Finite Group Databasebeta

← #46839846

Subgroups — #46839846

Order 34688.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576145C211yesmaximal, characteristic
2C2×C4×D1084C211yesmaximal, characteristic
3#31773837C211yesmaximal, characteristic, complemented
4#31773827C211yesmaximal, characteristic, complemented
5#31773826C211yesmaximal, characteristic, complemented
6#31773826C211yesmaximal, characteristic, complemented
7#31773781C211yesmaximal, characteristic, complemented
8#31773775C211yesmaximal, characteristic
9#31773768C211yesmaximal, characteristic, complemented
10#31773768C211yesmaximal, characteristic, complemented
11#31773759C211yesmaximal, characteristic, complemented
12#31773759C211yesmaximal, characteristic, complemented
13#31773756C211yesmaximal, characteristic, complemented
14#95×D271C211yesmaximal, characteristic, complemented
15#387×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21#22576145—24no—
22C4×D1084—24no—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C42×C271C2211yescharacteristic
49#95×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53C23×D271C2311yescharacteristic
54C23×D271C2311yescharacteristic
55C2×D1084—22no—
56C2×D1084—22no—
57C2×D1084D414yes—
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C2×C4×D271—22no—
65C2×C4×D271—22no—
66C2×C4×D271—22no—
67C2×C4×D271—22no—
68C2×C4×D271—22no—
69C2×C4×D271—22no—
70C2×C4×D271—22no—
71C2×C4×D271—22no—
72C2×C4×D271—22no—
73C2×C4×D271—41no—
74C2×C4×D271—41no—
75C2×C4×D271—41no—
76C4⋊Dic271—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—41no—
80Dic271⋊C4—41no—
81(C2×D271)⋊C4—22no—
82(C2×D271)⋊C4—22no—
83(C2×D271)⋊C4—22no—
84(C2×D271)⋊C4—22no—
85(C2×D271)⋊C4—22no—
86(C2×D271)⋊C4—22no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—41no—
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C4⋊C4×C271—22no—
97C4⋊C4×C271—22no—
98C42×C271—22no—
99C22×D271—21no—
100C22×D271—21no—
101C22×D271—21no—
102C22×D271—21no—
103C22×D271—21no—
104C22×D271—21no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271C2×D412yes—
110C22×D271C2×D412yes—
111C22×D271#4112yes—
112C22×D271#4112yes—
113D1084—28no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271#4112yes—
132C4×D271—44no—
133C4×D271—44no—
134C4×D271—44no—
135C4×D271—44no—
136C23×C271C2411yescharacteristic
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271C2×D412yes—
157C2×C4×C271#4112yes—
158C22×C271C22×D411yescharacteristic
159C22×C271C2×#4111yescharacteristic
160C22×C271C2×#4111yescharacteristic
161C22×C271C2×#4111yescharacteristic
162C22×C271C2×#4111yescharacteristic
163C22×C271H(5,𝔽2)11yescharacteristic
164C22×C2712−1+411yescharacteristic
165C2×D271—22no—
166C2×D271—22no—
167C2×D271—22no—
168C2×D271—22no—
169C2×D271—22no—
170C2×D271—22no—
171C2×D271—22no—
172C2×D271—22no—
173C2×D271—22no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—41no—
178C2×D271#12314yes—
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271—41no—
184C4×C271—41no—
185C4×C271—41no—
186Dic271—22no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191Dic271—41no—
192C2×C271C2×#12311yescharacteristic
193C2×C271#52911yescharacteristic
194C2×C271#53911yescharacteristic
195C2×C271#54611yescharacteristic
196C2×C271#54711yescharacteristic
197C2×C271#55111yescharacteristic
198C2×C271#55211yescharacteristic
199D271—24no—
200D271—42no—
201C271#241911yescharacteristic, complemented, Sylow
202#2419—2711nomaximal, Sylow
203C2×#123—2711no—
204C2×C4×D4—2711no—
205#400—2711no—
206#398—2711no—
207#397—2711no—
208#397—2711no—
209#387—2711no—
210#387D27111yescharacteristic, complemented
211#386—2711no—
212#385—2711no—
213#385—2711no—
214#381—2711no—
215#381—2711no—
216#380—2711no—
217C2×#95—2711no—
218C22×D4—2711no—
219C23×C4—2711no—
220C23×C4—2711no—
221C23×C4—2711no—
222C23×C4—2711no—
223#123—5424no—
224C4×D4—5424no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4—2711no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230C2×C4⋊C4C2×D27111yescharacteristic
231C2×C4⋊C4C2×D27111yescharacteristic
232C2×C22⋊C4—2711no—
233C2×C22⋊C4—2711no—
234C2×C22⋊C4—2711no—
235C2×C22⋊C4—2711no—
236C2×C22⋊C4—2711no—
237C2×C22⋊C4—2711no—
238C2×C22⋊C4—2711no—
239C2×C22⋊C4—2711no—
240C2×C42C2×D27111yescharacteristic
241#95—2711no—
242#95—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95—2711no—
248#95—2711no—
249#95—2711no—
250#95—2711no—
251#95C2×D27111yescharacteristic
252#95C2×D27111yescharacteristic
253#95C2×D27111yescharacteristic
254#95C2×D27111yescharacteristic
255C24—2711no—
256C24—2711no—
257C2×D4—2714no—
258C2×D4—5422no—
259C2×D4—5422no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—2711no—
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5422no—
267C22×C4—5422no—
268C22×C4—5422no—
269C22×C4—5422no—
270C22×C4—5422no—
271C22×C4—5422no—
272C22×C4—5422no—
273C22×C4—5422no—
274C22×C4—5422no—
275C22×C4—10841no—
276C22×C4—10841no—
277C22×C4—10841no—
278C22×C4C22×D27111yescharacteristic
279C22×C4C22×D27111yescharacteristic
280C22×C4C22×D27111yescharacteristic
281C22×C4C22×D27111yescharacteristic
282C22×C4C22×D27111yescharacteristic
283C22×C4C22×D27111yescharacteristic
284C22×C4C22×D27111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—5422no—
288C4⋊C4—5422no—
289C4⋊C4—5422no—
290C4⋊C4—10841no—
291C4⋊C4—10841no—
292C22⋊C4—5422no—
293C22⋊C4—5422no—
294C22⋊C4—5422no—
295C22⋊C4—5422no—
296C22⋊C4—5422no—
297C22⋊C4—5422no—
298C22⋊C4—5422no—
299C22⋊C4—10841no—
300C42—22no—
301C23—2712no—
302C23—2712no—
303C23—2712no—
304C23—2712no—
305C23—5421no—
306C23—5421no—
307C23—5421no—
308C23—5421no—
309C23—5421no—
310C23—5421no—
311C23—5421no—
312C23—5421no—
313C23—5421no—
314C23—5421no—
315C23C23×D27111yescharacteristic
316D4—5428no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2712no—
337C2×C4—5421no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—10844no—
355C2×C4—10844no—
356C2×C4—10844no—
357C2×C4—10844no—
358C2×C4D4×D27112yes—
359C2×C4#1686465412yes—
360C22—2714no—
361C22—5422no—
362C22—5422no—
363C22—5422no—
364C22—5422no—
365C22—5422no—
366C22—5422no—
367C22—5422no—
368C22—5422no—
369C22—5422no—
370C22—10841no—
371C22—10841no—
372C22—10841no—
373C22—10841no—
374C22C2×D4×D27111yescharacteristic
375C22#41×D27111yescharacteristic
376C22#41×D27111yescharacteristic
377C22#41×D27111yescharacteristic
378C22C2×#1686465411yescharacteristic
379C22#2257620611yescharacteristic
380C22#2257620811yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—5422no—
389C4—10841no—
390C4—10841no—
391C4—10841no—
392C4—10841no—
393C4—10841no—
394C2—5424no—
395C2—10842no—
396C2#123×D27111yescharacteristic
397C2#3177442511yescharacteristic
398C2#3177449511yescharacteristic
399C2#3177454311yescharacteristic
400C2#3177454911yescharacteristic
401C2#3177457211yescharacteristic
402C2#3177458811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#387×C271C2
prime radicalC22×D1084C22
hypercenter#387D271
hypocenterC271#2419
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #31773837C2semidirect
4. #31773827C2semidirect
5. #31773826C2semidirect
6. #31773826C2semidirect
7. #31773781C2semidirect
9. #31773768C2semidirect
10. #31773768C2semidirect
11. #31773759C2semidirect
12. #31773759C2semidirect
13. #31773756C2semidirect
14. #95×D271C2semidirect
15. #387×C271C2semidirect
201. C271#2419semidirect
210. #387D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761452
C2×C4×D10842
#317738372
#317738272
#317738262
#317737812
#317737752
#317737682
#317737592
#317737562
#95×D2712
#387×C2712
#2419271