Group #34011764 — subgroups

Subgroups — #34011764

Order 18688.

220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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	#22626270	C₂	1	2	yes	maximal
2	#22624967	C₂	1	2	yes	maximal, complemented
3	#22624966	C₂	1	2	yes	maximal, complemented
4	#22611524	C₂	1	1	yes	maximal, characteristic
5	#17989643	—	2	2	no	—
6	#17989510	—	2	2	no	—
7	#17989509	C₂²	1	4	yes	—
8	C₂×#14556530	C₄	1	1	yes	characteristic
9	C₂×#14556530	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#17988363	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
14	#346×C₇₃	C₄	1	1	yes	characteristic
15	#14556666	—	4	2	no	—
16	#14556665	—	2	4	no	—
17	#14556664	—	4	2	no	—
18	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
20	C₄²×D₇₃	Q₈	1	1	yes	characteristic
21	#14556530	—	2	2	no	—
22	#14556530	C₈	1	4	yes	—
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	D₄	1	1	yes	characteristic
26	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#11924031	—	4	4	no	—
32	C₂×C₄×D₇₃	—	2	1	no	—
33	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
36	C₈×D₇₃	C₂×C₈	1	2	yes	—
37	C₈×D₇₃	M₁₆	1	2	yes	—
38	C₈×D₇₃	D₈	1	2	yes	—
39	C₈×D₇₃	Q₁₆	1	2	yes	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
42	#11924003	—	2	1	no	—
43	#11924003	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	1	no	—
48	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
49	C₂²×D₇₃	#108	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	—	2	1	no	—
53	C₂×Dic₇₃	#108	1	1	yes	characteristic
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
56	C₄×D₇₃	#107	1	2	yes	—
57	C₈×C₇₃	#107	1	2	yes	—
58	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	#10596831	C₄⋊C₈	1	2	yes	—
61	#10596831	C₂×D₈	1	1	yes	characteristic
62	#10596831	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
64	C₂×D₇₃	—	4	1	no	—
65	C₂×D₇₃	#335	1	2	yes	—
66	C₄×C₇₃	—	4	1	no	—
67	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
69	Dic₇₃	—	2	2	no	—
70	Dic₇₃	#335	1	2	yes	—
71	#12011	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
74	D₇₃	—	2	2	no	—
75	C₂×#364	—	73	2	no	—
76	C₄×M₃₂	—	73	1	no	—
77	#1373	—	73	2	no	—
78	#1373	—	73	2	no	—
79	C₇₃	#12011	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	73	1	no	—
81	C₂×M₃₂	—	73	1	no	—
82	C₂×M₃₂	—	146	2	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	#364	—	73	4	no	—
85	#364	—	146	2	no	—
86	#346	—	73	1	no	—
87	#346	#64408	1	1	yes	characteristic
88	C₄×C₁₆	—	73	1	no	—
89	C₄×C₁₆	—	73	1	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂×C₄²	—	73	1	no	—
93	M₃₂	—	292	2	no	—
94	M₃₂	—	73	4	no	—
95	M₃₂	—	146	2	no	—
96	M₃₂	—	146	4	no	—
97	C₂×C₁₆	—	292	2	no	—
98	C₂×C₁₆	—	73	1	no	—
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
101	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	C₂×#64408	1	1	yes	characteristic
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	146	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
115	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
116	C₄²	—	73	1	no	—
117	C₄²	—	146	1	no	—
118	C₄²	C₄⋊#64408	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	292	4	no	—
121	C₁₆	—	73	2	no	—
122	C₁₆	—	146	1	no	—
123	C₁₆	#11924036	1	2	yes	—
124	C₂³	—	73	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	146	1	no	—
131	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
132	C₈	—	73	2	no	—
133	C₈	—	73	1	no	—
134	C₈	—	73	1	no	—
135	C₈	#14556660	1	2	yes	—
136	C₈	C₂×#11924036	1	1	yes	characteristic
137	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
138	C₂²	—	292	1	no	—
139	C₂²	—	73	2	no	—
140	C₂²	D₇₃⋊#108	1	1	yes	characteristic
141	C₄	—	4	1	no	—
142	C₄	—	73	2	no	—
143	C₄	—	146	2	no	—
144	C₄	#17989588	1	1	yes	characteristic
145	C₄	C₂×#14556660	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	146	2	no	—
148	C₂	#22625476	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626270

C₂

yes

maximal

#22624967

C₂

yes

maximal, complemented

#22624966

C₂

yes

maximal, complemented

#22611524

C₂

yes

maximal, characteristic

#17989643

—

#17989510

—

#17989509

C₂²

yes

—

C₂×#14556530

C₄

yes

characteristic

C₂×#14556530

C₄

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

Q₈

yes

characteristic

#14556530

—

#14556530

C₈

yes

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

#11924003

—

#11924003

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

—

C₂²×D₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₄⋊C₈

yes

—

C₄×D₇₃

#107

yes

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

C₄⋊C₈

yes

—

#10596831

C₂×D₈

yes

characteristic

#10596831

C₂×Q₁₆

yes

characteristic

C₂²×C₇₃

C₂×#108

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#335

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×#107

yes

characteristic

Dic₇₃

—

Dic₇₃

#335

yes

—

#12011

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#335

yes

characteristic

D₇₃

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

C₇₃

#12011

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

146

—

C₂×C₄×C₈

—

#364

—

#364

—

146

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

292

—

M₃₂

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂×#64408

yes

characteristic

101

C₂×C₁₆

C₂×#64408

yes

characteristic

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×#64408

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

146

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊#64408

yes

characteristic

115

C₂×C₈

C₂²×#64408

yes

characteristic

116

C₄²

—

117

C₄²

—

146

—

118

C₄²

C₄⋊#64408

yes

characteristic

119

C₁₆

—

120

C₁₆

—

292

—

121

C₁₆

—

122

C₁₆

—

146

—

123

C₁₆

#11924036

yes

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#14556660

yes

—

136

C₈

C₂×#11924036

yes

characteristic

137

C₈

D₇₃⋊M₁₆

yes

characteristic

138

C₂²

—

292

—

139

C₂²

—

140

C₂²

D₇₃⋊#108

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

146

—

144

C₄

#17989588

yes

characteristic

145

C₄

C₂×#14556660

yes

characteristic

146

C₂

—

147

C₂

—

146

—

148

C₂

#22625476

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#14556660
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂×C₇₃	C₂×#335
prime radical	C₂²×D₇₃	#108
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12011
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

C₂×#14556660

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#346×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #22624967	C₂	semidirect
3. #22624966	C₂	semidirect
79. C₇₃	#12011	semidirect

H ≅ G/N

type

2. #22624967

C₂

semidirect

3. #22624966

C₂

semidirect

79. C₇₃

#12011

semidirect

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

subgroup	index
#22626270	2
#22624967	2
#22624966	2
#22611524	2
#12011	73

subgroup

index

Subgroups — #34011764

Conjugacy classes of subgroups

Complements

Maximal subgroups