Group #34012321 — subgroups

Subgroups — #34012321

Order 18688.

358 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 168 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	C₂×#17989475	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22624970	C₂	1	4	yes	maximal
4	#22609619	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₂²	1	2	yes	—
6	C₂²×#11924031	C₂²	1	2	yes	—
7	#17989475	C₄	1	4	yes	complemented
8	C₄×#11924031	C₄	1	4	yes	complemented
9	C₂×#14556501	C₄	1	2	yes	—
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
12	#17988433	C₄	1	1	yes	characteristic
13	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂×#11924031	—	2	4	no	—
16	C₂×#11924031	C₂×C₄	1	4	yes	—
17	C₂×#11924031	C₂×C₄	1	4	yes	—
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
21	#14556501	—	2	4	no	—
22	C₄×#10596831	C₂×C₄	1	2	yes	—
23	C₄×#10596831	C₂×C₄	1	2	yes	—
24	C₈×Dic₇₃	—	2	4	no	—
25	#14556498	C₈	1	4	yes	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	#97×C₇₃	C₈	1	4	yes	—
29	#11924031	—	2	4	no	—
30	#11924031	—	2	4	no	—
31	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
32	C₂×#10596831	C₄²	1	2	yes	—
33	C₂×#10596831	M₁₆	1	4	yes	—
34	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
35	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
36	C₂×#10596831	#41	1	4	yes	—
37	C₄×Dic₇₃	—	2	2	no	—
38	C₄×Dic₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
44	C₄²×C₇₃	C₄²	1	2	yes	—
45	C₄²×C₇₃	C₂×C₈	1	2	yes	—
46	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#117	1	2	yes	—
51	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
53	C₂×Dic₇₃	—	2	4	no	—
54	C₂×Dic₇₃	—	2	8	no	—
55	C₈×C₇₃	—	2	4	no	—
56	#10596831	#97	1	4	yes	—
57	#10596831	#117	1	2	yes	—
58	#10596831	#117	1	2	yes	—
59	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₇₃	#431	1	2	yes	—
61	C₂²×C₇₃	#504	1	4	yes	—
62	C₄×C₇₃	—	2	4	no	—
63	C₄×C₇₃	#376	1	1	yes	characteristic
64	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
65	C₄×C₇₃	#431	1	2	yes	—
66	Dic₇₃	—	4	4	no	—
67	#12592	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	#1744	1	1	yes	characteristic
69	C₂×C₇₃	#2108	1	2	yes	—
70	C₂×C₇₃	#2164	1	4	yes	—
71	C₂×#346	—	73	1	no	—
72	C₂×C₄×C₁₆	—	73	1	no	—
73	#1726	—	73	1	no	—
74	#1374	—	73	4	no	—
75	C₇₃	#12592	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₁₆	—	73	2	no	—
77	C₂²×C₁₆	—	73	2	no	—
78	C₂×#97	—	73	2	no	—
79	C₂×#97	#64408	1	1	yes	characteristic
80	C₂×C₄×C₈	—	73	1	no	—
81	C₂×C₄×C₈	—	73	2	no	—
82	#376	—	73	1	no	—
83	#346	—	73	4	no	—
84	C₄×C₁₆	—	73	4	no	—
85	C₂²×C₈	—	73	1	no	—
86	C₂²×C₈	—	73	1	no	—
87	C₂²×C₈	C₂×#64408	1	2	yes	—
88	C₂×C₄²	—	73	2	no	—
89	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
90	C₂×C₁₆	—	73	4	no	—
91	C₂×C₁₆	—	73	4	no	—
92	C₂×C₁₆	—	146	4	no	—
93	C₂×C₁₆	—	146	4	no	—
94	#97	—	146	4	no	—
95	#97	#10596833	1	4	yes	—
96	C₄×C₈	—	73	2	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	146	4	no	—
99	#95	—	73	4	no	—
100	C₂²×C₄	—	73	4	no	—
101	C₂²×C₄	#11924036	1	2	yes	—
102	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	4	no	—
110	C₂×C₈	C₂×#10596833	1	4	yes	—
111	C₄²	—	146	2	no	—
112	C₄²	—	146	2	no	—
113	C₄²	C₄×#64408	1	2	yes	—
114	C₄²	C₂×#10596833	1	2	yes	—
115	C₁₆	—	146	4	no	—
116	C₁₆	—	146	4	no	—
117	C₂³	C₂×#11924036	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	146	8	no	—
120	C₂×C₄	—	146	4	no	—
121	C₂×C₄	C₄×#10596833	1	4	yes	—
122	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
123	C₂×C₄	#14556670	1	2	yes	—
124	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
125	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	—	73	2	no	—
128	C₈	—	73	2	no	—
129	C₈	—	73	4	no	—
130	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
131	C₂²	#17989605	1	2	yes	—
132	C₂²	#17989646	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	292	4	no	—
135	C₄	#17989523	1	1	yes	characteristic
136	C₄	#17989546	1	1	yes	characteristic
137	C₄	#17989603	1	2	yes	—
138	C₂	#22625645	1	1	yes	characteristic
139	C₂	#22626142	1	2	yes	—
140	C₂	#22626298	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989475

C₂

yes

maximal, characteristic

C₂×C₄×#11924031

C₂

yes

maximal, characteristic

#22624970

C₂

yes

maximal

#22609619

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

C₂²×#11924031

C₂²

yes

—

#17989475

C₄

yes

complemented

C₄×#11924031

C₄

yes

complemented

C₂×#14556501

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

—

#17988433

C₄

yes

characteristic

C₂×#97×C₇₃

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

#14556501

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

—

#14556498

C₈

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

C₈

yes

—

#11924031

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

#97

yes

—

#10596831

#117

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

#431

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2164

yes

—

C₂×#346

—

C₂×C₄×C₁₆

—

#1726

—

#1374

—

C₇₃

#12592

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

#64408

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#346

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#97

—

146

—

#97

#10596833

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

#95

—

100

C₂²×C₄

—

101

C₂²×C₄

#11924036

yes

—

102

C₂²×C₄

C₂²×#64408

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×#10596833

yes

—

111

C₄²

—

146

—

112

C₄²

—

146

—

113

C₄²

C₄×#64408

yes

—

114

C₄²

C₂×#10596833

yes

—

115

C₁₆

—

146

—

116

C₁₆

—

146

—

117

C₂³

C₂×#11924036

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

C₄×#10596833

yes

—

122

C₂×C₄

C₂×C₄×#64408

yes

characteristic

123

C₂×C₄

#14556670

yes

—

124

C₂×C₄

C₂²×#10596833

yes

characteristic

125

C₂×C₄

D₇₃⋊M₁₆

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

C₂×C₄×#10596833

yes

characteristic

131

C₂²

#17989605

yes

—

132

C₂²

#17989646

yes

—

133

C₄

—

134

C₄

—

292

—

135

C₄

#17989523

yes

characteristic

136

C₄

#17989546

yes

characteristic

137

C₄

#17989603

yes

—

138

C₂

#22625645

yes

characteristic

139

C₂

#22626142

yes

—

140

C₂

#22626298

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#12592
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator

C₂²×C₇₃

C₂×C₄×C₈

Fitting

C₂×#97×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂×#97

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
7. #17989475	C₄	semidirect
8. C₄×#11924031	C₄	semidirect
75. C₇₃	#12592	semidirect

H ≅ G/N

type

7. #17989475

C₄

semidirect

8. C₄×#11924031

C₄

semidirect

75. C₇₃

#12592

semidirect

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

subgroup	index
C₂×#17989475	2
C₂×C₄×#11924031	2
#22624970	2
#22609619	2
#12592	73

subgroup

index

Subgroups — #34012321

Conjugacy classes of subgroups

Complements

Maximal subgroups