Group #32997777 — subgroups

Subgroups — #32997777

Order 18688.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22611833	C₂	1	2	yes	maximal
2	C₈×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22607909	C₂	1	2	yes	maximal
4	C₈×#11924003	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₄₆	C₂²	1	2	yes	—
7	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
8	#17988405	—	2	2	no	—
9	C₄×#11924003	C₂²	1	1	yes	characteristic
10	C₄×#11924003	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₇₃	—	2	2	no	—
12	#17988340	C₄	1	1	yes	characteristic
13	#17988319	C₄	1	2	yes	—
14	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₈²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic
17	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
18	C₂×#11924003	—	2	1	no	—
19	C₂×#11924003	D₄	1	1	yes	characteristic
20	C₂×#11924003	D₄	1	1	yes	characteristic
21	#14556525	C₈	1	1	yes	characteristic
22	C₄⋊#10596831	C₂×C₄	1	2	yes	—
23	#14556520	—	2	2	no	—
24	#14556516	C₈	1	2	yes	—
25	C₈×Dic₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	—	2	2	no	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
31	Dic₂₉₂	C₁₆	1	4	yes	complemented
32	C₂×#10596831	—	2	2	no	—
33	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
34	#11924007	—	2	2	no	—
35	C₄×Dic₇₃	—	2	2	no	—
36	#11924003	—	2	2	no	—
37	#11924003	—	2	1	no	—
38	#11924003	Q₁₆	1	2	yes	complemented
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
44	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	—	2	2	no	—
51	Dic₁₄₆	—	2	2	no	—
52	Dic₁₄₆	C₂×C₁₆	1	4	yes	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
57	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₇₃	#135	1	1	yes	characteristic
59	#10596831	—	4	2	no	—
60	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
63	C₄×C₇₃	#443	1	1	yes	characteristic
64	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₇₃	#504	1	1	yes	characteristic
66	Dic₇₃	—	2	4	no	—
67	Dic₇₃	—	4	2	no	—
68	C₁₆×Q₁₆	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₇₃	#2170	1	1	yes	characteristic
72	Q₈×C₁₆	—	73	2	no	—
73	C₈×Q₁₆	—	73	1	no	—
74	#1370	D₇₃	1	1	yes	characteristic
75	Q₈⋊C₁₆	—	73	2	no	—
76	C₈×C₁₆	—	73	1	no	—
77	C₇₃	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₈×Q₈	—	73	2	no	—
79	C₄×Q₁₆	—	73	1	no	—
80	C₄⋊C₁₆	—	146	2	no	—
81	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
82	C₄×C₁₆	—	73	1	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	146	2	no	—
85	#335	—	73	1	no	—
86	Q₈⋊C₈	—	73	2	no	—
87	C₈²	C₂×D₇₃	1	1	yes	characteristic
88	C₂×Q₁₆	—	73	1	no	—
89	C₄×Q₈	—	73	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	73	1	no	—
93	C₂×C₁₆	—	146	1	no	—
94	#107	—	73	1	no	—
95	C₄⋊C₈	—	73	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	#103	—	73	2	no	—
98	C₄×C₈	—	146	2	no	—
99	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
100	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
102	C₂×Q₈	—	73	2	no	—
103	Q₁₆	—	73	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	146	2	no	—
106	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
107	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
108	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
109	C₂×C₈	#11924029	1	1	yes	characteristic
110	C₄⋊C₄	—	73	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄²	—	146	2	no	—
113	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
114	C₁₆	—	4	2	no	—
115	C₁₆	—	73	2	no	—
116	C₁₆	—	146	2	no	—
117	C₁₆	—	146	1	no	—
118	Q₈	—	73	4	no	—
119	Q₈	—	146	2	no	—
120	C₂×C₄	—	146	2	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	#14556553	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
124	C₂×C₄	#14556599	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	292	2	no	—
129	C₈	C₁₆×D₇₃	1	2	yes	—
130	C₈	Q₁₆×D₇₃	1	1	yes	characteristic
131	C₈	#14556610	1	1	yes	characteristic
132	C₂²	#17988652	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	292	2	no	—
135	C₄	—	146	4	no	—
136	C₄	#17988681	1	1	yes	characteristic
137	C₄	#17988703	1	1	yes	characteristic
138	C₄	C₂×C₁₆×D₇₃	1	1	yes	characteristic
139	C₄	#17989005	1	1	yes	characteristic
140	C₂	#22608737	1	1	yes	characteristic
141	C₂	#22611750	1	1	yes	characteristic
142	C₂	#22611767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611833

C₂

yes

maximal

C₈×Dic₂₉₂

C₂

yes

maximal, characteristic

#22607909

C₂

yes

maximal

C₈×#11924003

C₂

yes

maximal, characteristic

C₇₃×#1370

C₂

yes

maximal, characteristic

C₈×Dic₁₄₆

C₂²

yes

—

C₄×Dic₂₉₂

C₄

yes

characteristic

#17988405

—

C₄×#11924003

C₂²

yes

characteristic

C₄×#11924003

C₂²

yes

characteristic

C₁₆×Dic₇₃

—

#17988340

C₄

yes

characteristic

#17988319

C₄

yes

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

C₈

yes

characteristic

C₄×Dic₁₄₆

C₂×C₄

yes

—

C₂×#11924003

—

C₂×#11924003

D₄

yes

characteristic

C₂×#11924003

D₄

yes

characteristic

#14556525

C₈

yes

characteristic

C₄⋊#10596831

C₂×C₄

yes

—

#14556520

—

#14556516

C₈

yes

—

C₈×Dic₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

C₂×C₈

yes

—

Dic₂₉₂

C₁₆

yes

complemented

C₂×#10596831

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924007

—

C₄×Dic₇₃

—

#11924003

—

#11924003

—

#11924003

Q₁₆

yes

complemented

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

C₂×C₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₁₆

yes

characteristic

C₄×C₇₃

#443

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₁₆×Q₁₆

—

maximal, Sylow

C₂×C₇₃

C₈×Q₁₆

yes

characteristic

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

#2170

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

D₇₃

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₇₃

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

C₂×D₇₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

146

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

—

C₄×C₈

—

146

—

C₄×C₈

C₄×D₇₃

yes

characteristic

100

C₄×C₈

C₄×D₇₃

yes

characteristic

101

C₄×C₈

C₂²×D₇₃

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₈×D₇₃

yes

characteristic

107

C₂×C₈

C₈×D₇₃

yes

characteristic

108

C₂×C₈

D₄×D₇₃

yes

characteristic

109

C₂×C₈

#11924029

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

146

—

113

C₄²

C₂×C₄×D₇₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

146

—

117

C₁₆

—

146

—

118

Q₈

—

119

Q₈

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

#14556553

yes

characteristic

123

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

124

C₂×C₄

#14556599

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

292

—

129

C₈

C₁₆×D₇₃

yes

—

130

C₈

Q₁₆×D₇₃

yes

characteristic

131

C₈

#14556610

yes

characteristic

132

C₂²

#17988652

yes

characteristic

133

C₄

—

134

C₄

—

292

—

135

C₄

—

146

—

136

C₄

#17988681

yes

characteristic

137

C₄

#17988703

yes

characteristic

138

C₄

C₂×C₁₆×D₇₃

yes

characteristic

139

C₄

#17989005

yes

characteristic

140

C₂

#22608737

yes

characteristic

141

C₂

#22611750

yes

characteristic

142

C₂

#22611767

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1370	C₂
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1370	D₇₃
hypocenter	C₇₃	C₁₆×Q₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₇₃

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1370

C₂

socle, prime radical

C₂²×C₇₃

C₈×D₄

cosocle

C₄×C₈×C₇₃

C₂³

hypercenter

#1370

D₇₃

hypocenter

C₇₃

C₁₆×Q₁₆

LCS

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
31. Dic₂₉₂	C₁₆	semidirect
38. #11924003	Q₁₆	semidirect
77. C₇₃	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

31. Dic₂₉₂

C₁₆

semidirect

38. #11924003

Q₁₆

semidirect

77. C₇₃

C₁₆×Q₁₆

semidirect

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

subgroup	index
#22611833	2
C₈×Dic₂₉₂	2
#22607909	2
C₈×#11924003	2
C₇₃×#1370	2
C₁₆×Q₁₆	73

subgroup

index

#22611833

C₈×Dic₂₉₂

#22607909

C₈×#11924003

C₇₃×#1370

C₁₆×Q₁₆

Subgroups — #32997777

Conjugacy classes of subgroups

Complements

Maximal subgroups