Finite Group Databasebeta

← #31784119

Subgroups — #31784119

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×#14555509C211yesmaximal, characteristic
2C2×#17984070C212yesmaximal, complemented
3C2×C4⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577324C211yesmaximal, characteristic, complemented
5#22577323C211yesmaximal, characteristic, complemented
6#22577317C211yesmaximal, characteristic, complemented
7#22577279C212yesmaximal, complemented
8#22577278C212yesmaximal, complemented
9#22577275C212yesmaximal, complemented
10#22577265C211yesmaximal, characteristic, complemented
11#394×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—28no—
13C2×#14555509—24no—
14C22×Dic274C2211yescharacteristic
15C22×C22⋊D137C2212yes—
16C23×Dic137C2212yes—
17C22×C4×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984070—28no—
21C2×C4⋊Dic137C2211yescharacteristic
22C2×C4⋊Dic137C2212yes—
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2212yes—
25C2×#14555489C2211yescharacteristic
26C4⋊C4×D137—24no—
27C2×C22⋊Dic137C2212yes—
28C2×C22⋊Dic137C2212yes—
29C2×#14555487C2212yes—
30#17983995C2212yes—
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
40#14555509—232no—
41C2×Dic274—24no—
42C2×Dic274—22no—
43C2×C22⋊D137—24no—
44C2×C22⋊D137—24no—
45C2×C22⋊D137D418yes—
46C22×Dic137—24no—
47C22×Dic137—24no—
48C22×Dic137—24no—
49C22×Dic137C2312yes—
50C22×Dic137C2311yescharacteristic
51C22×Dic137C2311yescharacteristic
52C22×Dic137C2312yes—
53C22×Dic137C2311yescharacteristic
54C2×C4×D137—22no—
55C2×C4×D137—24no—
56C2×C4×D137—22no—
57C2×C4×D137—22no—
58C2×C4×D137—24no—
59C2×C4×D137—22no—
60C2×C4×D137D414yes—
61C4⋊Dic137—22no—
62C4⋊Dic137—24no—
63#14555489—22no—
64#14555489—24no—
65#14555489—22no—
66C22⋊Dic137—24no—
67C22⋊Dic137—24no—
68#14555487—24no—
69C24×C137C2312yes—
70C2×D4×C137—24no—
71C2×D4×C137—22no—
72C22×C4×C137C2312yes—
73C22×C4×C137C2311yescharacteristic
74C22×C4×C137C2311yescharacteristic
75C22×C4×C137C2311yescharacteristic
76C4⋊C4×C137—22no—
77C22⋊C4×C137—24no—
78C22⋊C4×C137—24no—
79C22×D137—21no—
80C22×D137—21no—
81C22×D137—22no—
82C22×D137C2×D414yes—
83C22×D137C2×D412yes—
84Dic274—28no—
85C22⋊D137—216no—
86C2×Dic137—216no—
87C2×Dic137—21no—
88C2×Dic137—22no—
89C2×Dic137—22no—
90C2×Dic137—22no—
91C2×Dic137—21no—
92C2×Dic137—22no—
93C2×Dic137—22no—
94C2×Dic137—21no—
95C2×Dic137—22no—
96C2×Dic137—22no—
97C2×Dic137C2×D412yes—
98C2×Dic137C2×D414yes—
99C2×Dic137#4112yescomplemented
100C4×D137—28no—
101C4×D137—28no—
102C4×D137—44no—
103C23×C137—22no—
104C23×C137—22no—
105C23×C137—22no—
106C23×C137—22no—
107C23×C137—22no—
108C23×C137—22no—
109C23×C137C2×D414yes—
110C23×C137C2411yescharacteristic
111D4×C137—28no—
112C2×C4×C137—21no—
113C2×C4×C137—22no—
114C2×C4×C137—21no—
115C2×C4×C137—22no—
116C2×C4×C137—22no—
117C2×C4×C137—21no—
118C2×C4×C137—22no—
119C2×C4×C137—22no—
120C2×C4×C137C2×D412yes—
121C2×C4×C137#4112yescomplemented
122C22×C137—24no—
123C22×C137—24no—
124C22×C137—24no—
125C22×C137—42no—
126C22×C137—42no—
127C22×C137—42no—
128C22×C137—42no—
129C22×C137C22×D411yescharacteristic
130C22×C137C22×D412yes—
131C22×C137C2×#4111yescharacteristic
132C22×C137C2×#4111yescharacteristic
133C22×C1372−1+412yes—
134C2×D137—22no—
135C2×D137—22no—
136C2×D137—24no—
137C2×D137—24no—
138C2×D137#12014yes—
139C4×C137—22no—
140C4×C137—22no—
141C4×C137—41no—
142C4×C137—42no—
143Dic137—22no—
144Dic137—24no—
145Dic137—22no—
146Dic137—41no—
147Dic137—42no—
148C2×C137—44no—
149C2×C137C2×#12011yescharacteristic
150C2×C137#52511yescharacteristic
151C2×C137#53611yescharacteristic
152C2×C137#54712yes—
153C2×C137#54712yes—
154D137—24no—
155C137#240611yescharacteristic, complemented, Sylow
156#2406—1371nomaximal, Sylow
157C22×#41—1371no—
158C2×#123—1372no—
159C22×C4⋊C4—1371no—
160#395—1371no—
161#395—1371no—
162#394—1371no—
163#394D13711yescharacteristic, complemented
164#386—1372no—
165#385—1372no—
166#385—1372no—
167#381—1371no—
168C2×#41—2748no—
169C2×#41—2744no—
170C22×Q8—1371no—
171C22×D4—1372no—
172C22×D4C2×D13711yescharacteristic
173C23×C4—1372no—
174C23×C4—1371no—
175C23×C4—1371no—
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⋊C4—1372no—
184C2×C4⋊C4C2×D13711yescharacteristic
185C2×C22⋊C4—1372no—
186C2×C22⋊C4—1372no—
187C2×C22⋊C4—1372no—
188C2×C22⋊C4C2×D13712yes—
189C2×C22⋊C4C2×D13712yes—
190#95—1372no—
191#95—1371no—
192#95—1372no—
193#95—1372no—
194#95C2×D13711yescharacteristic
195C24—1371no—
196C24C22×D13712yes—
197#41—27432no—
198C2×Q8—2744no—
199C2×Q8—2742no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—2744no—
203C2×D4—2744no—
204C2×D4—1378no—
205C22×C4—2742no—
206C22×C4—2744no—
207C22×C4—2742no—
208C22×C4—2742no—
209C22×C4—2744no—
210C22×C4—2744no—
211C22×C4—2744no—
212C22×C4—2744no—
213C22×C4—2742no—
214C22×C4—1372no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4—1371no—
218C22×C4—1372no—
219C22×C4—1374no—
220C22×C4C22×D13712yes—
221C22×C4C22×D13711yescharacteristic
222C22×C4C22×D13711yescharacteristic
223C22×C4C22×D13711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—2742no—
226C4⋊C4—2744no—
227C4⋊C4—2742no—
228C4⋊C4—2744no—
229C4⋊C4—2742no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—2744no—
233C22⋊C4—2744no—
234C22⋊C4—2744no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—2741no—
242C23—2741no—
243C23—2742no—
244C23—1374no—
245C23—1372no—
246C23D4×D13714yes—
247C23C23×D13711yescharacteristic
248Q8—2748no—
249D4—28no—
250D4—27416no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2748no—
260C2×C4—2748no—
261C2×C4—27416no—
262C2×C4—2741no—
263C2×C4—2742no—
264C2×C4—2742no—
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—1374no—
275C2×C4—1372no—
276C2×C4D4×D13712yes—
277C2×C4#1455550912yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—2742no—
286C22—2744no—
287C22—2744no—
288C22—2742no—
289C22—1374no—
290C22C2×D4×D13711yescharacteristic
291C22C2×D4×D13712yes—
292C22#41×D13711yescharacteristic
293C22C2×#1455550911yescharacteristic
294C22#1798413512yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—2742no—
300C4—2744no—
301C4—2742no—
302C4—5482no—
303C4—5481no—
304C2—44no—
305C2—2744no—
306C2#120×D13711yescharacteristic
307C2#2257791311yescharacteristic
308C2#2257797311yescharacteristic
309C2#2257805112yes—
310C2#2257805312yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#394×C137C2
prime radicalC22×#14555509C2
hypercenter#394D137
hypocenterC137#2406
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#17984070C2semidirect
3. C2×C4⋊C4×D137C2semidirect
4. #22577324C2semidirect
5. #22577323C2semidirect
6. #22577317C2semidirect
7. #22577279C2semidirect
8. #22577278C2semidirect
9. #22577275C2semidirect
10. #22577265C2semidirect
11. #394×C137C2semidirect
99. C2×Dic137#41semidirect
121. C2×C4×C137#41semidirect
155. C137#2406semidirect
163. #394D137semidirect
277. 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×#179840702
C2×C4⋊C4×D1372
#225773242
#225773232
#225773172
#225772792
#225772782
#225772752
#225772652
#394×C1372
#2406137