Finite Group Databasebeta

← #46814023

Subgroups — #46814023

Order 34432.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22575421C211yesmaximal, characteristic
2#31770508C211yesmaximal, characteristic, complemented
3#31770505C212yesmaximal, complemented
4#31770499C212yesmaximal, complemented
5#31770497C212yesmaximal, complemented
6#31770494C212yesmaximal, complemented
7#31770484C212yesmaximal, complemented
8#31770460C212yesmaximal, complemented
9#386×C269C211yesmaximal, characteristic, complemented
10C24×D269C2211yescharacteristic
11C22×C22⋊D269C2212yes—
12#22575421C418yescomplemented
13C2×C4⋊Dic269C2211yescharacteristic, complemented
14C2×#16864381C2212yescomplemented
15C2×C22⋊Dic269C2212yes—
16C2×C22⋊Dic269C2211yescharacteristic
17C2×C22⋊Dic269C2212yescomplemented
18C2×#16864379C2212yes—
19C2×#16864379C2212yes—
20C2×#16864379C2212yes—
21C2×#16864379C2212yes—
22C2×#16864379C2212yes—
23C2×C4×Dic269C2211yescharacteristic, complemented
24C2×C4×Dic269C2212yescomplemented
25#22575360C2212yescomplemented
26#22575360C2212yescomplemented
27C22×D4×C269C2211yescharacteristic
28C23×C4×C269C2211yescharacteristic, complemented
29C2×C22⋊C4×C269C2212yes—
30C2×C22⋊C4×C269C2211yescharacteristic, complemented
31#95×C269C2212yescomplemented
32C23×D269—42no—
33C23×D269—41no—
34C23×D269—41no—
35C23×D269—42no—
36C23×D269C2×C414yescomplemented
37C23×D269C2312yes—
38C2×C22⋊D269—42no—
39C2×C22⋊D269—42no—
40C2×C22⋊D269C2×C418yes—
41C22×Dic269C2312yes—
42C22×Dic269C2312yes—
43C22×Dic269C2312yes—
44C4⋊Dic269—41no—
45#16864381—42no—
46C22⋊Dic269—42no—
47C22⋊Dic269—42no—
48C22⋊Dic269C2×C414yescomplemented
49#16864379—42no—
50#16864379—42no—
51#16864379—42no—
52#16864379—42no—
53#16864379C2×C418yescomplemented
54C4×Dic269—41no—
55C4×Dic269—42no—
56C24×C269C2311yescharacteristic
57C24×C269C2311yescharacteristic
58C2×D4×C269—41no—
59C2×D4×C269—41no—
60C2×D4×C269C2×C414yes—
61C22×C4×C269—42no—
62C22×C4×C269—41no—
63C22×C4×C269C2312yes—
64C22×C4×C269C2312yes—
65C22×C4×C269C2311yescharacteristic
66C22⋊C4×C269—42no—
67C22⋊C4×C269—41no—
68C22×D269—22no—
69C22×D269—22no—
70C22×D269—22no—
71C22×D269—22no—
72C22×D269—22no—
73C22×D269—22no—
74C22×D269—48no—
75C22×D269—44no—
76C22×D269—48no—
77C22×D269—44no—
78C22×D269—44no—
79C22×D269C22×C414yes—
80C22⋊D269—48no—
81C2×Dic269—22no—
82C2×Dic269—22no—
83C2×Dic269—22no—
84C2×Dic269—22no—
85C2×Dic269—22no—
86C2×Dic269—22no—
87C2×Dic269—22no—
88C2×Dic269—22no—
89C2×Dic269C22×C414yes—
90C23×C269—21no—
91C23×C269—22no—
92C23×C269—21no—
93C23×C269—22no—
94C23×C269—21no—
95C23×C269—21no—
96C23×C269—22no—
97C23×C269—22no—
98C23×C269C22×C412yes—
99C23×C269C22×C412yes—
100C23×C269C2411yescharacteristic
101D4×C269—44no—
102C2×C4×C269—21no—
103C2×C4×C269—21no—
104C2×C4×C269—22no—
105C2×C4×C269—22no—
106C2×C4×C269—22no—
107C2×C4×C269—22no—
108C2×C4×C269—22no—
109C2×C4×C269—22no—
110C2×C4×C269—44no—
111C2×C4×C269C22×C412yes—
112C22×C269—22no—
113C22×C269—22no—
114C22×C269—24no—
115C22×C269—24no—
116C22×C269—42no—
117C22×C269—41no—
118C22×C269—42no—
119C22×C269—41no—
120C22×C269—41no—
121C22×C269—41no—
122C22×C269C23×C411yescharacteristic
123C22×C269H(5,𝔽2)11yescharacteristic
124C22×C269H(5,𝔽2)12yes—
125C22×C269H(5,𝔽2)11yescharacteristic
126C22×C269H(5,𝔽2)12yes—
127C2×D269—24no—
128C2×D269—24no—
129C2×D269—24no—
130C2×D269—416no—
131C2×D269—42no—
132C2×D269—42no—
133C2×D269—42no—
134C2×D269—42no—
135C4×C269—22no—
136C4×C269—42no—
137C4×C269—42no—
138Dic269—24no—
139Dic269—42no—
140Dic269—42no—
141C2×C269—42no—
142C2×C269—42no—
143C2×C269#51812yes—
144C2×C269#51811yescharacteristic
145C2×C269#56012yes—
146C2×C269#56011yescharacteristic
147C2×C269H(3,𝔽4)11yescharacteristic
148D269—44no—
149C269#237511yescharacteristic, complemented, Sylow
150#2375—2691nomaximal, Sylow
151C2×#120—2691no—
152#390—2691no—
153#390—2692no—
154#388—2692no—
155#388—2692no—
156#388—2692no—
157#386—2692no—
158#386D26911yescharacteristic, complemented
159#379—2692no—
160C25—2691no—
161C22×D4—2692no—
162C22×D4C2×D26911yescharacteristic
163C23×C4C2×D26911yescharacteristic, complemented
164#120—2698no—
165C2×C4⋊C4—2692no—
166C2×C4⋊C4—2691no—
167C2×C22⋊C4—2691no—
168C2×C22⋊C4—2692no—
169C2×C22⋊C4—2692no—
170C2×C22⋊C4—2692no—
171C2×C22⋊C4—2692no—
172C2×C22⋊C4—2692no—
173C2×C22⋊C4—2692no—
174C2×C22⋊C4—2692no—
175C2×C22⋊C4C2×D26912yes—
176C2×C22⋊C4C2×D26911yescharacteristic, complemented
177C2×C42—2692no—
178C2×C42—2691no—
179#95—2692no—
180#95—2692no—
181#95C2×D26912yescomplemented
182C24—2692no—
183C24—2694no—
184C24—10762no—
185C24—10761no—
186C24—10761no—
187C24—10762no—
188C24C22×D26911yescharacteristic
189C24C22×D26911yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—2698no—
193C2×D4—10762no—
194C2×D4—10762no—
195C2×D4C4×D26914yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—2692no—
199C22×C4—2692no—
200C22×C4—2692no—
201C22×C4C22×D26912yes—
202C22×C4C22×D26912yes—
203C22×C4C22×D26911yescharacteristic
204C4⋊C4—10762no—
205C4⋊C4—10761no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—2694no—
209C22⋊C4—2698no—
210C22⋊C4—10762no—
211C22⋊C4—10762no—
212C22⋊C4—10762no—
213C22⋊C4—10762no—
214C22⋊C4—10762no—
215C22⋊C4—10762no—
216C42—10761no—
217C42—10762no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—2694no—
227C23—5382no—
228C23—5382no—
229C23—5382no—
230C23—5382no—
231C23—5382no—
232C23—5382no—
233C23—10768no—
234C23—10764no—
235C23—10768no—
236C23—10764no—
237C23—10764no—
238C23C2×C4×D26912yes—
239C23C2×C4×D26912yes—
240C23C23×D26911yescharacteristic
241D4—44no—
242D4—10768no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—2694no—
253C2×C4—5382no—
254C2×C4—5382no—
255C2×C4—5382no—
256C2×C4—5382no—
257C2×C4—5382no—
258C2×C4—5382no—
259C2×C4—5382no—
260C2×C4—5382no—
261C2×C4C2×C4×D26912yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—5384no—
273C22—5384no—
274C22—5384no—
275C22—107616no—
276C22—10762no—
277C22—10762no—
278C22—10762no—
279C22—10762no—
280C22C22×C4×D26911yescharacteristic
281C22#2257549812yes—
282C22#2257549811yescharacteristic
283C22#2257549811yescharacteristic
284C22#2257549912yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—5384no—
289C4—10762no—
290C4—10762no—
291C2—42no—
292C2—42no—
293C2—10764no—
294C2#3177107712yes—
295C2#3177107811yescharacteristic
296C2#3177134312yes—
297C2#3177134511yescharacteristic
298C2#3177134811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#386×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575421C2
hypercenter#386D269
hypocenterC269#2375
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #31770508C2semidirect
3. #31770505C2semidirect
4. #31770499C2semidirect
5. #31770497C2semidirect
6. #31770494C2semidirect
7. #31770484C2semidirect
8. #31770460C2semidirect
9. #386×C269C2semidirect
12. #22575421C4semidirect
13. C2×C4⋊Dic269C22semidirect
14. C2×#16864381C22semidirect
17. C2×C22⋊Dic269C22semidirect
23. C2×C4×Dic269C22semidirect
24. C2×C4×Dic269C22semidirect
25. #22575360C22semidirect
26. #22575360C22semidirect
28. C23×C4×C269C22semidirect
30. C2×C22⋊C4×C269C22semidirect
31. #95×C269C22semidirect
36. C23×D269C2×C4semidirect
48. C22⋊Dic269C2×C4semidirect
53. #16864379C2×C4semidirect
149. C269#2375semidirect
158. #386D269semidirect
163. C23×C4C2×D269semidirect
176. C2×C22⋊C4C2×D269semidirect
181. #95C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754212
#317705082
#317705052
#317704992
#317704972
#317704942
#317704842
#317704602
#386×C2692
#2375269