Group #30594119 — subgroups

Subgroups — #30594119

Order 15616.

242 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 60 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	#21460863	C₂	1	2	yes	maximal
2	#21446620	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519851	—	2	2	no	—
4	C₂×#14519777	C₄	1	1	yes	characteristic
5	C₂×#14519700	—	2	2	no	—
6	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
7	C₆₁×#384	C₄	1	1	yes	characteristic
8	#14519851	—	2	4	no	—
9	#14519851	—	4	2	no	—
10	C₂×#11758187	—	4	2	no	—
11	C₂²×Dic₁₂₂	—	2	1	no	—
12	C₂²×#92418	D₄	1	2	yes	—
13	#14519777	—	2	1	no	—
14	#14519777	—	4	1	no	—
15	#14519777	C₈	1	2	yes	—
16	C₂×C₄⋊Dic₆₁	—	4	1	no	—
17	#14519700	—	4	2	no	—
18	#14519700	—	4	2	no	—
19	C₄×#92418	—	4	1	no	—
20	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
21	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	—	2	1	no	—
24	C₂²×C₄×C₆₁	—	4	1	no	—
25	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	4	1	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×Dic₁₂₂	—	4	2	no	—
29	C₂×Dic₁₂₂	—	4	2	no	—
30	C₂×Dic₁₂₂	—	4	1	no	—
31	C₄⋊C₄×C₆₁	—	4	2	no	—
32	C₄⋊C₄×C₆₁	—	8	2	no	—
33	C₄²×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	D₈	1	2	yes	—
41	C₂×#92418	SD₁₆	1	2	yes	—
42	#11758187	—	4	4	no	—
43	C₂³×C₆₁	#100	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	—	2	1	no	—
47	C₂×C₄×C₆₁	—	2	1	no	—
48	C₂×C₄×C₆₁	—	8	2	no	—
49	C₂×C₄×C₆₁	—	8	1	no	—
50	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#102	1	2	yes	—
53	C₂×C₄×C₆₁	#104	1	2	yes	—
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	—	8	1	no	—
56	C₂×Dic₆₁	—	8	1	no	—
57	Dic₁₂₂	—	4	4	no	—
58	Dic₁₂₂	—	8	2	no	—
59	#92418	—	2	2	no	—
60	#92418	—	2	2	no	—
61	#7108	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	#323	1	1	yes	characteristic
65	C₂²×C₆₁	#329	1	1	yes	characteristic
66	C₂²×C₆₁	#331	1	1	yes	characteristic
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	16	1	no	—
70	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
71	C₄×C₆₁	#327	1	2	yes	—
72	Dic₆₁	—	8	2	no	—
73	#1956	—	61	1	no	—
74	#1376	—	61	2	no	—
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	#1270	1	1	yes	characteristic
78	C₂×C₆₁	#1358	1	1	yes	characteristic
79	C₂×C₆₁	#1359	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	61	1	no	—
81	C₂×M₃₂	—	122	2	no	—
82	C₂×#103	—	122	2	no	—
83	C₂×C₄×C₈	—	61	1	no	—
84	#384	#6923	1	1	yes	characteristic
85	C₆₁	#7108	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	122	1	no	—
87	C₂²×C₈	—	61	2	no	—
88	C₄⋊Q₈	—	61	2	no	—
89	C₄⋊Q₈	—	122	1	no	—
90	C₄⋊Q₈	—	244	1	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	4	1	no	—
93	C₂×C₄⋊C₄	—	244	1	no	—
94	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
95	M₃₂	—	122	4	no	—
96	M₃₂	—	244	2	no	—
97	C₂×C₁₆	—	244	2	no	—
98	#103	—	244	2	no	—
99	#103	—	244	2	no	—
100	C₄×C₈	—	244	1	no	—
101	C₂×Q₈	—	122	2	no	—
102	C₂×Q₈	—	244	2	no	—
103	C₂×Q₈	—	244	2	no	—
104	C₂×Q₈	—	244	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	244	1	no	—
108	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
109	C₂×C₈	—	61	2	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	122	2	no	—
112	C₂×C₈	—	122	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	244	2	no	—
116	C₄⋊C₄	—	244	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
119	C₄²	#11758213	1	1	yes	characteristic
120	C₁₆	—	244	4	no	—
121	C₂³	#14519837	1	1	yes	characteristic
122	Q₈	—	488	2	no	—
123	Q₈	—	244	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	488	1	no	—
131	C₂×C₄	—	488	1	no	—
132	C₂×C₄	—	244	2	no	—
133	C₂×C₄	#14519834	1	1	yes	characteristic
134	C₂×C₄	#14519836	1	1	yes	characteristic
135	C₂×C₄	#14519839	1	2	yes	—
136	C₂×C₄	#14519842	1	2	yes	—
137	C₈	—	122	2	no	—
138	C₈	—	122	2	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16840606	1	1	yes	characteristic
142	C₂²	#16840618	1	1	yes	characteristic
143	C₂²	#16840620	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	16	1	no	—
147	C₄	—	488	2	no	—
148	C₄	#16840610	1	2	yes	—
149	C₄	#16840614	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	#21460636	1	1	yes	characteristic
153	C₂	#21460828	1	1	yes	characteristic
154	C₂	#21460829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460863

C₂

yes

maximal

#21446620

C₂

yes

maximal, characteristic

C₂×#14519851

—

C₂×#14519777

C₄

yes

characteristic

C₂×#14519700

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₆₁×#384

C₄

yes

—

—

—

C₂²×Dic₁₂₂

—

C₂²×#92418

D₄

yes

—

#14519777

—

#14519777

—

#14519777

C₈

yes

—

C₂×C₄⋊Dic₆₁

—

#14519700

—

#14519700

—

C₄×#92418

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₈

yes

—

C₂×#92418

SD₁₆

yes

—

#11758187

—

C₂³×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

D₄⋊C₈

yes

—

C₄×C₆₁

#327

yes

—

Dic₆₁

—

#1956

—

#1376

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

122

—

C₂×#103

—

122

—

C₂×C₄×C₈

—

yes

C₆₁

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

244

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

244

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

122

—

M₃₂

—

244

—

C₂×C₁₆

—

244

—

#103

—

244

—

#103

—

244

—

100

C₄×C₈

—

244

—

101

C₂×Q₈

—

122

—

102

C₂×Q₈

—

244

—

103

C₂×Q₈

—

244

—

104

C₂×Q₈

—

244

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

244

—

108

C₂²×C₄

C₂²⋊#6923

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

122

—

112

C₂×C₈

—

122

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

244

—

116

C₄⋊C₄

—

244

—

117

C₄²

—

118

C₄²

D₆₁⋊C₈

yes

characteristic

119

C₄²

#11758213

yes

characteristic

120

C₁₆

—

244

—

121

C₂³

#14519837

yes

characteristic

122

Q₈

—

488

—

123

Q₈

—

244

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

488

—

131

C₂×C₄

—

488

—

132

C₂×C₄

—

244

—

133

C₂×C₄

#14519834

yes

characteristic

134

C₂×C₄

#14519836

yes

characteristic

135

C₂×C₄

#14519839

yes

—

136

C₂×C₄

#14519842

yes

—

137

C₈

—

122

—

138

C₈

—

122

—

139

C₂²

—

140

C₂²

—

141

C₂²

#16840606

yes

characteristic

142

C₂²

#16840618

yes

characteristic

143

C₂²

#16840620

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

488

—

148

C₄

#16840610

yes

—

149

C₄

#16840614

yes

—

150

C₂

—

151

C₂

—

152

C₂

#21460636

yes

characteristic

153

C₂

#21460828

yes

characteristic

154

C₂

#21460829

yes

characteristic

	subgroup	G/H
center	C₂²	#16840620
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₆₁×#384	C₄
socle	C₂²×C₆₁	#331
prime radical	C₂³×C₆₁	#100
cosocle	C₂×C₄×#92418	C₂²
hypercenter	#384	#6923
hypocenter	C₆₁	#7108
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂²

#16840620

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₆₁×#384

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×#92418

C₂²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
85. C₆₁	#7108	semidirect

H ≅ G/N

type

85. C₆₁

#7108

semidirect

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

subgroup	index
#21460863	2
#21446620	2
#7108	61

subgroup

index

#21460863

#21446620

#7108

Subgroups — #30594119

Conjugacy classes of subgroups

Complements

Maximal subgroups