Finite Group Databasebeta

← #31784125

Subgroups — #31784125

Order 17536.

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

normal subgroups only

This group has 864 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×#14555511C211yesmaximal, characteristic
2C2×#17984073C212yesmaximal, complemented
3C2×C4⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577322C211yesmaximal, characteristic, complemented
5#22577321C211yesmaximal, characteristic, complemented
6#22577320C211yesmaximal, characteristic, complemented
7#22577281C212yesmaximal, complemented
8#22577277C212yesmaximal, complemented
9#22577274C212yesmaximal, complemented
10#22577265C211yesmaximal, characteristic, complemented
11#395×C137C211yesmaximal, characteristic, complemented
12C2×#14555511—28no—
13C2×#14555511—24no—
14C22×D548C2211yescharacteristic
15C22×D548C2212yes—
16C22×C4×D137C2212yes—
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984073—28no—
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2212yes—
24C2×#14555489C2211yescharacteristic
25C4⋊C4×D137—24no—
26C2×#14555487C2212yes—
27C2×#14555487C2212yes—
28C2×#14555487C2212yes—
29C2×#14555487C2212yes—
30C2×#14555487C2212yes—
31#17983995C2212yes—
32#17983995C2211yescharacteristic
33#17983995C2212yes—
34C22×Q8×C137C2211yescharacteristic
35C2×C4⋊C4×C137C2212yes—
36C2×C4⋊C4×C137C2211yescharacteristic
37#95×C137C2211yescharacteristic
38#95×C137C2212yes—
39C23×D137C2312yes—
40C23×D137C2311yescharacteristic
41#14555511—232no—
42C2×D548—24no—
43C2×D548—24no—
44C2×D548—24no—
45C2×D548—22no—
46C2×D548D418yes—
47C22×Dic137C2312yes—
48C22×Dic137C2311yescharacteristic
49C22×Dic137C2311yescharacteristic
50C22×Dic137C2311yescharacteristic
51C2×C4×D137—22no—
52C2×C4×D137—24no—
53C2×C4×D137—24no—
54C2×C4×D137—24no—
55C2×C4×D137—22no—
56C2×C4×D137—22no—
57C2×C4×D137—24no—
58C2×C4×D137—24no—
59C2×C4×D137—22no—
60C2×C4×D137D414yes—
61C4⋊Dic137—22no—
62#14555489—22no—
63#14555489—22no—
64#14555489—24no—
65#14555487—24no—
66#14555487—24no—
67#14555487—24no—
68#14555487—24no—
69#14555487—24no—
70C2×Q8×C137—24no—
71C2×Q8×C137—22no—
72C22×C4×C137C2312yes—
73C22×C4×C137C2311yescharacteristic
74C22×C4×C137C2312yes—
75C22×C4×C137C2311yescharacteristic
76C22×C4×C137C2311yescharacteristic
77C4⋊C4×C137—22no—
78C4⋊C4×C137—24no—
79C22×D137—21no—
80C22×D137—21no—
81C22×D137—22no—
82C22×D137—22no—
83C22×D137—22no—
84C22×D137—22no—
85C22×D137—22no—
86C22×D137—22no—
87C22×D137—22no—
88C22×D137C2×D414yes—
89C22×D137C2×D414yes—
90C22×D137C2×D412yes—
91D548—28no—
92D548—216no—
93C2×Dic137—21no—
94C2×Dic137—22no—
95C2×Dic137—21no—
96C2×Dic137—22no—
97C2×Dic137—22no—
98C2×Dic137—21no—
99C2×Dic137—22no—
100C2×Dic137—22no—
101C2×Dic137C2×D412yes—
102C2×Dic137#4112yescomplemented
103C4×D137—28no—
104C4×D137—216no—
105C4×D137—28no—
106C4×D137—44no—
107C23×C137C2411yescharacteristic
108Q8×C137—28no—
109C2×C4×C137—21no—
110C2×C4×C137—22no—
111C2×C4×C137—22no—
112C2×C4×C137—22no—
113C2×C4×C137—21no—
114C2×C4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—21no—
117C2×C4×C137—22no—
118C2×C4×C137—22no—
119C2×C4×C137C2×D414yes—
120C2×C4×C137C2×D412yes—
121C2×C4×C137#4112yescomplemented
122C22×C137C22×D411yescharacteristic
123C22×C137C22×D412yes—
124C22×C137C2×#4111yescharacteristic
125C22×C137C2×#4111yescharacteristic
126C22×C1372−1+412yes—
127C2×D137—22no—
128C2×D137—22no—
129C2×D137—24no—
130C2×D137—24no—
131C2×D137—24no—
132C2×D137—24no—
133C2×D137—24no—
134C2×D137—42no—
135C2×D137—42no—
136C2×D137—42no—
137C2×D137—42no—
138C2×D137#12014yes—
139C4×C137—24no—
140C4×C137—22no—
141C4×C137—22no—
142C4×C137—41no—
143C4×C137—42no—
144Dic137—22no—
145Dic137—22no—
146Dic137—41no—
147Dic137—42no—
148C2×C137C2×#12011yescharacteristic
149C2×C137#52511yescharacteristic
150C2×C137#53611yescharacteristic
151C2×C137#54712yes—
152C2×C137#54712yes—
153D137—24no—
154D137—44no—
155C137#240611yescharacteristic, complemented, Sylow
156#2406—1371nomaximal, Sylow
157C22×#41—1371no—
158C2×#123—1372no—
159C22×C4⋊C4—1371no—
160#395—1371no—
161#395D13711yescharacteristic, complemented
162#394—1371no—
163#394—1371no—
164#386—1372no—
165#385—1372no—
166#385—1372no—
167#381—1371no—
168C2×#41—2748no—
169C2×#41—2744no—
170C22×Q8C2×D13711yescharacteristic
171C22×D4—1371no—
172C22×D4—1372no—
173C23×C4—1371no—
174C23×C4—1371no—
175C23×C4—1372no—
176C23×C4—1371no—
177#123—2748no—
178C2×C4⋊C4—2744no—
179C2×C4⋊C4—1371no—
180C2×C4⋊C4—1371no—
181C2×C4⋊C4—1371no—
182C2×C4⋊C4—1372no—
183C2×C4⋊C4C2×D13711yescharacteristic
184C2×C4⋊C4C2×D13712yes—
185C2×C22⋊C4—1372no—
186C2×C22⋊C4—1372no—
187C2×C22⋊C4—1372no—
188C2×C22⋊C4—1372no—
189C2×C22⋊C4—1372no—
190#95—1372no—
191#95—1371no—
192#95—1372no—
193#95C2×D13711yescharacteristic
194#95C2×D13712yes—
195C24—1372no—
196C24—1371no—
197#41—27432no—
198C2×Q8—24no—
199C2×Q8—22no—
200C2×D4—2744no—
201C2×D4—2744no—
202C2×D4—2744no—
203C2×D4—2742no—
204C2×D4—1378no—
205C22×C4—2742no—
206C22×C4—2744no—
207C22×C4—2744no—
208C22×C4—2742no—
209C22×C4—2744no—
210C22×C4—2742no—
211C22×C4—2744no—
212C22×C4—2744no—
213C22×C4—2742no—
214C22×C4—1372no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4—1371no—
218C22×C4—1374no—
219C22×C4C22×D13712yes—
220C22×C4C22×D13711yescharacteristic
221C22×C4C22×D13712yes—
222C22×C4C22×D13711yescharacteristic
223C22×C4C22×D13711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—2742no—
227C4⋊C4—2742no—
228C4⋊C4—2742no—
229C4⋊C4—2744no—
230C22⋊C4—2744no—
231C22⋊C4—2744no—
232C22⋊C4—2744no—
233C22⋊C4—2744no—
234C22⋊C4—2744no—
235C23—2741no—
236C23—2741no—
237C23—2742no—
238C23—2742no—
239C23—2742no—
240C23—2742no—
241C23—2742no—
242C23—2742no—
243C23—2742no—
244C23—1374no—
245C23—1374no—
246C23—1372no—
247C23C23×D13711yescharacteristic
248Q8—28no—
249D4—2748no—
250D4—27416no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—27416no—
262C2×C4—2748no—
263C2×C4—2748no—
264C2×C4—2741no—
265C2×C4—2742no—
266C2×C4—2741no—
267C2×C4—2742no—
268C2×C4—2742no—
269C2×C4—2741no—
270C2×C4—2742no—
271C2×C4—2742no—
272C2×C4—5484no—
273C2×C4—1372no—
274C2×C4—1372no—
275C2×C4D4×D13714yes—
276C2×C4D4×D13712yes—
277C2×C4#1455551112yescomplemented
278C22—2742no—
279C22—2744no—
280C22—2744no—
281C22—2742no—
282C22—2744no—
283C22—2744no—
284C22—2744no—
285C22—5482no—
286C22—5482no—
287C22—5482no—
288C22—5482no—
289C22—1374no—
290C22C2×D4×D13711yescharacteristic
291C22C2×D4×D13712yes—
292C22#41×D13711yescharacteristic
293C22C2×#1455551111yescharacteristic
294C22#1798413612yes—
295C4—24no—
296C4—22no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—2742no—
301C4—2742no—
302C4—5482no—
303C4—5481no—
304C2—2744no—
305C2—5484no—
306C2#120×D13711yescharacteristic
307C2#2257791611yescharacteristic
308C2#2257797811yescharacteristic
309C2#2257805512yes—
310C2#2257805612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#395×C137C2
prime radicalC22×#14555511C2
hypercenter#395D137
hypocenterC137#2406
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#17984073C2semidirect
3. C2×C4⋊C4×D137C2semidirect
4. #22577322C2semidirect
5. #22577321C2semidirect
6. #22577320C2semidirect
7. #22577281C2semidirect
8. #22577277C2semidirect
9. #22577274C2semidirect
10. #22577265C2semidirect
11. #395×C137C2semidirect
102. C2×Dic137#41semidirect
121. C2×C4×C137#41semidirect
155. C137#2406semidirect
161. #395D137semidirect
277. C2×C4#14555511semidirect

Maximal subgroups

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

subgroupindex
C22×#145555112
C2×#179840732
C2×C4⋊C4×D1372
#225773222
#225773212
#225773202
#225772812
#225772772
#225772742
#225772652
#395×C1372
#2406137