Finite Group Databasebeta

← #46845942

Subgroups — #46845942

Order 34688.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22576180C211yesmaximal, characteristic, complemented
2C23×C271⋊C8C211yesmaximal, characteristic, complemented
3#31774408C211yesmaximal, characteristic, complemented
4#31774136C212yesmaximal, complemented
5#31774131C212yesmaximal, complemented
6#31774128C212yesmaximal, complemented
7#31774124C212yesmaximal, complemented
8#31773933C211yesmaximal, characteristic, complemented
9#31773893C211yesmaximal, characteristic, complemented
10#31773891C211yesmaximal, characteristic, complemented
11#525×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13#22576180—24no—
14C2×#16864644C2211yescharacteristic, complemented
15C2×#16864642C2211yescharacteristic, complemented
16C2×#16864641C2211yescharacteristic, complemented
17C2×#16864639C2211yescharacteristic, complemented
18C22×C271⋊C8—24no—
19C22×C271⋊C8C2212yes—
20C22×C271⋊C8C2212yes—
21C22×C271⋊C8C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic
23#22576143—21no—
24#22576139C2212yes—
25#22576134C2212yes—
26#22576129—21no—
27C4×C22⋊D271—22no—
28#22576119C2211yescharacteristic
29#22576093C2212yescomplemented
30#22576091C2212yescomplemented
31#22576084C2212yescomplemented
32#22576083C2212yescomplemented
33#22576081C2212yescomplemented
34#22576080C2212yescomplemented
35C2×#41×C271C2211yescharacteristic
36C23×C4×C271C2211yescharacteristic
37#123×C271—21no—
38C22⋊Q8×C271C2212yes—
39#121×C271C2212yes—
40#120×C271—21no—
41C4×D4×C271—22no—
42#117×C271C2211yescharacteristic
43#16864654—22no—
44#16864654—22no—
45C2×Dic542C2311yescharacteristic
46C2×D1084C2311yescharacteristic
47C2×C22⋊D271—22no—
48C2×C22⋊D271—21no—
49C2×C4×D271—21no—
50#16864644—22no—
51#16864642—22no—
52#16864641—22no—
53#16864639—22no—
54C2×C271⋊C8—24no—
55C2×C271⋊C8—24no—
56C2×C271⋊C8—22no—
57C2×C271⋊C8D414yescomplemented
58C2×C271⋊C8C2312yes—
59C2×C271⋊C8C2311yescharacteristic
60C2×C271⋊C8C2311yescharacteristic
61C4⋊Dic271C2312yes—
62Dic271⋊C4—22no—
63C22⋊Dic271—22no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—22no—
66C4×Dic271—21no—
67C24×C271D411yescharacteristic
68#41×C271—22no—
69#41×C271—22no—
70C2×Q8×C271C2311yescharacteristic
71C2×D4×C271—22no—
72C2×D4×C271—21no—
73C2×D4×C271C2311yescharacteristic
74C22×C4×C271—22no—
75C22×C4×C271—22no—
76C22×C4×C271—21no—
77C22×C4×C271D411yescharacteristic
78C22×C4×C271D412yes—
79C22×C4×C271C2312yes—
80C22×C4×C271C2311yescharacteristic
81C4⋊C4×C271—22no—
82C4⋊C4×C271C2312yes—
83C22⋊C4×C271—21no—
84C22⋊C4×C271—22no—
85C22⋊C4×C271—22no—
86C42×C271—21no—
87C22×D271—21no—
88Dic542—21no—
89Dic542—21no—
90D1084—21no—
91D1084—21no—
92C22⋊D271—42no—
93C22⋊D271—44no—
94C2×Dic271—22no—
95C2×Dic271—21no—
96C4×D271—42no—
97C271⋊C8—22no—
98C271⋊C8C2×D412yes—
99C271⋊C8C2×D412yes—
100C23×C271—22no—
101C23×C271—24no—
102C23×C271—21no—
103C23×C271C2×D411yescharacteristic
104C23×C271C2×D412yes—
105Q8×C271—21no—
106Q8×C271—21no—
107D4×C271—21no—
108D4×C271—21no—
109D4×C271—42no—
110D4×C271—44no—
111C2×C4×C271—22no—
112C2×C4×C271—24no—
113C2×C4×C271—22no—
114C2×C4×C271—22no—
115C2×C4×C271—22no—
116C2×C4×C271—21no—
117C2×C4×C271—42no—
118C2×C4×C271C2×D412yes—
119C2×C4×C271C2×D412yes—
120C2×C4×C271C2×D411yescharacteristic
121C2×C4×C271#4112yes—
122C2×C4×C271C2411yescharacteristic
123C22×C271—28no—
124C22×C271—22no—
125C22×C271—22no—
126C22×C271—22no—
127C22×C271—41no—
128C22×C271—42no—
129C22×C271#12112yes—
130C22×C271#13514yescomplemented
131C22×C271C22×D411yescharacteristic
132C2×D271—41no—
133C2×D271—42no—
134C4×C271—22no—
135C4×C271—41no—
136C4×C271—42no—
137C4×C271#12112yes—
138C4×C271C22×D411yescharacteristic
139C4×C271C2×#4111yescharacteristic
140Dic271—41no—
141Dic271—42no—
142C2×C271—22no—
143C2×C271—24no—
144C2×C271—81no—
145C2×C271C2×#12111yescharacteristic
146C2×C271C2×#13512yes—
147D271—81no—
148C271#305011yescharacteristic, complemented, Sylow
149#3050—2711nomaximal, Sylow
150C2×#135—2711no—
151C23×C8—2711no—
152#525—2711no—
153#525D27111yescharacteristic, complemented
154#467—2712no—
155#466—2712no—
156#465—2712no—
157#465—2712no—
158#427—2711no—
159#416—2711no—
160#416—2711no—
161C2×#41—2711no—
162C2×#41C2×D27111yescharacteristic
163C23×C4C2×D27111yescharacteristic
164#135—5424no—
165C2×Q16—2711no—
166C2×SD16—2711no—
167C2×SD16—2711no—
168C2×D8—2711no—
169C22×C8—2711no—
170C22×C8—2711no—
171C22×C8—2712no—
172C22×C8—2712no—
173C22×C8—5424no—
174#123—21no—
175#123—5421no—
176C22⋊Q8—2712no—
177C22⋊Q8C2×D27112yes—
178#121—2712no—
179#121C2×D27112yes—
180#120—21no—
181#120—5421no—
182C4×D4—22no—
183C4×D4—5422no—
184#117—2711no—
185#117C2×D27111yescharacteristic
186#107—2712no—
187#106—2712no—
188#103—2712no—
189#103—2712no—
190#102—2712no—
191#102—2712no—
192C24C22⋊D27111yescharacteristic
193#41—22no—
194#41—22no—
195#41—5422no—
196#41—5422no—
197C2×Q8—2711no—
198C2×Q8C22×D27111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—2711no—
202C2×D4—5422no—
203C2×D4—5421no—
204C2×D4C22×D27111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—5421no—
209C22×C4C22⋊D27111yescharacteristic
210C22×C4C22⋊D27112yes—
211C22×C4C22×D27112yes—
212C22×C4C22×D27111yescharacteristic
213Q16—5422no—
214SD16—5422no—
215SD16—5422no—
216D8—5422no—
217C2×C8—2714no—
218C2×C8—2711no—
219C2×C8—2711no—
220C2×C8—2712no—
221C2×C8—5424no—
222C2×C8—5424no—
223C2×C8—5422no—
224C4⋊C4—22no—
225C4⋊C4—2712no—
226C4⋊C4—5422no—
227C4⋊C4C22×D27112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—5421no—
232C22⋊C4—5422no—
233C22⋊C4—5422no—
234C42—21no—
235C42—5421no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—5421no—
240C23C2×C22⋊D27111yescharacteristic
241C23C2×C22⋊D27112yes—
242Q8—21no—
243Q8—21no—
244Q8—5421no—
245Q8—5421no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—5421no—
251D4—5421no—
252D4—10842no—
253D4—10844no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—5422no—
262C2×C4—5421no—
263C2×C4—10842no—
264C2×C4D4×D27112yes—
265C2×C4C2×C22⋊D27112yes—
266C2×C4C2×C22⋊D27111yescharacteristic
267C2×C4#1686465312yes—
268C2×C4C23×D27111yescharacteristic
269C8—2712no—
270C8—2712no—
271C8—5422no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—10841no—
279C22—10842no—
280C22#2257613212yes—
281C22#2257618014yescomplemented
282C22C22×C22⋊D27111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—10841no—
287C4—10842no—
288C4#2257613212yes—
289C4C2×D4×D27111yescharacteristic
290C4C2×#1686465311yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—21681no—
295C2C2×#2257613211yescharacteristic
296C2C2×#2257618012yes—
subgroupG/H
centerC2×C4C2×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#525×C271C2
socleC22×C271C22×D4
hypercenter#525D271
hypocenterC271#3050
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#22576180C2semidirect
2. C23×C271⋊C8C2semidirect
3. #31774408C2semidirect
4. #31774136C2semidirect
5. #31774131C2semidirect
6. #31774128C2semidirect
7. #31774124C2semidirect
8. #31773933C2semidirect
9. #31773893C2semidirect
10. #31773891C2semidirect
11. #525×C271C2semidirect
14. C2×#16864644C22semidirect
15. C2×#16864642C22semidirect
16. C2×#16864641C22semidirect
17. C2×#16864639C22semidirect
29. #22576093C22semidirect
30. #22576091C22semidirect
31. #22576084C22semidirect
32. #22576083C22semidirect
33. #22576081C22semidirect
34. #22576080C22semidirect
57. C2×C271⋊C8D4semidirect
130. C22×C271#135semidirect
148. C271#3050semidirect
153. #525D271semidirect
281. C22#22576180semidirect

Maximal subgroups

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

subgroupindex
C2×#225761802
C23×C271⋊C82
#317744082
#317741362
#317741312
#317741282
#317741242
#317739332
#317738932
#317738912
#525×C2712
#3050271