Finite Group Databasebeta

← #31783835

Subgroups — #31783835

Order 17536.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984063C211yesmaximal, characteristic
2C2×C4×D548C211yesmaximal, characteristic, complemented
3C2×#17984050C211yesmaximal, characteristic, complemented
4#22577295C211yesmaximal, characteristic, complemented
5#22577293C211yesmaximal, characteristic, complemented
6#22577289C211yesmaximal, characteristic, complemented
7#22577281C211yesmaximal, characteristic, complemented
8#22577281C211yesmaximal, characteristic, complemented
9#22577281C211yesmaximal, characteristic, complemented
10#22577281C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic, complemented
12#22577250C211yesmaximal, characteristic, complemented
13#22577250C211yesmaximal, characteristic, complemented
14#95×D137C211yesmaximal, characteristic, complemented
15#387×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C22×D548C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2211yescharacteristic
22C22×C4×D137C2211yescharacteristic
23C22×C4×D137C2211yescharacteristic
24#17984063C418yescomplemented
25C4×D548—24no—
26#17984050—24no—
27C2×C4⋊Dic137C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×C4×Dic137C2211yescharacteristic
39C2×C4×Dic137C2211yescharacteristic
40C2×C4×Dic137C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C2×C4⋊C4×C137C2211yescharacteristic
48C2×C4⋊C4×C137C2211yescharacteristic
49C2×C42×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53#95×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C23×D137C2311yescharacteristic
56C23×D137C2311yescharacteristic
57C2×D548—22no—
58C2×D548—22no—
59C2×D548—22no—
60C2×D548—22no—
61C2×D548—41no—
62C2×D548—41no—
63C2×D548C2×C414yescomplemented
64C2×D548C2×C414yescomplemented
65C2×D548C2×C414yescomplemented
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C2×C4×D137—22no—
72C2×C4×D137—22no—
73C2×C4×D137—22no—
74C2×C4×D137—22no—
75C2×C4×D137—22no—
76C2×C4×D137—22no—
77C2×C4×D137—22no—
78C2×C4×D137—22no—
79C2×C4×D137—41no—
80C2×C4×D137—41no—
81C2×C4×D137—41no—
82C2×C4×D137—41no—
83C2×C4×D137—41no—
84C2×C4×D137—41no—
85C2×C4×D137C2×C414yescomplemented
86C4⋊Dic137—22no—
87#14555487—22no—
88#14555487—22no—
89#14555487—22no—
90#14555487—22no—
91#14555487—22no—
92#14555487—41no—
93#14555487—41no—
94#14555487C2×C414yescomplemented
95#14555487C2×C414yescomplemented
96#14555487D414yescomplemented
97C4×Dic137—22no—
98C4×Dic137—41no—
99C4×Dic137—41no—
100C22×C4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104C22×C4×C137C2311yescharacteristic
105C22×C4×C137C2311yescharacteristic
106C22×C4×C137C2311yescharacteristic
107C4⋊C4×C137—22no—
108C4⋊C4×C137C2×C414yescomplemented
109C42×C137—22no—
110C22×D137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—21no—
115C22×D137—21no—
116C22×D137—21no—
117C22×D137—21no—
118C22×D137—21no—
119C22×D137—21no—
120C22×D137—21no—
121C22×D137—21no—
122C22×D137—21no—
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137C22×C412yes—
127C22×D137C22×C412yes—
128C22×D137C22×C412yes—
129C22×D137C2×D412yes—
130C22×D137#4112yes—
131D548—28no—
132D548—28no—
133D548—44no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—21no—
138C2×Dic137—21no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—21no—
145C2×Dic137—21no—
146C2×Dic137C22×C412yes—
147C2×Dic137C2×D412yes—
148C2×Dic137#4112yes—
149C4×D137—44no—
150C4×D137—44no—
151C4×D137—44no—
152C4×D137—44no—
153C4×D137—44no—
154C23×C137C2411yescharacteristic
155C2×C4×C137—21no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137C22×C412yes—
172C2×C4×C137C22×C412yes—
173C2×C4×C137C22×C412yes—
174C2×C4×C137C2×D412yes—
175C2×C4×C137#4112yes—
176C22×C137C23×C411yescharacteristic
177C22×C137C22×D411yescharacteristic
178C22×C137C2×#4111yescharacteristic
179C22×C137C2×#4111yescharacteristic
180C22×C137C2×#4111yescharacteristic
181C22×C137H(5,𝔽2)11yescharacteristic
182C22×C137H(5,𝔽2)11yescharacteristic
183C2×D137—22no—
184C2×D137—22no—
185C2×D137—22no—
186C2×D137—22no—
187C2×D137—22no—
188C2×D137—22no—
189C2×D137—22no—
190C2×D137—22no—
191C2×D137—22no—
192C2×D137—22no—
193C2×D137—22no—
194C2×D137—22no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—41no—
203C2×D137C4×D414yes—
204C4×C137—22no—
205C4×C137—22no—
206C4×C137—22no—
207C4×C137—22no—
208C4×C137—22no—
209C4×C137—41no—
210C4×C137—41no—
211Dic137—22no—
212Dic137—22no—
213Dic137—22no—
214Dic137—41no—
215Dic137—41no—
216C2×C137C2×C4×D411yescharacteristic
217C2×C137C4×#4111yescharacteristic
218C2×C137#51811yescharacteristic
219C2×C137#54611yescharacteristic
220C2×C137#54611yescharacteristic
221C2×C137#55311yescharacteristic
222C2×C137#55311yescharacteristic
223D137—24no—
224D137—42no—
225D137—42no—
226C137#236411yescharacteristic, complemented, Sylow
227#2364—1371nomaximal, Sylow
228C2×#121—1371no—
229C2×C4×D4—1371no—
230C2×C4×D4—1371no—
231#388—1371no—
232#388—1371no—
233#387—1371no—
234#387D13711yescharacteristic, complemented
235#386—1371no—
236#386—1371no—
237#386—1371no—
238#386—1371no—
239C4×C22⋊C4—1371no—
240C4×C22⋊C4—1371no—
241C4×C22⋊C4—1371no—
242C2×#95—1371no—
243C22×D4—1371no—
244C22×D4—1371no—
245C22×D4—1371no—
246C23×C4—1371no—
247C23×C4—1371no—
248C23×C4—1371no—
249C23×C4—1371no—
250C23×C4—1371no—
251#121—1378no—
252C4×D4—2744no—
253C4×D4—2744no—
254C2×C4⋊C4—1371no—
255C2×C4⋊C4C2×D13711yescharacteristic
256C2×C4⋊C4C2×D13711yescharacteristic
257C2×C22⋊C4—1371no—
258C2×C22⋊C4—1371no—
259C2×C22⋊C4—1371no—
260C2×C22⋊C4—1371no—
261C2×C22⋊C4—1371no—
262C2×C22⋊C4—1371no—
263C2×C22⋊C4—1371no—
264C2×C22⋊C4—1371no—
265C2×C22⋊C4—1371no—
266C2×C22⋊C4—1371no—
267C2×C42—1371no—
268C2×C42—1371no—
269C2×C42—1371no—
270C2×C42C2×D13711yescharacteristic
271#95—1371no—
272#95—1371no—
273#95—1371no—
274#95—1371no—
275#95—1371no—
276#95—1371no—
277#95C2×D13711yescharacteristic
278#95C2×D13711yescharacteristic
279#95C2×D13711yescharacteristic
280#95C2×D13711yescharacteristic
281C24—1371no—
282C24—1371no—
283C24—1371no—
284C2×D4—2742no—
285C2×D4—2742no—
286C2×D4—2742no—
287C2×D4—2742no—
288C2×D4—5481no—
289C2×D4—5481no—
290C2×D4—1374no—
291C2×D4—1374no—
292C2×D4—1374no—
293C22×C4—2742no—
294C22×C4—2742no—
295C22×C4—2742no—
296C22×C4—2742no—
297C22×C4—2742no—
298C22×C4—2742no—
299C22×C4—2742no—
300C22×C4—2742no—
301C22×C4—5481no—
302C22×C4—5481no—
303C22×C4—5481no—
304C22×C4—5481no—
305C22×C4—5481no—
306C22×C4—5481no—
307C22×C4—1371no—
308C22×C4—1371no—
309C22×C4—1371no—
310C22×C4—1371no—
311C22×C4—1371no—
312C22×C4—1374no—
313C22×C4C22×D13711yescharacteristic
314C22×C4C22×D13711yescharacteristic
315C22×C4C22×D13711yescharacteristic
316C22×C4C22×D13711yescharacteristic
317C22×C4C22×D13711yescharacteristic
318C22×C4C22×D13711yescharacteristic
319C22×C4C22×D13711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—2742no—
322C4⋊C4C4×D13714yescomplemented
323C22⋊C4—2742no—
324C22⋊C4—2742no—
325C22⋊C4—2742no—
326C22⋊C4—2742no—
327C22⋊C4—2742no—
328C22⋊C4—5481no—
329C22⋊C4—5481no—
330C22⋊C4—1374no—
331C22⋊C4—1374no—
332C22⋊C4—1374no—
333C42—22no—
334C42—2742no—
335C42—5481no—
336C42—5481no—
337C23—2741no—
338C23—2741no—
339C23—2741no—
340C23—2741no—
341C23—2741no—
342C23—2741no—
343C23—2741no—
344C23—2741no—
345C23—2741no—
346C23—2741no—
347C23—2741no—
348C23—2741no—
349C23—2741no—
350C23—2741no—
351C23—2741no—
352C23—2741no—
353C23—1372no—
354C23—1372no—
355C23—1372no—
356C23—1372no—
357C23—1372no—
358C23C23×D13711yescharacteristic
359D4—2748no—
360D4—2748no—
361D4—5484no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—2741no—
379C2×C4—2741no—
380C2×C4—2741no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—2741no—
386C2×C4—2741no—
387C2×C4—2741no—
388C2×C4—2741no—
389C2×C4—2741no—
390C2×C4—5484no—
391C2×C4—5484no—
392C2×C4—5484no—
393C2×C4—5484no—
394C2×C4—5484no—
395C2×C4—1372no—
396C2×C4—1372no—
397C2×C4—1372no—
398C2×C4C2×C4×D13712yes—
399C2×C4C2×C4×D13712yes—
400C2×C4C2×C4×D13712yes—
401C2×C4D4×D13712yes—
402C2×C4#1455551112yes—
403C22—2742no—
404C22—2742no—
405C22—2742no—
406C22—2742no—
407C22—2742no—
408C22—2742no—
409C22—2742no—
410C22—2742no—
411C22—2742no—
412C22—2742no—
413C22—2742no—
414C22—2742no—
415C22—5481no—
416C22—5481no—
417C22—5481no—
418C22—5481no—
419C22—5481no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22—1374no—
424C22C22×C4×D13711yescharacteristic
425C22C2×D4×D13711yescharacteristic
426C22#41×D13711yescharacteristic
427C22#41×D13711yescharacteristic
428C22C2×#1455551111yescharacteristic
429C22#1798413411yescharacteristic
430C22#1798413411yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—2742no—
439C4—2742no—
440C4—2742no—
441C4—5481no—
442C4—5481no—
443C2—2744no—
444C2—5482no—
445C2—5482no—
446C2C4×D4×D13711yescharacteristic
447C2C4×#1455551111yescharacteristic
448C2#2257787711yescharacteristic
449C2#2257804411yescharacteristic
450C2#2257804511yescharacteristic
451C2#2257810411yescharacteristic
452C2#2257810611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#387×C137C2
socle, cosocleC23×C137C24
prime radicalC2×#17984063C2
hypercenter#387D137
hypocenterC137#2364
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D548C2semidirect
3. C2×#17984050C2semidirect
4. #22577295C2semidirect
5. #22577293C2semidirect
6. #22577289C2semidirect
7. #22577281C2semidirect
8. #22577281C2semidirect
9. #22577281C2semidirect
10. #22577281C2semidirect
11. C4×#14555487C2semidirect
12. #22577250C2semidirect
13. #22577250C2semidirect
14. #95×D137C2semidirect
15. #387×C137C2semidirect
24. #17984063C4semidirect
63. C2×D548C2×C4semidirect
64. C2×D548C2×C4semidirect
65. C2×D548C2×C4semidirect
85. C2×C4×D137C2×C4semidirect
94. #14555487C2×C4semidirect
95. #14555487C2×C4semidirect
96. #14555487D4semidirect
108. C4⋊C4×C137C2×C4semidirect
226. C137#2364semidirect
234. #387D137semidirect
322. C4⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840632
C2×C4×D5482
C2×#179840502
#225772952
#225772932
#225772892
#225772812
C4×#145554872
#225772502
#95×D1372
#387×C1372
#2364137