Finite Group Databasebeta

← #31783477

Subgroups — #31783477

Order 17536.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984061C211yesmaximal, characteristic
2C2×#17984048C211yesmaximal, characteristic, complemented
3C2×#17984046C211yesmaximal, characteristic, complemented
4#22577305C211yesmaximal, characteristic, complemented
5#22577303C211yesmaximal, characteristic, complemented
6#22577303C211yesmaximal, characteristic, complemented
7#22577300C211yesmaximal, characteristic, complemented
8#22577293C211yesmaximal, characteristic, complemented
9#22577281C211yesmaximal, characteristic, complemented
10#22577280C211yesmaximal, characteristic, complemented
11#22577278C211yesmaximal, characteristic, complemented
12#22577277C211yesmaximal, characteristic, complemented
13#22577262C211yesmaximal, characteristic, complemented
14C4×#14555489C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C22×C22⋊D137C2211yescharacteristic
19C23×Dic137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2211yescharacteristic
22#17984061C418yescomplemented
23#17984048—24no—
24#17984046—24no—
25C2×C4⋊Dic137C2211yescharacteristic
26C2×C4⋊Dic137C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×#14555489C2211yescharacteristic
29C2×#14555489C2211yescharacteristic
30C2×#14555489C2211yescharacteristic
31C2×C22⋊Dic137C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×#14555487C2211yescharacteristic
39C2×#14555487C2211yescharacteristic
40C2×C4×Dic137C2211yescharacteristic
41C2×C4×Dic137C2211yescharacteristic
42C2×C4×Dic137C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C2×C4⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C42×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53#95×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C23×D137C2311yescharacteristic
56C2×D548—41no—
57C2×D548—41no—
58C2×D548C2×C414yescomplemented
59C2×C22⋊D137—22no—
60C2×C22⋊D137—22no—
61C2×C22⋊D137—41no—
62C2×C22⋊D137—41no—
63C2×C22⋊D137C2×C414yes—
64C2×C22⋊D137C2×C414yescomplemented
65C22×Dic137—22no—
66C22×Dic137—41no—
67C22×Dic137—41no—
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C22×Dic137C2311yescharacteristic
72C22×Dic137C2311yescharacteristic
73C22×Dic137C2311yescharacteristic
74C2×C4×D137—22no—
75C2×C4×D137—22no—
76C2×C4×D137—22no—
77C2×C4×D137—41no—
78C2×C4×D137—41no—
79C2×C4×D137C2×C414yescomplemented
80C4⋊Dic137—22no—
81C4⋊Dic137—41no—
82#14555489—22no—
83#14555489—22no—
84#14555489—41no—
85#14555489C2×C414yescomplemented
86C22⋊Dic137—41no—
87#14555487—22no—
88#14555487—22no—
89#14555487—22no—
90#14555487—41no—
91#14555487—41no—
92#14555487—41no—
93#14555487—41no—
94#14555487C2×C414yescomplemented
95C4×Dic137—22no—
96C4×Dic137—41no—
97C4×Dic137D414yescomplemented
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—21no—
119C22×D137—21no—
120C22×D137—21no—
121C22×D137—21no—
122C22×D137C22×C412yes—
123C22×D137C22×C412yes—
124D548—44no—
125C22⋊D137—28no—
126C22⋊D137—44no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
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—44no—
141C2×Dic137C22×C412yes—
142C2×Dic137C22×C412yes—
143C2×Dic137C2×D412yes—
144C2×Dic137C2×D412yes—
145C2×Dic137#4112yes—
146C4×D137—28no—
147C4×D137—44no—
148C23×C137—21no—
149C23×C137—21no—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137—21no—
153C23×C137—21no—
154C23×C137C22×C412yes—
155C23×C137C2411yescharacteristic
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×C137C2×D412yes—
174C22×C137—22no—
175C22×C137—22no—
176C22×C137—22no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137C23×C411yescharacteristic
182C22×C137C22×D411yescharacteristic
183C22×C137C2×#4111yescharacteristic
184C22×C137H(5,𝔽2)11yescharacteristic
185C22×C137H(5,𝔽2)11yescharacteristic
186C22×C137H(5,𝔽2)11yescharacteristic
187C22×C1372−1+411yescharacteristic
188C2×D137—22no—
189C2×D137—22no—
190C2×D137—22no—
191C2×D137—22no—
192C2×D137—22no—
193C2×D137—22no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C4×C137—22no—
203C4×C137—22no—
204C4×C137—22no—
205C4×C137—41no—
206C4×C137—41no—
207C4×C137—41no—
208Dic137—22no—
209Dic137—22no—
210Dic137—41no—
211Dic137—41no—
212Dic137—41no—
213Dic137C4×D414yes—
214C2×C137—42no—
215C2×C137C2×C4×D411yescharacteristic
216C2×C137#51811yescharacteristic
217C2×C137#52011yescharacteristic
218C2×C137#53511yescharacteristic
219C2×C137#53711yescharacteristic
220C2×C137#54011yescharacteristic
221C2×C137#54211yescharacteristic
222D137—42no—
223D137—42no—
224C137#232411yescharacteristic, complemented, Sylow
225#2324—1371nomaximal, Sylow
226C2×#121—1371no—
227C2×C4×D4—1371no—
228C2×#117—1371no—
229#390—1371no—
230#390—1371no—
231#390—1371no—
232#389—1371no—
233#388—1371no—
234#388D13711yescharacteristic, complemented
235#386—1371no—
236#386—1371no—
237#385—1371no—
238#385—1371no—
239#380—1371no—
240C4×C4⋊C4—1371no—
241C22×D4—1371no—
242C22×D4—1371no—
243C22×D4—1371no—
244C23×C4—1371no—
245C23×C4—1371no—
246C23×C4—1371no—
247#121—1378no—
248C4×D4—2744no—
249#117—2744no—
250C2×C4⋊C4—1371no—
251C2×C4⋊C4—1371no—
252C2×C4⋊C4—1371no—
253C2×C4⋊C4—1371no—
254C2×C4⋊C4—1371no—
255C2×C4⋊C4—1371no—
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⋊C4C2×D13711yescharacteristic
267C2×C22⋊C4C2×D13711yescharacteristic
268C2×C22⋊C4C2×D13711yescharacteristic
269C2×C42—1371no—
270C2×C42—1371no—
271C2×C42—1371no—
272C2×C42C2×D13711yescharacteristic
273#95—1371no—
274#95—1371no—
275#95—1371no—
276#95—1371no—
277#95C2×D13711yescharacteristic
278#95C2×D13711yescharacteristic
279C24—1371no—
280C24—1371no—
281C24C22×D13711yescharacteristic
282C2×D4—2742no—
283C2×D4—2742no—
284C2×D4—5481no—
285C2×D4—5481no—
286C2×D4—5481no—
287C2×D4—5481no—
288C2×D4—1374no—
289C2×D4—1374no—
290C2×D4—1374no—
291C22×C4—2742no—
292C22×C4—2742no—
293C22×C4—2742no—
294C22×C4—2742no—
295C22×C4—5481no—
296C22×C4—5481no—
297C22×C4—5481no—
298C22×C4—5481no—
299C22×C4—1371no—
300C22×C4—1371no—
301C22×C4—1371no—
302C22×C4—1371no—
303C22×C4—1371no—
304C22×C4—1374no—
305C22×C4—1371no—
306C22×C4C22×D13711yescharacteristic
307C22×C4C22×D13711yescharacteristic
308C22×C4C22×D13711yescharacteristic
309C22×C4C22×D13711yescharacteristic
310C22×C4C22×D13711yescharacteristic
311C22×C4C22×D13711yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—2742no—
314C4⋊C4—2742no—
315C4⋊C4—2742no—
316C4⋊C4—5481no—
317C4⋊C4—5481no—
318C4⋊C4—1374no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—2742no—
322C22⋊C4—2742no—
323C22⋊C4—2742no—
324C22⋊C4—5481no—
325C22⋊C4—5481no—
326C22⋊C4—5481no—
327C22⋊C4—5481no—
328C22⋊C4—5481no—
329C22⋊C4—1374no—
330C22⋊C4C4×D13714yescomplemented
331C42—41no—
332C42—2742no—
333C42—5481no—
334C42—1374no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
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—
355C23C2×C4×D13712yes—
356C23C23×D13711yescharacteristic
357D4—2748no—
358D4—5484no—
359D4—5484no—
360C2×C4—21no—
361C2×C4—21no—
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—2748no—
376C2×C4—2741no—
377C2×C4—2741no—
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—5484no—
390C2×C4—5484no—
391C2×C4—1372no—
392C2×C4—1372no—
393C2×C4—1372no—
394C2×C4—1372no—
395C2×C4—1372no—
396C2×C4C2×C4×D13712yes—
397C2×C4C2×C4×D13712yes—
398C2×C4D4×D13712yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2742no—
407C22—2742no—
408C22—2742no—
409C22—2742no—
410C22—2742no—
411C22—2742no—
412C22—5481no—
413C22—5481no—
414C22—5481no—
415C22—5481no—
416C22—5481no—
417C22—5481no—
418C22—5481no—
419C22—5481no—
420C22C22×C4×D13711yescharacteristic
421C22C2×D4×D13711yescharacteristic
422C22#41×D13711yescharacteristic
423C22#1798413311yescharacteristic
424C22#1798413411yescharacteristic
425C22#1798413411yescharacteristic
426C22#1798413511yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—2742no—
434C4—2742no—
435C4—5481no—
436C4—5481no—
437C4—5481no—
438C4—1374no—
439C2—42no—
440C2—5482no—
441C2—5482no—
442C2C4×D4×D13711yescharacteristic
443C2#2257787511yescharacteristic
444C2#2257788311yescharacteristic
445C2#2257797111yescharacteristic
446C2#2257798011yescharacteristic
447C2#2257800511yescharacteristic
448C2#2257801911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC2×#17984061C2
hypercenter#388D137
hypocenterC137#2324
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#17984048C2semidirect
3. C2×#17984046C2semidirect
4. #22577305C2semidirect
5. #22577303C2semidirect
6. #22577303C2semidirect
7. #22577300C2semidirect
8. #22577293C2semidirect
9. #22577281C2semidirect
10. #22577280C2semidirect
11. #22577278C2semidirect
12. #22577277C2semidirect
13. #22577262C2semidirect
14. C4×#14555489C2semidirect
15. #388×C137C2semidirect
22. #17984061C4semidirect
58. C2×D548C2×C4semidirect
64. C2×C22⋊D137C2×C4semidirect
79. C2×C4×D137C2×C4semidirect
85. #14555489C2×C4semidirect
94. #14555487C2×C4semidirect
97. C4×Dic137D4semidirect
108. C22⋊C4×C137C2×C4semidirect
224. C137#2324semidirect
234. #388D137semidirect
330. C22⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840612
C2×#179840482
C2×#179840462
#225773052
#225773032
#225773002
#225772932
#225772812
#225772802
#225772782
#225772772
#225772622
C4×#145554892
#388×C1372
#2324137