Finite Group Databasebeta

← #31784103

Subgroups — #31784103

Order 17536.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14555509C211yesmaximal, characteristic, complemented
2C2×#17984056C212yesmaximal, complemented
3C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577316C211yesmaximal, characteristic, complemented
5#22577315C211yesmaximal, characteristic, complemented
6#22577311C211yesmaximal, characteristic, complemented
7#22577283C212yesmaximal, complemented
8#22577280C212yesmaximal, complemented
9#22577279C212yesmaximal, complemented
10#22577265C211yesmaximal, characteristic, complemented
11#392×C137C211yesmaximal, characteristic, complemented
12C24×D137C2211yescharacteristic
13C2×#14555509—28no—
14C2×#14555509—24no—
15C22×Dic274C2211yescharacteristic, complemented
16C22×C22⋊D137C2212yes—
17C22×C22⋊D137C2212yes—
18C23×Dic137C2212yescomplemented
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984056—28no—
22C2×C22⋊Dic137C2211yescharacteristic
23C2×C22⋊Dic137C2212yes—
24C2×C22⋊Dic137C2212yescomplemented
25C2×#14555487C2212yes—
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2212yes—
28C2×#14555487C2211yescharacteristic
29C22⋊C4×D137—24no—
30#17983995C2212yescomplemented
31#17983995C2211yescharacteristic, complemented
32#17983995C2212yescomplemented
33#17983995C2212yescomplemented
34C22×D4×C137C2212yes—
35C22×D4×C137C2211yescharacteristic, complemented
36C2×C22⋊C4×C137C2211yescharacteristic
37C2×C22⋊C4×C137C2212yescomplemented
38#95×C137C2211yescharacteristic, complemented
39C23×D137—22no—
40C23×D137—22no—
41C23×D137—24no—
42C23×D137—22no—
43C23×D137—42no—
44C23×D137C2311yescharacteristic
45C23×D137C2311yescharacteristic
46#14555509—232no—
47C2×Dic274—24no—
48C2×Dic274—22no—
49C2×C22⋊D137—24no—
50C2×C22⋊D137—24no—
51C2×C22⋊D137—24no—
52C2×C22⋊D137—24no—
53C2×C22⋊D137—42no—
54C2×C22⋊D137D418yescomplemented
55C22×Dic137—24no—
56C22×Dic137—24no—
57C22×Dic137—24no—
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2312yes—
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2312yes—
62C2×C4×D137—24no—
63C2×C4×D137—22no—
64C2×C4×D137—42no—
65C2×C4×D137D414yescomplemented
66C22⋊Dic137—22no—
67C22⋊Dic137—24no—
68C22⋊Dic137—42no—
69#14555487—24no—
70#14555487—22no—
71#14555487—24no—
72#14555487—22no—
73C24×C137C2312yes—
74C24×C137C2311yescharacteristic
75C2×D4×C137—24no—
76C2×D4×C137—24no—
77C2×D4×C137—24no—
78C2×D4×C137—22no—
79C2×D4×C137—42no—
80C22×C4×C137C2312yes—
81C22×C4×C137C2311yescharacteristic
82C22×C4×C137C2311yescharacteristic
83C22⋊C4×C137—22no—
84C22⋊C4×C137—42no—
85C22×D137—21no—
86C22×D137—21no—
87C22×D137—22no—
88C22×D137—28no—
89C22×D137—21no—
90C22×D137—22no—
91C22×D137—21no—
92C22×D137—22no—
93C22×D137—48no—
94C22×D137—48no—
95C22×D137—44no—
96C22×D137—44no—
97C22×D137C2×D414yes—
98C22×D137C2×D412yes—
99C22×D137#4112yescomplemented
100Dic274—28no—
101C22⋊D137—216no—
102C22⋊D137—48no—
103C2×Dic137—216no—
104C2×Dic137—22no—
105C2×Dic137—22no—
106C2×Dic137—22no—
107C2×Dic137—21no—
108C2×Dic137—22no—
109C2×Dic137—22no—
110C2×Dic137—22no—
111C2×Dic137—22no—
112C2×Dic137C2×D412yes—
113C2×Dic137C2×D414yes—
114C4×D137—28no—
115C4×D137—44no—
116C23×C137—21no—
117C23×C137—22no—
118C23×C137—22no—
119C23×C137—22no—
120C23×C137—22no—
121C23×C137—22no—
122C23×C137—21no—
123C23×C137—22no—
124C23×C137—22no—
125C23×C137—22no—
126C23×C137C2×D414yes—
127C23×C137#4112yescomplemented
128C23×C137C2411yescharacteristic
129D4×C137—28no—
130D4×C137—48no—
131C2×C4×C137—22no—
132C2×C4×C137—21no—
133C2×C4×C137—22no—
134C2×C4×C137—22no—
135C2×C4×C137—22no—
136C2×C4×C137—22no—
137C2×C4×C137C2×D412yes—
138C22×C137—24no—
139C22×C137—22no—
140C22×C137—24no—
141C22×C137—24no—
142C22×C137—24no—
143C22×C137—41no—
144C22×C137—42no—
145C22×C137—42no—
146C22×C137—41no—
147C22×C137—42no—
148C22×C137—42no—
149C22×C137—42no—
150C22×C137C22×D411yescharacteristic
151C22×C137C22×D412yes—
152C22×C137C2×#4111yescharacteristic
153C22×C137C2×#4111yescharacteristic
154C22×C137H(5,𝔽2)12yes—
155C2×D137—22no—
156C2×D137—22no—
157C2×D137—24no—
158C2×D137—24no—
159C2×D137—22no—
160C2×D137—24no—
161C2×D137—416no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—42no—
165C2×D137#12014yes—
166C4×C137—22no—
167C4×C137—41no—
168C4×C137—42no—
169Dic137—22no—
170Dic137—24no—
171Dic137—41no—
172Dic137—42no—
173C2×C137—42no—
174C2×C137—44no—
175C2×C137C2×#12011yescharacteristic
176C2×C137#52511yescharacteristic
177C2×C137#53511yescharacteristic
178C2×C137#54612yes—
179C2×C137#54612yes—
180D137—24no—
181D137—42no—
182C137#240411yescharacteristic, complemented, Sylow
183#2404—1371nomaximal, Sylow
184C22×#41—1371no—
185C2×#120—1372no—
186C22×C22⋊C4—1371no—
187#393—1371no—
188#393—1371no—
189#392—1371no—
190#392D13711yescharacteristic, complemented
191#386—1372no—
192#386—1372no—
193#386—1372no—
194#381—1371no—
195C25—1371no—
196C2×#41—2748no—
197C2×#41—2744no—
198C22×Q8—1371no—
199C22×D4—1372no—
200C22×D4—1372no—
201C22×D4C2×D13712yes—
202C22×D4C2×D13711yescharacteristic, complemented
203C23×C4—1372no—
204C23×C4—1371no—
205C23×C4—1371no—
206#120—2748no—
207C2×C22⋊C4—2744no—
208C2×C22⋊C4—1371no—
209C2×C22⋊C4—1372no—
210C2×C22⋊C4—1371no—
211C2×C22⋊C4—1371no—
212C2×C22⋊C4—1372no—
213C2×C22⋊C4—1372no—
214C2×C22⋊C4—1372no—
215C2×C22⋊C4C2×D13711yescharacteristic
216C2×C22⋊C4C2×D13712yescomplemented
217#95—1372no—
218#95—1371no—
219#95—1372no—
220#95—1372no—
221#95C2×D13711yescharacteristic, complemented
222C24—2742no—
223C24—2742no—
224C24—2744no—
225C24—2742no—
226C24—5482no—
227C24—1371no—
228C24—1371no—
229C24C22×D13712yes—
230C24C22×D13711yescharacteristic
231#41—27432no—
232C2×Q8—2744no—
233C2×Q8—2742no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—2744no—
240C2×D4—2744no—
241C2×D4—2744no—
242C2×D4—2744no—
243C2×D4—5482no—
244C2×D4—1378no—
245C22×C4—2744no—
246C22×C4—2742no—
247C22×C4—2744no—
248C22×C4—2744no—
249C22×C4—2744no—
250C22×C4—5482no—
251C22×C4—1372no—
252C22×C4—1371no—
253C22×C4—1371no—
254C22×C4—1372no—
255C22×C4—1374no—
256C22×C4C22×D13712yes—
257C22×C4C22×D13711yescharacteristic
258C22×C4C22×D13711yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—2744no—
262C22⋊C4—2742no—
263C22⋊C4—2744no—
264C22⋊C4—2742no—
265C22⋊C4—2744no—
266C22⋊C4—2742no—
267C22⋊C4—5482no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—2748no—
279C23—2741no—
280C23—2741no—
281C23—2742no—
282C23—2741no—
283C23—2742no—
284C23—2741no—
285C23—2742no—
286C23—5488no—
287C23—5488no—
288C23—5484no—
289C23—5484no—
290C23—1372no—
291C23—1374no—
292C23—1372no—
293C23D4×D13714yes—
294C23#1455550912yescomplemented
295C23C23×D13711yescharacteristic
296Q8—2748no—
297D4—28no—
298D4—48no—
299D4—27416no—
300D4—5488no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—2748no—
308C2×C4—27416no—
309C2×C4—2742no—
310C2×C4—2742no—
311C2×C4—2742no—
312C2×C4—2741no—
313C2×C4—2742no—
314C2×C4—2742no—
315C2×C4—2742no—
316C2×C4—2742no—
317C2×C4—5484no—
318C2×C4—1372no—
319C2×C4—1374no—
320C2×C4D4×D13712yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—2742no—
334C22—2744no—
335C22—2744no—
336C22—2742no—
337C22—2744no—
338C22—2742no—
339C22—54816no—
340C22—5481no—
341C22—5482no—
342C22—5481no—
343C22—1374no—
344C22C2×D4×D13711yescharacteristic
345C22C2×D4×D13712yes—
346C22#41×D13711yescharacteristic
347C22C2×#1455550911yescharacteristic
348C22#1798413312yes—
349C4—22no—
350C4—41no—
351C4—42no—
352C4—2742no—
353C4—2744no—
354C4—5481no—
355C4—5482no—
356C2—42no—
357C2—44no—
358C2—2744no—
359C2—5482no—
360C2#120×D13711yescharacteristic
361C2#2257791311yescharacteristic
362C2#2257796611yescharacteristic
363C2#2257804012yes—
364C2#2257804212yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#392×C137C2
hypercenter#392D137
hypocenterC137#2404
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#14555509C2semidirect
2. C2×#17984056C2semidirect
3. C2×C22⋊C4×D137C2semidirect
4. #22577316C2semidirect
5. #22577315C2semidirect
6. #22577311C2semidirect
7. #22577283C2semidirect
8. #22577280C2semidirect
9. #22577279C2semidirect
10. #22577265C2semidirect
11. #392×C137C2semidirect
15. C22×Dic274C22semidirect
18. C23×Dic137C22semidirect
24. C2×C22⋊Dic137C22semidirect
30. #17983995C22semidirect
31. #17983995C22semidirect
32. #17983995C22semidirect
33. #17983995C22semidirect
35. C22×D4×C137C22semidirect
37. C2×C22⋊C4×C137C22semidirect
38. #95×C137C22semidirect
54. C2×C22⋊D137D4semidirect
65. C2×C4×D137D4semidirect
99. C22×D137#41semidirect
127. C23×C137#41semidirect
182. C137#2404semidirect
190. #392D137semidirect
202. C22×D4C2×D137semidirect
216. C2×C22⋊C4C2×D137semidirect
221. #95C2×D137semidirect
294. C23#14555509semidirect

Maximal subgroups

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

subgroupindex
C22×#145555092
C2×#179840562
C2×C22⋊C4×D1372
#225773162
#225773152
#225773112
#225772832
#225772802
#225772792
#225772652
#392×C1372
#2404137