Group #28504105 — subgroups

Subgroups — #28504105

Order 15104.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#16837232	C₂	1	2	yes	maximal
2	C₄×#14273663	C₂	1	2	yes	maximal
3	#21422190	C₂	1	2	yes	maximal
4	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	C₂²	1	4	yes	—
6	C₄×#11745664	—	2	2	no	—
7	#16837232	C₄	1	8	yes	complemented
8	#16837167	—	2	2	no	—
9	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#346	C₄	1	4	yes	—
13	C₂×#11745664	—	2	4	no	—
14	#14273663	—	2	4	no	—
15	#14273663	C₂×C₄	1	8	yes	—
16	#14273638	—	4	2	no	—
17	C₄×C₅₉⋊C₈	—	4	2	no	—
18	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	—	2	2	no	—
22	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	2	yes	—
35	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₄²×C₅₉	D₈	1	2	yes	—
37	#11745664	—	2	8	no	—
38	#11745664	—	4	4	no	—
39	C₂×C₅₉⋊C₈	—	4	4	no	—
40	C₁₆×C₅₉	—	2	2	no	—
41	C₁₆×C₅₉	C₄²	1	4	yes	—
42	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₉	—	4	2	no	—
44	C₂×C₄×C₅₉	#107	1	4	yes	—
45	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#135	1	2	yes	—
49	C₈×C₅₉	—	2	2	no	—
50	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
51	C₈×C₅₉	#117	1	1	yes	characteristic
52	C₈×C₅₉	C₄×Q₈	1	2	yes	—
53	C₅₉⋊C₈	—	4	8	no	—
54	#12676	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
57	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
58	C₄×C₅₉	—	2	4	no	—
59	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₉	#427	1	1	yes	characteristic
61	C₄×C₅₉	C₄×D₈	1	2	yes	—
62	C₂×#368	—	59	2	no	—
63	C₂×#346	D₅₉	1	1	yes	characteristic
64	C₄×#108	—	59	2	no	—
65	#1381	—	59	2	no	—
66	C₂×C₅₉	—	2	2	no	—
67	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
68	C₂×C₅₉	#2161	1	2	yes	—
69	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
71	C₂×#108	—	59	4	no	—
72	C₄×M₁₆	—	118	2	no	—
73	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#368	—	59	8	no	—
75	#346	Dic₅₉	1	4	yes	—
76	#341	—	118	2	no	—
77	C₅₉	#12676	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	118	4	no	—
79	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
81	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄×D₅₉	1	4	yes	—
84	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
86	#108	—	59	8	no	—
87	#108	—	118	4	no	—
88	#97	—	236	2	no	—
89	C₄×C₈	—	236	2	no	—
90	C₄×C₈	Dic₁₁₈	1	2	yes	—
91	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
92	C₂²×C₄	Dic₂₃₆	1	2	yes	—
93	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	M₁₆	—	118	8	no	—
95	M₁₆	—	236	4	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	236	4	no	—
98	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
101	C₂×C₈	#11745685	1	2	yes	—
102	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
103	C₄²	D₄₇₂	1	2	yes	—
104	C₄²	C₄⋊Dic₅₉	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×Dic₅₉	1	4	yes	—
107	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#14273661	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
111	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
112	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
113	C₂×C₄	#14273749	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	236	8	no	—
116	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
117	C₈	#14273686	1	1	yes	characteristic
118	C₈	C₄×Dic₁₁₈	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#14273661	1	1	yes	characteristic
121	C₂²	C₄×Dic₂₃₆	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
124	C₄	#16837440	1	1	yes	characteristic
125	C₄	C₄×D₄₇₂	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#14273661	1	1	yes	characteristic
128	C₂	#21425891	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837232

C₂

yes

maximal

C₄×#14273663

C₂

yes

maximal

#21422190

C₂

yes

maximal

C₂×C₅₉×#346

C₂

yes

maximal, characteristic

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

#16837232

C₄

yes

complemented

#16837167

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₄

yes

—

C₂×#11745664

—

#14273663

—

#14273663

C₂×C₄

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

Q₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₄²

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄²

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

—

C₅₉⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

C₄×Q₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#427

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

—

C₂×#368

—

C₂×#346

D₅₉

yes

characteristic

C₄×#108

—

#1381

—

C₂×C₅₉

—

C₂×C₅₉

C₄×#107

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#368

—

#346

Dic₅₉

yes

—

#341

—

118

—

C₅₉

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

#108

—

#108

—

118

—

#97

—

236

—

C₄×C₈

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

Dic₂₃₆

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

100

C₂×C₈

C₂×C₄×D₅₉

yes

—

101

C₂×C₈

#11745685

yes

—

102

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

103

C₄²

D₄₇₂

yes

—

104

C₄²

C₄⋊Dic₅₉

yes

—

105

C₁₆

—

106

C₁₆

C₄×Dic₅₉

yes

—

107

C₂³

C₂×Dic₂₃₆

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#14273661

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

111

C₂×C₄

C₄×D₂₃₆

yes

—

112

C₂×C₄

C₂×D₄₇₂

yes

characteristic

113

C₂×C₄

#14273749

yes

—

114

C₈

—

115

C₈

—

236

—

116

C₈

C₂×C₄×Dic₅₉

yes

characteristic

117

C₈

#14273686

yes

characteristic

118

C₈

C₄×Dic₁₁₈

yes

—

119

C₂²

—

120

C₂²

C₂×#14273661

yes

characteristic

121

C₂²

C₄×Dic₂₃₆

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊Dic₅₉

yes

characteristic

124

C₄

#16837440

yes

characteristic

125

C₄

C₄×D₄₇₂

yes

—

126

C₂

—

127

C₂

C₄×#14273661

yes

characteristic

128

C₂

#21425891

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#12676
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator

C₈×C₅₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#346

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
7. #16837232	C₄	semidirect
77. C₅₉	#12676	semidirect

H ≅ G/N

type

7. #16837232

C₄

semidirect

77. C₅₉

#12676

semidirect

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

subgroup	index
C₂×#16837232	2
C₄×#14273663	2
#21422190	2
C₂×C₅₉×#346	2
#12676	59

subgroup

index

C₂×C₅₉×#346

Subgroups — #28504105

Conjugacy classes of subgroups

Complements

Maximal subgroups