Group #33000637 — subgroups

Subgroups — #33000637

Order 18688.

306 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22612206	C₂	1	1	yes	maximal, characteristic, complemented
2	#22612181	C₂	1	1	yes	maximal, characteristic
3	#22612155	C₂	1	2	yes	maximal, complemented
4	#22609523	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₆₈	—	2	2	no	—
7	#17988992	C₂²	1	2	yes	—
8	#17988964	C₂²	1	2	yes	—
9	#17988960	—	2	2	no	—
10	#17988934	—	2	2	no	—
11	#17988413	—	2	2	no	—
12	#17988389	—	2	4	no	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₈²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924017	—	4	4	no	—
17	C₂×D₅₈₄	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	#14556588	—	2	2	no	—
21	D₁₁₆₈	—	4	4	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556513	—	4	4	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₆	—	4	2	no	—
31	C₂×D₂₉₂	—	4	2	no	—
32	C₂×D₂₉₂	—	4	2	no	—
33	C₂×D₂₉₂	—	4	2	no	—
34	#11924017	—	8	8	no	—
35	D₅₈₄	—	4	4	no	—
36	D₅₈₄	—	4	4	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	D₈	1	2	yes	—
40	C₂×C₈×C₇₃	D₈	1	2	yes	—
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	2	no	—
45	C₁₆×C₇₃	SD₁₆	1	4	yes	complemented
46	C₂²×D₇₃	—	8	2	no	—
47	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	8	2	no	—
51	D₂₉₂	—	8	2	no	—
52	D₂₉₂	—	8	2	no	—
53	D₂₉₂	—	8	4	no	—
54	D₂₉₂	—	8	4	no	—
55	Dic₁₄₆	—	8	4	no	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	D₁₆	1	4	yes	complemented
58	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
59	C₈×C₇₃	#135	1	2	yes	—
60	C₂²×C₇₃	#493	1	1	yes	characteristic
61	C₂×D₇₃	—	16	2	no	—
62	C₂×D₇₃	—	16	2	no	—
63	C₂×D₇₃	—	16	2	no	—
64	C₄×C₇₃	#492	1	1	yes	characteristic
65	C₄×C₇₃	#495	1	1	yes	characteristic
66	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
67	C₄×C₇₃	#508	1	2	yes	—
68	Dic₇₃	—	16	2	no	—
69	#12503	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	#1706	1	1	yes	characteristic
71	C₂×C₇₃	#2246	1	1	yes	characteristic
72	C₂×C₇₃	#2249	1	1	yes	characteristic
73	D₇₃	—	32	2	no	—
74	#2252	—	73	1	no	—
75	#2246	—	73	1	no	—
76	#2240	—	73	2	no	—
77	#1706	—	73	2	no	—
78	C₈×C₁₆	D₇₃	1	1	yes	characteristic, complemented
79	C₇₃	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	146	2	no	—
81	#500	—	73	2	no	—
82	#493	—	73	2	no	—
83	#492	—	146	2	no	—
84	#486	—	146	2	no	—
85	#366	—	146	2	no	—
86	#357	—	146	4	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
89	C₈²	C₂×D₇₃	1	1	yes	characteristic
90	C₂×SD₁₆	—	292	4	no	—
91	C₂×D₈	—	146	2	no	—
92	C₂×D₈	—	146	2	no	—
93	C₄⋊Q₈	—	146	2	no	—
94	#127	—	146	2	no	—
95	D₁₆	—	292	4	no	—
96	C₂×C₁₆	D₂₉₂	1	2	yes	—
97	C₂×C₁₆	D₂₉₂	1	2	yes	—
98	#107	—	146	2	no	—
99	#107	—	146	2	no	—
100	#102	—	292	4	no	—
101	C₄×C₈	D₂₉₂	1	1	yes	characteristic
102	C₄×C₈	D₂₉₂	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×Q₈	—	292	2	no	—
105	C₂×D₄	—	292	2	no	—
106	C₂×D₄	—	292	2	no	—
107	C₂×D₄	—	292	2	no	—
108	SD₁₆	—	584	8	no	—
109	D₈	—	292	4	no	—
110	D₈	—	292	4	no	—
111	C₂×C₈	D₅₈₄	1	2	yes	—
112	C₂×C₈	D₅₈₄	1	2	yes	—
113	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
115	C₄⋊C₄	—	292	2	no	—
116	C₄⋊C₄	—	292	2	no	—
117	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	#11924017	1	4	yes	complemented
120	C₂³	—	584	2	no	—
121	Q₈	—	584	4	no	—
122	D₄	—	584	2	no	—
123	D₄	—	584	2	no	—
124	D₄	—	584	4	no	—
125	D₄	—	584	4	no	—
126	C₂×C₄	—	584	2	no	—
127	C₂×C₄	#14556588	1	1	yes	characteristic
128	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
129	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₁₁₆₈	1	4	yes	complemented
132	C₈	C₂×#11924017	1	2	yes	—
133	C₈	#14556613	1	2	yes	—
134	C₂²	—	1168	2	no	—
135	C₂²	—	1168	2	no	—
136	C₂²	—	1168	2	no	—
137	C₂²	#17988964	1	1	yes	characteristic
138	C₄	—	1168	2	no	—
139	C₄	#17988960	1	1	yes	characteristic
140	C₄	#17988971	1	1	yes	characteristic
141	C₄	C₂×D₁₁₆₈	1	2	yes	—
142	C₄	#17989019	1	2	yes	—
143	C₂	—	2336	2	no	—
144	C₂	#22609523	1	1	yes	characteristic
145	C₂	#22612181	1	1	yes	characteristic
146	C₂	#22612192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic, complemented

#22612181

C₂

yes

maximal, characteristic

#22612155

C₂

yes

maximal, complemented

#22609523

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₆₈

—

#17988992

C₂²

yes

—

#17988964

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₂²

yes

characteristic

C₂×#11924017

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556591

—

#14556588

—

D₁₁₆₈

—

#14556525

—

#14556525

—

#14556513

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924017

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

complemented

C₂²×D₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

D₁₆

yes

complemented

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×D₁₆

yes

—

C₄×C₇₃

#508

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

—

C₈×C₁₆

D₇₃

yes

characteristic, complemented

C₇₃

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

146

—

#500

—

#493

—

#492

—

146

—

#486

—

146

—

#366

—

146

—

#357

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

C₂×D₇₃

yes

characteristic

C₂×SD₁₆

—

292

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

146

—

#127

—

146

—

D₁₆

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

146

—

#107

—

146

—

100

#102

—

292

—

101

C₄×C₈

D₂₉₂

yes

characteristic

102

C₄×C₈

D₂₉₂

yes

characteristic

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

C₂×Q₈

—

292

—

105

C₂×D₄

—

292

—

106

C₂×D₄

—

292

—

107

C₂×D₄

—

292

—

108

SD₁₆

—

584

—

109

D₈

—

292

—

110

D₈

—

292

—

111

C₂×C₈

D₅₈₄

yes

—

112

C₂×C₈

D₅₈₄

yes

—

113

C₂×C₈

C₂×D₂₉₂

yes

characteristic

114

C₂×C₈

C₂×D₂₉₂

yes

characteristic

115

C₄⋊C₄

—

292

—

116

C₄⋊C₄

—

292

—

117

C₄²

C₂×D₂₉₂

yes

characteristic

118

C₁₆

—

119

C₁₆

#11924017

yes

complemented

120

C₂³

—

584

—

121

Q₈

—

584

—

122

D₄

—

584

—

123

D₄

—

584

—

124

D₄

—

584

—

125

D₄

—

584

—

126

C₂×C₄

—

584

—

127

C₂×C₄

#14556588

yes

characteristic

128

C₂×C₄

C₂×D₅₈₄

yes

characteristic

129

C₂×C₄

C₂×D₅₈₄

yes

characteristic

130

C₈

—

131

C₈

D₁₁₆₈

yes

complemented

132

C₈

C₂×#11924017

yes

—

133

C₈

#14556613

yes

—

134

C₂²

—

1168

—

135

C₂²

—

1168

—

136

C₂²

—

1168

—

137

C₂²

#17988964

yes

characteristic

138

C₄

—

1168

—

139

C₄

#17988960

yes

characteristic

140

C₄

#17988971

yes

characteristic

141

C₄

C₂×D₁₁₆₈

yes

—

142

C₄

#17989019

yes

—

143

C₂

—

2336

—

144

C₂

#22609523

yes

characteristic

145

C₂

#22612181

yes

characteristic

146

C₂

#22612192

yes

characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#493
prime radical	#22612181	C₂
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12503
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₈×C₁₆×C₇₃

C₂

C₂²×C₇₃

C₂

C₈×C₁₆

D₇₃

hypocenter

C₇₃

#12503

LCS

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #22612206	C₂	semidirect
3. #22612155	C₂	semidirect
4. #22609523	C₂	semidirect
5. C₈×C₁₆×C₇₃	C₂	semidirect
45. C₁₆×C₇₃	SD₁₆	semidirect
57. C₈×C₇₃	D₁₆	semidirect
78. C₈×C₁₆	D₇₃	semidirect
79. C₇₃	#12503	semidirect
119. C₁₆	#11924017	semidirect
131. C₈	D₁₁₆₈	semidirect

H ≅ G/N

type

1. #22612206

C₂

semidirect

3. #22612155

C₂

semidirect

4. #22609523

C₂

semidirect

5. C₈×C₁₆×C₇₃

C₂

semidirect

45. C₁₆×C₇₃

SD₁₆

semidirect

57. C₈×C₇₃

D₁₆

semidirect

78. C₈×C₁₆

D₇₃

semidirect

79. C₇₃

#12503

semidirect

119. C₁₆

#11924017

semidirect

131. C₈

D₁₁₆₈

semidirect

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

subgroup	index
#22612206	2
#22612181	2
#22612155	2
#22609523	2
C₈×C₁₆×C₇₃	2
#12503	73

subgroup

index

C₈×C₁₆×C₇₃

Subgroups — #33000637

Conjugacy classes of subgroups

Complements

Maximal subgroups