Group #34011936 — subgroups

Subgroups — #34011936

Order 18688.

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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	#22626310	C₂	1	2	yes	maximal, complemented
2	#22624953	C₂	1	1	yes	maximal, characteristic, complemented
3	#22624882	C₂	1	2	yes	maximal, complemented
4	#22624844	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608725	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	2	yes	—
8	#17989482	—	2	2	no	—
9	C₄×#11924031	C₂²	1	1	yes	characteristic
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	C₄×D₅₈₄	C₄	1	1	yes	characteristic
12	#17988652	C₂²	1	2	yes	—
13	#17988314	C₄	1	2	yes	—
14	C₈×#10596831	C₂²	1	1	yes	characteristic
15	#335×C₇₃	C₄	1	1	yes	characteristic
16	#14556664	—	4	4	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	1	no	—
19	C₂×#11924031	D₄	1	2	yes	—
20	C₂×D₅₈₄	C₈	1	1	yes	characteristic
21	C₂×C₈×D₇₃	—	2	2	no	—
22	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
23	#14556525	C₈	1	1	yes	characteristic
24	#14556513	C₈	1	2	yes	—
25	#14556503	—	2	2	no	—
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄×#10596831	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924031	—	2	2	no	—
31	#11924031	—	4	1	no	—
32	#11924031	—	4	2	no	—
33	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
34	C₂×C₄×D₇₃	—	2	2	no	—
35	D₅₈₄	C₁₆	1	4	yes	complemented
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
38	C₂×#10596831	#41	1	1	yes	characteristic
39	C₈×D₇₃	—	4	2	no	—
40	C₈×D₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
42	#11924005	—	2	2	no	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	2	no	—
52	D₂₉₂	—	2	2	no	—
53	D₂₉₂	C₂×C₁₆	1	4	yes	complemented
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	—	4	2	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	4	2	no	—
58	C₈×C₇₃	C₂×C₁₆	1	2	yes	complemented
59	#10596831	—	2	1	no	—
60	#10596831	—	4	1	no	—
61	#10596831	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
63	C₂×D₇₃	—	2	4	no	—
64	C₂×D₇₃	—	4	2	no	—
65	C₂×D₇₃	—	4	2	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	#442	1	1	yes	characteristic
68	C₄×C₇₃	#444	1	1	yes	characteristic
69	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₇₃	#504	1	1	yes	characteristic
71	Dic₇₃	—	4	2	no	—
72	#12062	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1580	1	1	yes	characteristic
74	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
75	C₂×C₇₃	#2171	1	1	yes	characteristic
76	D₇₃	—	4	4	no	—
77	D₄×C₁₆	—	73	2	no	—
78	C₈×D₈	—	73	1	no	—
79	#1367	—	73	1	no	—
80	D₄⋊C₁₆	—	73	2	no	—
81	#1312	—	73	1	no	—
82	C₇₃	#12062	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	146	2	no	—
84	C₄×D₈	—	73	1	no	—
85	C₈×D₄	—	73	2	no	—
86	C₄⋊C₁₆	—	73	2	no	—
87	C₂²⋊C₁₆	—	146	2	no	—
88	C₄×C₁₆	—	73	1	no	—
89	C₄×C₁₆	—	73	1	no	—
90	#335	#64408	1	1	yes	characteristic
91	D₄⋊C₈	—	73	2	no	—
92	C₈²	—	73	1	no	—
93	C₂×D₈	—	73	1	no	—
94	C₂²×C₈	—	146	2	no	—
95	C₄×D₄	—	73	2	no	—
96	C₂×C₁₆	—	292	4	no	—
97	C₂×C₁₆	—	73	2	no	—
98	C₂×C₁₆	—	146	1	no	—
99	C₂×C₁₆	—	146	2	no	—
100	#107	—	73	1	no	—
101	C₄⋊C₈	C₂×#64408	1	2	yes	—
102	#102	—	73	2	no	—
103	C₂²⋊C₈	—	146	2	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	C₂×#64408	1	1	yes	characteristic
107	C₂×D₄	—	73	2	no	—
108	C₂²×C₄	—	146	2	no	—
109	D₈	—	73	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	292	2	no	—
112	C₂×C₈	—	292	2	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₂×C₈	#11924036	1	1	yes	characteristic
117	C₂×C₈	#11924036	1	1	yes	characteristic
118	C₄⋊C₄	—	73	2	no	—
119	C₂²⋊C₄	—	146	2	no	—
120	C₄²	C₂²×#64408	1	1	yes	characteristic
121	C₁₆	—	292	2	no	—
122	C₁₆	—	292	1	no	—
123	C₁₆	—	146	2	no	—
124	C₂³	—	146	2	no	—
125	D₄	—	73	4	no	—
126	D₄	—	146	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	292	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	D₄×#64408	1	1	yes	characteristic
131	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
132	C₂×C₄	#14556679	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	4	2	no	—
135	C₈	—	292	1	no	—
136	C₈	—	73	2	no	—
137	C₈	—	146	1	no	—
138	C₈	#14556664	1	2	yes	complemented
139	C₂²	—	292	2	no	—
140	C₂²	—	292	2	no	—
141	C₂²	—	146	4	no	—
142	C₂²	#17989615	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	292	2	no	—
145	C₄	#17989628	1	1	yes	characteristic
146	C₄	#17989633	1	1	yes	characteristic
147	C₄	C₂×#14556664	1	1	yes	characteristic
148	C₄	#17989646	1	1	yes	characteristic
149	C₂	—	292	4	no	—
150	C₂	#22625529	1	1	yes	characteristic
151	C₂	#22626310	1	1	yes	characteristic
152	C₂	#22626323	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626310

C₂

yes

maximal, complemented

#22624953

C₂

yes

maximal, characteristic, complemented

#22624882

C₂

yes

maximal, complemented

#22624844

C₂

yes

maximal, characteristic, complemented

#22608725

C₂

yes

maximal, characteristic

C₂×#14556664

—

C₄⋊#11924031

C₂²

yes

—

#17989482

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

C₄×D₅₈₄

C₄

yes

characteristic

#17988652

C₂²

yes

—

#17988314

C₄

yes

—

C₈×#10596831

C₂²

yes

characteristic

#335×C₇₃

C₄

yes

—

—

—

D₄

yes

—

C₂×D₅₈₄

C₈

yes

characteristic

C₂×C₈×D₇₃

—

C₄×D₂₉₂

C₂×C₄

yes

—

#14556525

C₈

yes

characteristic

#14556513

C₈

yes

—

#14556503

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

C₂×D₂₉₂

C₂×C₈

yes

—

C₂×C₄×D₇₃

—

D₅₈₄

C₁₆

yes

complemented

C₂×#10596831

—

C₂×#10596831

C₂×D₄

yes

yes

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924005

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

C₂×C₁₆

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

—

—

yes

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

#2171

yes

characteristic

D₇₃

—

D₄×C₁₆

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₇₃

#12062

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#64408

yes

characteristic

D₄⋊C₈

—

C₈²

—

C₂×D₈

—

C₂²×C₈

—

146

—

C₄×D₄

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

100

#107

—

101

C₄⋊C₈

C₂×#64408

yes

—

102

#102

—

103

C₂²⋊C₈

—

146

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×#64408

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

146

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

292

—

112

C₂×C₈

—

292

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₂×C₈

#11924036

yes

characteristic

117

C₂×C₈

#11924036

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

146

—

120

C₄²

C₂²×#64408

yes

characteristic

121

C₁₆

—

292

—

122

C₁₆

—

292

—

123

C₁₆

—

146

—

124

C₂³

—

146

—

125

D₄

—

126

D₄

—

146

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

146

—

130

C₂×C₄

D₄×#64408

yes

characteristic

131

C₂×C₄

C₂×#11924036

yes

characteristic

132

C₂×C₄

#14556679

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

292

—

136

C₈

—

137

C₈

—

146

—

138

C₈

#14556664

yes

complemented

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

146

—

142

C₂²

#17989615

yes

characteristic

143

C₄

—

144

C₄

—

292

—

145

C₄

#17989628

yes

characteristic

146

C₄

#17989633

yes

characteristic

147

C₄

C₂×#14556664

yes

characteristic

148

C₄

#17989646

yes

characteristic

149

C₂

—

292

—

150

C₂

#22625529

yes

characteristic

151

C₂

#22626310

yes

characteristic

152

C₂

#22626323

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂×D₅₈₄	C₈
cosocle	C₄×#10596831	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	#12062
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

#335×C₇₃

C₄

socle

C₂²×C₇₃

C₈×D₄

prime radical

C₂×D₅₈₄

C₈

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #22626310	C₂	semidirect
2. #22624953	C₂	semidirect
3. #22624882	C₂	semidirect
4. #22624844	C₂	semidirect
35. D₅₈₄	C₁₆	semidirect
53. D₂₉₂	C₂×C₁₆	semidirect
58. C₈×C₇₃	C₂×C₁₆	semidirect
82. C₇₃	#12062	semidirect
138. C₈	#14556664	semidirect

H ≅ G/N

type

1. #22626310

C₂

semidirect

2. #22624953

C₂

semidirect

3. #22624882

C₂

semidirect

4. #22624844

C₂

semidirect

35. D₅₈₄

C₁₆

semidirect

53. D₂₉₂

C₂×C₁₆

semidirect

58. C₈×C₇₃

C₂×C₁₆

semidirect

82. C₇₃

#12062

semidirect

138. C₈

#14556664

semidirect

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

subgroup	index
#22626310	2
#22624953	2
#22624882	2
#22624844	2
#22608725	2
#12062	73

subgroup

index

Subgroups — #34011936

Conjugacy classes of subgroups

Complements

Maximal subgroups