Finite Group Databasebeta

← #30677961

Subgroups — #30677961

Order 16768.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22553983C211yesmaximal, characteristic, complemented
2#22553678C211yesmaximal, characteristic, complemented
3#22553636C211yesmaximal, characteristic, complemented
4#22553625C211yesmaximal, characteristic, complemented
5#22553616C211yesmaximal, characteristic, complemented
6#22553563C211yesmaximal, characteristic, complemented
7#22553546C211yesmaximal, characteristic, complemented
8#22553507C211yesmaximal, characteristic, complemented
9#22553455C211yesmaximal, characteristic, complemented
10#22553455C211yesmaximal, characteristic, complemented
11#22553412C211yesmaximal, characteristic, complemented
12#22553384C211yesmaximal, characteristic, complemented
13#22553311C211yesmaximal, characteristic, complemented
14#22553306C211yesmaximal, characteristic, complemented
15#553×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C2×#14523513C2211yescharacteristic
18C2×#14523501C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×#14523498C2211yescharacteristic
21C2×#14523497C2211yescharacteristic
22C22×C131⋊C8C2211yescharacteristic
23#16862243C2211yescharacteristic
24#16862242—21no—
25#16862220C2211yescharacteristic
26#16862220C2211yescharacteristic
27#16862219—21no—
28#16862219—21no—
29C4×Dic262C2211yescharacteristic
30#16862208—21no—
31#16862179C2211yescharacteristic
32#16862179C2211yescharacteristic
33#16862177C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic
35#16862169C2211yescharacteristic
36#16862169C2211yescharacteristic
37#16862169C2211yescharacteristic
38#16862167C2211yescharacteristic
39#16862166C2211yescharacteristic
40#16862166C2211yescharacteristic
41#16862166C2211yescharacteristic
42#16862166C2211yescharacteristic
43#16862166C2211yescharacteristic
44#16862166C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic
47#16862157C2211yescharacteristic
48#126×C131—21no—
49#125×C131C2211yescharacteristic
50#123×C131—21no—
51#121×C131—21no—
52#121×C131C2211yescharacteristic
53#121×C131C2211yescharacteristic
54C4×D4×C131—21no—
55C4×D4×C131C2211yescharacteristic
56C4×D4×C131C2211yescharacteristic
57#117×C131C2211yescharacteristic
58C2×C4⋊C4×C131C2211yescharacteristic
59#14523513—22no—
60#14523513—22no—
61#14523513—41no—
62#14523513—41no—
63C2×Dic262C2311yescharacteristic
64C2×D524—21no—
65C2×D524—21no—
66C2×D524C2311yescharacteristic
67C2×D524C2311yescharacteristic
68C2×C22⋊D131—21no—
69C2×C22⋊D131—21no—
70C2×C22⋊D131—21no—
71C2×C22⋊D131—21no—
72C2×C4×D131—21no—
73C2×C4×D131—21no—
74#14523501—22no—
75#14523498—41no—
76#14523498—41no—
77#14523497—41no—
78C2×C131⋊C8—41no—
79C2×C131⋊C8C2311yescharacteristic
80C2×C131⋊C8C2311yescharacteristic
81C2×C131⋊C8C2311yescharacteristic
82C2×C131⋊C8C2311yescharacteristic
83C4⋊Dic131C2311yescharacteristic
84Dic131⋊C4—21no—
85(C2×D131)⋊C4—21no—
86(C2×D131)⋊C4—21no—
87C4×Dic131—21no—
88C2×D4×C131—21no—
89C2×D4×C131—21no—
90C2×D4×C131D411yescharacteristic
91C2×D4×C131C2311yescharacteristic
92C22×C4×C131—21no—
93C22×C4×C131—21no—
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C4⋊C4×C131—21no—
97C4⋊C4×C131—21no—
98C4⋊C4×C131—21no—
99C4⋊C4×C131D411yescharacteristic
100C4⋊C4×C131C2311yescharacteristic
101C4⋊C4×C131C2311yescharacteristic
102C4⋊C4×C131C2311yescharacteristic
103C22⋊C4×C131—21no—
104C22⋊C4×C131—21no—
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131D411yescharacteristic
108C22⋊C4×C131D411yescharacteristic
109C42×C131—21no—
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112C22×D131—21no—
113Dic262—21no—
114Dic262#4112yes—
115D524—21no—
116D524—21no—
117D524—21no—
118D524—21no—
119D524—42no—
120D524—81no—
121C22⋊D131—42no—
122C22⋊D131—42no—
123C22⋊D131—81no—
124C22⋊D131—81no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C4×D131—42no—
128C4×D131—42no—
129C131⋊C8—21no—
130C131⋊C8—21no—
131C131⋊C8—21no—
132C131⋊C8—21no—
133C23×C131—21no—
134C23×C131C2×D411yescharacteristic
135C23×C131C2×D411yescharacteristic
136D4×C131—21no—
137D4×C131—21no—
138D4×C131—41no—
139D4×C131—42no—
140D4×C131—81no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—81no—
149C2×C4×C131—81no—
150C2×C4×C131C2×D411yescharacteristic
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2×D411yescharacteristic
153C2×C4×C131C2×D411yescharacteristic
154C2×C4×C131C2411yescharacteristic
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—21no—
160C22×C131—21no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—41no—
164C22×C131C22×D411yescharacteristic
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C4×C131—21no—
172C4×C131—21no—
173C4×C131—21no—
174C4×C131—21no—
175C4×C131—41no—
176C4×C131—41no—
177C4×C131—41no—
178C4×C131C2×#4111yescharacteristic
179C4×C131H(5,𝔽2)11yescharacteristic
180Dic131—22no—
181Dic131—41no—
182C2×C131—41no—
183C2×C131—41no—
184C2×C131—81no—
185C2×C131#54611yescharacteristic
186C2×C131#57511yescharacteristic
187C2×C131#57611yescharacteristic
188D131—81no—
189D131—81no—
190C131#330911yescharacteristic, complemented, Sylow
191#3309—1311nomaximal, Sylow
192#553D13111yescharacteristic, complemented
193#550—1311no—
194#490—1311no—
195#482—1311no—
196#480—1311no—
197#479—1311no—
198#469—1311no—
199#466—1311no—
200#459—1311no—
201#449—1311no—
202#449—1311no—
203#440—1311no—
204#435—1311no—
205#417—1311no—
206#416—1311no—
207C2×#41—1311no—
208C2×#41—1311no—
209C2×SD16—1311no—
210C2×D8—1311no—
211C2×D8—1311no—
212C2×M16—1311no—
213C22×C8—1311no—
214#127—2621no—
215#127—1311no—
216#126—21no—
217#125C2×D13111yescharacteristic
218#123—21no—
219#121—21no—
220#121—2621no—
221#121—2621no—
222#121—1311no—
223#121—1311no—
224#121C2×D13111yescharacteristic
225#121C2×D13111yescharacteristic
226C4×Q8—1311no—
227C4×D4—21no—
228C4×D4—2621no—
229C4×D4C2×D13111yescharacteristic
230C4×D4C2×D13111yescharacteristic
231#117C2×D13111yescharacteristic
232C2×C4⋊C4C2×D13111yescharacteristic
233#107—1311no—
234#107—1311no—
235#106—1311no—
236C4⋊C8—1311no—
237#103—1311no—
238#103—1311no—
239#103—1311no—
240#102—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102—1311no—
247C22⋊C8—1311no—
248C22⋊C8—1311no—
249#97—1311no—
250#41—2622no—
251#41—2622no—
252#41—5241no—
253#41—5241no—
254C2×Q8—1311no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2621no—
258C2×D4—2621no—
259C2×D4—2621no—
260C2×D4—2621no—
261C2×D4—2621no—
262C2×D4—2621no—
263C2×D4—1311no—
264C2×D4—1311no—
265C2×D4C22⋊D13111yescharacteristic
266C2×D4C22×D13111yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2621no—
270C22×C4—2621no—
271C22×C4C22×D13111yescharacteristic
272C22×C4C22×D13111yescharacteristic
273SD16—2622no—
274D8—5241no—
275D8—5241no—
276M16—5241no—
277C2×C8—5241no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8—1311no—
281C2×C8—1311no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—2621no—
286C4⋊C4—1311no—
287C4⋊C4C22⋊D13111yescharacteristic
288C4⋊C4C22×D13111yescharacteristic
289C4⋊C4C22×D13111yescharacteristic
290C4⋊C4C22×D13111yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4C22⋊D13111yescharacteristic
298C22⋊C4C22⋊D13111yescharacteristic
299C42—21no—
300C42—2621no—
301C42C22×D13111yescharacteristic
302C23—21no—
303C23—2621no—
304C23—2621no—
305C23C2×C22⋊D13111yescharacteristic
306C23C2×C22⋊D13111yescharacteristic
307Q8—2621no—
308Q8—1312no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—2621no—
314D4—2621no—
315D4—2621no—
316D4—2621no—
317D4—81no—
318D4—5242no—
319D4—5242no—
320D4—5242no—
321D4—10481no—
322D4—10481no—
323D4—10481no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—5242no—
336C2×C4—5242no—
337C2×C4C2×C22⋊D13111yescharacteristic
338C2×C4C2×C22⋊D13111yescharacteristic
339C2×C4C2×C22⋊D13111yescharacteristic
340C2×C4C2×C22⋊D13111yescharacteristic
341C2×C4C23×D13111yescharacteristic
342C8—2621no—
343C8—2621no—
344C8—2621no—
345C8—2621no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22C22×C22⋊D13111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—2622no—
370C4—5241no—
371C4#41×D13111yescharacteristic
372C4#1686229211yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—10481no—
377C2—10481no—
378C2#2255395011yescharacteristic
379C2#2255411011yescharacteristic
380C2#2255411511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#553×C131C2
socleC22×C131C22×D4
hypercenter#553D131
hypocenterC131#3309
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22553983C2semidirect
2. #22553678C2semidirect
3. #22553636C2semidirect
4. #22553625C2semidirect
5. #22553616C2semidirect
6. #22553563C2semidirect
7. #22553546C2semidirect
8. #22553507C2semidirect
9. #22553455C2semidirect
10. #22553455C2semidirect
11. #22553412C2semidirect
12. #22553384C2semidirect
13. #22553311C2semidirect
14. #22553306C2semidirect
15. #553×C131C2semidirect
190. C131#3309semidirect
192. #553D131semidirect

Maximal subgroups

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

subgroupindex
#225539832
#225536782
#225536362
#225536252
#225536162
#225535632
#225535462
#225535072
#225534552
#225534122
#225533842
#225533112
#225533062
#553×C1312
#3309131