Group #32979756 — subgroups

Subgroups — #32979756

Order 18688.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22610088	C₂	1	1	yes	maximal, characteristic
2	#22610058	C₂	1	4	yes	maximal
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556523	—	2	2	no	—
6	C₂×C₄×#10596831	C₂²	1	2	yes	—
7	#17988457	C₂²	1	2	yes	—
8	#17988349	—	2	8	no	—
9	C₈×#10596831	C₄	1	4	yes	complemented
10	C₂×#106×C₇₃	—	2	2	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#384×C₇₃	C₂²	1	2	yes	—
13	C₂²×#10596831	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₇₃	—	2	2	no	—
15	C₂×C₄⋊Dic₇₃	—	2	4	no	—
16	#14556523	—	4	4	no	—
17	C₄×#10596831	C₂×C₄	1	4	yes	—
18	C₄×#10596831	Q₈	1	4	yes	—
19	C₂²×C₈×C₇₃	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
22	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
23	#106×C₇₃	—	4	4	no	—
24	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₃	Q₈	1	2	yes	—
26	C₂²×Dic₇₃	—	4	2	no	—
27	C₂×#10596831	—	2	8	no	—
28	C₂×#10596831	C₄⋊C₄	1	8	yes	—
29	C₄⋊Dic₇₃	—	4	4	no	—
30	C₄⋊Dic₇₃	—	4	8	no	—
31	C₂²×C₄×C₇₃	—	4	2	no	—
32	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
36	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
37	C₄⋊C₄×C₇₃	—	4	4	no	—
38	C₄⋊C₄×C₇₃	—	4	8	no	—
39	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
41	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₇₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	8	2	no	—
44	C₂×C₄×C₇₃	—	8	4	no	—
45	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₃	#135	1	8	yes	—
52	C₂×Dic₇₃	—	8	2	no	—
53	C₂×Dic₇₃	—	8	4	no	—
54	C₈×C₇₃	#106	1	8	yes	—
55	#10596831	—	2	8	no	—
56	C₂²×C₇₃	#384	1	1	yes	characteristic
57	C₂²×C₇₃	#492	1	1	yes	characteristic
58	C₂²×C₇₃	#492	1	1	yes	characteristic
59	C₂²×C₇₃	#495	1	4	yes	—
60	C₄×C₇₃	—	16	2	no	—
61	C₄×C₇₃	C₂×#106	1	2	yes	—
62	C₄×C₇₃	#427	1	4	yes	—
63	C₄×C₇₃	#498	1	2	yes	—
64	C₄×C₇₃	#499	1	4	yes	—
65	Dic₇₃	—	16	2	no	—
66	#8815	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#1708	1	2	yes	—
68	C₂×C₇₃	#1708	1	2	yes	—
69	C₂×C₇₃	#1844	1	1	yes	characteristic
70	C₂×C₇₃	#1846	1	2	yes	—
71	#1844	—	73	1	no	—
72	#1844	D₇₃	1	1	yes	characteristic
73	#1834	—	73	4	no	—
74	C₂×C₈²	—	73	1	no	—
75	C₇₃	#8815	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	2	2	no	—
77	C₂×#106	—	146	2	no	—
78	C₂×C₄×C₈	—	73	2	no	—
79	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
80	#384	—	73	2	no	—
81	#384	C₂×D₇₃	1	2	yes	—
82	#340	—	146	8	no	—
83	C₈²	—	73	4	no	—
84	C₂²×C₈	—	73	4	no	—
85	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	146	4	no	—
89	C₂×C₄⋊C₄	—	146	2	no	—
90	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	292	4	no	—
93	C₄×C₈	—	73	4	no	—
94	C₄×C₈	—	73	4	no	—
95	C₄×C₈	Dic₁₄₆	1	2	yes	—
96	C₄×C₈	C₄×D₇₃	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	292	2	no	—
99	C₂²×C₄	D₄×D₇₃	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
101	C₂×C₈	—	73	8	no	—
102	C₂×C₈	—	146	8	no	—
103	C₂×C₈	#11924007	1	4	yes	—
104	C₂×C₈	#11924015	1	4	yes	—
105	C₂×C₈	#11924016	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	292	4	no	—
109	C₄⋊C₄	—	292	8	no	—
110	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
111	C₄²	Q₈×D₇₃	1	2	yes	—
112	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
113	C₂³	#14556587	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	584	2	no	—
117	C₂×C₄	—	584	4	no	—
118	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
119	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
120	C₂×C₄	#14556589	1	1	yes	characteristic
121	C₂×C₄	#14556590	1	2	yes	—
122	C₂×C₄	C₂×#11924015	1	2	yes	—
123	C₂×C₄	C₂×#11924016	1	2	yes	—
124	C₂×C₄	#14556612	1	8	yes	—
125	C₈	—	146	8	no	—
126	C₈	#14556522	1	8	yes	—
127	C₂²	#17988456	1	1	yes	characteristic
128	C₂²	#17988959	1	1	yes	characteristic
129	C₂²	#17988961	1	1	yes	characteristic
130	C₂²	#17988968	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	1168	2	no	—
133	C₄	C₂×#14556522	1	2	yes	—
134	C₄	#17988622	1	4	yes	—
135	C₄	#17988983	1	2	yes	—
136	C₄	#17988987	1	4	yes	—
137	C₂	#22609530	1	2	yes	—
138	C₂	#22609533	1	2	yes	—
139	C₂	#22610087	1	1	yes	characteristic
140	C₂	#22610092	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610088

C₂

yes

maximal, characteristic

#22610058

C₂

yes

maximal

C₂×C₈×#10596831

C₂

yes

maximal, characteristic

C₇₃×#1844

C₂

yes

maximal, characteristic

C₂×#14556523

—

C₂×C₄×#10596831

C₂²

yes

—

#17988457

C₂²

yes

—

#17988349

—

C₈×#10596831

C₄

yes

complemented

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

C₂²×#10596831

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556523

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

#106

yes

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

#427

yes

—

C₄×C₇₃

#498

yes

—

C₄×C₇₃

#499

yes

—

Dic₇₃

—

#8815

—

maximal, Sylow

C₂×C₇₃

#1708

yes

—

C₂×C₇₃

#1708

yes

—

C₂×C₇₃

#1844

yes

characteristic

C₂×C₇₃

#1846

yes

—

#1844

—

#1844

D₇₃

yes

characteristic

#1834

—

C₂×C₈²

—

C₇₃

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#384

—

#384

C₂×D₇₃

yes

—

#340

—

146

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

#106

—

#106

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

C₄×D₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

292

—

C₂²×C₄

D₄×D₇₃

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

146

—

103

C₂×C₈

#11924007

yes

—

104

C₂×C₈

#11924015

yes

—

105

C₂×C₈

#11924016

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

292

—

109

C₄⋊C₄

—

292

—

110

C₄²

C₂×C₄×D₇₃

yes

characteristic

111

C₄²

Q₈×D₇₃

yes

—

112

C₄²

C₂×Dic₁₄₆

yes

characteristic

113

C₂³

#14556587

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

584

—

117

C₂×C₄

—

584

—

118

C₂×C₄

C₄⋊C₄×D₇₃

yes

—

119

C₂×C₄

C₂×#11924007

yes

characteristic

120

C₂×C₄

#14556589

yes

characteristic

121

C₂×C₄

#14556590

yes

—

122

C₂×C₄

C₂×#11924015

yes

—

123

C₂×C₄

C₂×#11924016

yes

—

124

C₂×C₄

#14556612

yes

—

125

C₈

—

146

—

126

C₈

#14556522

yes

—

127

C₂²

#17988456

yes

characteristic

128

C₂²

#17988959

yes

characteristic

129

C₂²

#17988961

yes

characteristic

130

C₂²

#17988968

yes

—

131

C₄

—

132

C₄

—

1168

—

133

C₄

C₂×#14556522

yes

—

134

C₄

#17988622

yes

—

135

C₄

#17988983

yes

—

136

C₄

#17988987

yes

—

137

C₂

#22609530

yes

—

138

C₂

#22609533

yes

—

139

C₂

#22610087

yes

characteristic

140

C₂

#22610092

yes

—

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1844	C₂
socle, prime radical	C₂³×C₇₃	#127
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#8815
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14556587

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1844

C₂

socle, prime radical

C₂³×C₇₃

#127

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×#10596831	C₄	semidirect
75. C₇₃	#8815	semidirect

H ≅ G/N

type

9. C₈×#10596831

C₄

semidirect

75. C₇₃

#8815

semidirect

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

subgroup	index
#22610088	2
#22610058	2
C₂×C₈×#10596831	2
C₇₃×#1844	2
#8815	73

subgroup

index

C₇₃×#1844

Subgroups — #32979756

Conjugacy classes of subgroups

Complements

Maximal subgroups