Finite Group Databasebeta

← #31783648

Subgroups — #31783648

Order 17536.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984046C211yesmaximal, characteristic
2#22577289C211yesmaximal, characteristic, complemented
3#22577288C211yesmaximal, characteristic, complemented
4#22577286C211yesmaximal, characteristic, complemented
5#22577277C211yesmaximal, characteristic
6#22577274C211yesmaximal, characteristic
7#22577270C211yesmaximal, characteristic, complemented
8#22577268C211yesmaximal, characteristic, complemented
9#22577265C211yesmaximal, characteristic
10C4×#14555489C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic
12#22577250C211yesmaximal, characteristic
13#22577248C211yesmaximal, characteristic, complemented
14#95×D137C211yesmaximal, characteristic
15#387×C137C211yesmaximal, characteristic, complemented
16C22×C4×D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19#17984046C418yescomplemented
20C2×C4⋊Dic137C2211yescharacteristic
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555487C2211yescharacteristic
25C2×#14555487C2211yescharacteristic
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2211yescharacteristic
28C2×C4×Dic137C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
30C2×C4×Dic137C2211yescharacteristic
31C2×C4×Dic137C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33#17983995C2211yescharacteristic
34#17983995C2211yescharacteristic
35#17983995C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C2×C4⋊C4×C137C2211yescharacteristic
46C2×C4⋊C4×C137C2211yescharacteristic
47C2×C42×C137C2211yescharacteristic
48#95×C137C2211yescharacteristic
49#95×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C23×D137C2311yescharacteristic
53C22×Dic137C2311yescharacteristic
54C22×Dic137C2311yescharacteristic
55C22×Dic137C2311yescharacteristic
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C2×C4×D137—22no—
61C2×C4×D137—22no—
62C2×C4×D137—22no—
63C2×C4×D137—22no—
64C2×C4×D137—22no—
65C2×C4×D137—22no—
66C2×C4×D137—41no—
67C2×C4×D137—41no—
68C2×C4×D137C2×C414yes—
69C4⋊Dic137—41no—
70C4⋊Dic137C2×C414yescomplemented
71#14555489—22no—
72#14555489—22no—
73#14555487—22no—
74#14555487—22no—
75#14555487C2×C414yes—
76#14555487C2×C414yes—
77C4×Dic137—22no—
78C4×Dic137—22no—
79C4×Dic137—41no—
80C4×Dic137C2×C414yescomplemented
81C4×Dic137C2×C414yescomplemented
82C22×C4×C137C2311yescharacteristic
83C22×C4×C137C2311yescharacteristic
84C22×C4×C137C2311yescharacteristic
85C22×C4×C137C2311yescharacteristic
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C22×C4×C137C2311yescharacteristic
89C4⋊C4×C137—41no—
90C4⋊C4×C137C2×C414yescomplemented
91C42×C137—41no—
92C22×D137—21no—
93C22×D137—21no—
94C22×D137—21no—
95C22×D137—21no—
96C22×D137C22×C412yes—
97C22×D137#4112yes—
98C22×D137#4112yes—
99C2×Dic137—21no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137—21no—
104C2×Dic137—21no—
105C2×Dic137—21no—
106C2×Dic137—21no—
107C2×Dic137—21no—
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×Dic137C22×C412yes—
117C2×Dic137C22×C412yes—
118C2×Dic137C22×C412yes—
119C2×Dic137#4112yes—
120C4×D137—28no—
121C4×D137—44no—
122C4×D137—44no—
123C23×C137C2411yescharacteristic
124C2×C4×C137—21no—
125C2×C4×C137—21no—
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137C22×C412yes—
142C2×C4×C137C22×C412yes—
143C2×C4×C137C22×C412yes—
144C2×C4×C137#4112yes—
145C22×C137C23×C411yescharacteristic
146C22×C137C2×#4111yescharacteristic
147C22×C137C2×#4111yescharacteristic
148C22×C137C2×#4111yescharacteristic
149C22×C137C2×#4111yescharacteristic
150C22×C137H(5,𝔽2)11yescharacteristic
151C22×C1372−1+411yescharacteristic
152C2×D137—22no—
153C2×D137—22no—
154C2×D137—22no—
155C2×D137—22no—
156C2×D137—22no—
157C2×D137—22no—
158C2×D137#11714yes—
159C4×C137—22no—
160C4×C137—22no—
161C4×C137—22no—
162C4×C137—22no—
163C4×C137—41no—
164C4×C137—41no—
165C4×C137—41no—
166Dic137—22no—
167Dic137—22no—
168Dic137—22no—
169Dic137—22no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173C2×C137C2×#11711yescharacteristic
174C2×C137C4×#4111yescharacteristic
175C2×C137#52011yescharacteristic
176C2×C137#55111yescharacteristic
177C2×C137#55111yescharacteristic
178C2×C137#55211yescharacteristic
179C2×C137#55211yescharacteristic
180D137—24no—
181C137#234311yescharacteristic, complemented, Sylow
182#2343—1371nomaximal, Sylow
183C2×#117—1371no—
184#387—1371no—
185#387—1371no—
186#387—1371no—
187#387D13711yescharacteristic, complemented
188#385—1371no—
189#385—1371no—
190#383—1371no—
191#382—1371no—
192#381—1371no—
193C4×C4⋊C4—1371no—
194C4×C22⋊C4—1371no—
195C4×C22⋊C4—1371no—
196#376—1371no—
197C2×#95—1371no—
198C23×C4—1371no—
199C23×C4—1371no—
200C23×C4—1371no—
201#117—1378no—
202C2×C4⋊C4—1371no—
203C2×C4⋊C4—1371no—
204C2×C4⋊C4—1371no—
205C2×C4⋊C4—1371no—
206C2×C4⋊C4C2×D13711yescharacteristic
207C2×C4⋊C4C2×D13711yescharacteristic
208C2×C22⋊C4—1371no—
209C2×C22⋊C4—1371no—
210C2×C22⋊C4—1371no—
211C2×C22⋊C4—1371no—
212C2×C42—1371no—
213C2×C42—1371no—
214C2×C42—1371no—
215C2×C42—1371no—
216C2×C42—1371no—
217C2×C42C2×D13711yescharacteristic
218#95—1371no—
219#95—1371no—
220#95—1371no—
221#95—1371no—
222#95—1371no—
223#95—1371no—
224#95—1371no—
225#95—1371no—
226#95—1371no—
227#95—1371no—
228#95—1371no—
229#95—1371no—
230#95C2×D13711yescharacteristic
231#95C2×D13711yescharacteristic
232#95C2×D13711yescharacteristic
233#95C2×D13711yescharacteristic
234C24—1371no—
235C22×C4—2742no—
236C22×C4—2742no—
237C22×C4—2742no—
238C22×C4—2742no—
239C22×C4—2742no—
240C22×C4—2742no—
241C22×C4—5481no—
242C22×C4—5481no—
243C22×C4—1371no—
244C22×C4—1371no—
245C22×C4—1371no—
246C22×C4—1371no—
247C22×C4—1371no—
248C22×C4—1371no—
249C22×C4—1371no—
250C22×C4—1374no—
251C22×C4C22×D13711yescharacteristic
252C22×C4C22×D13711yescharacteristic
253C22×C4C22×D13711yescharacteristic
254C22×C4C22×D13711yescharacteristic
255C22×C4C22×D13711yescharacteristic
256C22×C4C22×D13711yescharacteristic
257C22×C4C22×D13711yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—2742no—
260C4⋊C4—2742no—
261C4⋊C4—5481no—
262C4⋊C4—1374no—
263C4⋊C4C4×D13714yescomplemented
264C22⋊C4—2742no—
265C22⋊C4—2742no—
266C22⋊C4—1374no—
267C22⋊C4—1374no—
268C42—41no—
269C42—2742no—
270C42—2742no—
271C42—5481no—
272C42—1374no—
273C42—1374no—
274C23—2741no—
275C23—2741no—
276C23—2741no—
277C23—2741no—
278C23—1372no—
279C23—1372no—
280C23—1372no—
281C23C23×D13711yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—2748no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—5484no—
318C2×C4—5484no—
319C2×C4—1372no—
320C2×C4—1372no—
321C2×C4—1372no—
322C2×C4—1372no—
323C2×C4C2×C4×D13712yes—
324C2×C4C2×C4×D13712yes—
325C2×C4C2×C4×D13712yes—
326C2×C4#1455551112yes—
327C22—2742no—
328C22—2742no—
329C22—2742no—
330C22—2742no—
331C22—2742no—
332C22—2742no—
333C22—1374no—
334C22C22×C4×D13711yescharacteristic
335C22#41×D13711yescharacteristic
336C22#41×D13711yescharacteristic
337C22#41×D13711yescharacteristic
338C22C2×#1455551111yescharacteristic
339C22#1798413311yescharacteristic
340C22#1798413511yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2742no—
349C4—2742no—
350C4—2742no—
351C4—2742no—
352C4—5481no—
353C4—5481no—
354C4—5481no—
355C2—2744no—
356C2#117×D13711yescharacteristic
357C2C4×#1455551111yescharacteristic
358C2#2257788211yescharacteristic
359C2#2257808011yescharacteristic
360C2#2257808111yescharacteristic
361C2#2257808911yescharacteristic
362C2#2257809111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#387×C137C2
socle, cosocleC23×C137C24
prime radicalC23×D137C23
hypercenter#387D137
hypocenterC137#2343
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22577289C2semidirect
3. #22577288C2semidirect
4. #22577286C2semidirect
7. #22577270C2semidirect
8. #22577268C2semidirect
10. C4×#14555489C2semidirect
13. #22577248C2semidirect
15. #387×C137C2semidirect
19. #17984046C4semidirect
70. C4⋊Dic137C2×C4semidirect
80. C4×Dic137C2×C4semidirect
81. C4×Dic137C2×C4semidirect
90. C4⋊C4×C137C2×C4semidirect
181. C137#2343semidirect
187. #387D137semidirect
263. C4⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840462
#225772892
#225772882
#225772862
#225772772
#225772742
#225772702
#225772682
#225772652
C4×#145554892
C4×#145554872
#225772502
#225772482
#95×D1372
#387×C1372
#2343137