Finite Group Databasebeta

← #31783834

Subgroups — #31783834

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×#17984060C211yesmaximal, characteristic
2C2×#17984048C211yesmaximal, characteristic, complemented
3#22577297C211yesmaximal, characteristic, complemented
4#22577293C211yesmaximal, characteristic, complemented
5#22577287C211yesmaximal, characteristic, complemented
6#22577284C211yesmaximal, characteristic, complemented
7#22577283C211yesmaximal, characteristic, complemented
8#22577282C211yesmaximal, characteristic, complemented
9#22577282C211yesmaximal, characteristic, complemented
10#22577279C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic137C211yesmaximal, characteristic, complemented
12C4×#14555487C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic137C211yesmaximal, characteristic, complemented
14C2×#17983995C211yesmaximal, characteristic, complemented
15C2×C4×D4×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C23×Dic137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984060C418yescomplemented
22#17984048—24no—
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2211yescharacteristic
28C2×C22⋊Dic137C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46C22×D4×C137C2211yescharacteristic
47C23×C4×C137C2211yescharacteristic
48C23×C4×C137C2211yescharacteristic
49C4×D4×C137—24no—
50C2×C4⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic
52C2×C22⋊C4×C137C2211yescharacteristic
53C2×C42×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C2×C22⋊D137—22no—
56C2×C22⋊D137—22no—
57C2×C22⋊D137—41no—
58C2×C22⋊D137—41no—
59C2×C22⋊D137C2×C414yescomplemented
60C2×C22⋊D137C2×C414yescomplemented
61C22×Dic137—22no—
62C22×Dic137—22no—
63C22×Dic137—41no—
64C22×Dic137—41no—
65C22×Dic137—41no—
66C22×Dic137C2×C414yescomplemented
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C22×Dic137C2311yescharacteristic
72C22×Dic137C2311yescharacteristic
73C22×Dic137C2311yescharacteristic
74C2×C4×D137—22no—
75C2×C4×D137—22no—
76C2×C4×D137—41no—
77#14555489—22no—
78#14555489C2×C414yescomplemented
79C22⋊Dic137—22no—
80C22⋊Dic137—22no—
81C22⋊Dic137—41no—
82C22⋊Dic137C2×C414yescomplemented
83#14555487—22no—
84#14555487—22no—
85#14555487—41no—
86#14555487C2×C414yescomplemented
87C4×Dic137—22no—
88C4×Dic137—41no—
89C4×Dic137—41no—
90C24×C137C2311yescharacteristic
91C24×C137C2311yescharacteristic
92C2×D4×C137—22no—
93C2×D4×C137—22no—
94C2×D4×C137C2×C414yescomplemented
95C22×C4×C137—22no—
96C22×C4×C137—22no—
97C22×C4×C137—22no—
98C22×C4×C137—22no—
99C22×C4×C137—41no—
100C22×C4×C137—41no—
101C22×C4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104C22×C4×C137C2311yescharacteristic
105C22×C4×C137C2311yescharacteristic
106C4⋊C4×C137—22no—
107C22⋊C4×C137—22no—
108C22⋊C4×C137D414yescomplemented
109C42×C137—22no—
110C22×D137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—21no—
115C22×D137—21no—
116C22×D137C22×C412yes—
117C22⋊D137—28no—
118C22⋊D137—44no—
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×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—44no—
137C2×Dic137—44no—
138C2×Dic137C22×C412yes—
139C2×Dic137C22×C412yes—
140C2×Dic137C22×C412yes—
141C2×Dic137#4112yes—
142C4×D137—44no—
143C23×C137—21no—
144C23×C137—21no—
145C23×C137—21no—
146C23×C137—21no—
147C23×C137—21no—
148C23×C137—21no—
149C23×C137—21no—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137—21no—
153C23×C137C22×C412yes—
154C23×C137C22×C412yes—
155C23×C137C2×D412yes—
156C23×C137#4112yes—
157C23×C137C2411yescharacteristic
158D4×C137—28no—
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—44no—
171C2×C4×C137—44no—
172C2×C4×C137C22×C412yes—
173C2×C4×C137C2×D412yes—
174C2×C4×C137C2×D412yes—
175C2×C4×C137#4112yes—
176C22×C137—22no—
177C22×C137—22no—
178C22×C137—22no—
179C22×C137—22no—
180C22×C137—22no—
181C22×C137—22no—
182C22×C137—22no—
183C22×C137—22no—
184C22×C137—22no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137C4×D414yes—
190C22×C137C23×C411yescharacteristic
191C22×C137C22×D411yescharacteristic
192C22×C137C2×#4111yescharacteristic
193C22×C137C2×#4111yescharacteristic
194C22×C137C2×#4111yescharacteristic
195C22×C137H(5,𝔽2)11yescharacteristic
196C22×C137H(5,𝔽2)11yescharacteristic
197C2×D137—22no—
198C2×D137—22no—
199C2×D137—22no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—41no—
203C2×D137—41no—
204C4×C137—22no—
205C4×C137—22no—
206C4×C137—22no—
207C4×C137—22no—
208C4×C137—41no—
209Dic137—22no—
210Dic137—22no—
211Dic137—22no—
212Dic137—22no—
213Dic137—41no—
214Dic137—41no—
215Dic137—41no—
216C2×C137—24no—
217C2×C137—42no—
218C2×C137C2×C4×D411yescharacteristic
219C2×C137C4×#4111yescharacteristic
220C2×C137#51811yescharacteristic
221C2×C137#54611yescharacteristic
222C2×C137#54611yescharacteristic
223C2×C137#55311yescharacteristic
224C2×C137#55311yescharacteristic
225D137—42no—
226C137#236411yescharacteristic, complemented, Sylow
227#2364—1371nomaximal, Sylow
228C2×#121—1371no—
229C2×C4×D4—1371no—
230C2×C4×D4D13711yescharacteristic, complemented
231#388—1371no—
232#388—1371no—
233#387—1371no—
234#387—1371no—
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×D4C2×D13711yescharacteristic
246C23×C4—1371no—
247C23×C4—1371no—
248C23×C4—1371no—
249C23×C4C2×D13711yescharacteristic
250C23×C4C2×D13711yescharacteristic
251#121—1378no—
252C4×D4—24no—
253C4×D4—2744no—
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⋊C4C2×D13711yescharacteristic
266C2×C22⋊C4C2×D13711yescharacteristic
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#95—1371no—
278#95—1371no—
279#95—1371no—
280#95—1371no—
281C24—1371no—
282C24C22×D13711yescharacteristic
283C24C22×D13711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—2742no—
287C2×D4—2742no—
288C2×D4—5481no—
289C2×D4—5481no—
290C2×D4—1374no—
291C2×D4—1374no—
292C2×D4C4×D13714yescomplemented
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—2742no—
300C22×C4—2742no—
301C22×C4—2742no—
302C22×C4—2742no—
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—1371no—
313C22×C4—1371no—
314C22×C4—1374no—
315C22×C4C22×D13711yescharacteristic
316C22×C4C22×D13711yescharacteristic
317C22×C4C22×D13711yescharacteristic
318C22×C4C22×D13711yescharacteristic
319C22×C4C22×D13711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—2742no—
322C4⋊C4—1374no—
323C22⋊C4—22no—
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⋊C4C22⋊D13714yescomplemented
333C42—22no—
334C42—2742no—
335C42—5481no—
336C42—5481no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2741no—
348C23—2741no—
349C23—2741no—
350C23—2741no—
351C23—2741no—
352C23—2741no—
353C23—1372no—
354C23C2×C4×D13712yes—
355C23C2×C4×D13712yes—
356C23C2×C22⋊D13712yes—
357C23#1455551012yes—
358C23C23×D13711yescharacteristic
359D4—28no—
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—44no—
374C2×C4—44no—
375C2×C4—2741no—
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—2741no—
390C2×C4—2741no—
391C2×C4—2741no—
392C2×C4—5484no—
393C2×C4—5484no—
394C2×C4—5484no—
395C2×C4—1372no—
396C2×C4—1372no—
397C2×C4—1372no—
398C2×C4—1372no—
399C2×C4C2×C4×D13712yes—
400C2×C4C2×C22⋊D13712yes—
401C2×C4C2×C22⋊D13712yes—
402C2×C4#1455550912yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2742no—
417C22—2742no—
418C22—2742no—
419C22—5481no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22C4×C22⋊D13714yes—
424C22C22×C4×D13711yescharacteristic
425C22C22×C22⋊D13711yescharacteristic
426C22#41×D13711yescharacteristic
427C22C2×#1455550911yescharacteristic
428C22C2×#1455551011yescharacteristic
429C22#1798413311yescharacteristic
430C22#1798413411yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—2742no—
437C4—2742no—
438C4—2742no—
439C4—2742no—
440C4—5481no—
441C4—5481no—
442C4—5481no—
443C2—24no—
444C2—42no—
445C2—5482no—
446C2C2×C4×C22⋊D13711yescharacteristic
447C2C4×#1455550911yescharacteristic
448C2#2257787511yescharacteristic
449C2#2257804111yescharacteristic
450C2#2257804711yescharacteristic
451C2#2257809911yescharacteristic
452C2#2257810211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
FittingC2×C4×D4×C137C2
socle, cosocleC23×C137C24
prime radicalC2×#17984060C2
hypercenterC2×C4×D4D137
hypocenterC137#2364
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#17984048C2semidirect
3. #22577297C2semidirect
4. #22577293C2semidirect
5. #22577287C2semidirect
6. #22577284C2semidirect
7. #22577283C2semidirect
8. #22577282C2semidirect
9. #22577282C2semidirect
10. #22577279C2semidirect
11. C22⋊C4×Dic137C2semidirect
12. C4×#14555487C2semidirect
13. C4×C22⋊Dic137C2semidirect
14. C2×#17983995C2semidirect
15. C2×C4×D4×C137C2semidirect
21. #17984060C4semidirect
59. C2×C22⋊D137C2×C4semidirect
60. C2×C22⋊D137C2×C4semidirect
66. C22×Dic137C2×C4semidirect
78. #14555489C2×C4semidirect
82. C22⋊Dic137C2×C4semidirect
86. #14555487C2×C4semidirect
94. C2×D4×C137C2×C4semidirect
108. C22⋊C4×C137D4semidirect
226. C137#2364semidirect
230. C2×C4×D4D137semidirect
292. C2×D4C4×D137semidirect
332. C22⋊C4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840602
C2×#179840482
#225772972
#225772932
#225772872
#225772842
#225772832
#225772822
#225772792
C22⋊C4×Dic1372
C4×#145554872
C4×C22⋊Dic1372
C2×#179839952
C2×C4×D4×C1372
#2364137