Finite Group Databasebeta

← #31783728

Subgroups — #31783728

Order 17536.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984050C211yesmaximal, characteristic
2#22577295C211yesmaximal, characteristic, complemented
3#22577293C211yesmaximal, characteristic, complemented
4#22577289C211yesmaximal, characteristic, complemented
5#22577281C211yesmaximal, characteristic
6#22577281C211yesmaximal, characteristic
7#22577270C211yesmaximal, characteristic, complemented
8#22577265C211yesmaximal, characteristic, complemented
9#22577260C211yesmaximal, characteristic, complemented
10C4×#14555487C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic, complemented
12#22577250C211yesmaximal, characteristic, complemented
13#22577248C211yesmaximal, characteristic, complemented
14#95×D137C211yesmaximal, characteristic, complemented
15#387×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984050C418yescomplemented
22C2×C4⋊Dic137C2211yescharacteristic
23C2×#14555487C2211yescharacteristic
24C2×#14555487C2211yescharacteristic
25C2×#14555487C2211yescharacteristic
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×C4×Dic137C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35#17983995C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C2×C4⋊C4×C137C2211yescharacteristic
46C2×C4⋊C4×C137C2211yescharacteristic
47C2×C42×C137C2211yescharacteristic
48#95×C137C2211yescharacteristic
49#95×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C23×D137C2311yescharacteristic
53C23×D137C2311yescharacteristic
54C2×D548—22no—
55C2×D548—22no—
56C2×D548C2×C414yes—
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C2×C4×D137—22no—
64C2×C4×D137—22no—
65C2×C4×D137—22no—
66C2×C4×D137—22no—
67C2×C4×D137—41no—
68C2×C4×D137—41no—
69C2×C4×D137—41no—
70C2×C4×D137—41no—
71C2×C4×D137—41no—
72C2×C4×D137—41no—
73C2×C4×D137C2×C414yes—
74C2×C4×D137C2×C414yescomplemented
75C4⋊Dic137—41no—
76#14555487—22no—
77#14555487—22no—
78#14555487—22no—
79#14555487—22no—
80#14555487—41no—
81#14555487—41no—
82#14555487C2×C414yes—
83#14555487C2×C414yescomplemented
84C4×Dic137—22no—
85C4×Dic137—22no—
86C4×Dic137—41no—
87C4×Dic137C2×C414yescomplemented
88C22×C4×C137C2311yescharacteristic
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C22×C4×C137C2311yescharacteristic
94C22×C4×C137C2311yescharacteristic
95C4⋊C4×C137—41no—
96C4⋊C4×C137C2×C414yescomplemented
97C42×C137—41no—
98C22×D137—21no—
99C22×D137—21no—
100C22×D137—21no—
101C22×D137—21no—
102C22×D137—21no—
103C22×D137—21no—
104C22×D137—21no—
105C22×D137—21no—
106C22×D137—21no—
107C22×D137—21no—
108C22×D137C22×C412yes—
109C22×D137C22×C412yes—
110C22×D137#4112yes—
111C22×D137#4112yes—
112D548—28no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137C22×C412yes—
129C2×Dic137C22×C412yes—
130C2×Dic137#4112yes—
131C4×D137—44no—
132C4×D137—44no—
133C4×D137—44no—
134C4×D137—44no—
135C23×C137C2411yescharacteristic
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137C22×C412yes—
154C2×C4×C137C22×C412yes—
155C2×C4×C137C22×C412yes—
156C2×C4×C137#4112yes—
157C22×C137C23×C411yescharacteristic
158C22×C137C2×#4111yescharacteristic
159C22×C137C2×#4111yescharacteristic
160C22×C137C2×#4111yescharacteristic
161C22×C137C2×#4111yescharacteristic
162C22×C137H(5,𝔽2)11yescharacteristic
163C22×C137H(5,𝔽2)11yescharacteristic
164C2×D137—22no—
165C2×D137—22no—
166C2×D137—22no—
167C2×D137—22no—
168C2×D137—22no—
169C2×D137—22no—
170C2×D137—22no—
171C2×D137—22no—
172C2×D137—22no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—41no—
177C2×D137#11714yes—
178C4×C137—22no—
179C4×C137—22no—
180C4×C137—22no—
181C4×C137—22no—
182C4×C137—41no—
183C4×C137—41no—
184C4×C137—41no—
185Dic137—22no—
186Dic137—22no—
187Dic137—22no—
188Dic137—41no—
189Dic137—41no—
190Dic137—41no—
191C2×C137C2×#11711yescharacteristic
192C2×C137C4×#4111yescharacteristic
193C2×C137#51811yescharacteristic
194C2×C137#55111yescharacteristic
195C2×C137#55111yescharacteristic
196C2×C137#55311yescharacteristic
197C2×C137#55311yescharacteristic
198D137—24no—
199D137—42no—
200C137#235311yescharacteristic, complemented, Sylow
201#2353—1371nomaximal, Sylow
202C2×C4×D4—1371no—
203#388—1371no—
204#388—1371no—
205#387—1371no—
206#387D13711yescharacteristic, complemented
207#386—1371no—
208#386—1371no—
209#383—1371no—
210#381—1371no—
211#380—1371no—
212C4×C22⋊C4—1371no—
213C4×C22⋊C4—1371no—
214C4×C22⋊C4—1371no—
215#376—1371no—
216C2×#95—1371no—
217C22×D4—1371no—
218C23×C4—1371no—
219C23×C4—1371no—
220C23×C4—1371no—
221C23×C4—1371no—
222C4×D4—1378no—
223C2×C4⋊C4—1371no—
224C2×C4⋊C4C2×D13711yescharacteristic
225C2×C4⋊C4C2×D13711yescharacteristic
226C2×C22⋊C4—1371no—
227C2×C22⋊C4—1371no—
228C2×C22⋊C4—1371no—
229C2×C22⋊C4—1371no—
230C2×C22⋊C4—1371no—
231C2×C22⋊C4—1371no—
232C2×C22⋊C4—1371no—
233C2×C22⋊C4—1371no—
234C2×C42—1371no—
235C2×C42—1371no—
236C2×C42—1371no—
237C2×C42—1371no—
238C2×C42C2×D13711yescharacteristic
239#95—1371no—
240#95—1371no—
241#95—1371no—
242#95—1371no—
243#95—1371no—
244#95—1371no—
245#95—1371no—
246#95—1371no—
247#95—1371no—
248#95—1371no—
249#95C2×D13711yescharacteristic
250#95C2×D13711yescharacteristic
251#95C2×D13711yescharacteristic
252#95C2×D13711yescharacteristic
253C24—1371no—
254C24—1371no—
255C2×D4—2742no—
256C2×D4—2742no—
257C2×D4—1374no—
258C22×C4—2742no—
259C22×C4—2742no—
260C22×C4—2742no—
261C22×C4—2742no—
262C22×C4—5481no—
263C22×C4—5481no—
264C22×C4—5481no—
265C22×C4—5481no—
266C22×C4—5481no—
267C22×C4—5481no—
268C22×C4—1371no—
269C22×C4—1371no—
270C22×C4—1371no—
271C22×C4—1371no—
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1374no—
275C22×C4—1374no—
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278C22×C4C22×D13711yescharacteristic
279C22×C4C22×D13711yescharacteristic
280C22×C4C22×D13711yescharacteristic
281C22×C4C22×D13711yescharacteristic
282C22×C4C22×D13711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—5481no—
285C4⋊C4C4×D13714yescomplemented
286C22⋊C4—2742no—
287C22⋊C4—2742no—
288C22⋊C4—2742no—
289C22⋊C4—2742no—
290C22⋊C4—5481no—
291C22⋊C4—5481no—
292C22⋊C4—1374no—
293C22⋊C4—1374no—
294C42—41no—
295C42—2742no—
296C42—2742no—
297C42—5481no—
298C42—1374no—
299C23—2741no—
300C23—2741no—
301C23—2741no—
302C23—2741no—
303C23—2741no—
304C23—2741no—
305C23—2741no—
306C23—2741no—
307C23—2741no—
308C23—2741no—
309C23—1372no—
310C23—1372no—
311C23—1372no—
312C23—1372no—
313C23C23×D13711yescharacteristic
314D4—2748no—
315C2×C4—21no—
316C2×C4—21no—
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—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—5484no—
348C2×C4—5484no—
349C2×C4—5484no—
350C2×C4—5484no—
351C2×C4—1372no—
352C2×C4—1372no—
353C2×C4—1372no—
354C2×C4C2×C4×D13712yes—
355C2×C4C2×C4×D13712yes—
356C2×C4C2×C4×D13712yes—
357C2×C4#1455551112yes—
358C22—2742no—
359C22—2742no—
360C22—2742no—
361C22—2742no—
362C22—2742no—
363C22—2742no—
364C22—2742no—
365C22—2742no—
366C22—2742no—
367C22—5481no—
368C22—5481no—
369C22—5481no—
370C22—5481no—
371C22—1374no—
372C22C22×C4×D13711yescharacteristic
373C22#41×D13711yescharacteristic
374C22#41×D13711yescharacteristic
375C22#41×D13711yescharacteristic
376C22C2×#1455551111yescharacteristic
377C22#1798413311yescharacteristic
378C22#1798413411yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—2742no—
387C4—2742no—
388C4—2742no—
389C4—5481no—
390C4—5481no—
391C4—5481no—
392C2—2744no—
393C2—5482no—
394C2#117×D13711yescharacteristic
395C2C4×#1455551111yescharacteristic
396C2#2257787511yescharacteristic
397C2#2257807811yescharacteristic
398C2#2257808011yescharacteristic
399C2#2257809811yescharacteristic
400C2#2257810611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#387×C137C2
socle, cosocleC23×C137C24
prime radicalC22×D548C22
hypercenter#387D137
hypocenterC137#2353
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22577295C2semidirect
3. #22577293C2semidirect
4. #22577289C2semidirect
7. #22577270C2semidirect
8. #22577265C2semidirect
9. #22577260C2semidirect
10. C4×#14555487C2semidirect
11. C4×#14555487C2semidirect
12. #22577250C2semidirect
13. #22577248C2semidirect
14. #95×D137C2semidirect
15. #387×C137C2semidirect
21. #17984050C4semidirect
74. C2×C4×D137C2×C4semidirect
83. #14555487C2×C4semidirect
87. C4×Dic137C2×C4semidirect
96. C4⋊C4×C137C2×C4semidirect
200. C137#2353semidirect
206. #387D137semidirect
285. C4⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840502
#225772952
#225772932
#225772892
#225772812
#225772702
#225772652
#225772602
C4×#145554872
#225772502
#225772482
#95×D1372
#387×C1372
#2353137