Group #34013324 — subgroups

Subgroups — #34013324

Order 18688.

278 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 104 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	#22626274	C₂	1	2	yes	maximal
2	#22626272	C₂	1	1	yes	maximal, characteristic
3	#22626268	C₂	1	1	yes	maximal, characteristic
4	#22626266	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	#17989510	—	2	4	no	—
10	#17989509	—	2	2	no	—
11	#17989508	—	2	2	no	—
12	C₄⋊#11924031	—	2	4	no	—
13	#17988545	C₂²	1	1	yes	characteristic
14	#17988543	C₂²	1	2	yes	—
15	#17988345	C₄	1	1	yes	characteristic
16	#17988343	C₄	1	2	yes	—
17	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
18	#14556665	—	2	2	no	—
19	#14556665	—	2	2	no	—
20	#14556665	—	2	2	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	—	2	1	no	—
23	C₂×#11924031	—	2	2	no	—
24	C₂×#11924031	—	2	1	no	—
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	Q₈	1	4	yes	—
29	#14556501	Q₈	1	2	yes	—
30	C₄×#10596831	D₄	1	4	yes	—
31	C₈×Dic₇₃	D₄	1	2	yes	—
32	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#117×C₇₃	C₂×C₄	1	2	yes	—
34	#11924031	—	4	1	no	—
35	#11924031	—	4	2	no	—
36	#11924031	—	4	1	no	—
37	#11924012	—	2	2	no	—
38	#11924012	—	2	2	no	—
39	C₂×#10596831	C₄⋊C₄	1	4	yes	—
40	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
41	C₂×#10596831	C₂×D₄	1	2	yes	—
42	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
43	C₂×#10596831	C₂×Q₈	1	2	yes	—
44	C₄⋊Dic₇₃	—	2	1	no	—
45	C₂²⋊Dic₇₃	—	2	1	no	—
46	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
50	C₄⋊C₄×C₇₃	—	2	2	no	—
51	C₂²⋊C₄×C₇₃	—	2	2	no	—
52	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
53	C₂³×C₇₃	#131	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	2	4	no	—
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#131	1	2	yes	—
58	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	2	no	—
60	C₈×C₇₃	—	2	2	no	—
61	#10596831	—	2	4	no	—
62	#10596831	#127	1	1	yes	characteristic
63	#10596831	C₄⋊Q₈	1	1	yes	characteristic
64	#10596831	C₄⋊Q₈	1	2	yes	—
65	C₂²×C₇₃	—	4	1	no	—
66	C₂²×C₇₃	#424	1	2	yes	—
67	C₂²×C₇₃	#424	1	1	yes	characteristic
68	C₄×C₇₃	—	4	4	no	—
69	C₄×C₇₃	#384	1	1	yes	characteristic
70	C₄×C₇₃	#424	1	1	yes	characteristic
71	C₄×C₇₃	#424	1	2	yes	—
72	Dic₇₃	—	4	2	no	—
73	#12902	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	#1843	1	1	yes	characteristic
77	#2153	—	73	1	no	—
78	#2153	—	73	2	no	—
79	#2151	—	73	2	no	—
80	#2151	—	73	1	no	—
81	#1747	—	73	1	no	—
82	C₇₃	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	73	2	no	—
84	C₂×M₃₂	—	73	1	no	—
85	C₂²×C₁₆	—	73	1	no	—
86	#405	—	73	1	no	—
87	#405	—	73	2	no	—
88	#364	—	146	2	no	—
89	#364	—	146	4	no	—
90	C₄⋊C₁₆	—	146	4	no	—
91	C₄⋊C₁₆	—	146	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	—	146	2	no	—
101	M₃₂	—	146	2	no	—
102	M₃₂	—	146	2	no	—
103	C₂×C₁₆	—	146	2	no	—
104	C₂×C₁₆	—	146	1	no	—
105	C₂×C₁₆	—	146	2	no	—
106	C₂×C₁₆	—	146	1	no	—
107	#97	—	73	2	no	—
108	#97	—	73	4	no	—
109	C₄×C₈	—	73	2	no	—
110	C₄×C₈	—	73	4	no	—
111	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	146	2	no	—
114	M₁₆	—	146	2	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	73	4	no	—
120	C₂×C₈	C₄⋊#64408	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	146	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	146	1	no	—
125	C₄²	—	73	2	no	—
126	C₄²	C₄⋊#64408	1	4	yes	—
127	C₁₆	—	292	1	no	—
128	C₁₆	—	292	2	no	—
129	C₁₆	—	292	1	no	—
130	C₂³	#14556678	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
135	C₂×C₄	#14556678	1	2	yes	—
136	C₂×C₄	#14556679	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	73	1	no	—
139	C₈	—	73	1	no	—
140	C₈	—	73	2	no	—
141	C₈	—	146	4	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#17989593	1	2	yes	—
144	C₂²	#17989594	1	1	yes	characteristic
145	C₄	—	4	4	no	—
146	C₄	—	292	2	no	—
147	C₄	#17989533	1	1	yes	characteristic
148	C₄	#17989593	1	1	yes	characteristic
149	C₄	#17989594	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	—	2	1	no	—
152	C₂	#22625865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626274

C₂

yes

maximal

#22626272

C₂

yes

maximal, characteristic

#22626268

C₂

yes

maximal, characteristic

#22626266

C₂

yes

maximal

#22609685

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

C₂×#14556665

C₂²

yes

characteristic

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

Q₈

yes

—

#14556501

Q₈

yes

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

C₂×Q₈

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

C₄⋊C₄

yes

—

C₂³×C₇₃

#131

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

—

—

yes

C₄⋊Q₈

yes

characteristic

#10596831

C₄⋊Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#424

yes

—

C₂²×C₇₃

#424

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12902

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

—

—

—

C₇₃

#12902

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

146

—

#364

—

146

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

C₂×#64408

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#64408

yes

—

100

M₃₂

—

146

—

101

M₃₂

—

146

—

102

M₃₂

—

146

—

103

C₂×C₁₆

—

146

—

104

C₂×C₁₆

—

146

—

105

C₂×C₁₆

—

146

—

106

C₂×C₁₆

—

146

—

107

#97

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₂²×C₄

C₂²×#64408

yes

characteristic

112

M₁₆

—

113

M₁₆

—

146

—

114

M₁₆

—

146

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄⋊#64408

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

146

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

146

—

125

C₄²

—

126

C₄²

C₄⋊#64408

yes

—

127

C₁₆

—

292

—

128

C₁₆

—

292

—

129

C₁₆

—

292

—

130

C₂³

#14556678

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

134

C₂×C₄

C₂×C₄⋊#64408

yes

—

135

C₂×C₄

#14556678

yes

—

136

C₂×C₄

#14556679

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

146

—

142

C₂²

—

143

C₂²

#17989593

yes

—

144

C₂²

#17989594

yes

characteristic

145

C₄

—

146

C₄

—

292

—

147

C₄

#17989533

yes

characteristic

148

C₄

#17989593

yes

characteristic

149

C₄

#17989594

yes

—

150

C₂

—

151

C₂

—

152

C₂

#22625865

yes

characteristic

	subgroup	G/H
center	C₄	#17989533
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	H(3,ℤ/4)×C₇₃	C₄
socle	C₂×C₇₃	#1843
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	H(3,ℤ/4)	#64408
hypocenter	C₇₃	#12902
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17989533

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
82. C₇₃	#12902	semidirect

H ≅ G/N

type

82. C₇₃

#12902

semidirect

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

subgroup	index
#22626274	2
#22626272	2
#22626268	2
#22626266	2
#22609685	2
#12902	73

subgroup

index

Subgroups — #34013324

Conjugacy classes of subgroups

Complements

Maximal subgroups