Finite Group Databasebeta

← #46846281

Subgroups — #46846281

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774409C211yesmaximal, characteristic
2#117×D271C211yesmaximal, characteristic, complemented
3#31774241C211yesmaximal, characteristic, complemented
4#31774240C211yesmaximal, characteristic, complemented
5#31774234C211yesmaximal, characteristic, complemented
6#31774229C211yesmaximal, characteristic, complemented
7#31774224C211yesmaximal, characteristic, complemented
8#31774223C211yesmaximal, characteristic, complemented
9#31774217C211yesmaximal, characteristic, complemented
10#31774212C211yesmaximal, characteristic, complemented
11#31773933C211yesmaximal, characteristic, complemented
12#31773894C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14C2×(C2×D271)⋊C8C211yesmaximal, characteristic, complemented
15#427×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19#22576144—21no—
20#22576140C2211yescharacteristic
21#22576140C2211yescharacteristic
22#22576135C2211yescharacteristic
23#22576135C2211yescharacteristic
24#22576130—21no—
25C4×D1084—21no—
26C4×D1084—21no—
27#22576119C2211yescharacteristic
28#22576118C2211yescharacteristic
29#22576118C2211yescharacteristic
30#22576117—21no—
31#22576116—21no—
32C4⋊C4×D271C2211yescharacteristic
33C4⋊C4×D271C2211yescharacteristic
34C22⋊C4×D271—21no—
35C42×D271—21no—
36#22576093C2211yescharacteristic, complemented
37#22576093C2211yescharacteristic, complemented
38#22576091C2211yescharacteristic, complemented
39#22576091C2211yescharacteristic, complemented
40#22576085C2211yescharacteristic, complemented
41#22576085C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic, complemented
47#22576080C2211yescharacteristic, complemented
48(C2×D271)⋊C8C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C22×C8×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54#117×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic, complemented
56#107×C271C2211yescharacteristic, complemented
57#106×C271C2211yescharacteristic, complemented
58#106×C271C2211yescharacteristic, complemented
59C23×D271D411yescharacteristic
60#16864654—22no—
61#16864654—22no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C22×Dic271D411yescharacteristic
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—22no—
72C2×C4×D271—22no—
73C2×C4×D271D411yescharacteristic
74C2×C4×D271D411yescharacteristic
75C2×C4×D271C2311yescharacteristic
76C2×C4×D271C2311yescharacteristic
77C2×C271⋊C8—22no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271—21no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C22×C4×C271C2311yescharacteristic
97C2×C8×C271—22no—
98C2×C8×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C4⋊C4×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C42×C271—21no—
107C42×C271—21no—
108C22×D271—22no—
109C22×D271—22no—
110C22×D271—21no—
111C22×D271—41no—
112C22×D271C2×D411yescharacteristic
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—21no—
116D1084—21no—
117D1084—21no—
118D1084—42no—
119D1084—42no—
120C22⋊D271—42no—
121C2×Dic271—22no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—22no—
128C4×D271—22no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—42no—
134C4×D271—42no—
135C4×D271—42no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271C2×D411yescharacteristic
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271#4112yes—
145C2×C4×C271#4112yes—
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271#12312yes—
152C22×C271C22×D411yescharacteristic
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—44no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271#13512yes—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271#12312yes—
168C4×C271C2×#4111yescharacteristic
169C4×C271C2×#4111yescharacteristic
170Dic271—21no—
171Dic271—21no—
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175C2×C271—22no—
176C2×C271C2×#12311yescharacteristic
177C2×C271C2×#13511yescharacteristic
178C2×C271#57511yescharacteristic
179D271—22no—
180D271—41no—
181D271—81no—
182C271#309111yescharacteristic, complemented, Sylow
183#3091—2711nomaximal, Sylow
184#525—2711no—
185C2×#117—2711no—
186#485—2711no—
187#485—2711no—
188#484—2711no—
189#483—2711no—
190#482—2711no—
191#482—2711no—
192#481—2711no—
193#480—2711no—
194#427—2711no—
195#427D27111yescharacteristic, complemented
196#416—2711no—
197#416—2711no—
198C2×C22⋊C8—2711no—
199C2×#41—2711no—
200C23×C4—2711no—
201C22×C8—2711no—
202C22×C8C2×D27111yescharacteristic
203#123—5421no—
204C22⋊Q8—2711no—
205C22⋊Q8—2711no—
206#121—2711no—
207#121—2711no—
208#120—5421no—
209C4×D4—5421no—
210C4×D4—5421no—
211#117—2711no—
212#117—2711no—
213#117—2711no—
214#117—5421no—
215#117—5421no—
216#117C2×D27111yescharacteristic
217#117C2×D27111yescharacteristic
218C2×C4⋊C4—2711no—
219C2×C4⋊C4—2711no—
220C2×C22⋊C4—5421no—
221C2×C42—5421no—
222#107—2711no—
223#107—2711no—
224#107C2×D27111yescharacteristic, complemented
225#107C2×D27111yescharacteristic, complemented
226#106—2711no—
227#106—2711no—
228#106C2×D27111yescharacteristic, complemented
229#106C2×D27111yescharacteristic, complemented
230#103—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237#102—2711no—
238C22⋊C8—2711no—
239C22⋊C8—2711no—
240C22⋊C8—2711no—
241C22⋊C8—2711no—
242C24—2711no—
243#41—5422no—
244#41—5422no—
245C2×Q8—2711no—
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4—5421no—
250C22×C4—2711no—
251C22×C4—2711no—
252C22×C4—2711no—
253C22×C4—2711no—
254C22×C4—2711no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4—5422no—
258C22×C4—5422no—
259C22×C4—5421no—
260C22×C4C22×D27111yescharacteristic
261C2×C8—22no—
262C2×C8—2711no—
263C2×C8—2711no—
264C2×C8—5422no—
265C2×C8C22×D27111yescharacteristic
266C2×C8C22×D27111yescharacteristic
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22×D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—21no—
287C42—21no—
288C42—5421no—
289C42—5421no—
290C42—5421no—
291C23—2711no—
292C23—2711no—
293C23—5422no—
294C23—5422no—
295C23—5421no—
296C23—10841no—
297C23D4×D27111yescharacteristic
298Q8—5421no—
299Q8—5421no—
300D4—5421no—
301D4—5421no—
302D4—10842no—
303D4—10842no—
304D4—10842no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—5422no—
312C2×C4—5422no—
313C2×C4—5422no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—10842no—
322C2×C4—10842no—
323C2×C4—10842no—
324C2×C4D4×D27111yescharacteristic
325C2×C4#1686465412yes—
326C2×C4#1686465512yes—
327C2×C4C23×D27111yescharacteristic
328C8—21no—
329C8—21no—
330C8—5421no—
331C8—5421no—
332C22—21no—
333C22—21no—
334C22—2712no—
335C22—5421no—
336C22—5421no—
337C22—5421no—
338C22—5421no—
339C22—5421no—
340C22—10844no—
341C22—10841no—
342C22—10841no—
343C22—10841no—
344C22#2257614512yes—
345C22C2×D4×D27111yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—5421no—
351C4—5421no—
352C4—10841no—
353C4—10841no—
354C4—10841no—
355C4#2257614512yes—
356C4C2×#1686465411yescharacteristic
357C4C2×#1686465511yescharacteristic
358C2—22no—
359C2—5422no—
360C2—10841no—
361C2—21681no—
362C2C2×#2257614511yescharacteristic
363C2#135×D27111yescharacteristic
364C2#3177469411yescharacteristic
subgroupG/H
centerC2×C4D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#427×C271C2
socleC22×C271C22×D4
prime radical#31774409C2
hypercenter#427D271
hypocenterC271#3091
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #117×D271C2semidirect
3. #31774241C2semidirect
4. #31774240C2semidirect
5. #31774234C2semidirect
6. #31774229C2semidirect
7. #31774224C2semidirect
8. #31774223C2semidirect
9. #31774217C2semidirect
10. #31774212C2semidirect
11. #31773933C2semidirect
12. #31773894C2semidirect
13. #31773891C2semidirect
14. C2×(C2×D271)⋊C8C2semidirect
15. #427×C271C2semidirect
36. #22576093C22semidirect
37. #22576093C22semidirect
38. #22576091C22semidirect
39. #22576091C22semidirect
40. #22576085C22semidirect
41. #22576085C22semidirect
42. #22576083C22semidirect
43. #22576083C22semidirect
44. #22576082C22semidirect
45. #22576082C22semidirect
46. #22576080C22semidirect
47. #22576080C22semidirect
55. #107×C271C22semidirect
56. #107×C271C22semidirect
57. #106×C271C22semidirect
58. #106×C271C22semidirect
182. C271#3091semidirect
195. #427D271semidirect
224. #107C2×D271semidirect
225. #107C2×D271semidirect
228. #106C2×D271semidirect
229. #106C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744092
#117×D2712
#317742412
#317742402
#317742342
#317742292
#317742242
#317742232
#317742172
#317742122
#317739332
#317738942
#317738912
C2×(C2×D271)⋊C82
#427×C2712
#3091271