Group #34013289 — subgroups

Subgroups — #34013289

Order 18688.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22626272	C₂	1	1	yes	maximal, characteristic
2	#22626268	C₂	1	1	yes	maximal, characteristic
3	#22624985	C₂	1	2	yes	maximal
4	#22624981	C₂	1	2	yes	maximal
5	#22609685	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	1	yes	characteristic
7	C₂×#14556665	C₂²	1	2	yes	—
8	C₂²×#11924031	C₂²	1	1	yes	characteristic
9	#17989509	—	2	2	no	—
10	#17989508	—	2	2	no	—
11	#17988545	C₂²	1	1	yes	characteristic
12	#17988543	C₂²	1	2	yes	—
13	#17988345	C₄	1	1	yes	characteristic
14	#17988343	C₄	1	2	yes	—
15	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
16	#14556665	—	2	2	no	—
17	#14556665	—	2	2	no	—
18	#14556665	—	2	2	no	—
19	C₂×#11924031	—	2	1	no	—
20	C₂×#11924031	—	2	1	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	—	2	2	no	—
23	C₂×#11924012	C₂×C₄	1	2	yes	—
24	C₂²×#10596831	C₂³	1	1	yes	characteristic
25	#14556550	C₂×C₄	1	1	yes	characteristic
26	#14556502	—	2	2	no	—
27	#14556501	Q₈	1	2	yes	—
28	C₄×#10596831	—	2	2	no	—
29	C₈×Dic₇₃	D₄	1	2	yes	—
30	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#117×C₇₃	C₂×C₄	1	2	yes	—
32	#11924031	—	4	1	no	—
33	#11924031	—	4	1	no	—
34	#11924031	—	4	2	no	—
35	#11924012	—	2	2	no	—
36	#11924012	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
39	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
40	C₂×#10596831	#41	1	2	yes	—
41	C₂×#10596831	#41	1	2	yes	—
42	C₄⋊Dic₇₃	—	2	1	no	—
43	C₂²⋊Dic₇₃	—	2	1	no	—
44	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₂²⋊C₄×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₂³×C₇₃	#131	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	4	2	no	—
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#117	1	2	yes	—
55	C₂×C₄×C₇₃	#131	1	2	yes	—
56	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	2	no	—
58	C₈×C₇₃	—	2	2	no	—
59	#10596831	—	4	2	no	—
60	#10596831	#124	1	1	yes	characteristic
61	#10596831	#124	1	1	yes	characteristic
62	#10596831	#125	1	2	yes	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	#424	1	1	yes	characteristic
65	C₂²×C₇₃	#433	1	2	yes	—
66	C₄×C₇₃	—	8	2	no	—
67	C₄×C₇₃	#382	1	1	yes	characteristic
68	C₄×C₇₃	#424	1	1	yes	characteristic
69	C₄×C₇₃	#433	1	2	yes	—
70	Dic₇₃	—	4	2	no	—
71	#12885	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	#1832	1	1	yes	characteristic
75	#2153	—	73	1	no	—
76	#2151	—	73	1	no	—
77	#1747	—	73	1	no	—
78	#1379	—	73	2	no	—
79	#1378	—	73	2	no	—
80	C₇₃	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	73	1	no	—
82	C₂×M₃₂	—	73	2	no	—
83	C₂²×C₁₆	—	73	1	no	—
84	#405	—	73	1	no	—
85	#405	—	73	2	no	—
86	#364	—	146	2	no	—
87	C₄⋊C₁₆	—	146	2	no	—
88	#338	—	73	1	no	—
89	H(3,ℤ/4)	—	73	2	no	—
90	H(3,ℤ/4)	#64408	1	1	yes	characteristic
91	C₂×M₁₆	—	73	2	no	—
92	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
93	C₂²×C₈	—	73	1	no	—
94	#117	—	73	1	no	—
95	#117	C₂×#64408	1	2	yes	—
96	M₃₂	—	146	2	no	—
97	M₃₂	—	146	2	no	—
98	M₃₂	—	146	2	no	—
99	C₂×C₁₆	—	146	1	no	—
100	C₂×C₁₆	—	146	1	no	—
101	C₂×C₁₆	—	146	2	no	—
102	C₂×C₁₆	—	146	2	no	—
103	#97	—	73	2	no	—
104	#97	—	146	2	no	—
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	146	2	no	—
107	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	146	2	no	—
110	M₁₆	—	146	2	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	C₄⋊#64408	1	2	yes	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	146	1	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	146	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	73	2	no	—
123	C₁₆	—	292	1	no	—
124	C₁₆	—	292	1	no	—
125	C₁₆	—	292	2	no	—
126	C₂³	#14556678	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
130	C₂×C₄	#14556670	1	2	yes	—
131	C₂×C₄	#14556678	1	2	yes	—
132	C₂×C₄	#14556679	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	292	2	no	—
135	C₈	—	73	1	no	—
136	C₈	—	73	1	no	—
137	C₈	—	73	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	#17989594	1	1	yes	characteristic
140	C₂²	#17989611	1	2	yes	—
141	C₄	—	8	2	no	—
142	C₄	—	292	2	no	—
143	C₄	#17989531	1	1	yes	characteristic
144	C₄	#17989593	1	1	yes	characteristic
145	C₄	#17989612	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	—	2	1	no	—
148	C₂	#22625849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626272

C₂

yes

maximal, characteristic

#22626268

C₂

yes

maximal, characteristic

#22624985

C₂

yes

maximal

#22624981

C₂

yes

maximal

#22609685

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

characteristic

C₂×#14556665

C₂²

yes

—

C₂²×#11924031

C₂²

yes

—

—

C₂²

yes

characteristic

#17988543

C₂²

yes

—

#17988345

C₄

yes

characteristic

#17988343

C₄

yes

—

H(3,ℤ/4)×C₇₃

C₄

yes

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

#14556550

C₂×C₄

yes

characteristic

#14556502

—

#14556501

Q₈

yes

—

C₄×#10596831

—

C₈×Dic₇₃

D₄

yes

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

C₂×#10596831

#41

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#131

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

—

—

yes

yes

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#424

yes

characteristic

C₂²×C₇₃

#433

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12885

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

—

—

—

C₇₃

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

146

—

C₄⋊C₁₆

—

146

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

C₂×#64408

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#64408

yes

—

M₃₂

—

146

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

146

—

100

C₂×C₁₆

—

146

—

101

C₂×C₁₆

—

146

—

102

C₂×C₁₆

—

146

—

103

#97

—

104

#97

—

146

—

105

C₄×C₈

—

106

C₄×C₈

—

146

—

107

C₂²×C₄

C₂²×#64408

yes

characteristic

108

M₁₆

—

109

M₁₆

—

146

—

110

M₁₆

—

146

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₂×C₈

C₄⋊#64408

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

146

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

146

—

121

C₄²

—

122

C₄²

—

123

C₁₆

—

292

—

124

C₁₆

—

292

—

125

C₁₆

—

292

—

126

C₂³

#14556678

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

130

C₂×C₄

#14556670

yes

—

131

C₂×C₄

#14556678

yes

—

132

C₂×C₄

#14556679

yes

characteristic

133

C₈

—

134

C₈

—

292

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

#17989594

yes

characteristic

140

C₂²

#17989611

yes

—

141

C₄

—

142

C₄

—

292

—

143

C₄

#17989531

yes

characteristic

144

C₄

#17989593

yes

characteristic

145

C₄

#17989612

yes

—

146

C₂

—

147

C₂

—

148

C₂

#22625849

yes

characteristic

	subgroup	G/H
center	C₄	#17989531
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	H(3,ℤ/4)×C₇₃	C₄
socle	C₂×C₇₃	#1832
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	H(3,ℤ/4)	#64408
hypocenter	C₇₃	#12885
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17989531

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

H(3,ℤ/4)×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
80. C₇₃	#12885	semidirect

H ≅ G/N

type

80. C₇₃

#12885

semidirect

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

subgroup	index
#22626272	2
#22626268	2
#22624985	2
#22624981	2
#22609685	2
#12885	73

subgroup

index

Subgroups — #34013289

Conjugacy classes of subgroups

Complements

Maximal subgroups