Group #34199905 — subgroups

Subgroups — #34199905

Order 20224.

232 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 123 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	#23762156	C₂	1	2	yes	maximal
2	#23761466	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098587	—	2	2	no	—
5	#19098314	—	2	2	no	—
6	#19098314	C₂²	1	4	yes	—
7	C₂×M₃₂×C₇₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	#346×C₇₉	C₂²	1	2	yes	—
10	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
12	C₂×#11929608	—	4	2	no	—
13	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	M₃₂×C₇₉	—	2	4	no	—
19	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	#11929608	—	8	4	no	—
26	C₂×C₇₉⋊C₈	—	4	4	no	—
27	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
29	C₂×C₈×C₇₉	—	2	2	no	—
30	C₂×C₈×C₇₉	—	2	2	no	—
31	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₇₉	SD₁₆	1	2	yes	—
37	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₇₉	—	2	2	no	—
39	C₁₆×C₇₉	M₁₆	1	4	yes	complemented
40	C₂³×C₇₉	#106	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	4	no	—
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#106	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
48	C₈×C₇₉	—	2	1	no	—
49	C₈×C₇₉	—	2	1	no	—
50	C₈×C₇₉	#108	1	2	yes	—
51	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
52	C₇₉⋊C₈	—	8	4	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	—	2	2	no	—
55	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
56	C₄×C₇₉	—	2	1	no	—
57	C₄×C₇₉	—	2	2	no	—
58	C₄×C₇₉	#365	1	4	yes	—
59	C₄×C₇₉	#423	1	1	yes	characteristic
60	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
61	#12020	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	4	1	no	—
63	C₂×C₇₉	#1566	1	1	yes	characteristic
64	C₂×C₇₉	C₂×#365	1	2	yes	—
65	C₄×M₃₂	D₇₉	1	1	yes	characteristic
66	#1566	—	79	2	no	—
67	#1368	—	79	4	no	—
68	C₇₉	#12020	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	Dic₇₉	1	2	yes	—
70	#423	—	158	2	no	—
71	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
72	#346	C₂×D₇₉	1	2	yes	—
73	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
74	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
75	#334	—	79	4	no	—
76	#334	—	158	2	no	—
77	C₂×M₁₆	—	316	2	no	—
78	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
80	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
84	C₄⋊C₈	—	316	2	no	—
85	C₄⋊C₈	—	158	4	no	—
86	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
87	C₄×C₈	D₃₁₆	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
91	C₂²×C₄	#11929613	1	2	yes	—
92	M₁₆	—	632	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	316	4	no	—
96	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
98	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
99	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
101	C₄²	#11929613	1	2	yes	—
102	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11929608	1	4	yes	complemented
105	C₂³	#14579093	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	#14579093	1	1	yes	characteristic
110	C₂×C₄	#14579093	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
112	C₂×C₄	C₂×#11929613	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	632	4	no	—
116	C₈	#14579097	1	2	yes	—
117	C₈	C₂×#11929608	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#14579093	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	#19098386	1	4	yes	—
124	C₄	#19098587	1	1	yes	characteristic
125	C₄	C₂×#14579097	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#23762156	1	1	yes	characteristic
128	C₂	C₂×#19098386	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762156

C₂

yes

maximal

#23761466

C₂

yes

maximal, complemented

C₄×M₃₂×C₇₉

C₂

yes

maximal, characteristic

#19098587

—

#19098314

—

#19098314

C₂²

yes

—

C₂×M₃₂×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₄

yes

characteristic

C₄×C₁₆×C₇₉

C₄

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

M₁₆

yes

complemented

C₂³×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

#106

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#108

yes

—

C₈×C₇₉

C₂×M₁₆

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#365

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1566

yes

characteristic

C₂×C₇₉

C₂×#365

yes

—

C₄×M₃₂

D₇₉

yes

characteristic

#1566

—

#1368

—

C₇₉

#12020

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₉

yes

—

#423

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#346

C₂×D₇₉

yes

—

C₄×C₁₆

Dic₇₉

yes

characteristic

C₄×C₁₆

Dic₇₉

yes

characteristic

#334

—

#334

—

158

—

C₂×M₁₆

—

316

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

D₃₁₆

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₂×C₁₆

C₂×Dic₇₉

yes

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂²×C₄

#11929613

yes

—

M₁₆

—

632

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

100

C₄²

C₄⋊Dic₇₉

yes

characteristic

101

C₄²

#11929613

yes

—

102

C₄²

C₂×D₃₁₆

yes

characteristic

103

C₁₆

—

104

C₁₆

#11929608

yes

complemented

105

C₂³

#14579093

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14579093

yes

characteristic

110

C₂×C₄

#14579093

yes

—

111

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

112

C₂×C₄

C₂×#11929613

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

632

—

116

C₈

#14579097

yes

—

117

C₈

C₂×#11929608

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14579093

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#19098386

yes

—

124

C₄

#19098587

yes

characteristic

125

C₄

C₂×#14579097

yes

characteristic

126

C₂

—

127

C₂

#23762156

yes

characteristic

128

C₂

C₂×#19098386

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579093
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#12020
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#14579093

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₄×M₃₂×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₄×C₈×C₇₉

C₂³

hypercenter

C₄×M₃₂

D₇₉

hypocenter

C₇₉

#12020

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #23761466	C₂	semidirect
39. C₁₆×C₇₉	M₁₆	semidirect
68. C₇₉	#12020	semidirect
104. C₁₆	#11929608	semidirect

H ≅ G/N

type

2. #23761466

C₂

semidirect

39. C₁₆×C₇₉

M₁₆

semidirect

68. C₇₉

#12020

semidirect

104. C₁₆

#11929608

semidirect

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

subgroup	index
#23762156	2
#23761466	2
C₄×M₃₂×C₇₉	2
#12020	79

subgroup

index

#23762156

#23761466

C₄×M₃₂×C₇₉

#12020

Subgroups — #34199905

Conjugacy classes of subgroups

Complements

Maximal subgroups