Group #32997730 — subgroups

Subgroups — #32997730

Order 18688.

222 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 99 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	Q₁₆×#10596831	C₂	1	2	yes	maximal
2	#22608005	C₂	1	1	yes	maximal, characteristic
3	#22607895	C₂	1	2	yes	maximal
4	#22607847	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596831	—	2	2	no	—
7	C₄×#11924003	C₂²	1	1	yes	characteristic
8	#17988340	C₂²	1	2	yes	—
9	Q₈⋊#10596831	—	2	2	no	—
10	C₈×#10596831	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
13	#366×C₇₃	C₄	1	1	yes	characteristic
14	#358×C₇₃	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924003	—	2	1	no	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	4	2	no	—
19	C₄×#10596831	—	4	2	no	—
20	C₄×#10596831	D₄	1	1	yes	characteristic
21	C₄×#10596831	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₇₃	—	2	2	no	—
24	Q₃₂×C₇₃	C₈	1	4	yes	complemented
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	#107×C₇₃	C₂×C₄	1	2	yes	—
28	#103×C₇₃	—	2	2	no	—
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	—	4	2	no	—
32	C₂×#10596831	D₈	1	2	yes	—
33	#11924003	—	2	2	no	—
34	C₂×Q₈×C₇₃	—	2	2	no	—
35	Q₁₆×C₇₃	—	2	2	no	—
36	Q₁₆×C₇₃	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄⋊C₄×C₇₃	—	4	2	no	—
41	C₄²×C₇₃	—	4	2	no	—
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	1	no	—
44	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
45	Q₈×C₇₃	—	2	4	no	—
46	Q₈×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₇₃	#131	1	1	yes	characteristic
55	#10596831	—	2	2	no	—
56	#10596831	—	4	1	no	—
57	#10596831	—	8	2	no	—
58	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	4	4	no	—
61	C₄×C₇₃	—	8	2	no	—
62	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
63	C₄×C₇₃	#443	1	1	yes	characteristic
64	C₄×C₇₃	#510	1	2	yes	—
65	#12044	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
67	C₂×C₇₃	#2176	1	1	yes	characteristic
68	C₂×C₇₃	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	D₇₃	1	1	yes	characteristic
70	C₈×Q₁₆	—	73	2	no	—
71	#1372	—	73	1	no	—
72	#1332	—	73	2	no	—
73	#1311	—	73	1	no	—
74	C₇₃	#12044	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	Dic₇₃	1	1	yes	characteristic
76	C₈×Q₈	—	146	2	no	—
77	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
78	#366	Dic₇₃	1	1	yes	characteristic
79	#358	Dic₇₃	1	2	yes	—
80	C₄×C₁₆	—	73	1	no	—
81	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	#335	—	73	2	no	—
83	Q₈⋊C₈	—	146	2	no	—
84	C₈²	—	73	1	no	—
85	C₂×Q₁₆	C₂×Dic₇₃	1	2	yes	—
86	C₄×Q₈	—	2	2	no	—
87	Q₃₂	#10596831	1	4	yes	complemented
88	C₂×C₁₆	—	146	1	no	—
89	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
91	#107	C₂×Dic₇₃	1	2	yes	—
92	C₄⋊C₈	—	292	2	no	—
93	C₄⋊C₈	—	146	2	no	—
94	#103	—	2	2	no	—
95	C₄×C₈	—	292	2	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×#10596831	1	4	yes	—
102	C₂×C₈	—	292	2	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
106	C₂×C₈	#11924027	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	D₄×D₇₃	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	146	2	no	—
113	C₁₆	C₂×#10596831	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
119	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
120	C₂×C₄	#14556610	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	292	1	no	—
123	C₈	—	584	2	no	—
124	C₈	—	146	2	no	—
125	C₈	C₂²×#10596831	1	1	yes	characteristic
126	C₈	#14556598	1	1	yes	characteristic
127	C₂²	D₈×Dic₇₃	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	2	no	—
131	C₄	D₄×#10596831	1	1	yes	characteristic
132	C₄	#17988704	1	1	yes	characteristic
133	C₄	#17989029	1	2	yes	—
134	C₂	D₈×#10596831	1	1	yes	characteristic
135	C₂	#22611802	1	1	yes	characteristic
136	C₂	#22611817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#10596831

C₂

yes

maximal

#22608005

C₂

yes

maximal, characteristic

#22607895

C₂

yes

maximal

#22607847

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₃

C₂

yes

maximal, characteristic

Q₈×#10596831

—

C₄×#11924003

C₂²

yes

characteristic

#17988340

C₂²

yes

—

Q₈⋊#10596831

—

C₈×#10596831

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₄

yes

characteristic

C₄×Q₁₆×C₇₃

C₂²

yes

—

#366×C₇₃

C₄

yes

characteristic

#358×C₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×Q₁₆×C₇₃

C₂×C₄

yes

—

C₄×Q₈×C₇₃

—

Q₃₂×C₇₃

C₈

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₂×C₄

yes

—

#103×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

—

D₈

yes

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

—

—

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12044

—

maximal, Sylow

C₂×C₇₃

C₈×D₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×Q₃₂

D₇₃

yes

characteristic

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₇₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₃

yes

characteristic

C₈×Q₈

—

146

—

C₄×Q₁₆

C₂×D₇₃

yes

—

#366

Dic₇₃

yes

characteristic

#358

Dic₇₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

C₂×Q₁₆

C₂×Dic₇₃

yes

—

C₄×Q₈

—

Q₃₂

#10596831

yes

complemented

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

#107

C₂×Dic₇₃

yes

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×#10596831

yes

—

102

C₂×C₈

—

292

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

106

C₂×C₈

#11924027

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

D₄×D₇₃

yes

characteristic

111

C₁₆

—

112

C₁₆

—

146

—

113

C₁₆

C₂×#10596831

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₄×Dic₇₃

yes

characteristic

119

C₂×C₄

D₈×D₇₃

yes

characteristic

120

C₂×C₄

#14556610

yes

characteristic

121

C₈

—

122

C₈

—

292

—

123

C₈

—

584

—

124

C₈

—

146

—

125

C₈

C₂²×#10596831

yes

characteristic

126

C₈

#14556598

yes

characteristic

127

C₂²

D₈×Dic₇₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₄×#10596831

yes

characteristic

132

C₄

#17988704

yes

characteristic

133

C₄

#17989029

yes

—

134

C₂

D₈×#10596831

yes

characteristic

135

C₂

#22611802

yes

characteristic

136

C₂

#22611817

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₄×Q₃₂×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	C₄×D₈
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₄×Q₃₂	D₇₃
hypocenter	C₇₃	#12044
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₇₃

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₄×Q₃₂×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

C₄×D₈

cosocle

C₄×C₈×C₇₃

C₂³

hypercenter

C₄×Q₃₂

D₇₃

hypocenter

C₇₃

#12044

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
24. Q₃₂×C₇₃	C₈	semidirect
74. C₇₃	#12044	semidirect
87. Q₃₂	#10596831	semidirect

H ≅ G/N

type

24. Q₃₂×C₇₃

C₈

semidirect

74. C₇₃

#12044

semidirect

87. Q₃₂

#10596831

semidirect

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

subgroup	index
Q₁₆×#10596831	2
#22608005	2
#22607895	2
#22607847	2
C₄×Q₃₂×C₇₃	2
#12044	73

subgroup

index

Q₁₆×#10596831

C₄×Q₃₂×C₇₃

Subgroups — #32997730

Conjugacy classes of subgroups

Complements

Maximal subgroups