Finite Group Databasebeta

← #31784003

Subgroups — #31784003

Order 17536.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#17984050C211yesmaximal, characteristic, complemented
3C2×#17984048C212yesmaximal, complemented
4#22577323C211yesmaximal, characteristic, complemented
5#22577318C212yesmaximal, complemented
6#22577316C211yesmaximal, characteristic, complemented
7#22577312C212yesmaximal, complemented
8#22577311C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic137C212yesmaximal, complemented
10#22577250C211yesmaximal, characteristic, complemented
11#394×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—28no—
13C2×#14555509—24no—
14C22×Dic274C2211yescharacteristic
15C22×D548C2211yescharacteristic
16C22×C22⋊D137C2212yes—
17C22×C22⋊D137C2212yes—
18C23×Dic137C2212yes—
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984050—24no—
22#17984048—28no—
23C2×#14555489C2212yes—
24C2×C22⋊Dic137C2212yes—
25C2×#14555487C2212yes—
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2212yes—
28C2×#14555487C2211yescharacteristic
29C2×C4×Dic137C2212yes—
30C2×C4×Dic137C2211yescharacteristic
31#17983995C2211yescharacteristic
32#17983995C2212yes—
33#17983995C2212yes—
34C22×D4×C137C2211yescharacteristic
35C2×C4⋊C4×C137C2211yescharacteristic
36C2×C22⋊C4×C137C2212yes—
37C2×C22⋊C4×C137C2212yes—
38#95×C137C2211yescharacteristic
39C23×D137C2311yescharacteristic
40C23×D137C2311yescharacteristic
41#14555509—232no—
42C2×Dic274—24no—
43C2×Dic274—22no—
44C2×D548—22no—
45C2×D548—22no—
46C2×D548—22no—
47C2×C22⋊D137—24no—
48C2×C22⋊D137—24no—
49C2×C22⋊D137—24no—
50C2×C22⋊D137—24no—
51C2×C22⋊D137—24no—
52C2×C22⋊D137—24no—
53C22×Dic137—24no—
54C22×Dic137—24no—
55C22×Dic137D418yescomplemented
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2312yes—
58C22×Dic137C2312yes—
59C22×Dic137C2311yescharacteristic
60C2×C4×D137—24no—
61C2×C4×D137—24no—
62C2×C4×D137—22no—
63C2×C4×D137D414yescomplemented
64#14555489—24no—
65C22⋊Dic137—24no—
66#14555487—22no—
67#14555487—24no—
68#14555487—24no—
69#14555487—22no—
70C4×Dic137—24no—
71C4×Dic137—22no—
72C24×C137C2312yes—
73C2×D4×C137—24no—
74C2×D4×C137—22no—
75C22×C4×C137C2312yes—
76C22×C4×C137C2311yescharacteristic
77C22×C4×C137C2311yescharacteristic
78C22×C4×C137C2311yescharacteristic
79C4⋊C4×C137—22no—
80C22⋊C4×C137—24no—
81C22⋊C4×C137—24no—
82C22×D137—22no—
83C22×D137—21no—
84C22×D137—22no—
85C22×D137—22no—
86C22×D137—22no—
87C22×D137—21no—
88C22×D137—21no—
89C22×D137—21no—
90C22×D137C2×D412yes—
91C22×D137#4112yescomplemented
92Dic274—28no—
93D548—28no—
94C22⋊D137—216no—
95C22⋊D137—216no—
96C2×Dic137—22no—
97C2×Dic137—22no—
98C2×Dic137—22no—
99C2×Dic137—22no—
100C2×Dic137—216no—
101C2×Dic137—21no—
102C2×Dic137—22no—
103C2×Dic137C2×D414yes—
104C2×Dic137C2×D414yes—
105C2×Dic137C2×D412yes—
106C2×Dic137#4114yescomplemented
107C4×D137—28no—
108C4×D137—44no—
109C23×C137—22no—
110C23×C137—22no—
111C23×C137—22no—
112C23×C137—22no—
113C23×C137—22no—
114C23×C137—22no—
115C23×C137C2×D414yes—
116C23×C137C2411yescharacteristic
117D4×C137—28no—
118C2×C4×C137—21no—
119C2×C4×C137—22no—
120C2×C4×C137—21no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—21no—
124C2×C4×C137—22no—
125C2×C4×C137—22no—
126C2×C4×C137C2×D412yes—
127C2×C4×C137#4112yescomplemented
128C22×C137—24no—
129C22×C137—24no—
130C22×C137—24no—
131C22×C137—42no—
132C22×C137—42no—
133C22×C137—42no—
134C22×C137—42no—
135C22×C137C22×D412yes—
136C22×C137C22×D411yescharacteristic
137C22×C137C2×#4111yescharacteristic
138C22×C137C2×#4112yes—
139C22×C137C2×#4111yescharacteristic
140C2×D137—22no—
141C2×D137—24no—
142C2×D137—22no—
143C2×D137—22no—
144C2×D137—22no—
145C2×D137—41no—
146C2×D137—42no—
147C2×D137—42no—
148C2×D137—42no—
149C2×D137—41no—
150C4×C137—22no—
151C4×C137—22no—
152C4×C137—41no—
153C4×C137—42no—
154Dic137—24no—
155Dic137—24no—
156Dic137—41no—
157Dic137#12014yes—
158C2×C137—44no—
159C2×C137C2×#12011yescharacteristic
160C2×C137#52512yes—
161C2×C137#52511yescharacteristic
162C2×C137#53212yes—
163C2×C137#53211yescharacteristic
164D137—42no—
165D137—42no—
166C137#238811yescharacteristic, complemented, Sylow
167#2388—1371nomaximal, Sylow
168C22×#41—1371no—
169C2×C4×D4—1371no—
170C2×C4×D4—1372no—
171#395—1371no—
172#394—1372no—
173#394D13711yescharacteristic, complemented
174#393—1371no—
175#392—1371no—
176#392—1372no—
177C4×C22⋊C4—1372no—
178C4×C22⋊C4—1371no—
179C2×#41—2748no—
180C2×#41—2744no—
181C22×Q8—1371no—
182C22×D4—1372no—
183C22×D4—1372no—
184C22×D4—1371no—
185C22×D4C2×D13711yescharacteristic
186C23×C4—1371no—
187C23×C4—1372no—
188C23×C4—1371no—
189C4×D4—2748no—
190C4×D4—2744no—
191C2×C4⋊C4—1372no—
192C2×C4⋊C4C2×D13711yescharacteristic
193C2×C22⋊C4—1372no—
194C2×C22⋊C4—1372no—
195C2×C22⋊C4—1371no—
196C2×C22⋊C4—1371no—
197C2×C22⋊C4—1372no—
198C2×C22⋊C4C2×D13712yes—
199C2×C22⋊C4C2×D13712yes—
200C2×C42—1371no—
201C2×C42—1372no—
202#95—1371no—
203#95—1372no—
204#95—1372no—
205#95C2×D13711yescharacteristic
206C24—1371no—
207C24—1371no—
208C24C22×D13712yes—
209#41—27432no—
210C2×Q8—2744no—
211C2×Q8—2742no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—2742no—
215C2×D4—2742no—
216C2×D4—2744no—
217C2×D4—2744no—
218C2×D4—2744no—
219C2×D4—2744no—
220C2×D4—2744no—
221C2×D4—2744no—
222C2×D4—2742no—
223C22×C4—2744no—
224C22×C4—2744no—
225C22×C4—2742no—
226C22×C4—2744no—
227C22×C4—2744no—
228C22×C4—1371no—
229C22×C4—1378no—
230C22×C4—1372no—
231C22×C4—1372no—
232C22×C4—1374no—
233C22×C4—1371no—
234C22×C4C22×D13712yes—
235C22×C4C22×D13711yescharacteristic
236C22×C4C22×D13711yescharacteristic
237C22×C4C22×D13711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2744no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—2742no—
243C22⋊C4—2744no—
244C22⋊C4—2744no—
245C22⋊C4—2744no—
246C22⋊C4—2742no—
247C42—2742no—
248C42—2744no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—2742no—
256C23—2741no—
257C23—2742no—
258C23—2742no—
259C23—2742no—
260C23—2741no—
261C23—2741no—
262C23—2741no—
263C23—1372no—
264C23—1372no—
265C23D4×D13714yes—
266C23C23×D13711yescharacteristic
267Q8—2748no—
268D4—28no—
269D4—2748no—
270D4—27416no—
271D4—27416no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—2748no—
281C2×C4—2742no—
282C2×C4—2742no—
283C2×C4—2742no—
284C2×C4—2742no—
285C2×C4—27416no—
286C2×C4—2741no—
287C2×C4—2742no—
288C2×C4—5484no—
289C2×C4—1374no—
290C2×C4—1374no—
291C2×C4—1374no—
292C2×C4—1372no—
293C2×C4D4×D13712yes—
294C2×C4#1455550912yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—2742no—
303C22—2742no—
304C22—2744no—
305C22—2742no—
306C22—2742no—
307C22—5482no—
308C22—5482no—
309C22—5481no—
310C22—5481no—
311C22—5482no—
312C22C2×D4×D13712yes—
313C22C2×D4×D13711yescharacteristic
314C22#41×D13712yes—
315C22#41×D13711yescharacteristic
316C22C2×#1455550911yescharacteristic
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—2744no—
322C4—2744no—
323C4—5481no—
324C4—1374no—
325C2—44no—
326C2—5482no—
327C2—5482no—
328C2#120×D13711yescharacteristic
329C2#2257791112yes—
330C2#2257791311yescharacteristic
331C2#2257794812yes—
332C2#2257795211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#394×C137C2
hypercenter#394D137
hypocenterC137#2388
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#14555509C2semidirect
2. C2×#17984050C2semidirect
3. C2×#17984048C2semidirect
4. #22577323C2semidirect
5. #22577318C2semidirect
6. #22577316C2semidirect
7. #22577312C2semidirect
8. #22577311C2semidirect
9. C22⋊C4×Dic137C2semidirect
10. #22577250C2semidirect
11. #394×C137C2semidirect
55. C22×Dic137D4semidirect
63. C2×C4×D137D4semidirect
91. C22×D137#41semidirect
106. C2×Dic137#41semidirect
127. C2×C4×C137#41semidirect
166. C137#2388semidirect
173. #394D137semidirect
294. C2×C4#14555509semidirect

Maximal subgroups

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

subgroupindex
C22×#145555092
C2×#179840502
C2×#179840482
#225773232
#225773182
#225773162
#225773122
#225773112
C22⋊C4×Dic1372
#225772502
#394×C1372
#2388137