Finite Group Databasebeta

← #31784167

Subgroups — #31784167

Order 17536.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984070C211yesmaximal, characteristic, complemented
2C2×#17984056C211yesmaximal, characteristic
3C2×#17984046C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
5#22577326C211yesmaximal, characteristic, complemented
6#22577325C211yesmaximal, characteristic, complemented
7#22577319C211yesmaximal, characteristic, complemented
8#22577306C211yesmaximal, characteristic, complemented
9#22577303C211yesmaximal, characteristic, complemented
10#22577296C211yesmaximal, characteristic, complemented
11#22577295C211yesmaximal, characteristic, complemented
12#22577279C211yesmaximal, characteristic, complemented
13#22577277C211yesmaximal, characteristic, complemented
14#22577258C211yesmaximal, characteristic, complemented
15#394×C137C211yesmaximal, characteristic, complemented
16C24×D137C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C22×C22⋊D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984070—24no—
22#17984056—24no—
23#17984046—24no—
24C2×C4⋊Dic137C2211yescharacteristic, complemented
25C2×C4⋊Dic137C2211yescharacteristic, complemented
26C2×C4⋊Dic137C2211yescharacteristic, complemented
27C2×#14555489C2211yescharacteristic, complemented
28C2×#14555489C2211yescharacteristic, complemented
29C2×C22⋊Dic137C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×C22⋊Dic137C2211yescharacteristic
32C2×C22⋊Dic137C2211yescharacteristic, complemented
33C2×C22⋊Dic137C2211yescharacteristic, complemented
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×#14555487C2211yescharacteristic
39C2×#14555487C2211yescharacteristic
40C2×#14555487C2211yescharacteristic
41C2×#14555487C2211yescharacteristic
42C22⋊C4×D137—24no—
43C2×C4×Dic137C2211yescharacteristic, complemented
44C2×C4×Dic137C2211yescharacteristic, complemented
45#17983995C2211yescharacteristic, complemented
46#17983995C2211yescharacteristic, complemented
47#17983995C2211yescharacteristic, complemented
48C22×D4×C137C2211yescharacteristic
49C2×C4⋊C4×C137C2211yescharacteristic, complemented
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic
52C2×C22⋊C4×C137C2211yescharacteristic, complemented
53C2×C22⋊C4×C137C2211yescharacteristic, complemented
54#95×C137C2211yescharacteristic, complemented
55C23×D137—22no—
56C23×D137—22no—
57C23×D137—22no—
58C23×D137—22no—
59C23×D137—22no—
60C23×D137—41no—
61C23×D137—41no—
62C23×D137C2311yescharacteristic
63C23×D137C2311yescharacteristic
64C2×C22⋊D137—22no—
65C2×C22⋊D137—22no—
66C2×C22⋊D137—22no—
67C2×C22⋊D137—22no—
68C2×C22⋊D137—22no—
69C2×C22⋊D137—41no—
70C22×Dic137C2311yescharacteristic
71C22×Dic137C2311yescharacteristic
72C22×Dic137C2311yescharacteristic
73C22×Dic137C2311yescharacteristic
74C22×Dic137C2311yescharacteristic
75C22×Dic137C2311yescharacteristic
76C2×C4×D137—22no—
77C2×C4×D137—22no—
78C2×C4×D137—22no—
79C2×C4×D137—41no—
80C2×C4×D137—41no—
81C2×C4×D137D414yescomplemented
82C4⋊Dic137—22no—
83C4⋊Dic137—22no—
84C4⋊Dic137—41no—
85#14555489—22no—
86#14555489—41no—
87C22⋊Dic137—22no—
88C22⋊Dic137—22no—
89C22⋊Dic137—22no—
90C22⋊Dic137—41no—
91C22⋊Dic137—41no—
92#14555487—22no—
93#14555487—22no—
94#14555487—22no—
95#14555487—22no—
96#14555487—22no—
97#14555487—22no—
98#14555487—41no—
99#14555487D414yes—
100C4×Dic137—22no—
101C4×Dic137—22no—
102C24×C137C2311yescharacteristic
103C24×C137C2311yescharacteristic
104C2×D4×C137—22no—
105C2×D4×C137—22no—
106C2×D4×C137—41no—
107C22×C4×C137C2311yescharacteristic
108C22×C4×C137C2311yescharacteristic
109C22×C4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C4⋊C4×C137—22no—
113C22⋊C4×C137—22no—
114C22⋊C4×C137—22no—
115C22⋊C4×C137—22no—
116C22⋊C4×C137—41no—
117C22×D137—21no—
118C22×D137—21no—
119C22×D137—21no—
120C22×D137—21no—
121C22×D137—28no—
122C22×D137—21no—
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—21no—
128C22×D137—44no—
129C22×D137—44no—
130C22×D137—44no—
131C22×D137—44no—
132C22×D137—44no—
133C22×D137—44no—
134C22×D137C2×D412yes—
135C22×D137C2×D412yes—
136C22×D137#4112yes—
137C22×D137#4112yes—
138C22⋊D137—44no—
139C22⋊D137—44no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—21no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—21no—
148C2×Dic137—21no—
149C2×Dic137—21no—
150C2×Dic137—21no—
151C2×Dic137—21no—
152C2×Dic137—21no—
153C2×Dic137—21no—
154C2×Dic137—21no—
155C2×Dic137—21no—
156C2×Dic137C2×D412yes—
157C2×Dic137C2×D412yes—
158C4×D137—28no—
159C4×D137—44no—
160C23×C137—21no—
161C23×C137—21no—
162C23×C137—21no—
163C23×C137—21no—
164C23×C137—21no—
165C23×C137—21no—
166C23×C137—21no—
167C23×C137—21no—
168C23×C137—21no—
169C23×C137—21no—
170C23×C137—21no—
171C23×C137—21no—
172C23×C137—21no—
173C23×C137#4112yes—
174C23×C137C2411yescharacteristic
175D4×C137—44no—
176C2×C4×C137—21no—
177C2×C4×C137—21no—
178C2×C4×C137—21no—
179C2×C4×C137—21no—
180C2×C4×C137—21no—
181C2×C4×C137—21no—
182C2×C4×C137—21no—
183C2×C4×C137—21no—
184C2×C4×C137—21no—
185C2×C4×C137—21no—
186C2×C4×C137—21no—
187C2×C4×C137—21no—
188C2×C4×C137—21no—
189C2×C4×C137C2×D412yes—
190C2×C4×C137C2×D412yes—
191C22×C137—22no—
192C22×C137—22no—
193C22×C137—22no—
194C22×C137—41no—
195C22×C137—41no—
196C22×C137—41no—
197C22×C137—41no—
198C22×C137—41no—
199C22×C137—41no—
200C22×C137—41no—
201C22×C137—41no—
202C22×C137—41no—
203C22×C137—41no—
204C22×C137—41no—
205C22×C137C22×D411yescharacteristic
206C22×C137C22×D411yescharacteristic
207C22×C137C2×#4111yescharacteristic
208C22×C137C2×#4111yescharacteristic
209C22×C137C2×#4111yescharacteristic
210C22×C137H(5,𝔽2)11yescharacteristic
211C22×C137H(5,𝔽2)11yescharacteristic
212C2×D137—22no—
213C2×D137—22no—
214C2×D137—22no—
215C2×D137—22no—
216C2×D137—22no—
217C2×D137—22no—
218C2×D137—22no—
219C2×D137—22no—
220C2×D137—22no—
221C2×D137—416no—
222C2×D137—41no—
223C2×D137—41no—
224C2×D137—41no—
225C2×D137—41no—
226C2×D137#12114yes—
227C4×C137—22no—
228C4×C137—22no—
229C4×C137—41no—
230C4×C137—41no—
231C4×C137—41no—
232Dic137—22no—
233Dic137—22no—
234Dic137—41no—
235Dic137—41no—
236Dic137—41no—
237Dic137—41no—
238C2×C137—42no—
239C2×C137—81no—
240C2×C137C2×#12111yescharacteristic
241C2×C137#52511yescharacteristic
242C2×C137#53511yescharacteristic
243C2×C137#54611yescharacteristic
244C2×C137#54611yescharacteristic
245C2×C137#55111yescharacteristic
246C2×C137#55111yescharacteristic
247D137—24no—
248D137—42no—
249C137#241111yescharacteristic, complemented, Sylow
250#2411—1371nomaximal, Sylow
251C2×#123—1371no—
252C2×#120—1371no—
253C2×#117—1371no—
254C22×C22⋊C4—1371no—
255#396—1371no—
256#396—1371no—
257#394—1371no—
258#394D13711yescharacteristic, complemented
259#390—1371no—
260#390—1371no—
261#388—1371no—
262#388—1371no—
263#386—1371no—
264#385—1371no—
265#379—1371no—
266C25—1371no—
267C22×D4—1371no—
268C22×D4—1371no—
269C22×D4C2×D13711yescharacteristic
270C23×C4—1371no—
271C23×C4—1371no—
272#123—2744no—
273#120—2744no—
274#117—2744no—
275C2×C4⋊C4—1371no—
276C2×C4⋊C4—1371no—
277C2×C4⋊C4—1371no—
278C2×C4⋊C4—1371no—
279C2×C4⋊C4—1371no—
280C2×C4⋊C4C2×D13711yescharacteristic, complemented
281C2×C22⋊C4—2744no—
282C2×C22⋊C4—1371no—
283C2×C22⋊C4—1371no—
284C2×C22⋊C4—1371no—
285C2×C22⋊C4—1371no—
286C2×C22⋊C4—1371no—
287C2×C22⋊C4—1371no—
288C2×C22⋊C4—1371no—
289C2×C22⋊C4—1371no—
290C2×C22⋊C4—1371no—
291C2×C22⋊C4—1371no—
292C2×C22⋊C4—1371no—
293C2×C22⋊C4—1371no—
294C2×C22⋊C4—1371no—
295C2×C22⋊C4C2×D13711yescharacteristic
296C2×C22⋊C4C2×D13711yescharacteristic
297C2×C22⋊C4C2×D13711yescharacteristic, complemented
298C2×C22⋊C4C2×D13711yescharacteristic, complemented
299C2×C42—1371no—
300C2×C42—1371no—
301#95—1371no—
302#95—1371no—
303#95—1371no—
304#95C2×D13711yescharacteristic, complemented
305C24—2742no—
306C24—2742no—
307C24—2742no—
308C24—2742no—
309C24—2742no—
310C24—5481no—
311C24—5481no—
312C24—1371no—
313C24—1371no—
314C24C22×D13711yescharacteristic
315C24C22×D13711yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—2742no—
320C2×D4—2742no—
321C2×D4—2742no—
322C2×D4—2742no—
323C2×D4—2742no—
324C2×D4—5481no—
325C22×C4—2742no—
326C22×C4—2742no—
327C22×C4—2742no—
328C22×C4—5481no—
329C22×C4—5481no—
330C22×C4—1371no—
331C22×C4—1371no—
332C22×C4—1371no—
333C22×C4—1371no—
334C22×C4—1371no—
335C22×C4—1371no—
336C22×C4—1374no—
337C22×C4C22×D13711yescharacteristic
338C22×C4C22×D13711yescharacteristic
339C22×C4C22×D13711yescharacteristic
340C22×C4C22×D13711yescharacteristic
341C22×C4C22×D13711yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—2742no—
344C4⋊C4—2742no—
345C4⋊C4—2742no—
346C4⋊C4—5481no—
347C4⋊C4—5481no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—2742no—
353C22⋊C4—2742no—
354C22⋊C4—2742no—
355C22⋊C4—2742no—
356C22⋊C4—2742no—
357C22⋊C4—2742no—
358C22⋊C4—2742no—
359C22⋊C4—2742no—
360C22⋊C4—2742no—
361C22⋊C4—5481no—
362C22⋊C4—5481no—
363C22⋊C4—5481no—
364C22⋊C4—1374no—
365C42—2742no—
366C42—2742no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—2748no—
381C23—2741no—
382C23—2741no—
383C23—2741no—
384C23—2741no—
385C23—2741no—
386C23—2741no—
387C23—2741no—
388C23—2741no—
389C23—2741no—
390C23—2741no—
391C23—5484no—
392C23—5484no—
393C23—5484no—
394C23—5484no—
395C23—5484no—
396C23—5484no—
397C23—1372no—
398C23—1372no—
399C23—1372no—
400C23—1372no—
401C23#1455550912yes—
402C23C23×D13711yescharacteristic
403D4—44no—
404D4—5484no—
405D4—5484no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—2748no—
420C2×C4—2741no—
421C2×C4—2741no—
422C2×C4—2741no—
423C2×C4—2741no—
424C2×C4—2741no—
425C2×C4—2741no—
426C2×C4—2741no—
427C2×C4—2741no—
428C2×C4—2741no—
429C2×C4—2741no—
430C2×C4—2741no—
431C2×C4—2741no—
432C2×C4—2741no—
433C2×C4—2741no—
434C2×C4—2741no—
435C2×C4—2741no—
436C2×C4—5484no—
437C2×C4—1372no—
438C2×C4—1372no—
439C2×C4D4×D13712yes—
440C2×C4D4×D13712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—2742no—
456C22—2742no—
457C22—2742no—
458C22—2742no—
459C22—2742no—
460C22—2742no—
461C22—2742no—
462C22—2742no—
463C22—2742no—
464C22—54816no—
465C22—5481no—
466C22—5481no—
467C22—5481no—
468C22—5481no—
469C22—1374no—
470C22C2×D4×D13711yescharacteristic
471C22C2×D4×D13711yescharacteristic
472C22#41×D13711yescharacteristic
473C22#41×D13711yescharacteristic
474C22C2×#1455550911yescharacteristic
475C22#1798413311yescharacteristic
476C22#1798413411yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—2742no—
483C4—2742no—
484C4—5481no—
485C4—5481no—
486C4—5481no—
487C4—5481no—
488C2—42no—
489C2—81no—
490C2—2744no—
491C2—5482no—
492C2#121×D13711yescharacteristic
493C2#2257791311yescharacteristic
494C2#2257796911yescharacteristic
495C2#2257804011yescharacteristic
496C2#2257804611yescharacteristic
497C2#2257807611yescharacteristic
498C2#2257808111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#394×C137C2
prime radicalC2×#17984056C2
hypercenter#394D137
hypocenterC137#2411
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#17984070C2semidirect
3. C2×#17984046C2semidirect
4. C2×C22⋊C4×D137C2semidirect
5. #22577326C2semidirect
6. #22577325C2semidirect
7. #22577319C2semidirect
8. #22577306C2semidirect
9. #22577303C2semidirect
10. #22577296C2semidirect
11. #22577295C2semidirect
12. #22577279C2semidirect
13. #22577277C2semidirect
14. #22577258C2semidirect
15. #394×C137C2semidirect
24. C2×C4⋊Dic137C22semidirect
25. C2×C4⋊Dic137C22semidirect
26. C2×C4⋊Dic137C22semidirect
27. C2×#14555489C22semidirect
28. C2×#14555489C22semidirect
32. C2×C22⋊Dic137C22semidirect
33. C2×C22⋊Dic137C22semidirect
43. C2×C4×Dic137C22semidirect
44. C2×C4×Dic137C22semidirect
45. #17983995C22semidirect
46. #17983995C22semidirect
47. #17983995C22semidirect
49. C2×C4⋊C4×C137C22semidirect
52. C2×C22⋊C4×C137C22semidirect
53. C2×C22⋊C4×C137C22semidirect
54. #95×C137C22semidirect
81. C2×C4×D137D4semidirect
249. C137#2411semidirect
258. #394D137semidirect
280. C2×C4⋊C4C2×D137semidirect
297. C2×C22⋊C4C2×D137semidirect
298. C2×C22⋊C4C2×D137semidirect
304. #95C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840702
C2×#179840562
C2×#179840462
C2×C22⋊C4×D1372
#225773262
#225773252
#225773192
#225773062
#225773032
#225772962
#225772952
#225772792
#225772772
#225772582
#394×C1372
#2411137