Finite Group Databasebeta

← #37535389

Subgroups — #37535389

Order 23168.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041847C211yesmaximal, characteristic
2#99×D181C212yesmaximal, complemented
3#26041074C211yesmaximal, characteristic, complemented
4#26041071C212yesmaximal, complemented
5#354×C181C211yesmaximal, characteristic, complemented
6C2×#15668450C411yescharacteristic, complemented
7C2×D4×D181C2211yescharacteristic
8#19153813C411yescharacteristic, complemented
9#19153812C2211yescharacteristic
10#19153805C411yescharacteristic, complemented
11#19153789—22no—
12#19153786—22no—
13#19153773C411yescharacteristic, complemented
14C22⋊C4×D181—22no—
15#19153731C2212yes—
16#19153730—22no—
17#124×C181C2211yescharacteristic
18#99×C181C2212yes—
19C23×D181D412yes—
20#15668450—42no—
21C2×D724—21no—
22C2×D724—22no—
23C2×D724C2×C411yescharacteristic
24C2×D724C2×C411yescharacteristic, complemented
25C2×C22⋊D181—22no—
26C2×C22⋊D181D412yes—
27D4×D181—42no—
28C2×C4×D181—21no—
29C2×C4×D181—42no—
30C2×C4×D181C2×C411yescharacteristic
31#15668428—21no—
32C22⋊Dic181—22no—
33#15668426—21no—
34#15668426—22no—
35#15668426—42no—
36C4×Dic181C2×C411yescharacteristic, complemented
37C2×Q8×C181C2×C411yescharacteristic, complemented
38C2×D4×C181C2311yescharacteristic
39C22⋊C4×C181—22no—
40C22⋊C4×C181—22no—
41C42×C181C2×C411yescharacteristic, complemented
42C22×D181—22no—
43C22×D181—22no—
44C22×D181—21no—
45C22×D181—82no—
46C22×D181C22⋊C412yes—
47C22×D181C22⋊C411yescharacteristic
48D724—41no—
49D724—41no—
50D724—42no—
51D724—82no—
52C22⋊D181—42no—
53C22⋊D181—82no—
54C2×Dic181—21no—
55C2×Dic181—42no—
56C2×Dic181C22⋊C411yescharacteristic
57C4×D181—22no—
58C4×D181—42no—
59C4×D181—84no—
60C23×C181C2×D412yes—
61Q8×C181—41no—
62D4×C181—41no—
63C2×C4×C181—21no—
64C2×C4×C181—21no—
65C2×C4×C181—42no—
66C2×C4×C181C22×C411yescharacteristic
67C22×C181—22no—
68C22×C181—42no—
69C22×C181C2×C22⋊C411yescharacteristic
70C2×D181—22no—
71C2×D181—41no—
72C2×D181—44no—
73C2×D181—42no—
74C2×D181—42no—
75C2×D181—41no—
76C2×D181—82no—
77C2×D181#9912yes—
78C4×C181—21no—
79C4×C181—41no—
80C4×C181—41no—
81C4×C181—82no—
82Dic181—21no—
83Dic181—41no—
84Dic181—82no—
85C2×C181—21no—
86C2×C181—42no—
87C2×C181C2×#9911yescharacteristic
88D181—22no—
89D181—42no—
90D181—81no—
91C181#212711yescharacteristic, complemented, Sylow
92#2127—1811nomaximal, Sylow
93#535—1811no—
94C2×#99—1812no—
95#354—1811no—
96#354D18111yescharacteristic, complemented
97#353—1812no—
98C2×#41—1811no—
99C22×D4—1811no—
100#127—1811no—
101#127—1811no—
102#124—1811no—
103#124C2×D18111yescharacteristic
104#121—3622no—
105#120—3622no—
106#117—1811no—
107C2×C22⋊C4—3622no—
108#99—3622no—
109#99—1812no—
110#99C2×D18112yes—
111C24—1812no—
112#41—7242no—
113C2×Q8C4×D18111yescharacteristic, complemented
114C2×D4—3622no—
115C2×D4—3621no—
116C2×D4—3622no—
117C2×D4—1812no—
118C2×D4—1811no—
119C2×D4—1811no—
120C2×D4—7242no—
121C2×D4C22×D18111yescharacteristic
122C22×C4—3621no—
123C22×C4—1811no—
124C22×C4—7242no—
125C4⋊C4—3621no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—3621no—
129C22⋊C4—3622no—
130C22⋊C4—3622no—
131C22⋊C4—7242no—
132C42—1811no—
133C42C4×D18111yescharacteristic, complemented
134C23—3622no—
135C23—3622no—
136C23—3621no—
137C23—14482no—
138C23—1812no—
139C23—1811no—
140C23D4×D18112yes—
141Q8—41no—
142D4—41no—
143D4—14482no—
144D4—14482no—
145D4—7241no—
146D4—7241no—
147D4—7242no—
148D4—7242no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—3622no—
153C2×C4—3621no—
154C2×C4—14484no—
155C2×C4—1811no—
156C2×C4—7242no—
157C2×C4—7242no—
158C2×C4C2×C4×D18111yescharacteristic
159C22—22no—
160C22—42no—
161C22—3622no—
162C22—14482no—
163C22—1812no—
164C22—7241no—
165C22—7242no—
166C22—7242no—
167C22—7244no—
168C22—7241no—
169C22C22⋊C4×D18111yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—3621no—
175C4—14482no—
176C4—7241no—
177C2—21no—
178C2—42no—
179C2—3622no—
180C2—14481no—
181C2—7242no—
182C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC2×C4×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#354×C181C2
socleC2×C181C2×#99
prime radical#26041847C2
cosocleC2×D4×C181C23
hypercenter#354D181
hypocenterC181#2127
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D181C2semidirect
3. #26041074C2semidirect
4. #26041071C2semidirect
5. #354×C181C2semidirect
6. C2×#15668450C4semidirect
8. #19153813C4semidirect
10. #19153805C4semidirect
13. #19153773C4semidirect
24. C2×D724C2×C4semidirect
36. C4×Dic181C2×C4semidirect
37. C2×Q8×C181C2×C4semidirect
41. C42×C181C2×C4semidirect
91. C181#2127semidirect
96. #354D181semidirect
113. C2×Q8C4×D181semidirect
133. C42C4×D181semidirect

Maximal subgroups

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

subgroupindex
#260418472
#99×D1812
#260410742
#260410712
#354×C1812
#2127181