Group #33006980 — subgroups

Subgroups — #33006980

Order 18688.

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

normal subgroups only

This group has 230 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	C₂×Dic₁₁₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988427	C₂	1	2	yes	maximal
3	#22612182	C₂	1	2	yes	maximal
4	C₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	C₂²	1	2	yes	—
7	C₂×#14556537	—	2	2	no	—
8	#17988965	—	2	2	no	—
9	Dic₁₁₆₈	—	2	2	no	—
10	#17988427	—	2	4	no	—
11	C₂×#14556497	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
14	Q₆₄×C₇₃	—	2	2	no	—
15	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	—	2	2	no	—
17	C₂×#11924018	—	4	2	no	—
18	#14556589	—	4	2	no	—
19	Dic₅₈₄	—	2	2	no	—
20	Dic₅₈₄	—	2	2	no	—
21	#14556537	—	4	4	no	—
22	C₂×#11924003	D₄	1	1	yes	characteristic
23	C₈×Dic₇₃	D₄	1	1	yes	characteristic
24	#14556497	D₄	1	2	yes	—
25	C₂×Q₁₆×C₇₃	—	2	2	no	—
26	Q₃₂×C₇₃	—	2	2	no	—
27	Q₃₂×C₇₃	—	2	2	no	—
28	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
29	C₃₂×C₇₃	D₄	1	2	yes	—
30	C₂×Dic₁₄₆	—	4	2	no	—
31	Dic₂₉₂	—	4	2	no	—
32	Dic₂₉₂	—	4	2	no	—
33	#11924018	—	8	4	no	—
34	C₂×#10596831	D₈	1	1	yes	characteristic
35	#11924007	—	8	4	no	—
36	C₄×Dic₇₃	D₈	1	1	yes	characteristic
37	#11924003	D₈	1	2	yes	—
38	C₂×Q₈×C₇₃	—	4	2	no	—
39	Q₁₆×C₇₃	—	4	2	no	—
40	Q₁₆×C₇₃	—	4	2	no	—
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₇₃	—	8	2	no	—
45	Q₈×C₇₃	—	8	2	no	—
46	C₂×C₄×C₇₃	—	8	2	no	—
47	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₃	—	8	2	no	—
49	C₂×Dic₇₃	D₁₆	1	2	yes	—
50	Dic₁₄₆	—	8	2	no	—
51	Dic₁₄₆	—	8	2	no	—
52	C₈×C₇₃	#127	1	1	yes	characteristic
53	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
54	#10596831	D₁₆	1	2	yes	—
55	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₃	—	16	2	no	—
57	C₄×C₇₃	#493	1	1	yes	characteristic
58	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
59	Dic₇₃	—	16	2	no	—
60	Dic₇₃	Q₆₄	1	4	yes	complemented
61	#13726	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	#2246	1	1	yes	characteristic
63	C₂×C₇₃	C₂×Q₆₄	1	2	yes	—
64	C₂×Q₆₄	—	73	2	no	—
65	C₂×Q₆₄	—	73	1	no	—
66	C₂×Q₆₄	D₇₃	1	1	yes	characteristic
67	#2247	—	73	2	no	—
68	C₄×C₃₂	—	73	1	no	—
69	C₇₃	#13726	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₃₂	—	73	2	no	—
71	C₂×Q₃₂	—	146	2	no	—
72	C₂×Q₃₂	C₂×D₇₃	1	2	yes	—
73	#494	—	146	2	no	—
74	Q₆₄	—	2	2	no	—
75	Q₆₄	—	146	4	no	—
76	Q₆₄	—	146	2	no	—
77	C₂×C₃₂	—	73	1	no	—
78	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
79	C₄×C₁₆	—	73	1	no	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	292	2	no	—
82	C₂×Q₁₆	—	146	2	no	—
83	C₄⋊Q₈	—	292	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	292	4	no	—
87	Q₃₂	—	146	2	no	—
88	Q₃₂	—	146	2	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
91	C₄×C₈	—	73	1	no	—
92	C₃₂	—	73	2	no	—
93	C₃₂	C₂²⋊D₇₃	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	292	2	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	Q₁₆	—	292	2	no	—
99	Q₁₆	—	292	2	no	—
100	Q₁₆	—	584	4	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
103	C₄⋊C₄	—	584	4	no	—
104	C₄²	—	73	1	no	—
105	C₁₆	—	73	2	no	—
106	C₁₆	D₄×D₇₃	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	Q₈	—	8	2	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	584	2	no	—
111	Q₈	—	584	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	584	2	no	—
114	C₂×C₄	—	73	2	no	—
115	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
116	C₈	—	73	2	no	—
117	C₈	#14556587	1	1	yes	characteristic
118	C₈	D₈×D₇₃	1	1	yes	characteristic
119	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	73	4	no	—
122	C₄	—	1168	2	no	—
123	C₄	#17988962	1	1	yes	characteristic
124	C₄	D₁₆×D₇₃	1	1	yes	characteristic
125	C₂	#22612179	1	1	yes	characteristic
126	C₂	Q₆₄×D₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₆₈

C₂

yes

maximal, characteristic

C₂×#17988427

C₂

yes

maximal

#22612182

C₂

yes

maximal

C₃₂×Dic₇₃

C₂

yes

maximal, characteristic

C₂×Q₆₄×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

C₂²

yes

—

C₂×#14556537

—

#17988965

—

Dic₁₁₆₈

—

#17988427

—

C₂×#14556497

C₂²

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₂²

yes

—

Q₆₄×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×#11924018

—

#14556589

—

Dic₅₈₄

—

Dic₅₈₄

—

#14556537

—

C₂×#11924003

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

#14556497

D₄

yes

—

C₂×Q₁₆×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₃₂×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924018

—

C₂×#10596831

D₈

yes

characteristic

#11924007

—

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

D₈

yes

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#127

yes

characteristic

C₈×C₇₃

C₂×D₈

yes

characteristic

#10596831

D₁₆

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#493

yes

characteristic

C₄×C₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

Q₆₄

yes

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×Q₆₄

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

D₇₃

yes

characteristic

#2247

—

C₄×C₃₂

—

C₇₃

#13726

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

C₂×Q₃₂

C₂×D₇₃

yes

—

#494

—

146

—

Q₆₄

—

Q₆₄

—

146

—

Q₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

292

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

292

—

Q₃₂

—

146

—

Q₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₇₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

292

—

Q₁₆

—

292

—

100

Q₁₆

—

584

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₇₃

yes

characteristic

103

C₄⋊C₄

—

584

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

D₄×D₇₃

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₇₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

584

—

111

Q₈

—

584

—

112

C₂×C₄

—

113

C₂×C₄

—

584

—

114

C₂×C₄

—

115

C₂×C₄

D₈×D₇₃

yes

characteristic

116

C₈

—

117

C₈

#14556587

yes

characteristic

118

C₈

D₈×D₇₃

yes

characteristic

119

C₂²

D₁₆×D₇₃

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

1168

—

123

C₄

#17988962

yes

characteristic

124

C₄

D₁₆×D₇₃

yes

characteristic

125

C₂

#22612179

yes

characteristic

126

C₂

Q₆₄×D₇₃

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, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	C₂×Q₆₄	D₇₃
hypocenter	C₇₃	#13726
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, prime radical

C₂²×C₇₃

C₂×D₁₆

hypercenter

C₂×Q₆₄

D₇₃

hypocenter

C₇₃

#13726

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
60. Dic₇₃	Q₆₄	semidirect
69. C₇₃	#13726	semidirect

H ≅ G/N

type

60. Dic₇₃

Q₆₄

semidirect

69. C₇₃

#13726

semidirect

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

subgroup	index
C₂×Dic₁₁₆₈	2
C₂×#17988427	2
#22612182	2
C₃₂×Dic₇₃	2
C₂×Q₆₄×C₇₃	2
#13726	73

subgroup

index

C₂×Dic₁₁₆₈

C₂×#17988427

#22612182

C₃₂×Dic₇₃

C₂×Q₆₄×C₇₃

#13726

Subgroups — #33006980

Conjugacy classes of subgroups

Complements

Maximal subgroups