Group #34011963 — subgroups

Subgroups — #34011963

Order 18688.

246 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22625529	C₂	1	2	yes	maximal, complemented
2	#22624955	C₂	1	1	yes	maximal, characteristic, complemented
3	#22624890	C₂	1	2	yes	maximal, complemented
4	#22624848	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611806	C₂	1	1	yes	maximal, characteristic
6	#17989615	—	2	2	no	—
7	#17989476	C₂²	1	1	yes	characteristic
8	#17989448	C₂²	1	2	yes	—
9	#17989435	—	2	2	no	—
10	#17989428	C₂²	1	1	yes	characteristic
11	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic
12	#17988669	C₂²	1	2	yes	—
13	#17988387	C₄	1	2	yes	—
14	#17988362	C₂²	1	1	yes	characteristic
15	#367×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924036	—	4	2	no	—
17	C₂×#11924031	—	2	1	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	#14556648	—	4	2	no	—
20	C₄×#10596833	D₄	1	2	yes	—
21	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
22	#14556553	—	2	2	no	—
23	D₁₁₆₈	C₈	1	4	yes	complemented
24	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
25	#14556511	—	2	2	no	—
26	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
28	#107×C₇₃	C₂×C₄	1	2	yes	—
29	#11924036	—	8	4	no	—
30	C₂×#10596833	—	2	2	no	—
31	C₂×#10596833	—	4	2	no	—
32	#11924031	—	4	1	no	—
33	C₂×D₂₉₂	—	2	2	no	—
34	C₂×C₄×D₇₃	—	4	2	no	—
35	D₅₈₄	—	2	2	no	—
36	D₅₈₄	C₂×C₈	1	4	yes	complemented
37	C₂×#10596831	#41	1	1	yes	characteristic
38	#11924005	—	4	2	no	—
39	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
40	#11924003	—	2	1	no	—
41	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	2	2	no	—
43	C₁₆×C₇₃	C₂×C₈	1	2	yes	complemented
44	C₂²×D₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
48	D₂₉₂	—	2	4	no	—
49	D₂₉₂	—	4	2	no	—
50	C₄×D₇₃	—	8	2	no	—
51	C₄×D₇₃	—	8	2	no	—
52	#10596833	—	4	2	no	—
53	#10596833	—	8	2	no	—
54	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₇₃	#131	1	1	yes	characteristic
56	#10596831	—	2	1	no	—
57	C₂²×C₇₃	#442	1	1	yes	characteristic
58	C₂×D₇₃	—	4	4	no	—
59	C₂×D₇₃	—	8	2	no	—
60	C₂×D₇₃	—	8	2	no	—
61	C₄×C₇₃	—	8	2	no	—
62	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
63	C₄×C₇₃	#444	1	1	yes	characteristic
64	Dic₇₃	—	2	2	no	—
65	#12067	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	#1580	1	1	yes	characteristic
67	C₂×C₇₃	AGL(1,ℤ/16)	1	2	yes	—
68	D₇₃	—	8	4	no	—
69	#2177	—	73	1	no	—
70	#1580	—	73	2	no	—
71	#1368	—	73	1	no	—
72	#1333	—	73	2	no	—
73	#1313	—	73	1	no	—
74	C₇₃	#12067	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	73	1	no	—
76	C₄×D₈	—	73	2	no	—
77	C₈×D₄	—	146	2	no	—
78	#367	#64408	1	1	yes	characteristic
79	#357	—	73	2	no	—
80	#346	—	73	1	no	—
81	#346	—	73	1	no	—
82	#334	—	73	2	no	—
83	D₄⋊C₈	—	146	2	no	—
84	#322	—	73	1	no	—
85	C₂×D₈	—	73	2	no	—
86	C₂²×C₈	—	292	2	no	—
87	C₄×D₄	—	146	2	no	—
88	D₁₆	—	73	4	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	146	1	no	—
91	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
92	#107	C₂×#64408	1	2	yes	—
93	C₄⋊C₈	—	146	2	no	—
94	#102	—	146	2	no	—
95	C₂²⋊C₈	—	292	2	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	73	2	no	—
98	C₂×D₄	—	146	2	no	—
99	C₂²×C₄	—	292	2	no	—
100	D₈	—	73	4	no	—
101	D₈	—	146	2	no	—
102	C₂×C₈	—	292	2	no	—
103	C₂×C₈	—	584	4	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	146	2	no	—
106	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	292	2	no	—
109	C₄²	—	73	1	no	—
110	C₁₆	—	292	1	no	—
111	C₁₆	—	146	1	no	—
112	C₁₆	#11924036	1	2	yes	complemented
113	C₂³	—	292	2	no	—
114	D₄	—	292	2	no	—
115	D₄	—	146	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	584	2	no	—
118	C₂×C₄	—	584	2	no	—
119	C₂×C₄	—	73	2	no	—
120	C₂×C₄	D₄×#64408	1	1	yes	characteristic
121	C₈	—	292	2	no	—
122	C₈	—	584	2	no	—
123	C₈	—	146	1	no	—
124	C₈	C₂×#11924036	1	1	yes	characteristic
125	C₈	#14556679	1	1	yes	characteristic
126	C₂²	—	292	4	no	—
127	C₂²	—	584	2	no	—
128	C₂²	—	584	2	no	—
129	C₂²	#17989628	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	146	2	no	—
132	C₄	#17989615	1	1	yes	characteristic
133	C₄	#17989633	1	1	yes	characteristic
134	C₂	—	584	4	no	—
135	C₂	#22625529	1	1	yes	characteristic
136	C₂	#22626343	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625529

C₂

yes

maximal, complemented

#22624955

C₂

yes

maximal, characteristic, complemented

#22624890

C₂

yes

maximal, complemented

#22624848

C₂

yes

maximal, characteristic, complemented

#22611806

C₂

yes

maximal, characteristic

#17989615

—

#17989476

C₂²

yes

characteristic

#17989448

C₂²

yes

—

#17989435

—

#17989428

C₂²

yes

characteristic

C₂×D₁₁₆₈

C₄

yes

characteristic

#17988669

C₂²

yes

—

#17988387

C₄

yes

—

#17988362

C₂²

yes

characteristic

#367×C₇₃

C₄

yes

—

—

—

—

D₄

yes

—

C₂×D₅₈₄

C₂×C₄

yes

—

#14556553

—

D₁₁₆₈

C₈

yes

complemented

C₂×#11924003

C₂×C₄

yes

characteristic

#14556511

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×D₂₉₂

—

C₂×C₄×D₇₃

—

D₅₈₄

—

D₅₈₄

C₂×C₈

yes

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

complemented

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#442

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#444

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

—

C₇₃

#12067

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

146

—

yes

—

—

—

—

D₄⋊C₈

—

146

—

#322

—

C₂×D₈

—

C₂²×C₈

—

292

—

C₄×D₄

—

146

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

146

—

#102

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

100

D₈

—

101

D₈

—

146

—

102

C₂×C₈

—

292

—

103

C₂×C₈

—

584

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂²×#64408

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

292

—

109

C₄²

—

110

C₁₆

—

292

—

111

C₁₆

—

146

—

112

C₁₆

#11924036

yes

complemented

113

C₂³

—

292

—

114

D₄

—

292

—

115

D₄

—

146

—

116

C₂×C₄

—

117

C₂×C₄

—

584

—

118

C₂×C₄

—

584

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#64408

yes

characteristic

121

C₈

—

292

—

122

C₈

—

584

—

123

C₈

—

146

—

124

C₈

C₂×#11924036

yes

characteristic

125

C₈

#14556679

yes

characteristic

126

C₂²

—

292

—

127

C₂²

—

584

—

128

C₂²

—

584

—

129

C₂²

#17989628

yes

characteristic

130

C₄

—

131

C₄

—

146

—

132

C₄

#17989615

yes

characteristic

133

C₄

#17989633

yes

characteristic

134

C₂

—

584

—

135

C₂

#22625529

yes

characteristic

136

C₂

#22626343

yes

—

	subgroup	G/H
center	C₂²	#17989628
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#367×C₇₃	C₄
socle	C₂²×C₇₃	#442
prime radical	C₂×D₁₁₆₈	C₄
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#367	#64408
hypocenter	C₇₃	#12067
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#17989628

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#367×C₇₃

C₄

socle

C₂²×C₇₃

#442

prime radical

C₂×D₁₁₆₈

C₄

cosocle

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #22625529	C₂	semidirect
2. #22624955	C₂	semidirect
3. #22624890	C₂	semidirect
4. #22624848	C₂	semidirect
23. D₁₁₆₈	C₈	semidirect
36. D₅₈₄	C₂×C₈	semidirect
43. C₁₆×C₇₃	C₂×C₈	semidirect
74. C₇₃	#12067	semidirect
112. C₁₆	#11924036	semidirect

H ≅ G/N

type

1. #22625529

C₂

semidirect

2. #22624955

C₂

semidirect

3. #22624890

C₂

semidirect

4. #22624848

C₂

semidirect

23. D₁₁₆₈

C₈

semidirect

36. D₅₈₄

C₂×C₈

semidirect

43. C₁₆×C₇₃

C₂×C₈

semidirect

74. C₇₃

#12067

semidirect

112. C₁₆

#11924036

semidirect

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

subgroup	index
#22625529	2
#22624955	2
#22624890	2
#22624848	2
#22611806	2
#12067	73

subgroup

index

Subgroups — #34011963

Conjugacy classes of subgroups

Complements

Maximal subgroups