Finite Group Databasebeta

← #31784147

Subgroups — #31784147

Order 17536.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×Q8×D137C211yesmaximal, characteristic, complemented
2C2×#17984065C212yesmaximal, complemented
3C2×#17984048C211yesmaximal, characteristic
4#22577324C211yesmaximal, characteristic, complemented
5#22577322C211yesmaximal, characteristic, complemented
6#22577315C211yesmaximal, characteristic, complemented
7#22577307C212yesmaximal, complemented
8#22577279C212yesmaximal
9#22577277C212yesmaximal, complemented
10#22577262C211yesmaximal, characteristic, complemented
11#393×C137C211yesmaximal, characteristic, complemented
12C22×Dic274C2211yescharacteristic
13C22×Dic274C2212yes—
14C2×Q8×D137—28no—
15C2×Q8×D137—24no—
16C22×C22⋊D137C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18C22×C4×D137C2212yes—
19C22×C4×D137C2211yescharacteristic
20#17984065—28no—
21#17984048—24no—
22C2×C4⋊Dic137C2212yes—
23C2×#14555489C2212yes—
24C2×#14555489C2212yes—
25C2×#14555489C2211yescharacteristic
26C2×C22⋊Dic137C2212yes—
27C2×#14555487C2212yes—
28C2×#14555487C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
30#17983995C2211yescharacteristic
31#17983995C2212yes—
32#17983995C2212yes—
33#17983995C2212yes—
34C22×Q8×C137C2211yescharacteristic
35C2×C22⋊C4×C137C2212yes—
36C2×C22⋊C4×C137C2211yescharacteristic
37#95×C137C2211yescharacteristic
38#95×C137C2212yes—
39C23×D137C2311yescharacteristic
40C2×Dic274—24no—
41C2×Dic274—24no—
42C2×Dic274—24no—
43C2×Dic274—22no—
44C2×Dic274D418yescomplemented
45Q8×D137—232no—
46C2×C22⋊D137—22no—
47C2×C22⋊D137—22no—
48C2×C22⋊D137Q814yescomplemented
49C22×Dic137—22no—
50C22×Dic137—24no—
51C22×Dic137C2312yes—
52C22×Dic137C2311yescharacteristic
53C22×Dic137C2312yes—
54C22×Dic137C2311yescharacteristic
55C22×Dic137C2311yescharacteristic
56C2×C4×D137—24no—
57C2×C4×D137—24no—
58C2×C4×D137—24no—
59C2×C4×D137—24no—
60C2×C4×D137D414yes—
61C4⋊Dic137—24no—
62#14555489—24no—
63#14555489—22no—
64#14555489—24no—
65C22⋊Dic137—24no—
66#14555487—22no—
67#14555487—24no—
68C4×Dic137—22no—
69C24×C137C2311yescharacteristic
70C2×Q8×C137—24no—
71C2×Q8×C137—22no—
72C22×C4×C137C2311yescharacteristic
73C22×C4×C137C2312yes—
74C22×C4×C137C2312yes—
75C22×C4×C137C2311yescharacteristic
76C22⋊C4×C137—22no—
77C22⋊C4×C137—24no—
78C22×D137—22no—
79C22×D137—22no—
80C22×D137C2×D412yes—
81C22×D137C2×Q812yes—
82C22×D137#4112yes—
83Dic274—28no—
84Dic274—216no—
85C22⋊D137—28no—
86C2×Dic137—21no—
87C2×Dic137—22no—
88C2×Dic137—22no—
89C2×Dic137—21no—
90C2×Dic137—21no—
91C2×Dic137—22no—
92C2×Dic137—22no—
93C2×Dic137—22no—
94C2×Dic137—22no—
95C2×Dic137—44no—
96C2×Dic137C2×D414yes—
97C2×Dic137C2×D412yes—
98C2×Dic137C2×D414yes—
99C2×Dic137C2×Q812yescomplemented
100C4×D137—216no—
101C4×D137—28no—
102C23×C137—22no—
103C23×C137—21no—
104C23×C137—21no—
105C23×C137—22no—
106C23×C137C2×Q812yescomplemented
107C23×C137C2411yescharacteristic
108Q8×C137—28no—
109C2×C4×C137—22no—
110C2×C4×C137—22no—
111C2×C4×C137—22no—
112C2×C4×C137—21no—
113C2×C4×C137—22no—
114C2×C4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—22no—
117C2×C4×C137C2×D414yes—
118C2×C4×C137C2×D412yes—
119C22×C137—22no—
120C22×C137—24no—
121C22×C137—41no—
122C22×C137—41no—
123C22×C137—42no—
124C22×C137C22×D411yescharacteristic
125C22×C137C22×D412yes—
126C22×C137C22×Q811yescharacteristic
127C22×C137C2×#4111yescharacteristic
128C22×C1372−1+412yes—
129C2×D137—22no—
130C2×D137—22no—
131C2×D137—24no—
132C2×D137—24no—
133C2×D137C22⋊Q814yescomplemented
134C4×C137—24no—
135C4×C137—22no—
136C4×C137—41no—
137C4×C137—42no—
138Dic137—24no—
139Dic137—22no—
140Dic137—41no—
141Dic137—42no—
142Dic137#12014yes—
143C2×C137—42no—
144C2×C137C2×#12011yescharacteristic
145C2×C137C2×C22⋊Q811yescharacteristic
146C2×C137#53611yescharacteristic
147C2×C137#54812yes—
148C2×C137D4×Q812yes—
149D137—24no—
150C137#240911yescharacteristic, complemented, Sylow
151#2409—1371nomaximal, Sylow
152C23×Q8—1371no—
153C2×C22⋊Q8—1372no—
154C2×C4×D4—1371no—
155#395—1371no—
156#395—1371no—
157#393—1371no—
158#393D13711yescharacteristic, complemented
159#391—1372no—
160#386—1372no—
161#385—1372no—
162#380—1371no—
163C22×Q8—2748no—
164C22×Q8—2744no—
165C22×Q8—1371no—
166C22×Q8—1372no—
167C22×Q8C2×D13711yescharacteristic
168C22×D4—1371no—
169C23×C4—1371no—
170C23×C4—1372no—
171C23×C4—1371no—
172C22⋊Q8—2748no—
173C4×D4—2744no—
174C2×C4⋊C4—1371no—
175C2×C4⋊C4—1372no—
176C2×C4⋊C4—1372no—
177C2×C4⋊C4—1372no—
178C2×C22⋊C4—1372no—
179C2×C22⋊C4—1371no—
180C2×C22⋊C4—1372no—
181C2×C22⋊C4C2×D13711yescharacteristic
182C2×C22⋊C4C2×D13712yes—
183C2×C42—1371no—
184#95—1371no—
185#95—1372no—
186#95—1372no—
187#95—1372no—
188#95C2×D13711yescharacteristic
189#95C2×D13712yes—
190C24—1371no—
191C24C22×D13711yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—2744no—
195C2×Q8—2744no—
196C2×Q8—2744no—
197C2×Q8—27432no—
198C2×Q8—2742no—
199C2×Q8—1378no—
200C2×D4—2742no—
201C2×D4—2742no—
202C2×D4—1374no—
203C22×C4—2744no—
204C22×C4—2744no—
205C22×C4—2744no—
206C22×C4—2744no—
207C22×C4—2742no—
208C22×C4—2744no—
209C22×C4—1372no—
210C22×C4—1371no—
211C22×C4—1371no—
212C22×C4—1372no—
213C22×C4—1374no—
214C22×C4—1371no—
215C22×C4C22×D13711yescharacteristic
216C22×C4C22×D13712yes—
217C22×C4C22×D13712yes—
218C22×C4C22×D13711yescharacteristic
219C4⋊C4—2744no—
220C4⋊C4—2744no—
221C4⋊C4—2742no—
222C4⋊C4—2744no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—2744no—
226C22⋊C4—2742no—
227C22⋊C4—2744no—
228C42—2742no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—2742no—
234C23—2742no—
235C23—1372no—
236C23—1372no—
237C23—1372no—
238C23Q8×D13712yescomplemented
239C23C23×D13711yescharacteristic
240Q8—28no—
241Q8—2748no—
242Q8—27416no—
243D4—2748no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—27416no—
253C2×C4—2748no—
254C2×C4—2741no—
255C2×C4—2742no—
256C2×C4—2742no—
257C2×C4—2741no—
258C2×C4—2742no—
259C2×C4—2741no—
260C2×C4—2742no—
261C2×C4—2742no—
262C2×C4—2742no—
263C2×C4—5484no—
264C2×C4—1374no—
265C2×C4—1372no—
266C2×C4—1372no—
267C2×C4—1374no—
268C2×C4D4×D13714yes—
269C2×C4D4×D13712yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—2742no—
276C22—2744no—
277C22—2744no—
278C22—2742no—
279C22—1374no—
280C22C2×D4×D13711yescharacteristic
281C22C2×D4×D13712yes—
282C22C2×Q8×D13711yescharacteristic
283C22#41×D13711yescharacteristic
284C22#1798413512yes—
285C4—24no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—2744no—
290C4—2742no—
291C4—5482no—
292C4—5481no—
293C4—1374no—
294C2—42no—
295C2—2744no—
296C2#120×D13711yescharacteristic
297C2C22⋊Q8×D13711yescharacteristic
298C2#2257797311yescharacteristic
299C2#2257806112yes—
300C2#2257806612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#393×C137C2
prime radicalC22×C22⋊D137C22
hypercenter#393D137
hypocenterC137#2409
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D137C2semidirect
2. C2×#17984065C2semidirect
4. #22577324C2semidirect
5. #22577322C2semidirect
6. #22577315C2semidirect
7. #22577307C2semidirect
9. #22577277C2semidirect
10. #22577262C2semidirect
11. #393×C137C2semidirect
44. C2×Dic274D4semidirect
48. C2×C22⋊D137Q8semidirect
99. C2×Dic137C2×Q8semidirect
106. C23×C137C2×Q8semidirect
133. C2×D137C22⋊Q8semidirect
150. C137#2409semidirect
158. #393D137semidirect
238. C23Q8×D137semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1372
C2×#179840652
C2×#179840482
#225773242
#225773222
#225773152
#225773072
#225772792
#225772772
#225772622
#393×C1372
#2409137