Finite Group Databasebeta

← #352×D181 #37535355

Subgroups — #352×D181 #37535355

Order 23168.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D181C211yesmaximal, characteristic
2#100×D181C211yesmaximal, characteristic
3#99×D181C211yesmaximal, characteristic
4#26041070C211yesmaximal, characteristic, complemented
5#26041069C211yesmaximal, characteristic, complemented
6#26041068C211yesmaximal, characteristic, complemented
7#352×C181C211yesmaximal, characteristic, complemented
8C2×D4×D181C2211yescharacteristic
9C22×C4×D181C411yescharacteristic, complemented
10M16×D181—21no—
11#19153801—21no—
12#19153800C411yescharacteristic, complemented
13#19153799C411yescharacteristic, complemented
14#19153798—21no—
15#19153797C2211yescharacteristic
16C4⋊C4×D181—21no—
17C22⋊C4×D181—21no—
18C22⋊C4×D181—21no—
19C22⋊C4×D181C411yescharacteristic, complemented
20#19153733C2211yescharacteristic
21#19153732—21no—
22#19153731C2211yescharacteristic
23#19153730—21no—
24#123×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic
26#99×C181C2211yescharacteristic
27C23×D181C2×C411yescharacteristic
28C23×D181D411yescharacteristic
29C2×D724D411yescharacteristic
30C2×C22⋊D181C2×C411yescharacteristic
31C2×C22⋊D181D411yescharacteristic
32D4×D181—41no—
33D4×D181—41no—
34C22×Dic181C2×C411yescharacteristic, complemented
35C2×C4×D181—21no—
36C2×C4×D181—21no—
37C2×C4×D181—21no—
38C2×C4×D181—41no—
39C2×C4×D181—41no—
40C2×C4×D181—41no—
41C2×C4×D181D411yescharacteristic
42#15668433—21no—
43#15668432—41no—
44C8×D181—41no—
45C4⋊Dic181—21no—
46#15668428—21no—
47#15668428—21no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181C2×C411yescharacteristic, complemented
51#15668426—21no—
52#15668426—21no—
53#15668426—21no—
54#15668426—41no—
55C2×D4×C181C2311yescharacteristic
56C22×C4×C181C2×C411yescharacteristic, complemented
57M16×C181—21no—
58C4⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic, complemented
62C22×D181—21no—
63C22×D181—21no—
64C22×D181—21no—
65C22×D181—21no—
66C22×D181—42no—
67C22×D181—42no—
68C22×D181C22⋊C411yescharacteristic
69C22×D181C22⋊C411yescharacteristic
70C22×D181C22⋊C411yescharacteristic
71D724—41no—
72C22⋊D181—41no—
73C22⋊D181—41no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181—41no—
77C2×Dic181—41no—
78C2×Dic181C22⋊C411yescharacteristic
79C4×D181—21no—
80C4×D181—21no—
81C4×D181—42no—
82C4×D181—42no—
83C4×D181—41no—
84C4×D181—41no—
85C4×D181—82no—
86#13089678—41no—
87C23×C181C22×C411yescharacteristic
88C23×C181C2×D411yescharacteristic
89D4×C181—41no—
90C2×C4×C181—21no—
91C2×C4×C181—21no—
92C2×C4×C181—41no—
93C2×C4×C181—41no—
94C2×C4×C181C2×D411yescharacteristic
95C8×C181—41no—
96C22×C181—21no—
97C22×C181—21no—
98C22×C181—41no—
99C22×C181—41no—
100C22×C181C2×C22⋊C411yescharacteristic
101C2×D181—22no—
102C2×D181—21no—
103C2×D181—21no—
104C2×D181—42no—
105C2×D181—42no—
106C2×D181—42no—
107C2×D181—41no—
108C2×D181—41no—
109C2×D181—81no—
110C2×D181#9911yescharacteristic
111C2×D181#9911yescharacteristic
112C4×C181—21no—
113C4×C181—41no—
114C4×C181—41no—
115C4×C181—81no—
116Dic181—21no—
117Dic181—41no—
118Dic181—41no—
119Dic181—81no—
120C2×C181—21no—
121C2×C181—41no—
122C2×C181—41no—
123C2×C181C2×#9911yescharacteristic
124D181—21no—
125D181—41no—
126D181—41no—
127D181#35212yesdirect product factor
128C181C2×#35211yescharacteristic, complemented, Sylow
129C2×#352—1811nomaximal, Sylow
130C2×#123—1811no—
131C2×#100—1811no—
132C2×#99—1811no—
133#352—1811no—
134#352—1811no—
135#352—1811no—
136#352D18111yescharacteristic, direct product factor
137C22×D4—1811no—
138C23×C4—1811no—
139C2×M16—3621no—
140#123—3621no—
141#123—3621no—
142#123—1811no—
143#123—1811no—
144#123—1811no—
145#123C2×D18111yescharacteristic
146C2×C4⋊C4—3621no—
147C2×C22⋊C4—3621no—
148C2×C22⋊C4—3621no—
149C2×C22⋊C4—1811no—
150#100—3621no—
151#100—1811no—
152#100C2×D18111yescharacteristic
153#99—3621no—
154#99—1811no—
155#99C2×D18111yescharacteristic
156C24—1811no—
157C24—1811no—
158C2×D4—1811no—
159C2×D4—1811no—
160C2×D4—1811no—
161C2×D4—7241no—
162C2×D4—7241no—
163C2×D4C22×D18111yescharacteristic
164C22×C4—3621no—
165C22×C4—3621no—
166C22×C4—3621no—
167C22×C4—1811no—
168C22×C4—1811no—
169C22×C4—7241no—
170C22×C4—7241no—
171C22×C4—7241no—
172C22×C4C4×D18111yescharacteristic, complemented
173M16—21no—
174M16—3621no—
175M16—7241no—
176C2×C8—7241no—
177C4⋊C4—21no—
178C4⋊C4—3621no—
179C4⋊C4—3621no—
180C4⋊C4—3621no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—3621no—
184C22⋊C4—3621no—
185C22⋊C4—3621no—
186C22⋊C4—3621no—
187C22⋊C4—3621no—
188C22⋊C4—1811no—
189C22⋊C4—7241no—
190C22⋊C4C4×D18111yescharacteristic, complemented
191C23—3621no—
192C23—3621no—
193C23—3621no—
194C23—3621no—
195C23—1811no—
196C23—1811no—
197C23—1811no—
198C23—7242no—
199C23—7242no—
200C23C2×C4×D18111yescharacteristic
201C23D4×D18111yescharacteristic
202D4—41no—
203D4—7241no—
204D4—7241no—
205D4—7241no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3621no—
211C2×C4—3621no—
212C2×C4—3621no—
213C2×C4—3621no—
214C2×C4—14482no—
215C2×C4—1811no—
216C2×C4—7242no—
217C2×C4—7242no—
218C2×C4—7241no—
219C2×C4—7241no—
220C2×C4—7241no—
221C2×C4—7241no—
222C2×C4D4×D18111yescharacteristic
223C8—41no—
224C8—7241no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—3622no—
230C22—3621no—
231C22—3621no—
232C22—14481no—
233C22—1811no—
234C22—1811no—
235C22—7241no—
236C22—7241no—
237C22—7242no—
238C22—7242no—
239C22—7242no—
240C22C22⋊C4×D18111yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—3621no—
246C4—14481no—
247C4—7241no—
248C4—7241no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—3621no—
253C2—1812no—
254C2—7241no—
255C2—7241no—
256C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC23×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#352×C181C2
socleC2×C181C2×#99
prime radicalC2×D4×D181C22
cosocleC2×D4×C181C23
hypercenter#352D181
hypocenterC181C2×#352
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #26041070C2semidirect
5. #26041069C2semidirect
6. #26041068C2semidirect
7. #352×C181C2semidirect
9. C22×C4×D181C4semidirect
12. #19153800C4semidirect
13. #19153799C4semidirect
19. C22⋊C4×D181C4semidirect
34. C22×Dic181C2×C4semidirect
50. C22⋊Dic181C2×C4semidirect
56. C22×C4×C181C2×C4semidirect
61. C22⋊C4×C181C2×C4semidirect
127. D181#352direct (N × H)
128. C181C2×#352semidirect
136. #352D181direct (N × H)
172. C22×C4C4×D181semidirect
190. C22⋊C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
#123×D1812
#100×D1812
#99×D1812
#260410702
#260410692
#260410682
#352×C1812
C2×#352181