇₉

C₄

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

N	H ≅ G/N	type
1. #23762314	C₂	semidirect
2. #23761466	C₂	semidirect
3. #23761367	C₂	semidirect
4. #23761320	C₂	semidirect
43. C₁₆×C₇₉	M₁₆	semidirect
74. C₇₉	#12089	semidirect
112. C₁₆	#11929608	semidirect

H ≅ G/N

type

1. #23762314

C₂

semidirect

2. #23761466

C₂

semidirect

3. #23761367

C₂

semidirect

4. #23761320

C₂

semidirect

43. C₁₆×C₇₉

M₁₆

semidirect

74. C₇₉

#12089

semidirect

112. C₁₆

#11929608

semidirect

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

subgroup	index
#23762314	2
#23761466	2
#23761367	2
#23761320	2
C₇₉×#2177	2
#12089	79

subgroup

index

C₇₉×#2177

Subgroups — #34200294

Conjugacy classes of subgroups

Complements

Maximal subgroups