Group #29594372 — subgroups

Subgroups — #29594372

Order 15616.

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	#21448094	C₂	1	1	yes	maximal, characteristic
2	#21445213	C₂	1	2	yes	maximal
3	#21443891	C₂	1	1	yes	maximal, characteristic
4	#21443873	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840167	C₂²	1	2	yes	—
7	#16839972	—	2	2	no	—
8	#16839588	—	2	1	no	—
9	#16839588	—	2	1	no	—
10	#16839515	C₂²	1	2	yes	—
11	#16839507	—	2	2	no	—
12	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×C₄⋊C₁₆	—	2	1	no	—
14	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₁	—	2	1	no	—
16	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758202	—	4	2	no	—
18	#14519777	—	2	2	no	—
19	#14519711	—	2	2	no	—
20	#14519711	—	2	2	no	—
21	C₄⋊#92418	—	2	2	no	—
22	#14519700	—	4	2	no	—
23	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	—	2	1	no	—
25	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	1	no	—
28	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	—	2	1	no	—
30	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
32	C₂×Dic₁₂₂	—	4	2	no	—
33	C₂×C₈×C₆₁	—	2	1	no	—
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
38	C₄²×C₆₁	—	2	1	no	—
39	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
40	#11758202	—	8	4	no	—
41	C₄⋊Dic₆₁	—	4	2	no	—
42	C₄⋊Dic₆₁	—	4	2	no	—
43	C₂×#92418	—	4	2	no	—
44	C₁₆×C₆₁	—	2	1	no	—
45	C₁₆×C₆₁	—	4	1	no	—
46	C₁₆×C₆₁	Q₁₆	1	2	yes	complemented
47	Q₈×C₆₁	—	2	1	no	—
48	Q₈×C₆₁	D₁₆	1	2	yes	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
54	C₂×Dic₆₁	—	8	2	no	—
55	Dic₁₂₂	—	8	2	no	—
56	Dic₁₂₂	—	8	2	no	—
57	C₈×C₆₁	—	2	1	no	—
58	C₈×C₆₁	—	4	1	no	—
59	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₆₁	#137	1	1	yes	characteristic
61	#92418	—	8	2	no	—
62	#12457	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	#466	1	1	yes	characteristic
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	#462	1	1	yes	characteristic
68	C₄×C₆₁	#473	1	1	yes	characteristic
69	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₆₁	#508	1	1	yes	characteristic
71	Dic₆₁	—	16	2	no	—
72	#2252	—	61	1	no	—
73	Q₈×C₁₆	D₆₁	1	1	yes	characteristic
74	#1674	—	61	2	no	—
75	#1371	—	61	1	no	—
76	#1364	—	61	2	no	—
77	C₂×C₆₁	#1674	1	1	yes	characteristic
78	C₂×C₆₁	#2215	1	1	yes	characteristic
79	C₂×C₆₁	#2222	1	1	yes	characteristic
80	#500	—	61	2	no	—
81	#462	—	122	2	no	—
82	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
83	#366	—	122	1	no	—
84	#366	—	122	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
89	#335	—	61	2	no	—
90	#332	—	122	2	no	—
91	C₆₁	#12457	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	244	2	no	—
93	C₄⋊Q₈	—	122	2	no	—
94	C₄×Q₈	D₂₄₄	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
98	#107	—	122	2	no	—
99	#107	—	122	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	122	2	no	—
102	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
103	#103	—	244	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
106	C₂×Q₈	—	244	2	no	—
107	C₂×Q₈	D₄₈₈	1	1	yes	characteristic
108	Q₁₆	—	488	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	244	2	no	—
111	C₂×C₈	#11758222	1	1	yes	characteristic
112	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	244	2	no	—
115	C₄⋊C₄	—	244	2	no	—
116	C₄⋊C₄	D₄₈₈	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11758202	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	488	2	no	—
124	Q₈	—	488	2	no	—
125	Q₈	D₉₇₆	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	488	2	no	—
129	C₂×C₄	#14519751	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
131	C₂×C₄	#14519799	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	488	2	no	—
135	C₈	C₂×#11758202	1	1	yes	characteristic
136	C₈	#14519807	1	1	yes	characteristic
137	C₂²	#16839998	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	976	2	no	—
142	C₄	#16839972	1	1	yes	characteristic
143	C₄	#16840037	1	1	yes	characteristic
144	C₄	C₂×D₉₇₆	1	1	yes	characteristic
145	C₄	#16840194	1	1	yes	characteristic
146	C₂	#21445213	1	1	yes	characteristic
147	C₂	#21447905	1	1	yes	characteristic
148	C₂	#21447944	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448094

C₂

yes

maximal, characteristic

#21445213

C₂

yes

maximal

#21443891

C₂

yes

maximal, characteristic

#21443873

C₂

yes

maximal

Q₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic

#16840167

C₂²

yes

—

—

—

—

C₂²

yes

—

#16839507

—

C₈×Q₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₆₁×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

—

C₄×C₁₆×C₆₁

C₂²

yes

—

—

—

—

—

—

C₄×Q₈×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

D₈

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

Q₁₆

yes

complemented

Q₈×C₆₁

—

Q₈×C₆₁

D₁₆

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

#508

yes

characteristic

Dic₆₁

—

#2252

—

Q₈×C₁₆

D₆₁

yes

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

122

—

C₈×Q₈

C₂×D₆₁

yes

characteristic

#366

—

122

—

#366

—

122

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₆₁

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

—

#332

—

122

—

C₆₁

#12457

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

244

—

C₄⋊Q₈

—

122

—

C₄×Q₈

D₂₄₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

#107

—

122

—

#107

—

122

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

122

—

102

C₄⋊C₈

D₂₄₄

yes

characteristic

103

#103

—

244

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₆₁

yes

characteristic

106

C₂×Q₈

—

244

—

107

C₂×Q₈

D₄₈₈

yes

characteristic

108

Q₁₆

—

488

—

109

C₂×C₈

—

110

C₂×C₈

—

244

—

111

C₂×C₈

#11758222

yes

characteristic

112

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

244

—

115

C₄⋊C₄

—

244

—

116

C₄⋊C₄

D₄₈₈

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₄₄

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11758202

yes

complemented

122

Q₈

—

123

Q₈

—

488

—

124

Q₈

—

488

—

125

Q₈

D₉₇₆

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

488

—

129

C₂×C₄

#14519751

yes

characteristic

130

C₂×C₄

C₂×D₄₈₈

yes

characteristic

131

C₂×C₄

#14519799

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

488

—

135

C₈

C₂×#11758202

yes

characteristic

136

C₈

#14519807

yes

characteristic

137

C₂²

#16839998

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

976

—

142

C₄

#16839972

yes

characteristic

143

C₄

#16840037

yes

characteristic

144

C₄

C₂×D₉₇₆

yes

characteristic

145

C₄

#16840194

yes

characteristic

146

C₂

#21445213

yes

characteristic

147

C₂

#21447905

yes

characteristic

148

C₂

#21447944

yes

characteristic

	subgroup	G/H
center	C₂²	#16839998
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	Q₈×C₁₆×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	#466
hypercenter	Q₈×C₁₆	D₆₁
hypocenter	C₆₁	#12457
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16839998

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

Q₈×C₁₆×C₆₁

C₂

socle, prime radical

C₂²×C₆₁

#466

hypercenter

Q₈×C₁₆

D₆₁

hypocenter

C₆₁

#12457

LCS

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₆₁	Q₁₆	semidirect
91. C₆₁	#12457	semidirect
121. C₁₆	#11758202	semidirect

H ≅ G/N

type

46. C₁₆×C₆₁

Q₁₆

semidirect

91. C₆₁

#12457

semidirect

121. C₁₆

#11758202

semidirect

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

subgroup	index
#21448094	2
#21445213	2
#21443891	2
#21443873	2
Q₈×C₁₆×C₆₁	2
#12457	61

subgroup

index

Q₈×C₁₆×C₆₁

Subgroups — #29594372

Conjugacy classes of subgroups

Complements

Maximal subgroups