Finite Group Databasebeta

← #37535677

Subgroups — #37535677

Order 23168.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042080C211yesmaximal, characteristic, complemented
2#26041715C211yesmaximal, characteristic, complemented
3#26041546C211yesmaximal, characteristic, complemented
4#26041092C211yesmaximal, characteristic, complemented
5#26041087C211yesmaximal, characteristic, complemented
6#26041062C211yesmaximal, characteristic, complemented
7C2×D16×C181C211yesmaximal, characteristic, complemented
8#19153861—21no—
9#19153842—21no—
10#19153842—21no—
11#19153840—22no—
12#19153838C2211yescharacteristic
13#19153835—21no—
14D8×D181—21no—
15#19153822C2211yescharacteristic
16#19153760—21no—
17#19153757—21no—
18#19153756C2211yescharacteristic, complemented
19#19153752C2211yescharacteristic, complemented
20#19153733—21no—
21C2×D8×C181C2211yescharacteristic
22C2×D8×C181C2211yescharacteristic, complemented
23D16×C181—22no—
24C2×C16×C181C2211yescharacteristic, complemented
25#15668449—21no—
26#15668449D411yescharacteristic
27#15668448—22no—
28#15668448—41no—
29C2×C22⋊D181—42no—
30D4×D181—22no—
31D4×D181—41no—
32Dic724D411yescharacteristic
33#15668438—21no—
34#15668437—22no—
35#15668437—41no—
36D1448D411yescharacteristic
37#15668434—22no—
38#15668434—41no—
39#15668433—21no—
40#15668433D411yescharacteristic
41#15668432—21no—
42C8×D181—21no—
43#15668424—21no—
44C2×D4×C181—21no—
45C2×D4×C181—21no—
46D8×C181—21no—
47D8×C181—21no—
48D8×C181—21no—
49D8×C181D412yes—
50C2×C8×C181C2311yescharacteristic
51C16×C181—21no—
52C22×D181—42no—
53Dic362—21no—
54Dic362D811yescharacteristic
55D724—21no—
56D724D811yescharacteristic
57C22⋊D181—21no—
58C22⋊D181—42no—
59C22⋊D181—42no—
60C22⋊D181—41no—
61C2×Dic181—42no—
62C4×D181—21no—
63C4×D181—41no—
64#13089678—21no—
65#13089678—41no—
66C23×C181—41no—
67C23×C181—41no—
68D4×C181—22no—
69D4×C181—41no—
70D4×C181—41no—
71D4×C181—41no—
72C2×C4×C181C2×D411yescharacteristic
73C8×C181C2×D411yescharacteristic
74C8×C181C2×D411yescharacteristic
75C22×C181—42no—
76C22×C181—81no—
77C22×C181—161no—
78C22×C181—161no—
79C22×C181AGL(1,ℤ/8)11yescharacteristic
80C2×D181—21no—
81C2×D181—41no—
82C2×D181—162no—
83C4×C181#12011yescharacteristic
84C4×C181C2×D811yescharacteristic
85Dic181—21no—
86Dic181—41no—
87C2×C181—21no—
88C2×C181—82no—
89C2×C181—161no—
90C2×C181#44711yescharacteristic
91D181—41no—
92D181—81no—
93C181#219211yescharacteristic, complemented, Sylow
94#2192—1811nomaximal, Sylow
95#576—1811no—
96#509—1811no—
97C2×D16D18111yescharacteristic, complemented
98#472—1811no—
99#361—1811no—
100#359—1811no—
101#350—1811no—
102H(5,𝔽2)—3621no—
103AGL(1,ℤ/8)—3622no—
104AGL(1,ℤ/8)—3621no—
105AGL(1,ℤ/8)—3621no—
106#135—3621no—
107#135—1811no—
108C2×D8—3621no—
109C2×D8C2×D18111yescharacteristic
110C2×D8C2×D18111yescharacteristic, complemented
111#131—1811no—
112SD32—3621no—
113D16—22no—
114D16—3621no—
115M32—1811no—
116C2×C16C2×D18111yescharacteristic, complemented
117#108—1811no—
118#100—3621no—
119#41—3622no—
120#41—3621no—
121#41—1811no—
122#41—7241no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—3622no—
126C2×D4—7241no—
127C2×D4—7242no—
128Q16—1811no—
129SD16—3622no—
130SD16—3621no—
131SD16—7241no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—3622no—
136D8—1811no—
137D8—7241no—
138D8C22⋊D18112yes—
139M16—3621no—
140M16—3621no—
141M16—1811no—
142C2×C8—3621no—
143C2×C8C22×D18111yescharacteristic
144C16—21no—
145C16—3621no—
146C23—41no—
147C23—41no—
148C23—7242no—
149Q8—3621no—
150Q8—1811no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—3621no—
156D4—3621no—
157D4—1811no—
158D4—7242no—
159D4—7242no—
160D4—7241no—
161C2×C4—3621no—
162C2×C4—7242no—
163C2×C4—7241no—
164C2×C4D4×D18111yescharacteristic
165C8—3621no—
166C8—7241no—
167C8D4×D18111yescharacteristic
168C8C2×C22⋊D18111yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—28962no—
174C22—3621no—
175C22—7241no—
176C22#1915384011yescharacteristic
177C4—3621no—
178C4—7241no—
179C4#1915378411yescharacteristic
180C4D8×D18111yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—14481no—
185C2—7241no—
186C2#2604141111yescharacteristic
subgroupG/H
centerC2#26041411
commutator, cosocleC2×C8×C181C23
FrattiniC2×C8C22×D181
FittingC2×D16×C181C2
socleC2×C181#447
hypercenterC2×D16D181
hypocenterC181#2192
LCSG ▷ C2×C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26042080C2semidirect
2. #26041715C2semidirect
3. #26041546C2semidirect
4. #26041092C2semidirect
5. #26041087C2semidirect
6. #26041062C2semidirect
7. C2×D16×C181C2semidirect
18. #19153756C22semidirect
19. #19153752C22semidirect
22. C2×D8×C181C22semidirect
24. C2×C16×C181C22semidirect
93. C181#2192semidirect
97. C2×D16D181semidirect
110. C2×D8C2×D181semidirect
116. C2×C16C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420802
#260417152
#260415462
#260410922
#260410872
#260410622
C2×D16×C1812
#2192181