Group #34010035 — subgroups

Subgroups — #34010035

Order 18688.

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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	#22626288	C₂	1	2	yes	maximal
2	#22624869	C₂	1	4	yes	maximal
3	#22618866	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	2	no	—
5	C₂²⋊#11924031	C₂²	1	4	yes	—
6	C₄×#11924031	—	2	2	no	—
7	#17988718	—	2	2	no	—
8	Q₈×#10596831	—	2	2	no	—
9	#17988614	C₂²	1	2	yes	—
10	#17988611	C₄	1	1	yes	characteristic
11	#17988543	C₄	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	#533×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂×#11924012	C₂×C₄	1	2	yes	—
16	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10596831	C₂³	1	1	yes	characteristic
18	C₄⋊#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	#14556502	—	2	2	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	D₄	1	2	yes	—
23	C₄×#10596831	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₃	C₈	1	2	yes	—
25	#125×C₇₃	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₇₃	—	2	2	no	—
27	C₄×Q₈×C₇₃	—	2	2	no	—
28	#117×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924031	—	4	4	no	—
31	#11924012	—	4	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	4	1	no	—
35	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×#10596831	C₂×D₄	1	2	yes	—
38	C₂×Q₈×C₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	2	4	no	—
42	C₄⋊C₄×C₇₃	—	2	2	no	—
43	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
44	C₂²⋊C₄×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	—	2	2	no	—
46	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	4	no	—
51	C₂×C₄×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	4	no	—
55	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	#10596831	—	2	2	no	—
60	#10596831	—	2	1	no	—
61	#10596831	—	4	2	no	—
62	#10596831	#101	1	2	yes	—
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	4	4	no	—
69	C₄×C₇₃	#350	1	4	yes	—
70	C₄×C₇₃	#410	1	1	yes	characteristic
71	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
72	#11496	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	4	1	no	—
74	C₂×C₇₃	#1461	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#350	1	2	yes	—
76	#2960	—	73	1	no	—
77	#2162	—	73	2	no	—
78	#1323	—	73	4	no	—
79	C₇₃	#11496	1	1	yes	characteristic, complemented, Sylow
80	#533	#64408	1	1	yes	characteristic
81	#446	—	146	2	no	—
82	C₈×Q₈	—	146	2	no	—
83	#424	—	73	2	no	—
84	#423	—	73	1	no	—
85	#405	—	73	2	no	—
86	C₂×C₄×C₈	—	73	1	no	—
87	C₄⋊C₁₆	—	146	2	no	—
88	C₂²⋊C₁₆	—	73	4	no	—
89	C₄×C₁₆	—	146	2	no	—
90	C₂×M₁₆	—	73	2	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₄⋊Q₈	#10596833	1	2	yes	—
94	#125	#10596833	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×#64408	1	2	yes	—
98	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
99	C₂×C₁₆	—	146	4	no	—
100	C₄⋊C₈	—	146	4	no	—
101	C₄⋊C₈	—	146	2	no	—
102	#97	—	146	2	no	—
103	C₄×C₈	—	73	2	no	—
104	C₄×C₈	—	73	2	no	—
105	C₄×C₈	—	146	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#11924037	1	2	yes	—
108	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
109	M₁₆	—	292	2	no	—
110	C₂×C₈	—	292	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×#10596833	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊#64408	1	2	yes	—
122	C₄²	C₂×#10596833	1	1	yes	characteristic
123	C₄²	C₂×#10596833	1	1	yes	characteristic
124	C₁₆	—	292	4	no	—
125	C₂³	C₂×#11924037	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
132	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
134	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
135	C₈	—	292	2	no	—
136	C₈	—	73	2	no	—
137	C₈	—	146	2	no	—
138	C₈	—	146	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊#10596833	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	#17989485	1	4	yes	—
146	C₄	#17989568	1	1	yes	characteristic
147	C₄	D₇₃⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#22625174	1	1	yes	characteristic
150	C₂	C₂×#17989485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626288

C₂

yes

maximal

#22624869

C₂

yes

maximal

#22618866

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₂²⋊#11924031

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988611

C₄

yes

characteristic

#17988543

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

#533×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×#11924012

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₄⋊Q₈×C₇₃

C₈

yes

—

#125×C₇₃

C₈

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₈

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#350

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1461

yes

characteristic

C₂×C₇₃

C₂×#350

yes

—

#2960

—

#2162

—

#1323

—

C₇₃

#11496

yes

characteristic, complemented, Sylow

yes

—

146

—

C₈×Q₈

—

146

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

146

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

#10596833

yes

—

#125

#10596833

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

146

—

100

C₄⋊C₈

—

146

—

101

C₄⋊C₈

—

146

—

102

#97

—

146

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

146

—

106

C₂×Q₈

—

107

C₂²×C₄

#11924037

yes

—

108

C₂²×C₄

C₂²×#64408

yes

characteristic

109

M₁₆

—

292

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×#10596833

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊#64408

yes

—

122

C₄²

C₂×#10596833

yes

characteristic

123

C₄²

C₂×#10596833

yes

characteristic

124

C₁₆

—

292

—

125

C₂³

C₂×#11924037

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊#10596833

yes

—

132

C₂×C₄

C₂²⋊#10596833

yes

—

133

C₂×C₄

C₂×C₂²⋊#64408

yes

characteristic

134

C₂×C₄

C₂²×#10596833

yes

characteristic

135

C₈

—

292

—

136

C₈

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#10596833

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#17989485

yes

—

146

C₄

#17989568

yes

characteristic

147

C₄

D₇₃⋊#101

yes

characteristic

148

C₂

—

149

C₂

#22625174

yes

characteristic

150

C₂

C₂×#17989485

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#533×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	#533	#64408
hypocenter	C₇₃	#11496
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#533×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
79. C₇₃	#11496	semidirect

H ≅ G/N

type

79. C₇₃

#11496

semidirect

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

subgroup	index
#22626288	2
#22624869	2
#22618866	2
#11496	73

subgroup

index

Subgroups — #34010035

Conjugacy classes of subgroups

Complements

Maximal subgroups