Group #34012345 — subgroups

Subgroups — #34012345

Order 18688.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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	#22626154	C₂	1	1	yes	maximal, characteristic
2	#22626142	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	2	yes	—
7	C₂×#14556665	C₂²	1	1	yes	characteristic
8	C₂²×#11924031	C₂²	1	1	yes	characteristic
9	#17989480	C₄	1	4	yes	complemented
10	#17989476	C₄	1	2	yes	complemented
11	#17989473	C₄	1	2	yes	complemented
12	#17988545	C₂²	1	1	yes	characteristic
13	#17988543	C₂²	1	2	yes	—
14	#17988345	C₄	1	1	yes	characteristic
15	#17988343	C₄	1	2	yes	—
16	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
17	#14556665	—	2	2	no	—
18	#14556665	—	2	2	no	—
19	#14556665	C₂×C₄	1	4	yes	—
20	C₂×#11924031	—	2	2	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924031	C₂×C₄	1	2	yes	—
25	C₂×#11924012	C₂×C₄	1	2	yes	—
26	C₂²×#10596831	C₂³	1	1	yes	characteristic
27	#14556550	C₂×C₄	1	1	yes	characteristic
28	#14556502	—	2	2	no	—
29	#14556501	C₂×C₄	1	2	yes	—
30	C₄×#10596831	—	2	2	no	—
31	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#117×C₇₃	C₂×C₄	1	2	yes	—
34	#11924031	—	2	1	no	—
35	#11924031	—	2	1	no	—
36	#11924031	—	2	4	no	—
37	#11924031	—	2	2	no	—
38	#11924012	—	2	2	no	—
39	#11924012	—	2	2	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
43	C₂×#10596831	#41	1	2	yes	—
44	C₂×#10596831	#41	1	2	yes	—
45	C₄⋊Dic₇₃	—	2	1	no	—
46	C₂²⋊Dic₇₃	—	2	1	no	—
47	C₄×Dic₇₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	C₄²	1	2	yes	—
51	C₄⋊C₄×C₇₃	—	2	2	no	—
52	C₂²⋊C₄×C₇₃	—	2	2	no	—
53	C₄²×C₇₃	—	2	2	no	—
54	C₂³×C₇₃	#131	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#117	1	2	yes	—
58	C₂×C₄×C₇₃	#131	1	2	yes	—
59	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
60	C₂×Dic₇₃	—	2	2	no	—
61	C₈×C₇₃	—	2	2	no	—
62	#10596831	—	4	2	no	—
63	#10596831	#117	1	1	yes	characteristic
64	#10596831	#117	1	1	yes	characteristic
65	#10596831	#117	1	2	yes	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	#405	1	1	yes	characteristic
68	C₂²×C₇₃	#433	1	2	yes	—
69	C₄×C₇₃	—	8	2	no	—
70	C₄×C₇₃	#376	1	1	yes	characteristic
71	C₄×C₇₃	#405	1	1	yes	characteristic
72	C₄×C₇₃	#433	1	2	yes	—
73	Dic₇₃	—	4	2	no	—
74	#12597	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	—	2	2	no	—
77	C₂×C₇₃	#1746	1	1	yes	characteristic
78	#2110	—	73	1	no	—
79	#2108	—	73	1	no	—
80	#1747	—	73	1	no	—
81	#1379	—	73	2	no	—
82	#1378	—	73	2	no	—
83	C₇₃	#12597	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₃₂	—	73	1	no	—
85	C₂×M₃₂	—	73	2	no	—
86	C₂²×C₁₆	—	73	1	no	—
87	#405	—	73	1	no	—
88	#405	—	73	2	no	—
89	#347	—	73	4	no	—
90	#346	—	73	2	no	—
91	C₄×C₁₆	—	73	2	no	—
92	#338	—	73	1	no	—
93	H(3,ℤ/4)	—	73	2	no	—
94	H(3,ℤ/4)	#64408	1	1	yes	characteristic
95	C₂×M₁₆	—	73	2	no	—
96	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
97	C₂²×C₈	—	73	1	no	—
98	#117	—	73	1	no	—
99	#117	C₂×#64408	1	2	yes	—
100	M₃₂	—	73	4	no	—
101	M₃₂	—	146	2	no	—
102	M₃₂	—	146	2	no	—
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	—	73	2	no	—
105	C₂×C₁₆	—	73	1	no	—
106	C₂×C₁₆	—	146	2	no	—
107	C₂×C₁₆	—	146	2	no	—
108	#97	—	73	2	no	—
109	#97	—	146	2	no	—
110	C₄×C₈	—	73	2	no	—
111	C₄×C₈	—	146	2	no	—
112	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	146	2	no	—
115	M₁₆	—	146	2	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	73	2	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	C₄×#64408	1	2	yes	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	146	1	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	73	2	no	—
128	C₁₆	—	146	1	no	—
129	C₁₆	—	146	1	no	—
130	C₁₆	—	146	2	no	—
131	C₁₆	—	146	4	no	—
132	C₂³	#14556678	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
136	C₂×C₄	#14556670	1	2	yes	—
137	C₂×C₄	#14556678	1	2	yes	—
138	C₂×C₄	#14556679	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	292	2	no	—
141	C₈	—	73	1	no	—
142	C₈	—	73	1	no	—
143	C₈	—	73	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	#17989552	1	1	yes	characteristic
146	C₂²	#17989611	1	2	yes	—
147	C₄	—	8	2	no	—
148	C₄	—	292	2	no	—
149	C₄	#17989523	1	1	yes	characteristic
150	C₄	#17989551	1	1	yes	characteristic
151	C₄	#17989612	1	2	yes	—
152	C₂	—	2	2	no	—
153	C₂	—	2	1	no	—
154	C₂	#22625650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626154

C₂

yes

maximal, characteristic

#22626142

C₂

yes

maximal, characteristic

#22624985

C₂

yes

maximal

#22624981

C₂

yes

maximal

#22609685

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

C₂×#14556665

C₂²

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

#17989480

C₄

yes

complemented

#17989476

C₄

yes

complemented

#17989473

C₄

yes

complemented

#17988545

C₂²

yes

characteristic

#17988543

C₂²

yes

—

#17988345

C₄

yes

characteristic

#17988343

C₄

yes

—

H(3,ℤ/4)×C₇₃

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924031

—

C₂×#11924031

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924012

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

#14556550

C₂×C₄

yes

characteristic

#14556502

—

#14556501

C₂×C₄

yes

—

C₄×#10596831

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

#41

yes

—

C₂×#10596831

#41

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₄²

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#131

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

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

#405

yes

characteristic

C₂²×C₇₃

#433

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12597

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

—

—

—

C₇₃

#12597

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

C₄×C₁₆

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

C₂×#64408

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#64408

yes

—

100

M₃₂

—

101

M₃₂

—

146

—

102

M₃₂

—

146

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

146

—

107

C₂×C₁₆

—

146

—

108

#97

—

109

#97

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

146

—

112

C₂²×C₄

C₂²×#64408

yes

characteristic

113

M₁₆

—

114

M₁₆

—

146

—

115

M₁₆

—

146

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

C₄×#64408

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

146

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

146

—

126

C₄²

—

127

C₄²

—

128

C₁₆

—

146

—

129

C₁₆

—

146

—

130

C₁₆

—

146

—

131

C₁₆

—

146

—

132

C₂³

#14556678

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

C₂×C₄×#64408

yes

characteristic

136

C₂×C₄

#14556670

yes

—

137

C₂×C₄

#14556678

yes

—

138

C₂×C₄

#14556679

yes

characteristic

139

C₈

—

140

C₈

—

292

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

#17989552

yes

characteristic

146

C₂²

#17989611

yes

—

147

C₄

—

148

C₄

—

292

—

149

C₄

#17989523

yes

characteristic

150

C₄

#17989551

yes

characteristic

151

C₄

#17989612

yes

—

152

C₂

—

153

C₂

—

154

C₂

#22625650

yes

characteristic

	subgroup	G/H
center	C₄	#17989523
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	H(3,ℤ/4)×C₇₃	C₄
socle	C₂×C₇₃	#1746
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	H(3,ℤ/4)	#64408
hypocenter	C₇₃	#12597
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17989523

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×#64408

Fitting

H(3,ℤ/4)×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #17989480	C₄	semidirect
10. #17989476	C₄	semidirect
11. #17989473	C₄	semidirect
83. C₇₃	#12597	semidirect

H ≅ G/N

type

9. #17989480

C₄

semidirect

10. #17989476

C₄

semidirect

11. #17989473

C₄

semidirect

83. C₇₃

#12597

semidirect

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

subgroup	index
#22626154	2
#22626142	2
#22624985	2
#22624981	2
#22609685	2
#12597	73

subgroup

index

Subgroups — #34012345

Conjugacy classes of subgroups

Complements

Maximal subgroups