Finite Group Databasebeta

← #31784130

Subgroups — #31784130

Order 17536.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14555509C211yesmaximal, characteristic, complemented
2C2×#17984067C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D137C211yesmaximal, characteristic, complemented
4#22577318C212yesmaximal, complemented
5#22577311C212yesmaximal, complemented
6#22577308C211yesmaximal, characteristic, complemented
7#22577306C211yesmaximal, characteristic, complemented
8#22577283C212yesmaximal, complemented
9#22577275C212yesmaximal, complemented
10#22577260C211yesmaximal, characteristic, complemented
11C2×#121×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—24no—
13C2×#14555509—24no—
14C2×#14555509—24no—
15C22×Dic274C2211yescharacteristic
16C22×C22⋊D137C2212yes—
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2212yes—
19C22×C4×D137C2211yescharacteristic
20#17984067—24no—
21C4×C22⋊D137—24no—
22C2×C4⋊Dic137C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2212yes—
25C2×C22⋊Dic137C2212yes—
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2212yes—
28C2×C22⋊Dic137C2212yes—
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2212yes—
31C2×C4×Dic137C2211yescharacteristic
32#17983995C2212yes—
33#17983995C2212yes—
34#17983995C2212yes—
35C22×D4×C137C2212yes—
36C22×D4×C137C2211yescharacteristic
37C23×C4×C137C2211yescharacteristic
38#121×C137—24no—
39C2×C4⋊C4×C137C2211yescharacteristic
40C2×C22⋊C4×C137C2212yes—
41C23×D137C2311yescharacteristic
42#14555509—232no—
43C2×Dic274—22no—
44C2×Dic274—22no—
45C2×Dic274D414yescomplemented
46C2×C22⋊D137—22no—
47C2×C22⋊D137—24no—
48C2×C22⋊D137—24no—
49C2×C22⋊D137—22no—
50C2×C22⋊D137—24no—
51C2×C22⋊D137—22no—
52C22×Dic137—24no—
53C22×Dic137—24no—
54C22×Dic137—24no—
55C22×Dic137C2311yescharacteristic
56C22×Dic137C2312yes—
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2312yes—
59C22×Dic137C2311yescharacteristic
60C2×C4×D137—22no—
61C2×C4×D137—22no—
62C2×C4×D137D414yescomplemented
63C4⋊Dic137—22no—
64#14555489—22no—
65#14555489—24no—
66C22⋊Dic137—22no—
67C22⋊Dic137—24no—
68C22⋊Dic137—24no—
69C22⋊Dic137—42no—
70#14555487—22no—
71#14555487—42no—
72C4×Dic137—22no—
73C24×C137C2312yes—
74C24×C137C2311yescharacteristic
75C2×D4×C137—24no—
76C2×D4×C137—24no—
77C2×D4×C137—22no—
78C2×D4×C137—22no—
79C2×D4×C137—42no—
80C2×D4×C137D414yescomplemented
81C22×C4×C137—22no—
82C22×C4×C137—22no—
83C22×C4×C137—22no—
84C22×C4×C137C2311yescharacteristic
85C22×C4×C137C2312yes—
86C22×C4×C137C2311yescharacteristic
87C4⋊C4×C137—22no—
88C22⋊C4×C137—24no—
89C22×D137—21no—
90C22×D137—21no—
91C22×D137—21no—
92C22×D137—21no—
93C22×D137—21no—
94C22×D137—21no—
95C22×D137C2×D412yes—
96Dic274—28no—
97C22⋊D137—28no—
98C22⋊D137—216no—
99C2×Dic137—216no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137—22no—
104C2×Dic137—22no—
105C2×Dic137—21no—
106C2×Dic137—21no—
107C2×Dic137—22no—
108C2×Dic137—22no—
109C2×Dic137—22no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137C2×D412yes—
113C2×Dic137C2×D414yes—
114C2×Dic137#4112yescomplemented
115C4×D137—28no—
116C23×C137—21no—
117C23×C137—21no—
118C23×C137—21no—
119C23×C137—22no—
120C23×C137—22no—
121C23×C137—22no—
122C23×C137—22no—
123C23×C137—21no—
124C23×C137—21no—
125C23×C137—22no—
126C23×C137—22no—
127C23×C137—21no—
128C23×C137C2×D414yes—
129C23×C137#4112yescomplemented
130C23×C137C2411yescharacteristic
131D4×C137—28no—
132D4×C137—48no—
133C2×C4×C137—21no—
134C2×C4×C137—22no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—22no—
138C2×C4×C137—22no—
139C2×C4×C137—44no—
140C2×C4×C137C2×D412yes—
141C2×C4×C137C2×D412yes—
142C2×C4×C137C2×D412yes—
143C22×C137—24no—
144C22×C137—22no—
145C22×C137—22no—
146C22×C137—24no—
147C22×C137—24no—
148C22×C137—22no—
149C22×C137—41no—
150C22×C137—42no—
151C22×C137—41no—
152C22×C137—42no—
153C22×C137—41no—
154C22×C137—42no—
155C22×C137—41no—
156C22×C137—42no—
157C22×C137C22×D411yescharacteristic
158C22×C137C22×D411yescharacteristic
159C22×C137C22×D411yescharacteristic
160C22×C137C2×#4111yescharacteristic
161C22×C137C2×#4111yescharacteristic
162C22×C137H(5,𝔽2)11yescharacteristic
163C22×C1372−1+411yescharacteristic
164C2×D137—22no—
165C2×D137—22no—
166C2×D137—22no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C4×C137—41no—
172C4×C137—42no—
173C4×C137#12014yes—
174Dic137—22no—
175Dic137—24no—
176Dic137—22no—
177Dic137—41no—
178Dic137—42no—
179C2×C137—44no—
180C2×C137—42no—
181C2×C137C2×#12011yescharacteristic
182C2×C137#52511yescharacteristic
183C2×C137#53711yescharacteristic
184C2×C137#54611yescharacteristic
185C2×C137#54711yescharacteristic
186C2×C137#54811yescharacteristic
187C2×C137#55011yescharacteristic
188D137—42no—
189C137#240711yescharacteristic, complemented, Sylow
190#2407—1371nomaximal, Sylow
191C22×#41—1371no—
192C2×C22⋊Q8—1371no—
193C2×#121D13711yescharacteristic, complemented
194C2×C4×D4—1371no—
195#394—1372no—
196#392—1372no—
197#391—1371no—
198#390—1371no—
199#386—1372no—
200#385—1372no—
201#380—1371no—
202C2×#41—2744no—
203C2×#41—2744no—
204C2×#41—2744no—
205C22×Q8—1371no—
206C22×D4—1371no—
207C22×D4—1372no—
208C22×D4C2×D13712yes—
209C22×D4C2×D13711yescharacteristic
210C23×C4—1372no—
211C23×C4—1371no—
212C23×C4C2×D13711yescharacteristic
213C22⋊Q8—2744no—
214#121—24no—
215C4×D4—2744no—
216C2×C4⋊C4—1371no—
217C2×C4⋊C4—1372no—
218C2×C4⋊C4—1371no—
219C2×C4⋊C4C2×D13711yescharacteristic
220C2×C22⋊C4—1371no—
221C2×C22⋊C4—1372no—
222C2×C22⋊C4—1372no—
223C2×C22⋊C4—1371no—
224C2×C22⋊C4—1372no—
225C2×C22⋊C4—1372no—
226C2×C22⋊C4C2×D13712yes—
227C2×C42—1371no—
228#95—1372no—
229#95—1372no—
230#95—1372no—
231C24—1371no—
232C24C22×D13712yes—
233C24C22×D13711yescharacteristic
234#41—27432no—
235C2×Q8—2742no—
236C2×Q8—2742no—
237C2×Q8—1374no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—2742no—
244C2×D4—2744no—
245C2×D4—2744no—
246C2×D4—2742no—
247C2×D4—2742no—
248C2×D4—2744no—
249C2×D4C22⋊D13714yescomplemented
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—2744no—
254C22×C4—2744no—
255C22×C4—2744no—
256C22×C4—2742no—
257C22×C4—2742no—
258C22×C4—1371no—
259C22×C4—1372no—
260C22×C4—1371no—
261C22×C4—1371no—
262C22×C4—1372no—
263C22×C4—1374no—
264C22×C4C22×D13712yes—
265C22×C4C22×D13711yescharacteristic
266C22×C4C22×D13711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—2742no—
269C4⋊C4—2744no—
270C4⋊C4—2742no—
271C22⋊C4—24no—
272C22⋊C4—2742no—
273C22⋊C4—2744no—
274C22⋊C4—2742no—
275C22⋊C4—2744no—
276C22⋊C4—5482no—
277C22⋊C4—5482no—
278C42—2742no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—2741no—
292C23—2741no—
293C23—2741no—
294C23—2741no—
295C23—2741no—
296C23—2741no—
297C23—1372no—
298C23C2×C22⋊D13714yes—
299C23#1455550912yescomplemented
300C23C23×D13711yescharacteristic
301Q8—2748no—
302D4—28no—
303D4—48no—
304D4—27416no—
305D4—2748no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—44no—
313C2×C4—27416no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2742no—
317C2×C4—2741no—
318C2×C4—2742no—
319C2×C4—2741no—
320C2×C4—2748no—
321C2×C4—2741no—
322C2×C4—2742no—
323C2×C4—2742no—
324C2×C4—2741no—
325C2×C4—2742no—
326C2×C4—2741no—
327C2×C4—1372no—
328C2×C4—1374no—
329C2×C4—1372no—
330C2×C4D4×D13712yes—
331C2×C4D4×D13712yes—
332C2×C4C2×C22⋊D13712yes—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—22no—
337C22—24no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—42no—
347C22—2742no—
348C22—2742no—
349C22—2742no—
350C22—5481no—
351C22—5481no—
352C22—5481no—
353C22—5481no—
354C22C2×D4×D13711yescharacteristic
355C22C2×D4×D13711yescharacteristic
356C22C22×C22⋊D13711yescharacteristic
357C22#41×D13711yescharacteristic
358C22C2×#1455550911yescharacteristic
359C22#1798413311yescharacteristic
360C22#1798413511yescharacteristic
361C4—42no—
362C4—41no—
363C4—2742no—
364C4—2744no—
365C4—2742no—
366C4—5481no—
367C4—5482no—
368C4#1798405614yes—
369C2—42no—
370C2—44no—
371C2—5482no—
372C2C2×#1798405611yescharacteristic
373C2#2257791311yescharacteristic
374C2#2257798211yescharacteristic
375C2#2257804711yescharacteristic
376C2#2257805211yescharacteristic
377C2#2257805911yescharacteristic
378C2#2257807111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC2×#121×C137C2
hypercenterC2×#121D137
hypocenterC137#2407
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14555509C2semidirect
2. C2×#17984067C2semidirect
3. C2×C4×C22⋊D137C2semidirect
4. #22577318C2semidirect
5. #22577311C2semidirect
6. #22577308C2semidirect
7. #22577306C2semidirect
8. #22577283C2semidirect
9. #22577275C2semidirect
10. #22577260C2semidirect
11. C2×#121×C137C2semidirect
45. C2×Dic274D4semidirect
62. C2×C4×D137D4semidirect
80. C2×D4×C137D4semidirect
114. C2×Dic137#41semidirect
129. C23×C137#41semidirect
189. C137#2407semidirect
193. C2×#121D137semidirect
249. C2×D4C22⋊D137semidirect
299. 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×#179840672
C2×C4×C22⋊D1372
#225773182
#225773112
#225773082
#225773062
#225772832
#225772752
#225772602
C2×#121×C1372
#2407137