Finite Group Databasebeta

← #31784181

Subgroups — #31784181

Order 17536.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#17984056C211yesmaximal, characteristic
2C22×#14555487C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577329C212yesmaximal, complemented
5#22577319C212yesmaximal, complemented
6#22577283C211yesmaximal, characteristic, complemented
7#22577279C212yesmaximal, complemented
8#22577278C211yesmaximal, characteristic, complemented
9#22577266C211yesmaximal, characteristic, complemented
10#22577260C212yesmaximal, complemented
11C2×#123×C137C211yesmaximal, characteristic, complemented
12C24×D137C2211yescharacteristic
13C22×C22⋊D137C2211yescharacteristic
14C22×C22⋊D137C2211yescharacteristic
15C23×Dic137C2211yescharacteristic
16C22×C4×D137C2212yes—
17#17984056—24no—
18C2×C4⋊Dic137C2212yescomplemented
19C2×C22⋊Dic137C2211yescharacteristic
20C2×C22⋊Dic137C2211yescharacteristic
21C2×C22⋊Dic137C2212yescomplemented
22C2×#14555487—24no—
23C2×#14555487C2211yescharacteristic
24C2×#14555487C2212yes—
25C2×#14555487C2212yes—
26C2×#14555487C2212yes—
27C2×#14555487C2211yescharacteristic
28C22⋊C4×D137—24no—
29#17983995C2212yescomplemented
30#17983995C2212yescomplemented
31#17983995C2212yescomplemented
32#17983995C2212yescomplemented
33C22×D4×C137C2211yescharacteristic
34C23×C4×C137C2211yescharacteristic
35#123×C137—24no—
36C2×C4⋊C4×C137C2212yescomplemented
37C2×C22⋊C4×C137C2211yescharacteristic
38C2×C22⋊C4×C137C2212yescomplemented
39C23×D137—24no—
40C23×D137—22no—
41C23×D137—22no—
42C23×D137—42no—
43C23×D137D414yes—
44C23×D137C2311yescharacteristic
45C23×D137C2311yescharacteristic
46C2×C22⋊D137—22no—
47C2×C22⋊D137—22no—
48C2×C22⋊D137—24no—
49C2×C22⋊D137—42no—
50C22×Dic137—22no—
51C22×Dic137—42no—
52C22×Dic137C2312yes—
53C22×Dic137C2312yes—
54C22×Dic137C2311yescharacteristic
55C22×Dic137C2311yescharacteristic
56C2×C4×D137—24no—
57C2×C4×D137—24no—
58C2×C4×D137—42no—
59C4⋊Dic137—42no—
60C22⋊Dic137—22no—
61C22⋊Dic137—22no—
62C22⋊Dic137—42no—
63#14555487—24no—
64#14555487—24no—
65#14555487—22no—
66#14555487—22no—
67#14555487—24no—
68C24×C137C2311yescharacteristic
69C24×C137C2311yescharacteristic
70C2×D4×C137—22no—
71C2×D4×C137—22no—
72C2×D4×C137—22no—
73C22×C4×C137—22no—
74C22×C4×C137—22no—
75C22×C4×C137—22no—
76C22×C4×C137C2312yes—
77C22×C4×C137C2311yescharacteristic
78C22×C4×C137C2312yes—
79C4⋊C4×C137—24no—
80C22⋊C4×C137—24no—
81C22⋊C4×C137D414yescomplemented
82C22×D137—21no—
83C22×D137—21no—
84C22×D137—22no—
85C22×D137—28no—
86C22×D137—21no—
87C22×D137—22no—
88C22×D137—21no—
89C22×D137—22no—
90C22×D137—28no—
91C22×D137—44no—
92C22×D137—48no—
93C22×D137—48no—
94C22×D137C2×D412yes—
95C22×D137C2×D412yes—
96C22×D137#4114yes—
97C22⋊D137—44no—
98C22⋊D137—44no—
99C2×Dic137—22no—
100C2×Dic137—22no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137—22no—
104C2×Dic137—22no—
105C2×Dic137—22no—
106C2×Dic137—22no—
107C2×Dic137—22no—
108C2×Dic137—22no—
109C2×Dic137—44no—
110C4×D137—48no—
111C23×C137—22no—
112C23×C137—22no—
113C23×C137—21no—
114C23×C137—21no—
115C23×C137—21no—
116C23×C137—22no—
117C23×C137—22no—
118C23×C137C2×D412yes—
119C23×C137C2×D412yes—
120C23×C137#4112yes—
121C23×C137C2411yescharacteristic
122D4×C137—44no—
123C2×C4×C137—22no—
124C2×C4×C137—22no—
125C2×C4×C137—22no—
126C2×C4×C137—21no—
127C2×C4×C137—22no—
128C2×C4×C137—21no—
129C2×C4×C137—22no—
130C2×C4×C137—21no—
131C2×C4×C137—44no—
132C2×C4×C137C2×D414yes—
133C22×C137—22no—
134C22×C137—22no—
135C22×C137—24no—
136C22×C137—24no—
137C22×C137—41no—
138C22×C137—41no—
139C22×C137—41no—
140C22×C137—42no—
141C22×C137—42no—
142C22×C137#12114yes—
143C22×C137C22×D411yescharacteristic
144C22×C137C22×D411yescharacteristic
145C22×C137C2×#4112yes—
146C22×C137C2×#4111yescharacteristic
147C22×C137H(5,𝔽2)12yes—
148C2×D137—22no—
149C2×D137—22no—
150C2×D137—24no—
151C2×D137—24no—
152C2×D137—22no—
153C2×D137—22no—
154C2×D137—22no—
155C2×D137—416no—
156C2×D137—42no—
157C2×D137—42no—
158C2×D137#12314yescomplemented
159C4×C137—24no—
160C4×C137—42no—
161C4×C137—41no—
162Dic137—41no—
163Dic137—42no—
164Dic137—41no—
165Dic137—42no—
166C2×C137—24no—
167C2×C137—81no—
168C2×C137C2×#12111yescharacteristic
169C2×C137C2×#12311yescharacteristic
170C2×C137#53411yescharacteristic
171C2×C137#54612yes—
172C2×C137#55112yes—
173D137—24no—
174D137—42no—
175C137#241211yescharacteristic, complemented, Sylow
176#2412—1371nomaximal, Sylow
177C2×#123D13711yescharacteristic, complemented
178C2×#120—1371no—
179C22×C22⋊C4—1371no—
180C22×C22⋊C4—1371no—
181#397—1372no—
182#394—1372no—
183#386—1371no—
184#386—1372no—
185#385—1371no—
186#381—1371no—
187#380—1372no—
188C25—1371no—
189C22×D4—1371no—
190C22×D4—1371no—
191C22×D4C2×D13711yescharacteristic
192C23×C4—1372no—
193C23×C4—1371no—
194C23×C4C2×D13711yescharacteristic
195#123—24no—
196#120—2744no—
197C2×C4⋊C4—1372no—
198C2×C4⋊C4C2×D13712yescomplemented
199C2×C22⋊C4—2744no—
200C2×C22⋊C4—2744no—
201C2×C22⋊C4—1371no—
202C2×C22⋊C4—1371no—
203C2×C22⋊C4—1372no—
204C2×C22⋊C4—1372no—
205C2×C22⋊C4—1372no—
206C2×C22⋊C4—1371no—
207C2×C22⋊C4—1371no—
208C2×C22⋊C4—1372no—
209C2×C22⋊C4C2×D13711yescharacteristic
210C2×C22⋊C4C2×D13712yescomplemented
211#95—1372no—
212#95—1372no—
213#95—1372no—
214#95—1372no—
215C24—2744no—
216C24—2742no—
217C24—2742no—
218C24—5482no—
219C24—1371no—
220C24—1374no—
221C24—1371no—
222C24C22×D13711yescharacteristic
223C24C22×D13711yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—2742no—
228C2×D4—2742no—
229C2×D4—2744no—
230C2×D4—5482no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—2744no—
235C22×C4—2742no—
236C22×C4—2744no—
237C22×C4—5482no—
238C22×C4—5482no—
239C22×C4—1371no—
240C22×C4—1372no—
241C22×C4—1372no—
242C22×C4—1371no—
243C22×C4C22×D13712yes—
244C22×C4C22×D13711yescharacteristic
245C22×C4C22×D13712yes—
246C4⋊C4—24no—
247C4⋊C4—5482no—
248C22⋊C4—24no—
249C22⋊C4—2744no—
250C22⋊C4—2744no—
251C22⋊C4—2742no—
252C22⋊C4—2742no—
253C22⋊C4—2742no—
254C22⋊C4—2742no—
255C22⋊C4—2744no—
256C22⋊C4—5482no—
257C22⋊C4C22⋊D13714yescomplemented
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—2748no—
266C23—2741no—
267C23—2741no—
268C23—2742no—
269C23—2748no—
270C23—2741no—
271C23—2742no—
272C23—2741no—
273C23—2742no—
274C23—5484no—
275C23—5488no—
276C23—5488no—
277C23—1372no—
278C23—1372no—
279C23—1374no—
280C23D4×D13712yes—
281C23C2×C22⋊D13712yes—
282C23#1455550912yes—
283C23C23×D13711yescharacteristic
284D4—44no—
285D4—5484no—
286D4—5484no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—2742no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2742no—
300C2×C4—2742no—
301C2×C4—2742no—
302C2×C4—2742no—
303C2×C4—2742no—
304C2×C4—2742no—
305C2×C4—2742no—
306C2×C4—5488no—
307C2×C4—5484no—
308C2×C4C2×C22⋊D13714yes—
309C22—22no—
310C22—24no—
311C22—24no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—2742no—
319C22—2744no—
320C22—2744no—
321C22—2742no—
322C22—2742no—
323C22—2742no—
324C22—2742no—
325C22—54816no—
326C22—5482no—
327C22—5482no—
328C22—1374no—
329C22#1798405914yes—
330C22C2×D4×D13711yescharacteristic
331C22C22×C22⋊D13711yescharacteristic
332C22#41×D13712yes—
333C22C2×#1455550911yescharacteristic
334C22#1798413412yes—
335C4—24no—
336C4—42no—
337C4—41no—
338C4—5482no—
339C4—5481no—
340C4—5482no—
341C4—5481no—
342C2—24no—
343C2—81no—
344C2—2744no—
345C2—5482no—
346C2C2×#1798405911yescharacteristic
347C2#123×D13711yescharacteristic
348C2#2257796411yescharacteristic
349C2#2257804112yes—
350C2#2257807612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC2×#123×C137C2
prime radicalC2×#17984056C2
hypercenterC2×#123D137
hypocenterC137#2412
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×#14555487C2semidirect
3. C2×C22⋊C4×D137C2semidirect
4. #22577329C2semidirect
5. #22577319C2semidirect
6. #22577283C2semidirect
7. #22577279C2semidirect
8. #22577278C2semidirect
9. #22577266C2semidirect
10. #22577260C2semidirect
11. C2×#123×C137C2semidirect
18. C2×C4⋊Dic137C22semidirect
21. C2×C22⋊Dic137C22semidirect
29. #17983995C22semidirect
30. #17983995C22semidirect
31. #17983995C22semidirect
32. #17983995C22semidirect
36. C2×C4⋊C4×C137C22semidirect
38. C2×C22⋊C4×C137C22semidirect
81. C22⋊C4×C137D4semidirect
158. C2×D137#123semidirect
175. C137#2412semidirect
177. C2×#123D137semidirect
198. C2×C4⋊C4C2×D137semidirect
210. C2×C22⋊C4C2×D137semidirect
257. C22⋊C4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840562
C22×#145554872
C2×C22⋊C4×D1372
#225773292
#225773192
#225772832
#225772792
#225772782
#225772662
#225772602
C2×#123×C1372
#2412137