Group #33006946 — subgroups

Subgroups — #33006946

Order 18688.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22611726	C₂	1	1	yes	maximal, characteristic
2	#22608136	C₂	1	2	yes	maximal, complemented
3	#22608124	C₂	1	2	yes	maximal, complemented
4	#22608071	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2219	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	—	2	1	no	—
7	#17988421	C₂²	1	2	yes	—
8	#17988417	—	2	1	no	—
9	#17988417	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
12	#467×C₇₃	—	2	1	no	—
13	#367×C₇₃	C₂²	1	1	yes	characteristic
14	#358×C₇₃	—	2	1	no	—
15	C₂×#11924012	—	4	1	no	—
16	#14556527	—	2	2	no	—
17	#14556527	—	4	1	no	—
18	#14556497	—	2	2	no	—
19	C₂×Q₁₆×C₇₃	—	2	1	no	—
20	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
21	C₂²⋊Q₈×C₇₃	—	4	1	no	—
22	Q₃₂×C₇₃	—	2	2	no	—
23	M₃₂×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
26	#107×C₇₃	—	2	1	no	—
27	#103×C₇₃	—	4	1	no	—
28	#11924012	—	4	2	no	—
29	#11924012	—	8	1	no	—
30	C₂×#10596831	—	8	1	no	—
31	C₂×Q₈×C₇₃	—	4	1	no	—
32	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
33	Q₁₆×C₇₃	—	4	1	no	—
34	Q₁₆×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	4	1	no	—
40	C₄⋊C₄×C₇₃	—	8	1	no	—
41	C₂²⋊C₄×C₇₃	—	8	1	no	—
42	C₁₆×C₇₃	—	2	1	no	—
43	C₁₆×C₇₃	#41	1	2	yes	—
44	C₂³×C₇₃	SD₃₂	1	1	yes	characteristic
45	Q₈×C₇₃	—	8	1	no	—
46	Q₈×C₇₃	—	8	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	—	8	1	no	—
51	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	#123	1	1	yes	characteristic
55	C₈×C₇₃	#137	1	1	yes	characteristic
56	#10596831	—	8	2	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	16	1	no	—
62	C₄×C₇₃	—	16	1	no	—
63	C₄×C₇₃	#480	1	1	yes	characteristic
64	C₄×C₇₃	#509	1	1	yes	characteristic
65	#13718	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	1	no	—
67	C₂×C₇₃	—	4	1	no	—
68	C₂×C₇₃	#2233	1	1	yes	characteristic
69	#2219	D₇₃	1	1	yes	characteristic
70	#2161	—	73	1	no	—
71	#1426	—	73	2	no	—
72	#1420	—	73	2	no	—
73	#1400	—	73	1	no	—
74	C₇₃	#13718	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
76	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
77	#467	—	2	1	no	—
78	C₂×#108	—	146	1	no	—
79	M₆₄	—	73	2	no	—
80	#368	—	73	2	no	—
81	#368	—	146	1	no	—
82	#367	C₂×D₇₃	1	1	yes	characteristic
83	#358	—	2	1	no	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×M₁₆	—	292	1	no	—
86	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
87	C₂²⋊Q₈	—	4	1	no	—
88	Q₃₂	—	2	2	no	—
89	M₃₂	—	2	2	no	—
90	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
92	#108	—	292	1	no	—
93	#108	—	146	2	no	—
94	#107	—	2	1	no	—
95	#103	—	4	1	no	—
96	C₃₂	—	146	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂²×C₄	#11924013	1	1	yes	characteristic
99	Q₁₆	—	4	1	no	—
100	Q₁₆	—	4	1	no	—
101	M₁₆	—	292	2	no	—
102	M₁₆	—	584	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	584	1	no	—
106	C₂×C₈	#11924013	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	8	1	no	—
110	C₂²⋊C₄	—	8	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#11924027	1	2	yes	—
113	C₂³	#14556535	1	1	yes	characteristic
114	Q₈	—	8	1	no	—
115	Q₈	—	8	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	1	no	—
120	C₂×C₄	#14556535	1	1	yes	characteristic
121	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	584	2	no	—
124	C₈	#14556572	1	1	yes	characteristic
125	C₈	#14556621	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×#14556535	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	#17988898	1	1	yes	characteristic
133	C₄	#17989022	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#22612114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611726

C₂

yes

maximal, characteristic

#22608136

C₂

yes

maximal, complemented

#22608124

C₂

yes

maximal, complemented

#22608071

C₂

yes

maximal, characteristic

C₇₃×#2219

C₂

yes

maximal, characteristic

C₂×#14556527

—

#17988421

C₂²

yes

—

#17988417

—

#17988417

C₂²

yes

—

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

#467×C₇₃

—

#367×C₇₃

C₂²

yes

characteristic

#358×C₇₃

—

—

—

—

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

SD₃₂

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₃₂

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×SD₃₂

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2233

yes

characteristic

#2219

D₇₃

yes

—

—

—

—

C₇₃

#13718

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₃

yes

characteristic

C₂×M₃₂

C₂×D₇₃

yes

characteristic

#467

—

C₂×#108

—

146

—

M₆₄

—

#368

—

#368

—

146

—

#367

C₂×D₇₃

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂×M₁₆

—

292

—

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#108

—

292

—

#108

—

146

—

#107

—

#103

—

C₃₂

—

146

—

C₂×Q₈

—

C₂²×C₄

#11924013

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

M₁₆

—

292

—

102

M₁₆

—

584

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

584

—

106

C₂×C₈

#11924013

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₁₆

—

112

C₁₆

#11924027

yes

—

113

C₂³

#14556535

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#14556535

yes

characteristic

121

C₂×C₄

C₂×#11924013

yes

characteristic

122

C₈

—

123

C₈

—

584

—

124

C₈

#14556572

yes

characteristic

125

C₈

#14556621

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×#14556535

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#17988898

yes

characteristic

133

C₄

#17989022

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#22612114

yes

characteristic

	subgroup	G/H
center	C₂	#22612114
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2219	C₂
socle	C₂×C₇₃	#2233
prime radical	C₂³×C₇₃	SD₃₂
hypercenter	#2219	D₇₃
hypocenter	C₇₃	#13718
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2219

subgroup

G/H

center

C₂

#22612114

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2219

C₂

socle

C₂×C₇₃

#2233

prime radical

C₂³×C₇₃

SD₃₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2219

N	H ≅ G/N	type
2. #22608136	C₂	semidirect
3. #22608124	C₂	semidirect
74. C₇₃	#13718	semidirect

H ≅ G/N

type

2. #22608136

C₂

semidirect

3. #22608124

C₂

semidirect

74. C₇₃

#13718

semidirect

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

subgroup	index
#22611726	2
#22608136	2
#22608124	2
#22608071	2
C₇₃×#2219	2
#13718	73

subgroup

index

C₇₃×#2219

Subgroups — #33006946

Conjugacy classes of subgroups

Complements

Maximal subgroups