Group #34200155 — subgroups

Subgroups — #34200155

Order 20224.

246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23765277	C₂	1	1	yes	maximal, characteristic
2	#23762225	C₂	1	2	yes	maximal, complemented
3	#23761466	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761368	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₄	C₄	1	1	yes	characteristic
7	C₄×D₆₃₂	C₂²	1	2	yes	—
8	C₈×D₃₁₆	—	2	2	no	—
9	#19098391	C₄	1	1	yes	characteristic
10	#19098365	C₄	1	2	yes	—
11	#19098314	C₂²	1	2	yes	—
12	#19098292	—	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	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₇₉	—	4	2	no	—
18	C₄×D₃₁₆	—	2	2	no	—
19	D₁₂₆₄	C₈	1	4	yes	complemented
20	#14579095	C₂×C₄	1	2	yes	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	#14579083	—	2	2	no	—
23	(C₂×D₇₉)⋊C₈	—	4	2	no	—
24	C₂×C₁₆×C₇₉	—	2	1	no	—
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₁₆	—	2	2	no	—
30	C₂×C₄×D₇₉	—	4	2	no	—
31	D₆₃₂	—	2	2	no	—
32	D₆₃₂	C₂×C₈	1	4	yes	complemented
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₈×D₇₉	—	8	4	no	—
35	C₄⋊Dic₇₉	—	2	2	no	—
36	(C₂×D₇₉)⋊C₄	—	4	2	no	—
37	C₂×C₈×C₇₉	—	2	2	no	—
38	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₉	—	2	1	no	—
42	C₁₆×C₇₉	—	4	1	no	—
43	C₁₆×C₇₉	C₂×C₈	1	2	yes	complemented
44	C₂²×D₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₇₉	—	4	2	no	—
48	D₃₁₆	—	2	4	no	—
49	D₃₁₆	—	4	2	no	—
50	C₄×D₇₉	—	8	2	no	—
51	C₄×D₇₉	—	8	2	no	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	—	4	2	no	—
54	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₇₉	#131	1	1	yes	characteristic
56	C₇₉⋊C₈	—	8	2	no	—
57	C₂²×C₇₉	#442	1	1	yes	characteristic
58	C₂×D₇₉	—	4	4	no	—
59	C₂×D₇₉	—	8	2	no	—
60	C₂×D₇₉	—	8	2	no	—
61	C₄×C₇₉	—	2	2	no	—
62	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
63	C₄×C₇₉	#444	1	1	yes	characteristic
64	Dic₇₉	—	8	2	no	—
65	#12067	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	#1580	1	1	yes	characteristic
67	C₂×C₇₉	AGL(1,ℤ/16)	1	2	yes	—
68	D₇₉	—	8	4	no	—
69	#2177	—	79	1	no	—
70	#1580	—	79	2	no	—
71	#1368	—	79	1	no	—
72	#1333	—	79	2	no	—
73	#1313	D₇₉	1	1	yes	characteristic, complemented
74	C₇₉	#12067	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	79	1	no	—
76	C₄×D₈	—	79	2	no	—
77	C₈×D₄	—	158	2	no	—
78	#367	—	79	1	no	—
79	#357	—	79	2	no	—
80	#346	C₂×D₇₉	1	1	yes	characteristic
81	#346	C₂×D₇₉	1	1	yes	characteristic
82	#334	—	79	2	no	—
83	D₄⋊C₈	—	158	2	no	—
84	#322	C₂×D₇₉	1	1	yes	characteristic
85	C₂×D₈	—	79	2	no	—
86	C₂²×C₈	—	316	2	no	—
87	C₄×D₄	—	158	2	no	—
88	D₁₆	—	79	4	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
92	#107	—	79	2	no	—
93	C₄⋊C₈	—	158	2	no	—
94	#102	—	158	2	no	—
95	C₂²⋊C₈	—	316	2	no	—
96	C₄×C₈	D₃₁₆	1	2	yes	—
97	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×D₄	—	158	2	no	—
99	C₂²×C₄	—	316	2	no	—
100	D₈	—	79	4	no	—
101	D₈	—	158	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	316	2	no	—
104	C₂×C₈	—	632	4	no	—
105	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
106	C₂×C₈	#11929624	1	1	yes	characteristic
107	C₄⋊C₄	—	158	2	no	—
108	C₂²⋊C₄	—	316	2	no	—
109	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	C₈×D₇₉	1	2	yes	complemented
113	C₂³	—	316	2	no	—
114	D₄	—	316	2	no	—
115	D₄	—	158	4	no	—
116	C₂×C₄	—	316	2	no	—
117	C₂×C₄	—	632	2	no	—
118	C₂×C₄	—	632	2	no	—
119	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
120	C₂×C₄	#14579189	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	632	2	no	—
124	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
125	C₈	#14579167	1	1	yes	characteristic
126	C₂²	—	316	4	no	—
127	C₂²	—	632	2	no	—
128	C₂²	—	632	2	no	—
129	C₂²	#19098676	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	632	2	no	—
132	C₄	C₈×D₃₁₆	1	1	yes	characteristic
133	C₄	#19098686	1	1	yes	characteristic
134	C₂	—	632	4	no	—
135	C₂	#23762225	1	1	yes	characteristic
136	C₂	#23765300	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765277

C₂

yes

maximal, characteristic

#23762225

C₂

yes

maximal, complemented

#23761466

C₂

yes

maximal, characteristic, complemented

#23761368

C₂

yes

maximal, complemented

C₇₉×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₁₂₆₄

C₄

yes

characteristic

C₄×D₆₃₂

C₂²

yes

—

C₈×D₃₁₆

—

#19098391

C₄

yes

characteristic

#19098365

C₄

yes

—

#19098314

C₂²

yes

—

#19098292

—

#346×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₂²

yes

characteristic

C₂×D₆₃₂

C₂×C₄

yes

—

C₂×C₈×D₇₉

—

C₄×D₃₁₆

—

D₁₂₆₄

C₈

yes

complemented

#14579095

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

#14579083

—

(C₂×D₇₉)⋊C₈

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₃₁₆

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

C₂×C₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

complemented

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#442

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

#444

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#12067

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

158

—

#367

—

#357

—

#346

C₂×D₇₉

yes

characteristic

#346

C₂×D₇₉

yes

characteristic

#334

—

D₄⋊C₈

—

158

—

#322

C₂×D₇₉

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

316

—

C₄×D₄

—

158

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

characteristic

#107

—

C₄⋊C₈

—

158

—

#102

—

158

—

C₂²⋊C₈

—

316

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

100

D₈

—

101

D₈

—

158

—

102

C₂×C₈

—

103

C₂×C₈

—

316

—

104

C₂×C₈

—

632

—

105

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

106

C₂×C₈

#11929624

yes

characteristic

107

C₄⋊C₄

—

158

—

108

C₂²⋊C₄

—

316

—

109

C₄²

C₂×D₃₁₆

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₈×D₇₉

yes

complemented

113

C₂³

—

316

—

114

D₄

—

316

—

115

D₄

—

158

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

632

—

118

C₂×C₄

—

632

—

119

C₂×C₄

C₄×D₃₁₆

yes

characteristic

120

C₂×C₄

#14579189

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

632

—

124

C₈

C₂×C₈×D₇₉

yes

characteristic

125

C₈

#14579167

yes

characteristic

126

C₂²

—

316

—

127

C₂²

—

632

—

128

C₂²

—

632

—

129

C₂²

#19098676

yes

characteristic

130

C₄

—

131

C₄

—

632

—

132

C₄

C₈×D₃₁₆

yes

characteristic

133

C₄

#19098686

yes

characteristic

134

C₂

—

632

—

135

C₂

#23762225

yes

characteristic

136

C₂

#23765300

yes

—

	subgroup	G/H
center	C₂²	#19098676
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1313	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂×D₁₂₆₄	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1313	D₇₉
hypocenter	C₇₉	#12067
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#19098676

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1313

C₂

socle

C₂²×C₇₉

#442

prime radical

C₂×D₁₂₆₄

C₄

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #23762225	C₂	semidirect
3. #23761466	C₂	semidirect
4. #23761368	C₂	semidirect
5. C₇₉×#1313	C₂	semidirect
19. D₁₂₆₄	C₈	semidirect
32. D₆₃₂	C₂×C₈	semidirect
43. C₁₆×C₇₉	C₂×C₈	semidirect
73. #1313	D₇₉	semidirect
74. C₇₉	#12067	semidirect
112. C₁₆	C₈×D₇₉	semidirect

H ≅ G/N

type

2. #23762225

C₂

semidirect

3. #23761466

C₂

semidirect

4. #23761368

C₂

semidirect

5. C₇₉×#1313

C₂

semidirect

19. D₁₂₆₄

C₈

semidirect

32. D₆₃₂

C₂×C₈

semidirect

43. C₁₆×C₇₉

C₂×C₈

semidirect

73. #1313

D₇₉

semidirect

74. C₇₉

#12067

semidirect

112. C₁₆

C₈×D₇₉

semidirect

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

subgroup	index
#23765277	2
#23762225	2
#23761466	2
#23761368	2
C₇₉×#1313	2
#12067	79

subgroup

index

C₇₉×#1313

Subgroups — #34200155

Conjugacy classes of subgroups

Complements

Maximal subgroups