Group #33999994 — subgroups

Subgroups — #33999994

Order 18688.

194 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22624851	C₂	1	2	yes	maximal
2	#22610002	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11924031	—	2	2	no	—
4	C₂×C₂²⋊#10596831	C₄	1	1	yes	characteristic
5	C₂×C₂²⋊#10596831	C₂²	1	1	yes	characteristic
6	#17988342	—	2	2	no	—
7	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924031	—	4	2	no	—
9	C₂²×#10596831	—	2	1	no	—
10	C₂²×#10596831	D₄	1	2	yes	—
11	C₂²⋊#10596831	—	4	1	no	—
12	C₂²⋊#10596831	—	4	1	no	—
13	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
14	#117×C₇₃	—	4	1	no	—
15	#117×C₇₃	—	4	1	no	—
16	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
19	C₂×C₄²×C₇₃	—	2	1	no	—
20	#11924031	—	8	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	4	2	no	—
23	C₂×#10596831	—	4	2	no	—
24	C₂×#10596831	—	4	1	no	—
25	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
26	C₂²×C₄×C₇₃	—	2	1	no	—
27	C₂²×C₄×C₇₃	—	2	1	no	—
28	C₂²×C₄×C₇₃	—	2	1	no	—
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	—	4	2	no	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	4	1	no	—
34	C₄⋊C₄×C₇₃	—	4	1	no	—
35	C₂²⋊C₄×C₇₃	—	4	2	no	—
36	C₄²×C₇₃	—	4	1	no	—
37	C₄²×C₇₃	—	4	1	no	—
38	C₂³×C₇₃	—	2	1	no	—
39	C₂³×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	—	4	1	no	—
41	C₂³×C₇₃	—	4	1	no	—
42	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	2	no	—
52	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
54	#10596831	—	4	2	no	—
55	#10596831	—	8	1	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	—	4	1	no	—
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	—	8	2	no	—
62	C₂²×C₇₃	#323	1	1	yes	characteristic
63	C₂²×C₇₃	#350	1	2	yes	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	4	1	no	—
66	C₄×C₇₃	—	8	1	no	—
67	C₄×C₇₃	—	8	1	no	—
68	C₄×C₇₃	#352	1	2	yes	—
69	#7059	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	—	8	1	no	—
72	C₂×C₇₃	#1315	1	1	yes	characteristic
73	C₂×C₇₃	#1317	1	1	yes	characteristic
74	C₂×C₇₃	#1321	1	1	yes	characteristic
75	#1818	—	73	1	no	—
76	#1314	—	73	2	no	—
77	C₇₃	#7059	1	1	yes	characteristic, complemented, Sylow
78	C₂×#117	#64408	1	1	yes	characteristic
79	C₂×C₂²⋊C₈	—	73	1	no	—
80	C₂×C₂²⋊C₈	—	73	1	no	—
81	C₂²⋊C₁₆	—	146	2	no	—
82	#336	—	146	2	no	—
83	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
84	C₂²×C₈	—	73	2	no	—
85	C₂²×C₈	—	146	1	no	—
86	#117	—	4	1	no	—
87	#117	—	4	1	no	—
88	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
89	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₁₆	—	292	2	no	—
93	C₂²⋊C₈	—	292	1	no	—
94	C₂²⋊C₈	—	292	1	no	—
95	C₂⁴	#11924037	1	1	yes	characteristic
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
102	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
103	C₂×C₈	—	292	2	no	—
104	C₂×C₈	—	292	2	no	—
105	C₂×C₈	—	292	1	no	—
106	C₂×C₈	—	146	2	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₄²	—	4	1	no	—
111	C₄²	—	4	1	no	—
112	C₁₆	—	584	2	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂²⋊#10596833	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	#14556649	1	1	yes	characteristic
128	C₂×C₄	#14556651	1	1	yes	characteristic
129	C₈	—	292	2	no	—
130	C₈	—	584	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#17989430	1	1	yes	characteristic
138	C₂²	#17989485	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#17989489	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	8	1	no	—
146	C₂	#22624853	1	1	yes	characteristic
147	C₂	#22624857	1	1	yes	characteristic
148	C₂	#22624865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624851

C₂

yes

maximal

#22610002

C₂

yes

maximal, characteristic

C₂²⋊#11924031

—

C₂×C₂²⋊#10596831

C₄

yes

characteristic

C₂×C₂²⋊#10596831

C₂²

yes

characteristic

#17988342

—

C₂×#117×C₇₃

C₄

yes

—

—

D₄

yes

—

C₂²⋊#10596831

—

C₂²⋊#10596831

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

—

#117×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

—

—

—

—

—

C₂⁴×C₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#323

yes

characteristic

C₂²×C₇₃

#350

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#352

yes

—

#7059

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7059

yes

characteristic, complemented, Sylow

C₂×#117

#64408

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

—

C₂²⋊C₁₆

—

146

—

#336

—

146

—

C₂³×C₄

C₂×#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

146

—

#117

—

#117

—

C₂×C₄⋊C₄

yes

C₂×C₄⋊C₄

yes

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₁₆

—

292

—

C₂²⋊C₈

—

292

—

C₂²⋊C₈

—

292

—

C₂⁴

#11924037

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²⋊#64408

yes

characteristic

102

C₂²×C₄

C₂×#10596833

yes

characteristic

103

C₂×C₈

—

292

—

104

C₂×C₈

—

292

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

146

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₁₆

—

584

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂²⋊#10596833

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#14556649

yes

characteristic

128

C₂×C₄

#14556651

yes

characteristic

129

C₈

—

292

—

130

C₈

—

584

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#17989430

yes

characteristic

138

C₂²

#17989485

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#17989489

yes

—

144

C₂

—

145

C₂

—

146

C₂

#22624853

yes

characteristic

147

C₂

#22624857

yes

characteristic

148

C₂

#22624865

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556649
commutator	C₂²×C₄×C₇₃	C₂×C₈
Frattini	C₂³×C₄	C₂×#64408
Fitting	C₂×#117×C₇₃	C₄
socle	C₂²×C₇₃	#323
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂×C₂²⋊#10596831	C₂²
hypercenter	C₂×#117	#64408
hypocenter	C₇₃	#7059
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

#14556649

commutator

C₂²×C₄×C₇₃

C₂×C₈

Frattini

C₂³×C₄

C₂×#64408

Fitting

C₂×#117×C₇₃

C₄

socle

C₂²×C₇₃

#323

prime radical

C₂⁴×C₇₃

M₁₆

cosocle

C₂×C₂²⋊#10596831

C₂²

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
77. C₇₃	#7059	semidirect

H ≅ G/N

type

77. C₇₃

#7059

semidirect

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

subgroup	index
#22624851	2
#22610002	2
#7059	73

subgroup

index

#22624851

#22610002

#7059

Subgroups — #33999994

Conjugacy classes of subgroups

Complements

Maximal subgroups