Finite Group Databasebeta

← #31790252

Subgroups — #31790252

Order 17536.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984113C211yesmaximal, characteristic, complemented
3#22577949C211yesmaximal, characteristic, complemented
4C2×#17984050C211yesmaximal, characteristic, complemented
5#22577621C212yesmaximal, complemented
6#22577617C212yesmaximal, complemented
7#22577607C212yesmaximal, complemented
8#22577597C212yesmaximal, complemented
9#22577426C211yesmaximal, characteristic, complemented
10#22577402C211yesmaximal, characteristic, complemented
11#417×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×D548C2211yescharacteristic
14C22×C4×D137—21no—
15#17984116—24no—
16#17984113—24no—
17C2×#14555499C2212yes—
18C2×#14555497C2212yes—
19C2×D1096C2212yes—
20C2×#14555495C2212yes—
21C2×#14555494C2211yescharacteristic
22#17984086C2211yescharacteristic, complemented
23#17984084C2211yescharacteristic, complemented
24#17984077C2211yescharacteristic, complemented
25#17984075C2211yescharacteristic, complemented
26#17984060—21no—
27#17984060—21no—
28#17984050—21no—
29#17984050—21no—
30#17984050C2212yes—
31#17984050C2212yes—
32C4×C22⋊D137—21no—
33C4×C22⋊D137—21no—
34C2×#14555487—21no—
35C2×C4×Dic137C2211yescharacteristic
36#17984017C2212yescomplemented
37#17984017C2212yescomplemented
38#17984011C2212yescomplemented
39#17984008C2212yescomplemented
40C2×#41×C137C2211yescharacteristic
41C2×M16×C137C2211yescharacteristic
42C2×C4⋊C4×C137C2211yescharacteristic
43#103×C137C2212yescomplemented
44#102×C137C2212yescomplemented
45C23×D137—21no—
46#14555510—22no—
47#14555510—22no—
48C2×Dic274C2311yescharacteristic
49C2×D548—21no—
50C2×D548—22no—
51C2×D548D414yescomplemented
52C2×D548C2311yescharacteristic
53C2×D548C2312yes—
54C2×C22⋊D137—21no—
55C2×C22⋊D137—21no—
56C2×C22⋊D137—21no—
57C22×Dic137D411yescharacteristic
58C22×Dic137D411yescharacteristic
59C2×C4×D137—22no—
60C2×C4×D137—22no—
61C2×C4×D137—21no—
62C2×C4×D137—44no—
63#14555499—24no—
64#14555497—24no—
65D1096—24no—
66#14555495—24no—
67#14555494—22no—
68C2×#11764040C2312yes—
69#14555489—21no—
70#14555489—21no—
71C22⋊Dic137—21no—
72C22⋊Dic137—21no—
73#14555487—22no—
74#14555487—22no—
75#14555487—21no—
76#14555487—21no—
77C4×Dic137D411yescharacteristic
78C4×Dic137D411yescharacteristic
79C4×Dic137C2311yescharacteristic
80C4×Dic137C2311yescharacteristic
81#41×C137—22no—
82#41×C137—22no—
83C2×Q8×C137C2311yescharacteristic
84C2×D4×C137—21no—
85C2×D4×C137C2311yescharacteristic
86C22×C4×C137—21no—
87C22×C4×C137—21no—
88C22×C4×C137C2311yescharacteristic
89M16×C137—22no—
90C2×C8×C137C2312yes—
91C4⋊C4×C137—21no—
92C4⋊C4×C137C2312yes—
93C22×D137—24no—
94C22×D137—22no—
95C22×D137—21no—
96C22×D137—42no—
97C22×D137—42no—
98Dic274—21no—
99Dic274—21no—
100D548—24no—
101D548—21no—
102D548—22no—
103D548—21no—
104D548C2×D414yes—
105C22⋊D137—42no—
106C22⋊D137—42no—
107C22⋊D137—42no—
108C2×Dic137—22no—
109C2×Dic137—21no—
110C2×Dic137—41no—
111C2×Dic137C2×D411yescharacteristic
112C2×Dic137C2×D411yescharacteristic
113C2×Dic137C2×D411yescharacteristic
114C2×Dic137C2×D411yescharacteristic
115C4×D137—44no—
116C4×D137—42no—
117C4×D137—44no—
118#11764040—22no—
119C23×C137—21no—
120C23×C137C2×D411yescharacteristic
121Q8×C137—21no—
122Q8×C137—21no—
123D4×C137—21no—
124D4×C137—21no—
125D4×C137—42no—
126C2×C4×C137—22no—
127C2×C4×C137—21no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137C2×D411yescharacteristic
131C2×C4×C137C2×D412yes—
132C2×C4×C137#4112yes—
133C2×C4×C137C2411yescharacteristic
134C8×C137—22no—
135C22×C137—21no—
136C22×C137—21no—
137C22×C137—41no—
138C22×C137—41no—
139C22×C137—41no—
140C22×C137#12112yes—
141C22×C137C22×D411yescharacteristic
142C2×D137—24no—
143C2×D137—48no—
144C2×D137—42no—
145C2×D137—42no—
146C2×D137—41no—
147C2×D137—41no—
148C2×D137—41no—
149C4×C137—41no—
150C4×C137—42no—
151C4×C137#12112yes—
152C4×C137C22×D411yescharacteristic
153C4×C137C2×#4111yescharacteristic
154Dic137—21no—
155Dic137—21no—
156Dic137—21no—
157Dic137—41no—
158Dic137AGL(1,ℤ/8)12yescomplemented
159C2×C137—22no—
160C2×C137—81no—
161C2×C137C2×#12111yescharacteristic
162C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C137#57511yescharacteristic
164D137—44no—
165D137—81no—
166C137#303611yescharacteristic, complemented, Sylow
167#3036—1371nomaximal, Sylow
168C2×AGL(1,ℤ/8)—1371no—
169C2×AGL(1,ℤ/8)—1371no—
170#532—1371no—
171C2×C4×D4—1371no—
172#463—1372no—
173#463—1372no—
174#461—1372no—
175#459—1372no—
176#423—1371no—
177#417—1371no—
178#417D13711yescharacteristic, complemented
179C2×#41—1371no—
180C2×#41C2×D13711yescharacteristic
181C22×D4—1371no—
182C23×C4—2741no—
183AGL(1,ℤ/8)—2744no—
184AGL(1,ℤ/8)—2744no—
185C2×SD16—1372no—
186C2×SD16—1372no—
187C2×D8—1372no—
188C2×D8—1372no—
189C2×M16—1371no—
190C2×M16C2×D13711yescharacteristic
191C4⋊Q8—1371no—
192#127—1371no—
193#124—1371no—
194#124—1371no—
195#121—2741no—
196#121—2741no—
197C4×D4—2741no—
198C4×D4—2741no—
199C4×D4—2741no—
200C4×D4—2741no—
201C4×D4—1372no—
202C4×D4—1372no—
203C2×C4⋊C4C2×D13711yescharacteristic
204C2×C22⋊C4—2741no—
205C2×C42—1371no—
206C4⋊C8—1372no—
207C4⋊C8—1372no—
208#103—1372no—
209#103C2×D13712yescomplemented
210#102—1372no—
211#102C2×D13712yescomplemented
212C24—2741no—
213#41—22no—
214#41—22no—
215#41—2742no—
216#41—2742no—
217C2×Q8—1371no—
218C2×Q8C22×D13711yescharacteristic
219C2×D4—21no—
220C2×D4—2741no—
221C2×D4—2741no—
222C2×D4—2741no—
223C2×D4—2741no—
224C2×D4—2742no—
225C2×D4—1374no—
226C2×D4—1371no—
227C2×D4—1372no—
228C2×D4C22×D13711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—2742no—
232C22×C4—2742no—
233C22×C4—2741no—
234C22×C4—5484no—
235C22×C4—1371no—
236C22×C4—1371no—
237C22×C4C22×D13711yescharacteristic
238SD16—2744no—
239SD16—2744no—
240D8—2744no—
241D8—2744no—
242M16—22no—
243M16—2742no—
244C2×C8—1372no—
245C2×C8C22×D13712yes—
246C4⋊C4—21no—
247C4⋊C4—2741no—
248C4⋊C4—2741no—
249C4⋊C4C22×D13712yes—
250C22⋊C4—2742no—
251C22⋊C4—2742no—
252C22⋊C4—2741no—
253C22⋊C4—2741no—
254C22⋊C4—2741no—
255C22⋊C4—2741no—
256C42—1371no—
257C42—1371no—
258C42—1371no—
259C42—1371no—
260C23—21no—
261C23—2744no—
262C23—2742no—
263C23—2741no—
264C23—5482no—
265C23—5482no—
266C23D4×D13711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2741no—
270Q8—2741no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—2744no—
275D4—2741no—
276D4—2742no—
277D4—2741no—
278D4—5482no—
279D4—5482no—
280D4—5482no—
281D4—1374no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—2742no—
287C2×C4—2741no—
288C2×C4—5484no—
289C2×C4—5484no—
290C2×C4—5481no—
291C2×C4—5482no—
292C2×C4—1371no—
293C2×C4—1371no—
294C2×C4—1371no—
295C2×C4—1371no—
296C2×C4D4×D13711yescharacteristic
297C2×C4D4×D13712yes—
298C2×C4#1455551012yes—
299C2×C4C23×D13711yescharacteristic
300C8—22no—
301C8—2742no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—2744no—
308C22—5488no—
309C22—5482no—
310C22—5482no—
311C22—5481no—
312C22—5481no—
313C22—5481no—
314C22#1798406112yes—
315C22C2×D4×D13711yescharacteristic
316C4—42no—
317C4—41no—
318C4—2741no—
319C4—2741no—
320C4—2741no—
321C4—5481no—
322C4—1372no—
323C4#1798406112yes—
324C4C2×D4×D13711yescharacteristic
325C4C2×#1455551011yescharacteristic
326C2—22no—
327C2—81no—
328C2—5484no—
329C2—10961no—
330C2C2×#1798406111yescharacteristic
331C2AGL(1,ℤ/8)×D13711yescharacteristic
332C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
hypercenter#417D137
hypocenterC137#3036
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C2×#17984113C2semidirect
3. #22577949C2semidirect
4. C2×#17984050C2semidirect
5. #22577621C2semidirect
6. #22577617C2semidirect
7. #22577607C2semidirect
8. #22577597C2semidirect
9. #22577426C2semidirect
10. #22577402C2semidirect
11. #417×C137C2semidirect
22. #17984086C22semidirect
23. #17984084C22semidirect
24. #17984077C22semidirect
25. #17984075C22semidirect
36. #17984017C22semidirect
37. #17984017C22semidirect
38. #17984011C22semidirect
39. #17984008C22semidirect
43. #103×C137C22semidirect
44. #102×C137C22semidirect
51. C2×D548D4semidirect
158. Dic137AGL(1,ℤ/8)semidirect
166. C137#3036semidirect
178. #417D137semidirect
209. #103C2×D137semidirect
211. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179841132
#225779492
C2×#179840502
#225776212
#225776172
#225776072
#225775972
#225774262
#225774022
#417×C1372
#3036137