Group #28503048 — subgroups

Subgroups — #28503048

Order 15104.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21426371	C₂	1	1	yes	maximal, characteristic
2	#21423490	C₂	1	2	yes	maximal
3	#21422168	C₂	1	1	yes	maximal, characteristic
4	#21422150	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837612	—	2	2	no	—
8	#16837228	—	2	1	no	—
9	#16837228	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	#16837147	—	2	2	no	—
12	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×C₄⋊C₁₆	—	2	1	no	—
14	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₉	—	2	1	no	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745672	—	4	2	no	—
18	#14273727	—	2	2	no	—
19	#14273661	—	2	2	no	—
20	#14273661	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	#14273650	—	4	2	no	—
23	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₉	—	2	1	no	—
25	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	—	2	1	no	—
28	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	—	2	1	no	—
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
32	C₂×Dic₁₁₈	—	4	2	no	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
38	C₄²×C₅₉	—	2	1	no	—
39	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
40	#11745672	—	8	4	no	—
41	C₄⋊Dic₅₉	—	4	2	no	—
42	C₄⋊Dic₅₉	—	4	2	no	—
43	C₂×C₅₉⋊C₈	—	4	2	no	—
44	C₁₆×C₅₉	—	2	1	no	—
45	C₁₆×C₅₉	—	4	1	no	—
46	C₁₆×C₅₉	Q₁₆	1	2	yes	complemented
47	Q₈×C₅₉	—	2	1	no	—
48	Q₈×C₅₉	D₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
54	C₂×Dic₅₉	—	8	2	no	—
55	Dic₁₁₈	—	8	2	no	—
56	Dic₁₁₈	—	8	2	no	—
57	C₈×C₅₉	—	2	1	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₅₉	#137	1	1	yes	characteristic
61	C₅₉⋊C₈	—	8	2	no	—
62	#12457	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	#466	1	1	yes	characteristic
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	4	1	no	—
67	C₄×C₅₉	#462	1	1	yes	characteristic
68	C₄×C₅₉	#473	1	1	yes	characteristic
69	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₅₉	#508	1	1	yes	characteristic
71	Dic₅₉	—	16	2	no	—
72	#2252	—	59	1	no	—
73	Q₈×C₁₆	D₅₉	1	1	yes	characteristic
74	#1674	—	59	2	no	—
75	#1371	—	59	1	no	—
76	#1364	—	59	2	no	—
77	C₂×C₅₉	#1674	1	1	yes	characteristic
78	C₂×C₅₉	#2215	1	1	yes	characteristic
79	C₂×C₅₉	#2222	1	1	yes	characteristic
80	#500	—	59	2	no	—
81	#462	—	118	2	no	—
82	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#366	—	118	1	no	—
84	#366	—	118	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
89	#335	—	59	2	no	—
90	#332	—	118	2	no	—
91	C₅₉	#12457	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	236	2	no	—
93	C₄⋊Q₈	—	118	2	no	—
94	C₄×Q₈	D₂₃₆	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
98	#107	—	118	2	no	—
99	#107	—	118	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	118	2	no	—
102	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
103	#103	—	236	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
106	C₂×Q₈	—	236	2	no	—
107	C₂×Q₈	D₄₇₂	1	1	yes	characteristic
108	Q₁₆	—	472	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	236	2	no	—
111	C₂×C₈	#11745685	1	1	yes	characteristic
112	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	236	2	no	—
115	C₄⋊C₄	—	236	2	no	—
116	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11745672	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	472	2	no	—
124	Q₈	—	472	2	no	—
125	Q₈	D₉₄₄	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	472	2	no	—
129	C₂×C₄	#14273701	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
131	C₂×C₄	#14273749	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	472	2	no	—
135	C₈	C₂×#11745672	1	1	yes	characteristic
136	C₈	#14273757	1	1	yes	characteristic
137	C₂²	#16837638	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	944	2	no	—
142	C₄	#16837612	1	1	yes	characteristic
143	C₄	#16837677	1	1	yes	characteristic
144	C₄	C₂×D₉₄₄	1	1	yes	characteristic
145	C₄	#16837834	1	1	yes	characteristic
146	C₂	#21423490	1	1	yes	characteristic
147	C₂	#21426182	1	1	yes	characteristic
148	C₂	#21426221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic

#21423490

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic

#21422150

C₂

yes

maximal

Q₈×C₁₆×C₅₉

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

—

—

—

C₂²

yes

—

#16837147

—

C₈×Q₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₅₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

—

C₄×C₁₆×C₅₉

C₂²

yes

—

—

—

—

C₄⋊(C₅₉⋊C₈)

—

#14273650

—

C₄×Q₈×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

D₈

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

Q₁₆

yes

complemented

Q₈×C₅₉

—

Q₈×C₅₉

D₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

#2252

—

Q₈×C₁₆

D₅₉

yes

—

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

C₈×Q₈

C₂×D₅₉

yes

characteristic

#366

—

118

—

#366

—

118

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

#332

—

118

—

C₅₉

#12457

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

236

—

C₄⋊Q₈

—

118

—

C₄×Q₈

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

#107

—

118

—

#107

—

118

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

118

—

102

C₄⋊C₈

D₂₃₆

yes

characteristic

103

#103

—

236

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₅₉

yes

characteristic

106

C₂×Q₈

—

236

—

107

C₂×Q₈

D₄₇₂

yes

characteristic

108

Q₁₆

—

472

—

109

C₂×C₈

—

110

C₂×C₈

—

236

—

111

C₂×C₈

#11745685

yes

characteristic

112

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

236

—

115

C₄⋊C₄

—

236

—

116

C₄⋊C₄

D₄₇₂

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₃₆

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11745672

yes

complemented

122

Q₈

—

123

Q₈

—

472

—

124

Q₈

—

472

—

125

Q₈

D₉₄₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

472

—

129

C₂×C₄

#14273701

yes

characteristic

130

C₂×C₄

C₂×D₄₇₂

yes

characteristic

131

C₂×C₄

#14273749

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

472

—

135

C₈

C₂×#11745672

yes

characteristic

136

C₈

#14273757

yes

characteristic

137

C₂²

#16837638

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

944

—

142

C₄

#16837612

yes

characteristic

143

C₄

#16837677

yes

characteristic

144

C₄

C₂×D₉₄₄

yes

characteristic

145

C₄

#16837834

yes

characteristic

146

C₂

#21423490

yes

characteristic

147

C₂

#21426182

yes

characteristic

148

C₂

#21426221

yes

characteristic

	subgroup	G/H
center	C₂²	#16837638
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	Q₈×C₁₆×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	Q₈×C₁₆	D₅₉
hypocenter	C₅₉	#12457
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16837638

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

Q₈×C₁₆×C₅₉

C₂

socle, prime radical

C₂²×C₅₉

#466

hypercenter

Q₈×C₁₆

D₅₉

hypocenter

C₅₉

#12457

LCS

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₅₉	Q₁₆	semidirect
91. C₅₉	#12457	semidirect
121. C₁₆	#11745672	semidirect

H ≅ G/N

type

46. C₁₆×C₅₉

Q₁₆

semidirect

91. C₅₉

#12457

semidirect

121. C₁₆

#11745672

semidirect

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

subgroup	index
#21426371	2
#21423490	2
#21422168	2
#21422150	2
Q₈×C₁₆×C₅₉	2
#12457	59

subgroup

index

Q₈×C₁₆×C₅₉

Subgroups — #28503048

Conjugacy classes of subgroups

Complements

Maximal subgroups