Finite Group Databasebeta

← #31783637

Subgroups — #31783637

Order 17536.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984046C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D137C211yesmaximal, characteristic
3C2×C22⋊C4×D137C211yesmaximal, characteristic
4#22577297C211yesmaximal, characteristic, complemented
5#22577296C211yesmaximal, characteristic, complemented
6#22577291C211yesmaximal, characteristic, complemented
7#22577277C211yesmaximal, characteristic, complemented
8#22577274C211yesmaximal, characteristic, complemented
9#22577265C211yesmaximal, characteristic, complemented
10#22577262C211yesmaximal, characteristic, complemented
11#22577258C211yesmaximal, characteristic
12C4×#14555487C211yesmaximal, characteristic, complemented
13C4×#14555487C211yesmaximal, characteristic, complemented
14#22577250C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic, complemented
16C24×D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984046—24no—
22C2×C4⋊Dic137C2211yescharacteristic, complemented
23C2×#14555489C2211yescharacteristic, complemented
24C2×#14555489C2211yescharacteristic, complemented
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C22⋊C4×D137—24no—
37C22⋊C4×D137C418yescomplemented
38C2×C4×Dic137C2211yescharacteristic, complemented
39C2×C4×Dic137C2211yescharacteristic, complemented
40C2×C4×Dic137C2211yescharacteristic, complemented
41#17983995C2211yescharacteristic, complemented
42#17983995C2211yescharacteristic, complemented
43#17983995C2211yescharacteristic, complemented
44#17983995C2211yescharacteristic, complemented
45#17983995C2211yescharacteristic, complemented
46#17983995C2211yescharacteristic, complemented
47C2×C4⋊C4×C137C2211yescharacteristic, complemented
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C42×C137C2211yescharacteristic, complemented
52#95×C137C2211yescharacteristic, complemented
53#95×C137C2211yescharacteristic, complemented
54C23×D137—22no—
55C23×D137—22no—
56C23×D137—22no—
57C23×D137—22no—
58C23×D137—41no—
59C23×D137—41no—
60C23×D137C2×C414yes—
61C23×D137C2311yescharacteristic
62C23×D137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C2×C4×D137—22no—
70C2×C4×D137—22no—
71C2×C4×D137—22no—
72C2×C4×D137—22no—
73C2×C4×D137—22no—
74C2×C4×D137—22no—
75C2×C4×D137—41no—
76C2×C4×D137—41no—
77C2×C4×D137—41no—
78C2×C4×D137—41no—
79C2×C4×D137C2×C414yescomplemented
80C2×C4×D137C2×C414yescomplemented
81C4⋊Dic137—22no—
82#14555489—41no—
83#14555489—41no—
84C22⋊Dic137—22no—
85C22⋊Dic137—41no—
86C22⋊Dic137C2×C414yescomplemented
87#14555487—22no—
88#14555487—22no—
89#14555487—22no—
90#14555487—22no—
91#14555487—22no—
92#14555487C2×C414yes—
93#14555487C2×C414yes—
94#14555487D414yes—
95C4×Dic137—22no—
96C4×Dic137—22no—
97C4×Dic137—41no—
98C24×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104C22×C4×C137C2311yescharacteristic
105C4⋊C4×C137—22no—
106C22⋊C4×C137—22no—
107C22⋊C4×C137—41no—
108C22⋊C4×C137C2×C414yescomplemented
109C42×C137—41no—
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—28no—
119C22×D137—28no—
120C22×D137—44no—
121C22×D137—44no—
122C22×D137—44no—
123C22×D137—44no—
124C22×D137—44no—
125C22×D137C22×C412yes—
126C22×D137C22×C412yes—
127C22×D137C2×D412yes—
128C22×D137#4112yes—
129C22×D137#4112yes—
130C22×D137#4112yes—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137—21no—
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×Dic137C22×C412yes—
148C2×Dic137C2×D412yes—
149C4×D137—28no—
150C4×D137—44no—
151C4×D137—44no—
152C4×D137—44no—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137—21no—
156C23×C137—21no—
157C23×C137—21no—
158C23×C137—21no—
159C23×C137C22×C412yes—
160C23×C137C2411yescharacteristic
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×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137—21no—
174C2×C4×C137—21no—
175C2×C4×C137—21no—
176C2×C4×C137C22×C412yes—
177C2×C4×C137C22×C412yes—
178C2×C4×C137C2×D412yes—
179C22×C137—22no—
180C22×C137—22no—
181C22×C137—22no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137C23×C411yescharacteristic
187C22×C137C22×D411yescharacteristic
188C22×C137C2×#4111yescharacteristic
189C22×C137C2×#4111yescharacteristic
190C22×C137C2×#4111yescharacteristic
191C22×C137H(5,𝔽2)11yescharacteristic
192C22×C137H(5,𝔽2)11yescharacteristic
193C2×D137—22no—
194C2×D137—22no—
195C2×D137—22no—
196C2×D137—22no—
197C2×D137—22no—
198C2×D137—22no—
199C2×D137—22no—
200C2×D137—22no—
201C2×D137—22no—
202C2×D137—22no—
203C2×D137—22no—
204C2×D137—22no—
205C2×D137—416no—
206C2×D137#11714yescomplemented
207C2×D137C4×D414yes—
208C4×C137—22no—
209C4×C137—22no—
210C4×C137—22no—
211C4×C137—41no—
212C4×C137—41no—
213C4×C137—41no—
214Dic137—22no—
215Dic137—22no—
216Dic137—22no—
217Dic137—41no—
218Dic137—41no—
219Dic137—41no—
220C2×C137—42no—
221C2×C137C2×#11711yescharacteristic
222C2×C137C2×C4×D411yescharacteristic
223C2×C137#51811yescharacteristic
224C2×C137#54611yescharacteristic
225C2×C137#54611yescharacteristic
226C2×C137#55111yescharacteristic
227C2×C137#55111yescharacteristic
228D137—24no—
229D137—24no—
230C137#234211yescharacteristic, complemented, Sylow
231#2342—1371nomaximal, Sylow
232C2×#117—1371no—
233C22×C22⋊C4—1371no—
234C22×C22⋊C4—1371no—
235#388—1371no—
236#388—1371no—
237#388—1371no—
238#388D13711yescharacteristic, complemented
239#385—1371no—
240#385—1371no—
241#381—1371no—
242#380—1371no—
243#379—1371no—
244C4×C22⋊C4—1371no—
245C4×C22⋊C4—1371no—
246C4×C22⋊C4—1371no—
247C25—1371no—
248C23×C4—1371no—
249C23×C4—1371no—
250C23×C4—1371no—
251C23×C4—1371no—
252#117—2744no—
253C2×C4⋊C4—1371no—
254C2×C4⋊C4—1371no—
255C2×C4⋊C4—1371no—
256C2×C4⋊C4C2×D13711yescharacteristic, complemented
257C2×C22⋊C4—2744no—
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×C22⋊C4—1371no—
268C2×C22⋊C4—1371no—
269C2×C22⋊C4—1378no—
270C2×C22⋊C4C2×D13711yescharacteristic
271C2×C22⋊C4C2×D13711yescharacteristic
272C2×C22⋊C4C2×D13711yescharacteristic
273C2×C42—1371no—
274C2×C42—1371no—
275C2×C42—1371no—
276C2×C42C2×D13711yescharacteristic, complemented
277#95—1371no—
278#95—1371no—
279#95—1371no—
280#95—1371no—
281#95—1371no—
282#95—1371no—
283#95C2×D13711yescharacteristic, complemented
284#95C2×D13711yescharacteristic, complemented
285C24—2742no—
286C24—2742no—
287C24—2742no—
288C24—2742no—
289C24—5481no—
290C24—5481no—
291C24—1374no—
292C24—1371no—
293C24—1371no—
294C24C22×D13711yescharacteristic
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—1371no—
306C22×C4—1371no—
307C22×C4—1371no—
308C22×C4—1371no—
309C22×C4—1371no—
310C22×C4—1371no—
311C22×C4—1374no—
312C22×C4—1374no—
313C22×C4C22×D13711yescharacteristic
314C22×C4C22×D13711yescharacteristic
315C22×C4C22×D13711yescharacteristic
316C22×C4C22×D13711yescharacteristic
317C22×C4C22×D13711yescharacteristic
318C22×C4C22×D13711yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—2742no—
321C4⋊C4—5481no—
322C4⋊C4—5481no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—2742no—
326C22⋊C4—2742no—
327C22⋊C4—2742no—
328C22⋊C4—2742no—
329C22⋊C4—2742no—
330C22⋊C4—2742no—
331C22⋊C4—5481no—
332C22⋊C4—1374no—
333C22⋊C4—1374no—
334C22⋊C4—1374no—
335C22⋊C4—1374no—
336C22⋊C4C4×D13714yescomplemented
337C42—41no—
338C42—2742no—
339C42—2742no—
340C42—5481no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2748no—
348C23—2748no—
349C23—2741no—
350C23—2741no—
351C23—2741no—
352C23—2741no—
353C23—2741no—
354C23—2741no—
355C23—2741no—
356C23—2741no—
357C23—5484no—
358C23—5484no—
359C23—5484no—
360C23—5484no—
361C23—5484no—
362C23—1372no—
363C23—1372no—
364C23—1372no—
365C23—1372no—
366C23—1372no—
367C23—1372no—
368C23C2×C4×D13712yes—
369C23C23×D13711yescharacteristic
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—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—2748no—
386C2×C4—2741no—
387C2×C4—2741no—
388C2×C4—2741no—
389C2×C4—2741no—
390C2×C4—2741no—
391C2×C4—2741no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—2741no—
396C2×C4—2741no—
397C2×C4—2741no—
398C2×C4—2741no—
399C2×C4—2741no—
400C2×C4—2741no—
401C2×C4—5484no—
402C2×C4—5484no—
403C2×C4—5484no—
404C2×C4—1372no—
405C2×C4—1372no—
406C2×C4—1372no—
407C2×C4C2×C4×D13712yes—
408C2×C4C2×C4×D13712yes—
409C2×C4D4×D13712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—2742no—
418C22—2742no—
419C22—2742no—
420C22—2742no—
421C22—2742no—
422C22—2742no—
423C22—2742no—
424C22—2742no—
425C22—2742no—
426C22—2742no—
427C22—2742no—
428C22—2742no—
429C22—54816no—
430C22—1374no—
431C22—1374no—
432C22C22×C4×D13711yescharacteristic
433C22C2×D4×D13711yescharacteristic
434C22#41×D13711yescharacteristic
435C22#41×D13711yescharacteristic
436C22#41×D13711yescharacteristic
437C22#1798413311yescharacteristic
438C22#1798413411yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—2742no—
446C4—2742no—
447C4—2742no—
448C4—5481no—
449C4—5481no—
450C4—5481no—
451C2—42no—
452C2—2744no—
453C2—2744no—
454C2#117×D13711yescharacteristic
455C2C4×D4×D13711yescharacteristic
456C2#2257787511yescharacteristic
457C2#2257804211yescharacteristic
458C2#2257804611yescharacteristic
459C2#2257807811yescharacteristic
460C2#2257808111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC24×D137C22
hypercenter#388D137
hypocenterC137#2342
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#17984046C2semidirect
4. #22577297C2semidirect
5. #22577296C2semidirect
6. #22577291C2semidirect
7. #22577277C2semidirect
8. #22577274C2semidirect
9. #22577265C2semidirect
10. #22577262C2semidirect
12. C4×#14555487C2semidirect
13. C4×#14555487C2semidirect
14. #22577250C2semidirect
15. #388×C137C2semidirect
22. C2×C4⋊Dic137C22semidirect
23. C2×#14555489C22semidirect
24. C2×#14555489C22semidirect
37. C22⋊C4×D137C4semidirect
38. C2×C4×Dic137C22semidirect
39. C2×C4×Dic137C22semidirect
40. C2×C4×Dic137C22semidirect
41. #17983995C22semidirect
42. #17983995C22semidirect
43. #17983995C22semidirect
44. #17983995C22semidirect
45. #17983995C22semidirect
46. #17983995C22semidirect
47. C2×C4⋊C4×C137C22semidirect
51. C2×C42×C137C22semidirect
52. #95×C137C22semidirect
53. #95×C137C22semidirect
79. C2×C4×D137C2×C4semidirect
80. C2×C4×D137C2×C4semidirect
86. C22⋊Dic137C2×C4semidirect
108. C22⋊C4×C137C2×C4semidirect
206. C2×D137#117semidirect
230. C137#2342semidirect
238. #388D137semidirect
256. C2×C4⋊C4C2×D137semidirect
276. C2×C42C2×D137semidirect
283. #95C2×D137semidirect
284. #95C2×D137semidirect
336. C22⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840462
C2×C22⋊C4×D1372
#225772972
#225772962
#225772912
#225772772
#225772742
#225772652
#225772622
#225772582
C4×#145554872
#225772502
#388×C1372
#2342137