Finite Group Databasebeta

← #46846563

Subgroups — #46846563

Order 34688.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774502C211yesmaximal, characteristic, complemented
2#31774215C211yesmaximal, characteristic, complemented
3#31774210C211yesmaximal, characteristic, complemented
4#31774204C211yesmaximal, characteristic, complemented
5#31774197C211yesmaximal, characteristic, complemented
6#31774097C211yesmaximal, characteristic, complemented
7#31774092C211yesmaximal, characteristic, complemented
8#31774042C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774007C211yesmaximal, characteristic, complemented
11#31773995C211yesmaximal, characteristic, complemented
12C4×#16864641C211yesmaximal, characteristic, complemented
13C4×#16864639C211yesmaximal, characteristic, complemented
14#31773958C211yesmaximal, characteristic, complemented
15#520×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20#22576157C2211yescharacteristic
21#22576152C2211yescharacteristic
22#22576145—21no—
23#22576145—21no—
24#22576134C2211yescharacteristic
25#22576134C2211yescharacteristic
26#22576133—21no—
27#22576133—21no—
28C4×D1084C2211yescharacteristic
29C4×D1084C2211yescharacteristic
30#22576093C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35C4⋊(C271⋊C8)C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576081C2211yescharacteristic
39#22576081C2211yescharacteristic
40#22576080C2211yescharacteristic
41#22576080C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44C22⋊(C271⋊C8)C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46#22576071C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48C2×#41×C271C2211yescharacteristic
49C4×Q8×C271C2211yescharacteristic
50C4×Q8×C271C2211yescharacteristic
51C4×D4×C271—21no—
52C4×D4×C271—21no—
53C4×D4×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55#117×C271—21no—
56#117×C271C2211yescharacteristic
57C2×C4⋊C4×C271—21no—
58C2×C4⋊C4×C271C2211yescharacteristic
59C2×D1084—21no—
60C2×D1084—21no—
61C2×D1084C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C4×D271—21no—
66C2×C4×D271—21no—
67#16864641—22no—
68#16864641—22no—
69#16864639—22no—
70#16864639—22no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79(C2×D271)⋊C4—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83#41×C271—22no—
84#41×C271—22no—
85C2×Q8×C271C2311yescharacteristic
86C2×D4×C271—21no—
87C2×D4×C271C2311yescharacteristic
88C22×C4×C271—21no—
89C22×C4×C271—21no—
90C22×C4×C271—21no—
91C22×C4×C271—21no—
92C22×C4×C271C2311yescharacteristic
93C4⋊C4×C271—21no—
94C4⋊C4×C271—21no—
95C4⋊C4×C271—21no—
96C4⋊C4×C271D411yescharacteristic
97C4⋊C4×C271D411yescharacteristic
98C4⋊C4×C271C2311yescharacteristic
99C4⋊C4×C271C2311yescharacteristic
100C22⋊C4×C271—21no—
101C22⋊C4×C271—21no—
102C22⋊C4×C271D411yescharacteristic
103C22⋊C4×C271D411yescharacteristic
104C42×C271—21no—
105C42×C271—21no—
106C42×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—21no—
110D1084—21no—
111D1084—21no—
112D1084—21no—
113D1084—21no—
114D1084—81no—
115D1084—81no—
116C22⋊D271—81no—
117C22⋊D271—81no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C4×D271—81no—
121C4×D271—81no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C271⋊C8—21no—
125C271⋊C8—21no—
126C23×C271—21no—
127C23×C271C2×D411yescharacteristic
128Q8×C271—21no—
129Q8×C271#4112yes—
130D4×C271—24no—
131D4×C271—21no—
132D4×C271#4112yes—
133C2×C4×C271—24no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—81no—
140C2×C4×C271—81no—
141C2×C4×C271—81no—
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C22×C271—21no—
149C22×C271—22no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271C22×D411yescharacteristic
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C4×C271—21no—
162C4×C271—22no—
163C4×C271—21no—
164C4×C271—21no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271C2×#4111yescharacteristic
171Dic271—41no—
172Dic271—41no—
173C2×C271—41no—
174C2×C271—42no—
175C2×C271#52511yescharacteristic
176C2×C271#57611yescharacteristic
177C2×C271#57711yescharacteristic
178D271—81no—
179D271—81no—
180C271#311911yescharacteristic, complemented, Sylow
181#3119—2711nomaximal, Sylow
182#540—2711no—
183#520D27111yescharacteristic, complemented
184#481—2711no—
185#480—2711no—
186#479—2711no—
187#478—2711no—
188#460—2711no—
189#459—2711no—
190#449—2711no—
191#449—2711no—
192#442—2711no—
193#440—2711no—
194C4×SD16—2711no—
195C4×D8—2711no—
196#433—2711no—
197C2×#41C2×D27111yescharacteristic
198C2×SD16—2711no—
199C2×SD16—2711no—
200C2×D8—2711no—
201C2×D8—2711no—
202#127—2711no—
203#125—2711no—
204#123—5421no—
205#123—5421no—
206#121—2711no—
207#121—2711no—
208#121—5421no—
209#121—5421no—
210C4×Q8C2×D27111yescharacteristic
211C4×Q8C2×D27111yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4C2×D27111yescharacteristic
217C4×D4C2×D27111yescharacteristic
218#117—21no—
219#117C2×D27111yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D27111yescharacteristic
222#107—2711no—
223#107—2711no—
224#106—2711no—
225#106—2711no—
226C4⋊C8—2711no—
227C4⋊C8—2711no—
228#103—2711no—
229#103—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102—2711no—
236C22⋊C8—2711no—
237C22⋊C8—2711no—
238#97—2711no—
239C4×C8—2711no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D27111yescharacteristic
243C2×D4—21no—
244C2×D4—2711no—
245C2×D4—2711no—
246C2×D4—5421no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4—5421no—
250C2×D4C22×D27111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4C22×D27111yescharacteristic
258SD16—5422no—
259SD16—5422no—
260D8—5422no—
261D8—5422no—
262C2×C8—2711no—
263C2×C8—2711no—
264C2×C8—2711no—
265C2×C8—2711no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—2711no—
270C4⋊C4—2711no—
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⋊C4C22⋊D27111yescharacteristic
284C22⋊C4C22⋊D27111yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D27111yescharacteristic
288C42C22×D27111yescharacteristic
289C23—21no—
290C23—5421no—
291C23—5421no—
292C23C2×C22⋊D27111yescharacteristic
293Q8—21no—
294Q8#1686465412yes—
295D4—24no—
296D4—21no—
297D4—5421no—
298D4—5421no—
299D4—5421no—
300D4—5421no—
301D4—21681no—
302D4—21681no—
303D4—21681no—
304D4—21681no—
305D4#1686465412yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—21681no—
318C2×C4—21681no—
319C2×C4C2×C22⋊D27111yescharacteristic
320C2×C4C2×C22⋊D27111yescharacteristic
321C2×C4C2×C22⋊D27111yescharacteristic
322C2×C4C2×C22⋊D27111yescharacteristic
323C2×C4C2×C22⋊D27111yescharacteristic
324C2×C4C23×D27111yescharacteristic
325C8—5421no—
326C8—5421no—
327C8—5421no—
328C8—5421no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—10841no—
336C22—10841no—
337C22—10841no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22C22×C22⋊D27111yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—10841no—
351C4—10841no—
352C4C2×#1686465411yescharacteristic
353C4C2×#1686465411yescharacteristic
354C2—41no—
355C2—42no—
356C2—21681no—
357C2—21681no—
358C2#3177440811yescharacteristic
359C2#3177470011yescharacteristic
360C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#520×C271C2
socleC22×C271C22×D4
hypercenter#520D271
hypocenterC271#3119
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #31774502C2semidirect
2. #31774215C2semidirect
3. #31774210C2semidirect
4. #31774204C2semidirect
5. #31774197C2semidirect
6. #31774097C2semidirect
7. #31774092C2semidirect
8. #31774042C2semidirect
9. #31774042C2semidirect
10. #31774007C2semidirect
11. #31773995C2semidirect
12. C4×#16864641C2semidirect
13. C4×#16864639C2semidirect
14. #31773958C2semidirect
15. #520×C271C2semidirect
180. C271#3119semidirect
183. #520D271semidirect

Maximal subgroups

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

subgroupindex
#317745022
#317742152
#317742102
#317742042
#317741972
#317740972
#317740922
#317740422
#317740072
#317739952
C4×#168646412
C4×#168646392
#317739582
#520×C2712
#3119271