Group #33006960 — subgroups

Subgroups — #33006960

Order 18688.

180 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22612210	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608112	C₂	1	2	yes	maximal, complemented
3	#22608109	C₂	1	2	yes	maximal
4	C₄×#14556497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2249	C₂	1	1	yes	maximal, characteristic
6	#17988989	—	2	1	no	—
7	C₂×#14556497	C₂²	1	2	yes	—
8	#17988415	—	2	1	no	—
9	#17988413	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×SD₃₂×C₇₃	—	2	1	no	—
12	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	#495×C₇₃	—	2	1	no	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#14556583	—	4	1	no	—
16	#14556525	—	2	2	no	—
17	#14556523	—	4	1	no	—
18	#14556497	—	2	2	no	—
19	C₂×Q₁₆×C₇₃	—	2	1	no	—
20	C₂×SD₁₆×C₇₃	—	4	1	no	—
21	C₂×D₈×C₇₃	—	2	1	no	—
22	#124×C₇₃	—	4	1	no	—
23	Q₃₂×C₇₃	—	2	2	no	—
24	SD₃₂×C₇₃	—	4	2	no	—
25	D₁₆×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄⋊Dic₇₃	—	4	2	no	—
30	#11924007	—	8	2	no	—
31	C₂×Q₈×C₇₃	—	4	1	no	—
32	C₂×D₄×C₇₃	—	4	1	no	—
33	Q₁₆×C₇₃	—	4	1	no	—
34	Q₁₆×C₇₃	—	4	1	no	—
35	SD₁₆×C₇₃	—	8	2	no	—
36	D₈×C₇₃	—	4	1	no	—
37	D₈×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₃	—	8	2	no	—
41	C₄²×C₇₃	D₈	1	1	yes	characteristic
42	C₁₆×C₇₃	SD₁₆	1	2	yes	—
43	C₁₆×C₇₃	#41	1	2	yes	—
44	C₂³×C₇₃	—	8	1	no	—
45	Q₈×C₇₃	—	8	1	no	—
46	Q₈×C₇₃	—	8	1	no	—
47	D₄×C₇₃	—	8	1	no	—
48	D₄×C₇₃	—	8	1	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₇₃	—	8	2	no	—
53	C₈×C₇₃	SD₃₂	1	2	yes	—
54	C₈×C₇₃	#124	1	1	yes	characteristic
55	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₇₃	—	16	1	no	—
57	C₂²×C₇₃	—	16	2	no	—
58	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	—	16	1	no	—
61	C₄×C₇₃	#486	1	1	yes	characteristic
62	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
63	Dic₇₃	—	16	2	no	—
64	#13721	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	—	32	1	no	—
66	C₂×C₇₃	#2240	1	1	yes	characteristic
67	C₂×C₇₃	#2262	1	2	yes	—
68	#2253	—	73	1	no	—
69	#2249	D₇₃	1	1	yes	characteristic
70	#1416	—	73	2	no	—
71	#1415	—	73	2	no	—
72	C₄×C₃₂	—	73	1	no	—
73	C₇₃	#13721	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
75	C₂×SD₃₂	—	2	1	no	—
76	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
77	#499	—	146	1	no	—
78	#495	—	2	1	no	—
79	C₂×C₃₂	—	73	2	no	—
80	#367	—	146	1	no	—
81	#366	—	73	2	no	—
82	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	C₂×Q₁₆	—	2	1	no	—
84	C₂×SD₁₆	—	4	1	no	—
85	C₂×D₈	—	2	1	no	—
86	#125	—	292	1	no	—
87	#124	—	4	1	no	—
88	Q₃₂	—	2	2	no	—
89	SD₃₂	—	4	2	no	—
90	D₁₆	—	2	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
92	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
93	#107	—	146	2	no	—
94	#106	—	292	1	no	—
95	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
96	C₃₂	—	146	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×D₄	—	4	1	no	—
99	Q₁₆	—	4	1	no	—
100	Q₁₆	—	4	1	no	—
101	SD₁₆	—	8	2	no	—
102	D₈	—	4	1	no	—
103	D₈	—	4	1	no	—
104	C₂×C₈	#11924013	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
106	C₄⋊C₄	—	292	2	no	—
107	C₄⋊C₄	—	584	2	no	—
108	C₂²⋊C₄	—	8	2	no	—
109	C₄²	#11924013	1	1	yes	characteristic
110	C₁₆	#11924016	1	2	yes	—
111	C₁₆	#11924027	1	2	yes	—
112	C₂³	—	8	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	8	1	no	—
115	D₄	—	8	1	no	—
116	D₄	—	8	1	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	584	2	no	—
119	C₂×C₄	#14556532	1	2	yes	—
120	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
121	C₈	#14556535	1	2	yes	—
122	C₈	#14556578	1	1	yes	characteristic
123	C₈	C₂×#11924016	1	1	yes	characteristic
124	C₂²	—	16	2	no	—
125	C₂²	—	16	1	no	—
126	C₂²	C₂×#14556532	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	16	1	no	—
129	C₄	—	1168	2	no	—
130	C₄	#17988933	1	1	yes	characteristic
131	C₄	C₂×#14556535	1	1	yes	characteristic
132	C₂	—	32	1	no	—
133	C₂	#22612154	1	1	yes	characteristic
134	C₂	#22612238	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612210

C₂

yes

maximal, characteristic, complemented

#22608112

C₂

yes

maximal, complemented

#22608109

C₂

yes

maximal

C₄×#14556497

C₂

yes

maximal, characteristic, complemented

C₇₃×#2249

C₂

yes

maximal, characteristic

#17988989

—

C₂×#14556497

C₂²

yes

—

#17988415

—

#17988413

C₂²

yes

—

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂×SD₃₂×C₇₃

—

C₂×D₁₆×C₇₃

C₂²

yes

characteristic

#495×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

C₂×Q₁₆×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

—

#124×C₇₃

—

Q₃₂×C₇₃

—

SD₃₂×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#486

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

Dic₇₃

—

#13721

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2240

yes

characteristic

C₂×C₇₃

#2262

yes

—

#2253

—

#2249

D₇₃

yes

characteristic

#1416

—

#1415

—

C₄×C₃₂

—

C₇₃

#13721

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₃

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

C₂×D₇₃

yes

characteristic

#499

—

146

—

#495

—

C₂×C₃₂

—

#367

—

146

—

#366

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#125

—

292

—

#124

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#107

—

146

—

#106

—

292

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₃₂

—

146

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

D₈

—

104

C₂×C₈

#11924013

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

106

C₄⋊C₄

—

292

—

107

C₄⋊C₄

—

584

—

108

C₂²⋊C₄

—

109

C₄²

#11924013

yes

characteristic

110

C₁₆

#11924016

yes

—

111

C₁₆

#11924027

yes

—

112

C₂³

—

113

Q₈

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

584

—

119

C₂×C₄

#14556532

yes

—

120

C₂×C₄

C₂×#11924013

yes

characteristic

121

C₈

#14556535

yes

—

122

C₈

#14556578

yes

characteristic

123

C₈

C₂×#11924016

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×#14556532

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

1168

—

130

C₄

#17988933

yes

characteristic

131

C₄

C₂×#14556535

yes

characteristic

132

C₂

—

133

C₂

#22612154

yes

characteristic

134

C₂

#22612238

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556532
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2249	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂×D₁₆×C₇₃	C₂²
hypercenter	#2249	D₇₃
hypocenter	C₇₃	#13721
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

subgroup

G/H

center

C₂²

C₂×#14556532

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2249

C₂

socle

C₂²×C₇₃

C₂×D₁₆

prime radical

C₂×D₁₆×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

N	H ≅ G/N	type
1. #22612210	C₂	semidirect
2. #22608112	C₂	semidirect
4. C₄×#14556497	C₂	semidirect
73. C₇₃	#13721	semidirect

H ≅ G/N

type

1. #22612210

C₂

semidirect

2. #22608112

C₂

semidirect

4. C₄×#14556497

C₂

semidirect

73. C₇₃

#13721

semidirect

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

subgroup	index
#22612210	2
#22608112	2
#22608109	2
C₄×#14556497	2
C₇₃×#2249	2
#13721	73

subgroup

index

C₇₃×#2249

Subgroups — #33006960

Conjugacy classes of subgroups

Complements

Maximal subgroups