Group #28478032 — subgroups

Subgroups — #28478032

Order 15104.

414 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21425634	C₂	1	3	yes	maximal
2	#21421922	C₂	1	3	yes	maximal
3	C₅₉×#2811	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273660	—	2	3	no	—
5	C₂×#14273651	—	2	3	no	—
6	#16837272	C₂²	1	1	yes	characteristic
7	#16837157	C₂²	1	3	yes	—
8	C₅₉×#395	—	2	3	no	—
9	C₅₉×#391	—	2	3	no	—
10	C₅₉×#391	C₂²	1	3	yes	—
11	C₂²×C₅₉⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₅₉	—	2	3	no	—
13	#14273660	—	4	6	no	—
14	#14273651	—	4	6	no	—
15	C₂²×Q₈×C₅₉	—	2	1	no	—
16	C₂²×Q₈×C₅₉	D₄	1	3	yes	—
17	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
18	C₂×C₄⋊C₄×C₅₉	D₄	1	3	yes	—
19	C₂×C₄²×C₅₉	—	2	1	no	—
20	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	#95×C₅₉	—	2	3	no	—
22	#95×C₅₉	—	2	6	no	—
23	C₂×Q₈×C₅₉	—	2	3	no	—
24	C₂×Q₈×C₅₉	—	4	3	no	—
25	C₂×Q₈×C₅₉	—	4	3	no	—
26	C₂×Q₈×C₅₉	—	8	3	no	—
27	C₂×Q₈×C₅₉	Q₁₆	1	6	yes	—
28	C₂²×C₄×C₅₉	—	2	3	no	—
29	C₂²×C₄×C₅₉	—	2	3	no	—
30	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
31	C₂²×Dic₅₉	—	4	1	no	—
32	C₄⋊C₄×C₅₉	—	4	3	no	—
33	C₄⋊C₄×C₅₉	—	8	3	no	—
34	C₄²×C₅₉	—	4	1	no	—
35	C₄²×C₅₉	—	8	1	no	—
36	C₄⋊Dic₅₉	—	4	3	no	—
37	C₄⋊Dic₅₉	—	4	3	no	—
38	C₂×C₅₉⋊C₈	—	2	6	no	—
39	C₂×C₅₉⋊C₈	—	4	3	no	—
40	C₂×C₅₉⋊C₈	—	4	3	no	—
41	C₂³×C₅₉	#120	1	1	yes	characteristic
42	Q₈×C₅₉	—	4	6	no	—
43	Q₈×C₅₉	—	8	3	no	—
44	Q₈×C₅₉	—	8	4	no	—
45	C₂×C₄×C₅₉	—	2	3	no	—
46	C₂×C₄×C₅₉	—	2	3	no	—
47	C₂×C₄×C₅₉	—	2	6	no	—
48	C₂×C₄×C₅₉	—	4	3	no	—
49	C₂×C₄×C₅₉	—	4	3	no	—
50	C₂×C₄×C₅₉	—	4	3	no	—
51	C₂×C₄×C₅₉	—	4	6	no	—
52	C₂×C₄×C₅₉	C₂×Q₁₆	1	3	yes	—
53	C₂×C₄×C₅₉	#137	1	3	yes	—
54	C₂×Dic₅₉	—	8	3	no	—
55	C₅₉⋊C₈	—	4	6	no	—
56	#7837	—	59	1	no	maximal, Sylow
57	C₂²×C₅₉	#451	1	3	yes	—
58	C₂²×C₅₉	#453	1	3	yes	—
59	C₂²×C₅₉	#458	1	1	yes	characteristic
60	C₄×C₅₉	—	2	3	no	—
61	C₄×C₅₉	—	2	3	no	—
62	C₄×C₅₉	—	4	6	no	—
63	C₄×C₅₉	—	8	3	no	—
64	Dic₅₉	—	16	1	no	—
65	#2811	D₅₉	1	1	yes	characteristic
66	#2095	—	59	3	no	—
67	#1272	—	59	3	no	—
68	C₂×C₅₉	#1614	1	3	yes	—
69	C₂×C₅₉	#1633	1	3	yes	—
70	C₂×C₅₉	#1655	1	1	yes	characteristic
71	C₂×#107	—	118	3	no	—
72	C₂×#103	—	118	3	no	—
73	#395	—	2	3	no	—
74	#391	—	2	3	no	—
75	#391	C₂×D₅₉	1	3	yes	—
76	#384	—	59	1	no	—
77	#336	—	59	3	no	—
78	C₅₉	#7837	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×Q₈	C₂²⋊D₅₉	1	3	yes	—
81	C₂²×C₈	—	118	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	118	3	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
87	#107	—	236	6	no	—
88	#103	—	236	6	no	—
89	#95	—	2	3	no	—
90	#95	—	2	6	no	—
91	C₂×Q₈	—	2	3	no	—
92	C₂×Q₈	—	4	3	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂×Q₈	—	8	3	no	—
95	C₂×Q₈	#11745672	1	6	yes	—
96	C₂²×C₄	—	2	3	no	—
97	C₂²×C₄	—	2	3	no	—
98	C₂²×C₄	—	236	1	no	—
99	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
100	C₂×C₈	—	118	6	no	—
101	C₂×C₈	—	236	3	no	—
102	C₂×C₈	—	236	3	no	—
103	C₄⋊C₄	—	4	3	no	—
104	C₄⋊C₄	—	8	3	no	—
105	C₄⋊C₄	—	236	3	no	—
106	C₄⋊C₄	—	236	3	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₂³	#14273696	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	Q₈	—	8	4	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	4	3	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	6	no	—
120	C₂×C₄	—	472	3	no	—
121	C₂×C₄	C₂×#11745672	1	3	yes	—
122	C₂×C₄	#14273758	1	3	yes	—
123	C₈	—	236	6	no	—
124	C₂²	#16837558	1	3	yes	—
125	C₂²	#16837570	1	3	yes	—
126	C₂²	#16837592	1	1	yes	characteristic
127	C₄	—	2	3	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	6	no	—
130	C₄	—	8	3	no	—
131	C₄	—	944	1	no	—
132	C₂	#21423121	1	3	yes	—
133	C₂	#21423236	1	3	yes	—
134	C₂	#21423385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425634

C₂

yes

maximal

#21421922

C₂

yes

maximal

C₅₉×#2811

C₂

yes

maximal, characteristic

C₂×#14273660

—

C₂×#14273651

—

#16837272

C₂²

yes

characteristic

#16837157

C₂²

yes

—

C₅₉×#395

—

C₅₉×#391

—

C₅₉×#391

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

C₂×C₄⋊Dic₅₉

—

#14273660

—

#14273651

—

C₂²×Q₈×C₅₉

—

C₂²×Q₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₁₆

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

#7837

—

maximal, Sylow

C₂²×C₅₉

#451

yes

—

C₂²×C₅₉

#453

yes

—

C₂²×C₅₉

#458

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

#2811

D₅₉

yes

characteristic

#2095

—

#1272

—

C₂×C₅₉

#1614

yes

—

C₂×C₅₉

#1633

yes

—

C₂×C₅₉

#1655

yes

characteristic

C₂×#107

—

118

—

C₂×#103

—

118

—

#395

—

#391

—

#391

C₂×D₅₉

yes

—

#384

—

#336

—

C₅₉

#7837

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊D₅₉

yes

—

C₂²×C₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#107

—

236

—

#103

—

236

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#11745672

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

100

C₂×C₈

—

118

—

101

C₂×C₈

—

236

—

102

C₂×C₈

—

236

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

236

—

106

C₄⋊C₄

—

236

—

107

C₄²

—

108

C₄²

—

109

C₂³

#14273696

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

472

—

121

C₂×C₄

C₂×#11745672

yes

—

122

C₂×C₄

#14273758

yes

—

123

C₈

—

236

—

124

C₂²

#16837558

yes

—

125

C₂²

#16837570

yes

—

126

C₂²

#16837592

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

944

—

132

C₂

#21423121

yes

—

133

C₂

#21423236

yes

—

134

C₂

#21423385

yes

characteristic

	subgroup	G/H
center	C₂³	#14273696
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2811	C₂
socle, prime radical	C₂³×C₅₉	#120
hypercenter	#2811	D₅₉
hypocenter	C₅₉	#7837
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

subgroup

G/H

center

C₂³

#14273696

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2811

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂³×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

N	H ≅ G/N	type
78. C₅₉	#7837	semidirect

H ≅ G/N

type

78. C₅₉

#7837

semidirect

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

subgroup	index
#21425634	2
#21421922	2
C₅₉×#2811	2
#7837	59

subgroup

index

#21425634

#21421922

C₅₉×#2811

#7837

Subgroups — #28478032

Conjugacy classes of subgroups

Complements

Maximal subgroups