Group #34010039 — subgroups

Subgroups — #34010039

Order 18688.

288 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 90 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	#22626292	C₂	1	2	yes	maximal
2	#22624870	C₂	1	4	yes	maximal
3	#22618861	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	2	no	—
5	C₂²⋊#11924031	C₂²	1	4	yes	—
6	#17989475	—	2	2	no	—
7	#17989144	C₄	1	1	yes	characteristic
8	#17988714	—	2	2	no	—
9	#17988711	—	2	2	no	—
10	#17988615	C₂²	1	2	yes	—
11	#17988545	C₄	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	#423×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
16	C₂²×#10596831	C₂³	1	1	yes	characteristic
17	#14556591	C₈	1	2	yes	—
18	#14556583	C₈	1	2	yes	—
19	#14556570	—	2	2	no	—
20	C₄×Dic₁₄₆	—	2	2	no	—
21	#14556550	C₂×C₄	1	2	yes	—
22	#14556520	—	2	4	no	—
23	#14556501	—	2	2	no	—
24	C₄×#10596831	D₄	1	2	yes	—
25	C₄×#10596831	D₄	1	2	yes	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	#11924031	—	4	4	no	—
31	C₂×Dic₁₄₆	—	2	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	4	1	no	—
34	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×#10596831	C₂×D₄	1	2	yes	—
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
39	#11924007	—	2	4	no	—
40	C₂²⋊Dic₇₃	—	2	2	no	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	M₁₆×C₇₃	—	4	2	no	—
45	C₂×C₈×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	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	4	2	no	—
53	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	4	no	—
58	Dic₁₄₆	—	4	4	no	—
59	C₈×C₇₃	—	4	2	no	—
60	#10596831	—	2	2	no	—
61	#10596831	—	2	1	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	2	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	#410	1	1	yes	characteristic
70	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
71	Dic₇₃	—	4	4	no	—
72	#11497	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	4	1	no	—
74	C₂×C₇₃	#1461	1	1	yes	characteristic
75	C₂×C₇₃	#2117	1	2	yes	—
76	#2960	—	73	1	no	—
77	#2163	—	73	2	no	—
78	#1323	—	73	4	no	—
79	C₇₃	#11497	1	1	yes	characteristic, complemented, Sylow
80	#533	—	73	1	no	—
81	#446	—	146	2	no	—
82	C₈×Q₈	—	146	2	no	—
83	#424	—	73	2	no	—
84	#423	#64408	1	1	yes	characteristic
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	#346	—	146	2	no	—
90	C₂×M₁₆	C₂×#64408	1	2	yes	—
91	C₂²×C₈	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₄⋊Q₈	—	73	2	no	—
94	#125	—	73	2	no	—
95	C₂²⋊Q₈	—	146	2	no	—
96	C₄×Q₈	—	146	2	no	—
97	#117	—	73	2	no	—
98	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
99	C₂×C₁₆	—	146	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	146	4	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	—	146	2	no	—
107	C₂²×C₄	#11924038	1	2	yes	—
108	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	292	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₄⋊C₄	—	73	4	no	—
117	C₄⋊C₄	—	146	4	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₂²⋊C₄	—	146	2	no	—
120	C₄²	—	146	2	no	—
121	C₄²	C₂²⋊#64408	1	2	yes	—
122	C₄²	#11924036	1	1	yes	characteristic
123	C₄²	#11924036	1	1	yes	characteristic
124	C₁₆	—	292	4	no	—
125	C₂³	C₂×#11924038	1	1	yes	characteristic
126	Q₈	—	292	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	146	4	no	—
131	C₂×C₄	#14556648	1	2	yes	—
132	C₂×C₄	#14556648	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
134	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
135	C₈	—	4	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×#14556648	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	292	4	no	—
146	C₄	#17989567	1	1	yes	characteristic
147	C₄	D₇₃⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#22625175	1	1	yes	characteristic
150	C₂	#22626175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal

#22624870

C₂

yes

maximal

#22618861

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₂²⋊#11924031

C₂²

yes

—

#17989475

—

#17989144

C₄

yes

—

—

C₂²

yes

—

#17988545

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

#423×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

#14556591

C₈

yes

—

#14556583

C₈

yes

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

C₂×C₄

yes

—

#14556520

—

#14556501

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

#11924031

—

C₂×Dic₁₄₆

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

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

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#11497

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1461

yes

characteristic

C₂×C₇₃

#2117

yes

—

#2960

—

#2163

—

#1323

—

C₇₃

#11497

yes

characteristic, complemented, Sylow

#533

—

#446

—

146

—

C₈×Q₈

—

146

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

#346

—

146

—

C₂×M₁₆

C₂×#64408

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

146

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

146

—

102

#97

—

146

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

146

—

106

C₂×Q₈

—

146

—

107

C₂²×C₄

#11924038

yes

—

108

C₂²×C₄

C₂²×#64408

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

292

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

—

146

—

119

C₂²⋊C₄

—

146

—

120

C₄²

—

146

—

121

C₄²

C₂²⋊#64408

yes

—

122

C₄²

#11924036

yes

characteristic

123

C₄²

#11924036

yes

characteristic

124

C₁₆

—

292

—

125

C₂³

C₂×#11924038

yes

characteristic

126

Q₈

—

292

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

#14556648

yes

—

132

C₂×C₄

#14556648

yes

—

133

C₂×C₄

C₂×C₂²⋊#64408

yes

characteristic

134

C₂×C₄

C₂×#11924036

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×#14556648

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

292

—

146

C₄

#17989567

yes

characteristic

147

C₄

D₇₃⋊#101

yes

characteristic

148

C₂

—

149

C₂

#22625175

yes

characteristic

150

C₂

#22626175

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#423×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	#423	#64408
hypocenter	C₇₃	#11497
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#423×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 ◁ #423

N	H ≅ G/N	type
79. C₇₃	#11497	semidirect

H ≅ G/N

type

79. C₇₃

#11497

semidirect

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

subgroup	index
#22626292	2
#22624870	2
#22618861	2
#11497	73

subgroup

index

Subgroups — #34010039

Conjugacy classes of subgroups

Complements

Maximal subgroups