Group #34009673 — subgroups

Subgroups — #34009673

Order 18688.

302 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 140 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	#22626132	C₂	1	1	yes	maximal, characteristic
2	#22626132	C₂	1	1	yes	maximal, characteristic
3	#22624849	C₂	1	4	yes	maximal, complemented
4	#322×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	C₄	1	2	yes	—
6	C₂×#14556664	C₄	1	2	yes	—
7	#17989476	C₄	1	2	yes	—
8	#17989476	C₄	1	2	yes	—
9	#17989475	C₂²	1	2	yes	—
10	#17989475	C₂²	1	2	yes	—
11	C₄×C₈×D₇₃	C₄	1	2	yes	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988309	C₂²	1	2	yes	—
14	#17988308	C₄	1	1	yes	characteristic
15	#322×C₇₃	C₄	1	1	yes	characteristic
16	#14556664	—	2	4	no	—
17	#14556664	C₈	1	8	yes	complemented
18	C₂×#11924031	C₂×C₄	1	4	yes	—
19	C₂×#11924031	C₂×C₄	1	4	yes	—
20	C₂×C₈×D₇₃	—	2	2	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	2	yes	—
25	C₄×#10596831	C₂³	1	1	yes	characteristic
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#11924031	—	2	4	no	—
32	#11924031	C₂×C₈	1	8	yes	complemented
33	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
37	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
38	C₈×D₇₃	—	4	4	no	—
39	C₈×D₇₃	C₂×C₈	1	4	yes	—
40	C₈×D₇₃	M₁₆	1	4	yes	—
41	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
42	C₄×Dic₇₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₇₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
50	C₂×Dic₇₃	#97	1	2	yes	—
51	C₂×Dic₇₃	#97	1	1	yes	characteristic
52	C₄×D₇₃	—	2	4	no	—
53	C₄×D₇₃	C₄×C₈	1	2	yes	—
54	C₄×D₇₃	#97	1	2	yes	—
55	C₈×C₇₃	—	4	2	no	—
56	C₈×C₇₃	C₄×C₈	1	2	yes	—
57	C₈×C₇₃	#97	1	2	yes	—
58	#10596831	—	4	2	no	—
59	#10596831	C₂²×C₈	1	2	yes	—
60	#10596831	C₂×M₁₆	1	2	yes	—
61	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
62	C₂×D₇₃	#322	1	2	yes	—
63	C₂×D₇₃	#347	1	4	yes	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
67	Dic₇₃	—	2	2	no	—
68	Dic₇₃	#322	1	2	yes	—
69	C₂×#1313	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
71	C₂×C₇₃	C₂×#347	1	2	yes	—
72	D₇₃	#1313	1	4	yes	complemented
73	C₂×#346	—	73	1	no	—
74	C₂×#346	—	73	1	no	—
75	C₂×#322	—	73	1	no	—
76	#1313	—	73	4	no	—
77	C₇₃	C₂×#1313	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₁₆	—	73	2	no	—
79	C₂²×C₁₆	—	73	2	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	C₂×C₄×C₈	—	73	2	no	—
82	#346	—	73	2	no	—
83	#346	—	73	2	no	—
84	#346	—	73	2	no	—
85	#346	—	73	2	no	—
86	#322	—	73	2	no	—
87	#322	—	73	1	no	—
88	#322	#64408	1	1	yes	characteristic
89	C₂²×C₈	—	73	1	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂²×C₈	—	146	2	no	—
92	C₂×C₄²	—	73	1	no	—
93	C₂×C₁₆	—	73	4	no	—
94	C₂×C₁₆	—	73	8	no	—
95	C₂×C₁₆	—	73	4	no	—
96	C₂×C₁₆	—	146	4	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	2	no	—
101	C₄×C₈	—	146	2	no	—
102	C₄×C₈	C₂×#64408	1	2	yes	—
103	C₄×C₈	C₂×#64408	1	1	yes	characteristic
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	292	4	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	4	no	—
111	C₂×C₈	—	73	4	no	—
112	C₂×C₈	—	146	2	no	—
113	C₂×C₈	C₄×#64408	1	1	yes	characteristic
114	C₂×C₈	C₄×#64408	1	1	yes	characteristic
115	C₄²	—	73	2	no	—
116	C₄²	—	73	1	no	—
117	C₄²	C₂²×#64408	1	1	yes	characteristic
118	C₁₆	—	73	8	no	—
119	C₁₆	—	146	4	no	—
120	C₂³	—	73	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	73	2	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	146	4	no	—
126	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
127	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
128	C₈	—	4	2	no	—
129	C₈	—	292	2	no	—
130	C₈	—	73	2	no	—
131	C₈	—	73	2	no	—
132	C₈	C₈×#64408	1	2	yes	—
133	C₈	#14556645	1	2	yes	—
134	C₂²	—	73	4	no	—
135	C₂²	—	73	2	no	—
136	C₂²	#17989546	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	73	2	no	—
139	C₄	—	146	2	no	—
140	C₄	C₂×C₈×#64408	1	1	yes	characteristic
141	C₄	C₂×#14556645	1	1	yes	characteristic
142	C₂	—	73	4	no	—
143	C₂	#22625127	1	1	yes	characteristic
144	C₂	#22626148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626132

C₂

yes

maximal, characteristic

#22626132

C₂

yes

maximal, characteristic

#22624849

C₂

yes

maximal, complemented

#322×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

C₄

yes

—

C₂×#14556664

C₄

yes

—

#17989476

C₄

yes

—

#17989476

C₄

yes

—

#17989475

C₂²

yes

—

#17989475

C₂²

yes

—

C₄×C₈×D₇₃

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₂²

yes

—

#17988308

C₄

yes

characteristic

#322×C₇₃

C₄

yes

characteristic

#14556664

—

#14556664

C₈

yes

complemented

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

#11924031

C₂×C₈

yes

complemented

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

#97

yes

—

C₂×Dic₇₃

#97

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₄×C₈

yes

—

C₄×D₇₃

#97

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

—

C₈×C₇₃

#97

yes

—

#10596831

—

#10596831

C₂²×C₈

yes

—

#10596831

C₂×M₁₆

yes

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₂×D₇₃

#322

yes

—

C₂×D₇₃

#347

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

Dic₇₃

—

Dic₇₃

#322

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

—

C₇₃

C₂×#1313

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

—

—

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

146

—

102

C₄×C₈

C₂×#64408

yes

—

103

C₄×C₈

C₂×#64408

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂×C₈

—

107

C₂×C₈

—

292

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

C₄×#64408

yes

characteristic

114

C₂×C₈

C₄×#64408

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

C₂²×#64408

yes

characteristic

118

C₁₆

—

119

C₁₆

—

146

—

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

146

—

126

C₂×C₄

C₂×C₄×#64408

yes

characteristic

127

C₂×C₄

D₇₃⋊M₁₆

yes

—

128

C₈

—

129

C₈

—

292

—

130

C₈

—

131

C₈

—

132

C₈

C₈×#64408

yes

—

133

C₈

#14556645

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

#17989546

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

146

—

140

C₄

C₂×C₈×#64408

yes

characteristic

141

C₄

C₂×#14556645

yes

characteristic

142

C₂

—

143

C₂

#22625127

yes

characteristic

144

C₂

#22626148

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#64408
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂²×D₇₃	#97
cosocle	C₄×#10596831	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	C₂×#1313
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#64408

Fitting

#322×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

C₂²×D₇₃

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #322

N	H ≅ G/N	type
3. #22624849	C₂	semidirect
17. #14556664	C₈	semidirect
32. #11924031	C₂×C₈	semidirect
72. D₇₃	#1313	semidirect
77. C₇₃	C₂×#1313	semidirect

H ≅ G/N

type

3. #22624849

C₂

semidirect

17. #14556664

C₈

semidirect

32. #11924031

C₂×C₈

semidirect

72. D₇₃

#1313

semidirect

77. C₇₃

C₂×#1313

semidirect

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

subgroup	index
#22626132	2
#22624849	2
#322×D₇₃	2
C₂×#1313	73

subgroup

index

#22626132

#22624849

#322×D₇₃

C₂×#1313

Subgroups — #34009673

Conjugacy classes of subgroups

Complements

Maximal subgroups