Group #33000671 — subgroups

Subgroups — #33000671

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	#22612156	C₂	1	4	yes	maximal, complemented
2	#22609546	C₂	1	1	yes	maximal, characteristic
3	C₈×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988964	—	2	1	no	—
6	#17988964	—	2	1	no	—
7	#17988964	C₂²	1	2	yes	—
8	#17988387	—	2	8	no	—
9	C₄×#11924003	C₂²	1	2	yes	—
10	#500×C₇₃	—	2	1	no	—
11	#500×C₇₃	C₂²	1	2	yes	—
12	#494×C₇₃	—	2	1	no	—
13	C₈²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×D₅₈₄	—	2	4	no	—
15	C₂×D₅₈₄	—	4	2	no	—
16	C₂×D₅₈₄	—	4	2	no	—
17	#14556588	—	2	2	no	—
18	C₂×#11924003	D₄	1	4	yes	—
19	C₂×Q₁₆×C₇₃	—	4	2	no	—
20	C₄⋊Q₈×C₇₃	—	2	2	no	—
21	#107×C₇₃	—	2	4	no	—
22	#107×C₇₃	—	4	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×D₂₉₂	—	4	2	no	—
27	C₂×D₂₉₂	—	4	2	no	—
28	C₂×D₂₉₂	—	4	2	no	—
29	D₅₈₄	—	4	8	no	—
30	D₅₈₄	—	8	4	no	—
31	D₅₈₄	—	8	4	no	—
32	#11924003	—	2	4	no	—
33	C₂×Q₈×C₇₃	—	4	2	no	—
34	Q₁₆×C₇₃	—	8	4	no	—
35	C₂×C₈×C₇₃	D₈	1	2	yes	—
36	C₂×C₈×C₇₃	D₈	1	2	yes	—
37	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	4	2	no	—
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₇₃	—	8	2	no	—
42	Q₈×C₇₃	—	8	4	no	—
43	C₂×C₄×C₇₃	—	8	2	no	—
44	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	D₂₉₂	—	8	4	no	—
48	D₂₉₂	—	8	4	no	—
49	D₂₉₂	—	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	#493	1	1	yes	characteristic
54	C₂×D₇₃	—	16	2	no	—
55	C₂×D₇₃	—	16	2	no	—
56	C₂×D₇₃	—	16	2	no	—
57	C₄×C₇₃	—	16	2	no	—
58	C₄×C₇₃	#495	1	2	yes	—
59	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
60	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
61	#12509	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	#1709	1	1	yes	characteristic
63	C₂×C₇₃	#2246	1	1	yes	characteristic
64	C₂×C₇₃	#2248	1	1	yes	characteristic
65	D₇₃	—	32	2	no	—
66	#2240	—	73	4	no	—
67	#1713	D₇₃	1	1	yes	characteristic, complemented
68	#1712	—	73	1	no	—
69	C₈×C₁₆	—	73	1	no	—
70	C₇₃	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	2	1	no	—
72	#500	C₂×D₇₃	1	2	yes	—
73	#494	—	2	1	no	—
74	#493	—	73	2	no	—
75	#493	—	146	1	no	—
76	#493	—	146	1	no	—
77	#357	—	146	8	no	—
78	C₄×C₁₆	—	73	2	no	—
79	C₈²	C₂×D₇₃	1	1	yes	characteristic
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×D₈	—	292	2	no	—
82	C₂×D₈	—	292	2	no	—
83	C₂×D₈	—	146	4	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	146	2	no	—
86	C₂×C₁₆	—	73	4	no	—
87	#107	—	2	4	no	—
88	#107	—	4	2	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	—	4	2	no	—
93	C₂×D₄	—	292	2	no	—
94	C₂×D₄	—	292	2	no	—
95	C₂×D₄	—	292	2	no	—
96	Q₁₆	—	8	4	no	—
97	D₈	—	292	8	no	—
98	D₈	—	584	4	no	—
99	D₈	—	584	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	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
106	C₁₆	—	146	4	no	—
107	C₂³	—	584	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	584	4	no	—
110	D₄	—	584	4	no	—
111	D₄	—	584	4	no	—
112	C₂×C₄	—	8	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	#14556534	1	4	yes	—
118	C₈	#14556614	1	4	yes	—
119	C₂²	—	1168	2	no	—
120	C₂²	—	1168	2	no	—
121	C₂²	—	1168	2	no	—
122	C₂²	#17988963	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	#17988972	1	2	yes	—
125	C₄	C₂×#14556532	1	2	yes	—
126	C₄	C₂×#14556534	1	2	yes	—
127	C₂	—	2336	2	no	—
128	C₂	#22609538	1	1	yes	characteristic
129	C₂	#22612180	1	1	yes	characteristic
130	C₂	#22612188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612156

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1713

C₂

yes

maximal, characteristic, complemented

#17988964

—

#17988964

—

#17988964

C₂²

yes

—

#17988387

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

—

#500×C₇₃

C₂²

yes

—

#494×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

C₂×#11924003

D₄

yes

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

D₁₆

yes

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#495

yes

—

C₄×C₇₃

C₂×D₁₆

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2240

—

#1713

D₇₃

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₇₃

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₃

yes

—

#494

—

#493

—

#493

—

146

—

#493

—

146

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

292

—

C₂×D₈

—

292

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

146

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

D₈

—

292

—

D₈

—

584

—

D₈

—

584

—

100

C₂×C₈

#11924013

yes

—

101

C₂×C₈

#11924013

yes

—

102

C₂×C₈

D₄×D₇₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

106

C₁₆

—

146

—

107

C₂³

—

584

—

108

Q₈

—

109

D₄

—

584

—

110

D₄

—

584

—

111

D₄

—

584

—

112

C₂×C₄

—

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

#14556534

yes

—

118

C₈

#14556614

yes

—

119

C₂²

—

1168

—

120

C₂²

—

1168

—

121

C₂²

—

1168

—

122

C₂²

#17988963

yes

characteristic

123

C₄

—

124

C₄

#17988972

yes

—

125

C₄

C₂×#14556532

yes

—

126

C₄

C₂×#14556534

yes

—

127

C₂

—

2336

—

128

C₂

#22609538

yes

characteristic

129

C₂

#22612180

yes

characteristic

130

C₂

#22612188

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1713	C₂
socle	C₂²×C₇₃	#493
prime radical	#22609546	C₂
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#12509
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1713

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #22612156	C₂	semidirect
3. C₈×#11924003	C₂	semidirect
4. C₇₃×#1713	C₂	semidirect
67. #1713	D₇₃	semidirect
70. C₇₃	#12509	semidirect

H ≅ G/N

type

1. #22612156

C₂

semidirect

3. C₈×#11924003

C₂

semidirect

4. C₇₃×#1713

C₂

semidirect

67. #1713

D₇₃

semidirect

70. C₇₃

#12509

semidirect

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

subgroup	index
#22612156	2
#22609546	2
C₈×#11924003	2
C₇₃×#1713	2
#12509	73

subgroup

index

C₇₃×#1713

Subgroups — #33000671

Conjugacy classes of subgroups

Complements

Maximal subgroups