Group #33006834 — subgroups

Subgroups — #33006834

Order 18688.

190 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#17988420	C₂	1	1	yes	maximal, characteristic
2	#22612019	C₂	1	2	yes	maximal
3	#22608133	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608112	C₂	1	2	yes	maximal, complemented
5	C₂×Q₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
7	#17988825	—	2	2	no	—
8	#17988420	—	2	2	no	—
9	C₂×#14556497	C₂²	1	1	yes	characteristic
10	#17988413	C₂²	1	2	yes	—
11	#17988391	—	2	2	no	—
12	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
13	Q₆₄×C₇₃	—	2	2	no	—
14	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
16	#14556567	—	4	2	no	—
17	C₂×#11924003	D₄	1	1	yes	characteristic
18	C₁₆×D₇₃	—	2	1	no	—
19	C₁₆×D₇₃	—	2	1	no	—
20	#14556525	—	2	2	no	—
21	#14556515	—	4	2	no	—
22	#14556497	—	2	1	no	—
23	C₂×Q₁₆×C₇₃	—	2	2	no	—
24	Q₃₂×C₇₃	—	2	2	no	—
25	Q₃₂×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
27	C₃₂×C₇₃	D₄	1	2	yes	—
28	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
29	C₂×#10596831	D₈	1	1	yes	characteristic
30	C₈×D₇₃	—	2	1	no	—
31	C₈×D₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	4	2	no	—
33	#11924007	—	8	2	no	—
34	#11924005	—	8	2	no	—
35	#11924003	—	2	1	no	—
36	C₂×Q₈×C₇₃	—	4	2	no	—
37	Q₁₆×C₇₃	—	4	2	no	—
38	Q₁₆×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	#41	1	1	yes	characteristic
42	C₂²×D₇₃	D₁₆	1	1	yes	characteristic
43	Q₈×C₇₃	—	8	2	no	—
44	Q₈×C₇₃	—	8	2	no	—
45	C₂×C₄×C₇₃	—	8	2	no	—
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₃	—	8	2	no	—
48	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
49	C₄×D₇₃	—	2	1	no	—
50	C₄×D₇₃	—	2	1	no	—
51	C₈×C₇₃	#121	1	1	yes	characteristic
52	C₈×C₇₃	#135	1	1	yes	characteristic
53	#10596831	—	2	1	no	—
54	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
55	C₂×D₇₃	—	2	1	no	—
56	C₂×D₇₃	—	2	2	no	—
57	C₄×C₇₃	—	16	2	no	—
58	C₄×C₇₃	#466	1	1	yes	characteristic
59	C₄×C₇₃	#508	1	1	yes	characteristic
60	Dic₇₃	—	2	1	no	—
61	Dic₇₃	—	16	2	no	—
62	#13698	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	#2215	1	1	yes	characteristic
64	C₂×C₇₃	#2264	1	2	yes	—
65	D₇₃	—	4	1	no	—
66	C₂×Q₆₄	D₇₃	1	1	yes	characteristic, complemented
67	C₂×M₆₄	—	73	1	no	—
68	#2216	—	73	2	no	—
69	#1424	—	73	1	no	—
70	#1416	—	73	2	no	—
71	C₇₃	#13698	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	C₂×D₇₃	1	2	yes	—
73	C₂²×C₁₆	—	73	1	no	—
74	#467	—	146	2	no	—
75	Q₆₄	—	2	2	no	—
76	M₆₄	—	146	2	no	—
77	C₂×C₃₂	—	73	1	no	—
78	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
79	#366	—	73	2	no	—
80	#358	—	146	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²⋊Q₈	—	292	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	146	1	no	—
88	C₂×C₁₆	—	146	1	no	—
89	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
90	#107	—	146	2	no	—
91	#103	—	292	2	no	—
92	C₃₂	—	146	1	no	—
93	C₃₂	C₂²⋊D₇₃	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂²×C₄	—	73	1	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	146	1	no	—
100	C₂×C₈	—	146	1	no	—
101	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
102	C₄⋊C₄	—	292	2	no	—
103	C₄⋊C₄	—	584	2	no	—
104	C₂²⋊C₄	—	584	2	no	—
105	C₁₆	—	146	1	no	—
106	C₁₆	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
107	C₁₆	#11924027	1	1	yes	characteristic
108	C₂³	—	73	1	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	584	2	no	—
113	C₂×C₄	—	73	1	no	—
114	C₂×C₄	—	146	1	no	—
115	C₂×C₄	—	146	1	no	—
116	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
117	C₈	—	146	1	no	—
118	C₈	#14556562	1	1	yes	characteristic
119	C₈	#14556610	1	1	yes	characteristic
120	C₂²	—	146	2	no	—
121	C₂²	—	146	1	no	—
122	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	—	1168	2	no	—
125	C₄	—	146	1	no	—
126	C₄	#17988820	1	1	yes	characteristic
127	C₄	#17989016	1	1	yes	characteristic
128	C₂	—	292	1	no	—
129	C₂	#22612014	1	1	yes	characteristic
130	C₂	#22612250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988420

C₂

yes

maximal, characteristic

#22612019

C₂

yes

maximal

#22608133

C₂

yes

maximal, characteristic, complemented

#22608112

C₂

yes

maximal, complemented

C₂×Q₆₄×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₂²

yes

—

—

C₂²

yes

characteristic

#17988413

C₂²

yes

—

#17988391

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

Q₆₄×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

#14556567

—

C₂×#11924003

D₄

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

#14556525

—

#14556515

—

#14556497

—

C₂×Q₁₆×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₃₂×C₇₃

D₄

yes

—

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×#10596831

D₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂²×D₇₃

D₁₆

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×Q₆₄

D₇₃

yes

characteristic, complemented

C₂×M₆₄

—

#2216

—

#1424

—

#1416

—

C₇₃

#13698

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₃

yes

—

C₂²×C₁₆

—

#467

—

146

—

Q₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

#366

—

#358

—

146

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²⋊Q₈

—

292

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#107

—

146

—

#103

—

292

—

C₃₂

—

146

—

C₃₂

C₂²⋊D₇₃

yes

—

C₂×Q₈

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₂×C₈

D₄×D₇₃

yes

characteristic

102

C₄⋊C₄

—

292

—

103

C₄⋊C₄

—

584

—

104

C₂²⋊C₄

—

584

—

105

C₁₆

—

146

—

106

C₁₆

C₂×C₂²⋊D₇₃

yes

characteristic

107

C₁₆

#11924027

yes

characteristic

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

584

—

113

C₂×C₄

—

114

C₂×C₄

—

146

—

115

C₂×C₄

—

146

—

116

C₂×C₄

D₈×D₇₃

yes

characteristic

117

C₈

—

146

—

118

C₈

#14556562

yes

characteristic

119

C₈

#14556610

yes

characteristic

120

C₂²

—

146

—

121

C₂²

—

146

—

122

C₂²

D₁₆×D₇₃

yes

characteristic

123

C₄

—

124

C₄

—

1168

—

125

C₄

—

146

—

126

C₄

#17988820

yes

characteristic

127

C₄

#17989016

yes

characteristic

128

C₂

—

292

—

129

C₂

#22612014

yes

characteristic

130

C₂

#22612250

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₇₃
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₂×Q₆₄×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂²×D₇₃	D₁₆
hypercenter	C₂×Q₆₄	D₇₃
hypocenter	C₇₃	#13698
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₇₃

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₂×Q₆₄×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₁₆

prime radical

C₂²×D₇₃

D₁₆

hypercenter

C₂×Q₆₄

D₇₃

hypocenter

C₇₃

#13698

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
3. #22608133	C₂	semidirect
4. #22608112	C₂	semidirect
5. C₂×Q₆₄×C₇₃	C₂	semidirect
66. C₂×Q₆₄	D₇₃	semidirect
71. C₇₃	#13698	semidirect

H ≅ G/N

type

3. #22608133

C₂

semidirect

4. #22608112

C₂

semidirect

5. C₂×Q₆₄×C₇₃

C₂

semidirect

66. C₂×Q₆₄

D₇₃

semidirect

71. C₇₃

#13698

semidirect

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

subgroup	index
C₂×#17988420	2
#22612019	2
#22608133	2
#22608112	2
C₂×Q₆₄×C₇₃	2
#13698	73

subgroup

index

C₂×Q₆₄×C₇₃

Subgroups — #33006834

Conjugacy classes of subgroups

Complements

Maximal subgroups