Finite Group Databasebeta

← #31784076

Subgroups — #31784076

Order 17536.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic274C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D137C211yesmaximal, characteristic
3#22577345C211yesmaximal, characteristic, complemented
4#22577344C211yesmaximal, characteristic, complemented
5#22577332C211yesmaximal, characteristic, complemented
6#22577331C211yesmaximal, characteristic, complemented
7#22577319C211yesmaximal, characteristic
8#22577318C211yesmaximal, characteristic
9#22577316C211yesmaximal, characteristic, complemented
10#22577315C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic, complemented
12C4×#14555487C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic137C211yesmaximal, characteristic, complemented
14#22577249C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C4×Dic274—24no—
20C4×C22⋊D137—24no—
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
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
46C23×C4×C137C2211yescharacteristic
47C2×C22⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C42×C137C2211yescharacteristic
50C2×C42×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53C23×D137C2311yescharacteristic
54C2×Dic274—22no—
55C2×Dic274—22no—
56C2×Dic274—22no—
57C2×C22⋊D137—22no—
58C2×C22⋊D137—22no—
59C2×C22⋊D137—22no—
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C2×C4×D137—22no—
68C2×C4×D137—22no—
69C2×C4×D137—22no—
70C4⋊Dic137—22no—
71#14555489—22no—
72#14555489—22no—
73#14555489—22no—
74C22⋊Dic137—22no—
75C22⋊Dic137—22no—
76C22⋊Dic137—22no—
77#14555487—22no—
78#14555487—22no—
79#14555487—22no—
80#14555487—22no—
81#14555487—22no—
82C4×Dic137—22no—
83C4×Dic137—22no—
84C4×Dic137D414yes—
85C24×C137C2311yescharacteristic
86C22×C4×C137—22no—
87C22×C4×C137—22no—
88C22×C4×C137—22no—
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C22×C4×C137C2311yescharacteristic
94C22×C4×C137C2311yescharacteristic
95C22⋊C4×C137—22no—
96C22⋊C4×C137—22no—
97C42×C137—22no—
98C42×C137—22no—
99C22×D137—21no—
100C22×D137—21no—
101C22×D137—21no—
102C22×D137—21no—
103C22×D137—21no—
104C22×D137—21no—
105C22×D137#4112yes—
106Dic274—28no—
107C22⋊D137—28no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137C2×D412yes—
127C2×Dic137C2×D412yes—
128C2×Dic137#4112yes—
129C4×D137—44no—
130C23×C137—21no—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137—21no—
135C23×C137—21no—
136C23×C137#4112yes—
137C23×C137C2411yescharacteristic
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—44no—
153C2×C4×C137C2×D412yes—
154C2×C4×C137#4112yes—
155C2×C4×C137#4112yes—
156C2×C4×C137#4112yes—
157C22×C137—22no—
158C22×C137—22no—
159C22×C137—22no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137C22×D411yescharacteristic
165C22×C137C2×#4111yescharacteristic
166C22×C137C2×#4111yescharacteristic
167C22×C137C2×#4111yescharacteristic
168C22×C137C2×#4111yescharacteristic
169C22×C137C2×#4111yescharacteristic
170C22×C137C2×#4111yescharacteristic
171C2×D137—22no—
172C2×D137—22no—
173C2×D137—22no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—41no—
177C2×D137—41no—
178C4×C137—22no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137—41no—
182C4×C137—41no—
183C4×C137#12414yes—
184Dic137—22no—
185Dic137—22no—
186Dic137—22no—
187Dic137—41no—
188Dic137—41no—
189Dic137—41no—
190Dic137—41no—
191C2×C137—42no—
192C2×C137#52511yescharacteristic
193C2×C137#52511yescharacteristic
194C2×C137C2×#12411yescharacteristic
195C2×C137#52911yescharacteristic
196C2×C137#52911yescharacteristic
197C2×C137#52911yescharacteristic
198C2×C137#52911yescharacteristic
199D137—42no—
200C137#239811yescharacteristic, complemented, Sylow
201#2398—1371nomaximal, Sylow
202C2×C4×Q8—1371no—
203C2×C4×D4—1371no—
204#400—1371no—
205#400—1371no—
206#397—1371no—
207#397—1371no—
208#394—1371no—
209#394—1371no—
210#393—1371no—
211#393—1371no—
212C4×C22⋊C4—1371no—
213C4×C22⋊C4—1371no—
214C4×C22⋊C4—1371no—
215C4×C22⋊C4D13711yescharacteristic, complemented
216#376—1371no—
217C22×Q8—1371no—
218C22×D4—1371no—
219C23×C4—1371no—
220C23×C4C2×D13711yescharacteristic
221C4×Q8—2744no—
222C4×D4—2744no—
223C2×C4⋊C4—1371no—
224C2×C4⋊C4—1371no—
225C2×C4⋊C4—1371no—
226C2×C4⋊C4—1371no—
227C2×C22⋊C4—1371no—
228C2×C22⋊C4—1371no—
229C2×C22⋊C4—1371no—
230C2×C22⋊C4—1371no—
231C2×C22⋊C4—1371no—
232C2×C22⋊C4—1371no—
233C2×C22⋊C4—1371no—
234C2×C22⋊C4—1371no—
235C2×C22⋊C4C2×D13711yescharacteristic
236C2×C22⋊C4C2×D13711yescharacteristic
237C2×C42—1371no—
238C2×C42—1371no—
239C2×C42—1371no—
240C2×C42C2×D13711yescharacteristic
241C2×C42C2×D13711yescharacteristic
242#95—1371no—
243#95—1371no—
244#95—1371no—
245#95—1371no—
246#95—1371no—
247#95—1371no—
248#95—1371no—
249#95—1371no—
250#95—1371no—
251#95—1371no—
252#95C2×D13711yescharacteristic
253#95C2×D13711yescharacteristic
254C24—1371no—
255C24C22×D13711yescharacteristic
256C2×Q8—2742no—
257C2×Q8—2742no—
258C2×Q8—2742no—
259C2×D4—2742no—
260C2×D4—2742no—
261C2×D4—2742no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—2742no—
266C22×C4—2742no—
267C22×C4—2742no—
268C22×C4—1371no—
269C22×C4—1371no—
270C22×C4—1371no—
271C22×C4—1371no—
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1371no—
275C22×C4C22×D13711yescharacteristic
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278C22×C4C22×D13711yescharacteristic
279C22×C4C22×D13711yescharacteristic
280C22×C4C22×D13711yescharacteristic
281C4⋊C4—2742no—
282C4⋊C4—2742no—
283C4⋊C4—2742no—
284C4⋊C4—2742no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—2742no—
288C22⋊C4—2742no—
289C22⋊C4—2742no—
290C22⋊C4—2742no—
291C22⋊C4—2742no—
292C22⋊C4—2742no—
293C22⋊C4—2742no—
294C22⋊C4—2742no—
295C42—22no—
296C42—22no—
297C42—2742no—
298C42—2742no—
299C42—1374no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—2741no—
307C23—2741no—
308C23—2741no—
309C23—2741no—
310C23—2741no—
311C23—2741no—
312C23—1372no—
313C23#1455551012yes—
314C23C23×D13711yescharacteristic
315Q8—2748no—
316D4—2748no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—2741no—
348C2×C4—2741no—
349C2×C4—2741no—
350C2×C4—5484no—
351C2×C4—1372no—
352C2×C4—1372no—
353C2×C4—1372no—
354C2×C4D4×D13712yes—
355C2×C4#1455550912yes—
356C2×C4#1455551012yes—
357C2×C4#1455551012yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—2742no—
366C22—2742no—
367C22—2742no—
368C22—5481no—
369C22—5481no—
370C22—5481no—
371C22—5481no—
372C22C2×D4×D13711yescharacteristic
373C22#41×D13711yescharacteristic
374C22#41×D13711yescharacteristic
375C22C2×#1455550911yescharacteristic
376C22C2×#1455551011yescharacteristic
377C22C2×#1455551011yescharacteristic
378C22C2×#1455551011yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2742no—
385C4—2742no—
386C4—2742no—
387C4—5481no—
388C4—5481no—
389C4—5481no—
390C4—5481no—
391C4#1798407414yes—
392C2—42no—
393C2—5482no—
394C2#2257791211yescharacteristic
395C2#2257791211yescharacteristic
396C2C2×#1798407411yescharacteristic
397C2#2257793111yescharacteristic
398C2#2257793311yescharacteristic
399C2#2257793511yescharacteristic
400C2#2257793711yescharacteristic
subgroupG/H
centerC22×C4C22×D137
commutator, socle, cosocleC23×C137C24
FrattiniC23C23×D137
FittingC4×C22⋊C4×C137C2
prime radicalC22×C22⋊D137C22
hypercenterC4×C22⋊C4D137
hypocenterC137#2398
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic274C2semidirect
3. #22577345C2semidirect
4. #22577344C2semidirect
5. #22577332C2semidirect
6. #22577331C2semidirect
9. #22577316C2semidirect
10. #22577315C2semidirect
11. C4×#14555487C2semidirect
12. C4×#14555487C2semidirect
13. C4×C22⋊Dic137C2semidirect
14. #22577249C2semidirect
15. C4×C22⋊C4×C137C2semidirect
200. C137#2398semidirect
215. C4×C22⋊C4D137semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2742
C2×C4×C22⋊D1372
#225773452
#225773442
#225773322
#225773312
#225773192
#225773182
#225773162
#225773152
C4×#145554872
C4×C22⋊Dic1372
#225772492
C4×C22⋊C4×C1372
#2398137