Group #33000669 — subgroups

Subgroups — #33000669

Order 18688.

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

normal subgroups only

This group has 306 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	#22612154	C₂	1	4	yes	maximal, complemented
2	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1712	C₂	1	1	yes	maximal, characteristic
5	#17988992	—	2	1	no	—
6	#17988992	C₂²	1	2	yes	—
7	#17988967	—	2	1	no	—
8	#17988388	—	2	8	no	—
9	C₄×#11924003	C₂²	1	2	yes	—
10	#493×C₇₃	—	2	1	no	—
11	#493×C₇₃	—	2	1	no	—
12	#493×C₇₃	C₂²	1	2	yes	—
13	C₈²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₉₂	—	4	2	no	—
15	#14556591	—	2	2	no	—
16	C₂×#11924003	D₄	1	4	yes	—
17	#14556525	—	2	4	no	—
18	#14556525	—	4	2	no	—
19	C₂×D₈×C₇₃	—	2	4	no	—
20	C₂×D₈×C₇₃	—	4	2	no	—
21	C₂×D₈×C₇₃	—	4	2	no	—
22	#127×C₇₃	—	2	2	no	—
23	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₄₆	—	4	2	no	—
27	Dic₂₉₂	—	8	4	no	—
28	C₄⋊Dic₇₃	—	4	2	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	#11924003	—	2	4	no	—
31	C₂×D₄×C₇₃	—	4	2	no	—
32	C₂×D₄×C₇₃	—	4	2	no	—
33	C₂×D₄×C₇₃	—	4	2	no	—
34	D₈×C₇₃	—	4	8	no	—
35	D₈×C₇₃	—	8	4	no	—
36	D₈×C₇₃	—	8	4	no	—
37	C₂×C₈×C₇₃	D₈	1	2	yes	—
38	C₂×C₈×C₇₃	D₈	1	2	yes	—
39	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂³×C₇₃	—	8	2	no	—
42	D₄×C₇₃	—	8	4	no	—
43	D₄×C₇₃	—	8	4	no	—
44	D₄×C₇₃	—	8	4	no	—
45	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₃	—	8	2	no	—
49	Dic₁₄₆	—	8	4	no	—
50	C₈×C₇₃	D₁₆	1	4	yes	—
51	C₈×C₇₃	SD₃₂	1	4	yes	—
52	C₈×C₇₃	#135	1	4	yes	—
53	C₂²×C₇₃	—	16	2	no	—
54	C₂²×C₇₃	—	16	2	no	—
55	C₂²×C₇₃	—	16	2	no	—
56	C₂²×C₇₃	#493	1	1	yes	characteristic
57	C₄×C₇₃	#495	1	2	yes	—
58	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
59	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
60	Dic₇₃	—	16	2	no	—
61	#12509	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	32	2	no	—
63	C₂×C₇₃	#1709	1	1	yes	characteristic
64	C₂×C₇₃	#2246	1	1	yes	characteristic
65	C₂×C₇₃	#2248	1	1	yes	characteristic
66	#2240	—	73	4	no	—
67	#1713	—	73	1	no	—
68	#1712	D₇₃	1	1	yes	characteristic
69	C₈×C₁₆	—	73	1	no	—
70	C₇₃	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	73	2	no	—
72	#500	—	146	1	no	—
73	#494	—	146	1	no	—
74	#493	—	2	1	no	—
75	#493	—	2	1	no	—
76	#493	C₂×D₇₃	1	2	yes	—
77	#357	—	146	8	no	—
78	C₄×C₁₆	—	73	2	no	—
79	C₈²	C₂×D₇₃	1	1	yes	characteristic
80	C₂×Q₁₆	—	292	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	146	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	73	4	no	—
87	#107	—	292	2	no	—
88	#107	—	146	4	no	—
89	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
92	C₂×Q₈	—	292	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	584	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#11924013	1	2	yes	—
101	C₂×C₈	#11924013	1	2	yes	—
102	C₂×C₈	D₄×D₇₃	1	2	yes	—
103	C₄⋊C₄	—	292	2	no	—
104	C₄⋊C₄	—	292	2	no	—
105	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
106	C₁₆	—	146	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	584	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	584	2	no	—
113	C₂×C₄	#14556587	1	1	yes	characteristic
114	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
115	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
116	C₈	#14556532	1	4	yes	—
117	C₈	#14556535	1	4	yes	—
118	C₈	#14556610	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#17988963	1	1	yes	characteristic
123	C₄	—	1168	2	no	—
124	C₄	#17988970	1	2	yes	—
125	C₄	C₂×#14556532	1	2	yes	—
126	C₄	C₂×#14556535	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#22609536	1	1	yes	characteristic
129	C₂	#22612180	1	1	yes	characteristic
130	C₂	#22612186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612154

C₂

yes

maximal, complemented

#22609549

C₂

yes

maximal, characteristic, complemented

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic

#17988992

—

#17988992

C₂²

yes

—

#17988967

—

#17988388

—

C₄×#11924003

C₂²

yes

—

#493×C₇₃

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

#14556591

—

C₂×#11924003

D₄

yes

—

#14556525

—

#14556525

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

D₁₆

yes

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

#495

yes

—

C₄×C₇₃

C₂×D₁₆

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

Dic₇₃

—

#12509

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₈×C₁₆

—

C₇₃

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

#494

—

146

—

#493

—

#493

—

#493

C₂×D₇₃

yes

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

146

—

#127

—

C₂×C₁₆

—

#107

—

292

—

#107

—

146

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

584

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#11924013

yes

—

101

C₂×C₈

#11924013

yes

—

102

C₂×C₈

D₄×D₇₃

yes

—

103

C₄⋊C₄

—

292

—

104

C₄⋊C₄

—

292

—

105

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

106

C₁₆

—

146

—

107

C₂³

—

108

Q₈

—

584

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

584

—

113

C₂×C₄

#14556587

yes

characteristic

114

C₂×C₄

C₂×#11924013

yes

characteristic

115

C₂×C₄

C₂×#11924013

yes

characteristic

116

C₈

#14556532

yes

—

117

C₈

#14556535

yes

—

118

C₈

#14556610

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#17988963

yes

characteristic

123

C₄

—

1168

—

124

C₄

#17988970

yes

—

125

C₄

C₂×#14556532

yes

—

126

C₄

C₂×#14556535

yes

—

127

C₂

—

128

C₂

#22609536

yes

characteristic

129

C₂

#22612180

yes

characteristic

130

C₂

#22612186

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₇₃×#1712	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1712	D₇₃
hypocenter	C₇₃	#12509
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting, prime radical

C₇₃×#1712

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22612154	C₂	semidirect
2. #22609549	C₂	semidirect
3. C₈×#11924003	C₂	semidirect
70. C₇₃	#12509	semidirect

H ≅ G/N

type

1. #22612154

C₂

semidirect

2. #22609549

C₂

semidirect

3. C₈×#11924003

C₂

semidirect

70. C₇₃

#12509

semidirect

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

subgroup	index
#22612154	2
#22609549	2
C₈×#11924003	2
C₇₃×#1712	2
#12509	73

subgroup

index

C₇₃×#1712

Subgroups — #33000669

Conjugacy classes of subgroups

Complements

Maximal subgroups