Group #34014521 — subgroups

Subgroups — #34014521

Order 18688.

214 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×#17989518	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14556641	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625045	C₂	1	2	yes	maximal, complemented
4	C₄×#14556641	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611750	C₂	1	1	yes	maximal, characteristic
6	#17989518	—	2	4	no	—
7	C₂×#14556641	C₂²	1	1	yes	characteristic
8	C₂×#14556641	C₂²	1	2	yes	—
9	C₂×#14556641	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
11	C₈×D₂₉₂	C₄	1	1	yes	characteristic
12	#17988366	C₄	1	2	yes	—
13	C₄×#11924003	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic
15	#14556641	—	2	2	no	—
16	#14556641	—	2	1	no	—
17	#14556641	D₄	1	2	yes	complemented
18	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
19	C₄×D₂₉₂	C₈	1	1	yes	characteristic
20	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	C₂³	1	1	yes	characteristic
22	C₁₆×D₇₃	—	2	2	no	—
23	C₁₆×D₇₃	—	2	2	no	—
24	C₄⋊#10596831	C₈	1	1	yes	characteristic
25	#14556503	C₈	1	2	yes	—
26	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
30	C₂×#10596831	C₂×C₈	1	2	yes	—
31	C₈×D₇₃	—	2	2	no	—
32	C₈×D₇₃	—	2	2	no	—
33	C₄⋊Dic₇₃	C₁₆	1	1	yes	characteristic
34	#11924005	C₁₆	1	2	yes	—
35	#11924003	—	2	1	no	—
36	#11924003	C₂×D₄	1	1	yes	characteristic
37	#11924003	#41	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	2	no	—
42	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
47	D₂₉₂	C₃₂	1	4	yes	complemented
48	C₄×D₇₃	—	2	2	no	—
49	C₄×D₇₃	—	2	2	no	—
50	C₈×C₇₃	—	2	1	no	—
51	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
52	C₈×C₇₃	#131	1	1	yes	characteristic
53	#10596831	—	2	2	no	—
54	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×D₇₃	—	2	2	no	—
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	C₂×C₃₂	1	4	yes	complemented
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	C₂×C₃₂	1	2	yes	complemented
60	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	#504	1	1	yes	characteristic
62	Dic₇₃	—	2	2	no	—
63	D₄×C₃₂	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
65	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
66	C₂×C₇₃	#2258	1	1	yes	characteristic
67	D₇₃	—	2	4	no	—
68	C₂²×C₃₂	—	73	2	no	—
69	D₄×C₁₆	—	73	1	no	—
70	C₄⋊C₃₂	—	73	1	no	—
71	C₂²⋊C₃₂	—	73	2	no	—
72	C₄×C₃₂	—	73	1	no	—
73	C₇₃	D₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	73	2	no	—
75	C₈×D₄	—	73	1	no	—
76	C₂×C₃₂	—	73	1	no	—
77	C₂×C₃₂	—	73	1	no	—
78	C₂×C₃₂	—	73	2	no	—
79	C₂×C₃₂	—	146	4	no	—
80	C₄⋊C₁₆	#64408	1	1	yes	characteristic
81	C₂²⋊C₁₆	—	73	2	no	—
82	C₄×C₁₆	—	73	1	no	—
83	C₂²×C₈	—	73	2	no	—
84	C₄×D₄	—	73	1	no	—
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	146	2	no	—
88	C₂×C₁₆	—	146	2	no	—
89	C₂×C₁₆	C₂×#64408	1	2	yes	—
90	C₄⋊C₈	—	73	1	no	—
91	C₂²⋊C₈	—	73	2	no	—
92	C₄×C₈	C₂×#64408	1	1	yes	characteristic
93	C₃₂	—	73	2	no	—
94	C₃₂	—	146	2	no	—
95	C₃₂	—	146	1	no	—
96	C₂×D₄	—	73	1	no	—
97	C₂²×C₄	—	73	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	—	146	2	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	#11924036	1	1	yes	characteristic
102	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
103	C₄⋊C₄	—	73	1	no	—
104	C₂²⋊C₄	—	73	2	no	—
105	C₄²	#11924036	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	73	1	no	—
108	C₁₆	—	73	1	no	—
109	C₁₆	—	146	1	no	—
110	C₂³	—	73	2	no	—
111	D₄	—	73	4	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	146	2	no	—
114	C₂×C₄	—	146	2	no	—
115	C₂×C₄	#14556664	1	1	yes	characteristic
116	C₂×C₄	#14556664	1	1	yes	characteristic
117	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	146	2	no	—
120	C₈	D₄×#64408	1	1	yes	characteristic
121	C₈	#14556679	1	1	yes	characteristic
122	C₂²	—	73	4	no	—
123	C₂²	—	146	2	no	—
124	C₂²	—	146	2	no	—
125	C₂²	C₂×#14556664	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	146	2	no	—
128	C₄	#17989518	1	2	yes	complemented
129	C₄	#17989615	1	1	yes	characteristic
130	C₄	#17989646	1	1	yes	characteristic
131	C₂	—	146	4	no	—
132	C₂	#22626310	1	1	yes	characteristic
133	C₂	C₂×#17989518	1	1	yes	characteristic
134	C₂	#22626355	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989518

C₂

yes

maximal, complemented

C₄⋊#14556641

C₂

yes

maximal, characteristic, complemented

#22625045

C₂

yes

maximal, complemented

C₄×#14556641

C₂

yes

maximal, characteristic, complemented

#22611750

C₂

yes

maximal, characteristic

#17989518

—

C₂×#14556641

C₂²

yes

characteristic

C₂×#14556641

C₂²

yes

—

C₂×#14556641

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

—

C₈×D₂₉₂

C₄

yes

characteristic

#17988366

C₄

yes

—

C₄×#11924003

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

C₄

yes

—

—

D₄

yes

complemented

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄×D₂₉₂

C₈

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂³

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₄⋊#10596831

C₈

yes

characteristic

#14556503

C₈

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₂₉₂

C₁₆

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₁₆

yes

characteristic

#11924005

C₁₆

yes

—

#11924003

—

#11924003

C₂×D₄

yes

yes

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

D₂₉₂

C₃₂

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×C₃₂

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

C₂×C₃₂

yes

complemented

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

D₄×C₃₂

—

maximal, Sylow

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

C₂²×C₃₂

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

D₇₃

—

C₂²×C₃₂

—

D₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₇₃

D₄×C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₈×D₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

C₄⋊C₁₆

#64408

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₂×C₈

#11924036

yes

characteristic

102

C₂×C₈

C₂²×#64408

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

#11924036

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

146

—

110

C₂³

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

146

—

114

C₂×C₄

—

146

—

115

C₂×C₄

#14556664

yes

characteristic

116

C₂×C₄

#14556664

yes

characteristic

117

C₂×C₄

C₂×#11924036

yes

characteristic

118

C₈

—

119

C₈

—

146

—

120

C₈

D₄×#64408

yes

characteristic

121

C₈

#14556679

yes

characteristic

122

C₂²

—

123

C₂²

—

146

—

124

C₂²

—

146

—

125

C₂²

C₂×#14556664

yes

characteristic

126

C₄

—

127

C₄

—

146

—

128

C₄

#17989518

yes

complemented

129

C₄

#17989615

yes

characteristic

130

C₄

#17989646

yes

characteristic

131

C₂

—

146

—

132

C₂

#22626310

yes

characteristic

133

C₂

C₂×#17989518

yes

characteristic

134

C₂

#22626355

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂²×C₃₂
Fitting	C₄⋊C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂×D₂₉₂	C₁₆
cosocle	C₂×#11924003	C₂³
hypercenter	C₄⋊C₁₆	#64408
hypocenter	C₇₃	D₄×C₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator

C₂×C₇₃

C₂²×C₃₂

Fitting

C₄⋊C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₁₆

prime radical

C₂×D₂₉₂

C₁₆

C₂³

C₄⋊C₁₆

C₇₃

D₄×C₃₂

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×#17989518	C₂	semidirect
2. C₄⋊#14556641	C₂	semidirect
3. #22625045	C₂	semidirect
4. C₄×#14556641	C₂	semidirect
17. #14556641	D₄	semidirect
47. D₂₉₂	C₃₂	semidirect
57. C₂×D₇₃	C₂×C₃₂	semidirect
59. C₄×C₇₃	C₂×C₃₂	semidirect
73. C₇₃	D₄×C₃₂	semidirect
128. C₄	#17989518	semidirect

H ≅ G/N

type

1. C₂×#17989518

C₂

semidirect

2. C₄⋊#14556641

C₂

semidirect

3. #22625045

C₂

semidirect

4. C₄×#14556641

C₂

semidirect

17. #14556641

D₄

semidirect

47. D₂₉₂

C₃₂

semidirect

57. C₂×D₇₃

C₂×C₃₂

semidirect

59. C₄×C₇₃

C₂×C₃₂

semidirect

73. C₇₃

D₄×C₃₂

semidirect

128. C₄

#17989518

semidirect

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

subgroup	index
C₂×#17989518	2
C₄⋊#14556641	2
#22625045	2
C₄×#14556641	2
#22611750	2
D₄×C₃₂	73

subgroup

index

D₄×C₃₂

Subgroups — #34014521

Conjugacy classes of subgroups

Complements

Maximal subgroups