Group #34204659 — subgroups

Subgroups — #34204659

Order 20224.

282 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 155 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	#23763572	C₂	1	2	yes	maximal
2	#23761494	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579097	—	2	2	no	—
5	C₂×#14579097	C₂²	1	4	yes	—
6	#19098587	—	2	2	no	—
7	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₉	C₄	1	2	yes	—
10	C₄×C₁₆×C₇₉	C₄	1	2	yes	—
11	C₂×#11929608	—	2	4	no	—
12	#14579097	—	2	8	no	—
13	#14579097	—	4	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
15	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	—	2	4	no	—
19	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
24	#11929608	—	4	4	no	—
25	#11929608	—	4	4	no	—
26	C₂×C₇₉⋊C₈	—	4	4	no	—
27	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
32	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₉	#41	1	4	yes	—
35	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₉	D₈	1	2	yes	—
38	C₁₆×C₇₉	—	2	4	no	—
39	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₉	—	2	4	no	—
41	C₂×C₄×C₇₉	#107	1	2	yes	—
42	C₂×C₄×C₇₉	#107	1	2	yes	—
43	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	#124	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
47	C₈×C₇₉	#106	1	2	yes	—
48	C₈×C₇₉	#106	1	2	yes	—
49	C₈×C₇₉	#117	1	2	yes	—
50	C₈×C₇₉	#125	1	2	yes	—
51	C₇₉⋊C₈	—	8	4	no	—
52	C₂²×C₇₉	—	2	2	no	—
53	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
54	C₂²×C₇₉	#487	1	2	yes	—
55	C₄×C₇₉	—	2	2	no	—
56	C₄×C₇₉	#368	1	4	yes	—
57	C₄×C₇₉	#382	1	1	yes	characteristic
58	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
59	C₄×C₇₉	#486	1	2	yes	—
60	#12891	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	—	2	2	no	—
62	C₂×C₇₉	#1834	1	1	yes	characteristic
63	C₂×C₇₉	C₂×#368	1	2	yes	—
64	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
65	#1848	—	79	2	no	—
66	#1381	—	79	4	no	—
67	C₇₉	#12891	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
69	C₂×#108	—	79	4	no	—
70	C₂×#108	—	158	2	no	—
71	#423	—	158	2	no	—
72	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
73	C₄×C₁₆	Dic₇₉	1	2	yes	—
74	C₄×C₁₆	Dic₇₉	1	2	yes	—
75	C₂×M₁₆	—	158	4	no	—
76	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
78	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
81	#108	—	316	4	no	—
82	#108	—	158	8	no	—
83	C₄⋊C₈	—	316	4	no	—
84	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
85	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
88	C₂²×C₄	Dic₃₁₆	1	2	yes	—
89	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
90	M₁₆	—	316	4	no	—
91	M₁₆	—	316	4	no	—
92	C₂×C₈	—	316	4	no	—
93	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
95	C₂×C₈	#11929613	1	4	yes	—
96	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
97	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
98	C₂×C₈	#11929624	1	4	yes	—
99	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
101	C₄²	D₆₃₂	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Dic₃₁₆	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	#14579095	1	2	yes	—
106	C₂×C₄	#14579095	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
108	C₂×C₄	#14579149	1	2	yes	—
109	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
110	C₂×C₄	C₂×#11929613	1	2	yes	—
111	C₈	—	632	4	no	—
112	C₈	#14579093	1	2	yes	—
113	C₈	#14579093	1	2	yes	—
114	C₈	#14579120	1	2	yes	—
115	C₈	#14579153	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14579095	1	1	yes	characteristic
118	C₂²	#19098914	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#19098393	1	4	yes	—
121	C₄	#19098429	1	1	yes	characteristic
122	C₄	C₂×#14579093	1	1	yes	characteristic
123	C₄	#19098910	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	#23763528	1	1	yes	characteristic
126	C₂	C₂×#19098393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal

#23761494

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579097

—

C₂×#14579097

C₂²

yes

—

#19098587

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₄

yes

—

C₂×#11929608

—

#14579097

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

Q₁₆

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

D₈

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#106

yes

—

C₈×C₇₉

#106

yes

—

C₈×C₇₉

#117

yes

—

C₈×C₇₉

#125

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#107

yes

characteristic

C₂²×C₇₉

#487

yes

—

C₄×C₇₉

—

C₄×C₇₉

#368

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#12891

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1834

yes

characteristic

C₂×C₇₉

C₂×#368

yes

—

C₂×C₄×C₁₆

D₇₉

yes

characteristic

#1848

—

#1381

—

C₇₉

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₉

yes

—

C₂×#108

—

C₂×#108

—

158

—

#423

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄×C₁₆

Dic₇₉

yes

—

C₄×C₁₆

Dic₇₉

yes

—

C₂×M₁₆

—

158

—

C₂²×C₈

D₃₁₆

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

#108

—

316

—

#108

—

158

—

C₄⋊C₈

—

316

—

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

Dic₃₁₆

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

M₁₆

—

316

—

M₁₆

—

316

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

#11929613

yes

—

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

C₂×C₈

#11929624

yes

—

C₄²

C₄⋊Dic₇₉

yes

characteristic

100

C₄²

C₄⋊Dic₇₉

yes

characteristic

101

C₄²

D₆₃₂

yes

—

102

C₁₆

—

103

C₂³

C₂×Dic₃₁₆

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14579095

yes

—

106

C₂×C₄

#14579095

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

108

C₂×C₄

#14579149

yes

—

109

C₂×C₄

C₂×D₆₃₂

yes

characteristic

110

C₂×C₄

C₂×#11929613

yes

—

111

C₈

—

632

—

112

C₈

#14579093

yes

—

113

C₈

#14579093

yes

—

114

C₈

#14579120

yes

—

115

C₈

#14579153

yes

—

116

C₂²

—

117

C₂²

C₂×#14579095

yes

characteristic

118

C₂²

#19098914

yes

—

119

C₄

—

120

C₄

#19098393

yes

—

121

C₄

#19098429

yes

characteristic

122

C₄

C₂×#14579093

yes

characteristic

123

C₄

#19098910

yes

—

124

C₂

—

125

C₂

#23763528

yes

characteristic

126

C₂

C₂×#19098393

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#107
prime radical	C₂³×C₇₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12891
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×C₄×C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×#107

prime radical

C₂³×C₇₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₉

hypocenter

C₇₉

#12891

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
67. C₇₉	#12891	semidirect

H ≅ G/N

type

67. C₇₉

#12891

semidirect

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

subgroup	index
#23763572	2
#23761494	2
C₂×C₄×C₁₆×C₇₉	2
#12891	79

subgroup

index

#23763572

#23761494

C₂×C₄×C₁₆×C₇₉

#12891

Subgroups — #34204659

Conjugacy classes of subgroups

Complements

Maximal subgroups