Finite Group Databasebeta

← #31790739

Subgroups — #31790739

Order 17536.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17984068C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic
3#22577747C212yesmaximal, complemented
4#22577746C212yesmaximal, complemented
5#22577740C212yesmaximal, complemented
6#22577735C212yesmaximal, complemented
7#22577443C211yesmaximal, characteristic, complemented
8#22577413C211yesmaximal, characteristic, complemented
9#22577411C211yesmaximal, characteristic, complemented
10#22577366C211yesmaximal, characteristic
11#428×C137C211yesmaximal, characteristic, complemented
12C22×Dic274C2211yescharacteristic
13C22×C4×D137C2211yescharacteristic
14C2×#14555494C2211yescharacteristic
15#17984068—21no—
16#17984068—21no—
17#17984068C2212yes—
18#17984068C2212yes—
19#17984047C2211yescharacteristic
20#17984046C2211yescharacteristic
21#17984046C2211yescharacteristic
22#17984045—21no—
23#17984044—21no—
24C2×C4⋊Dic137—21no—
25C4⋊C4×D137C2211yescharacteristic
26C4⋊C4×D137C2211yescharacteristic
27C2×#14555487—21no—
28C22⋊C4×D137—21no—
29C42×D137—21no—
30#17984021C2212yescomplemented
31#17984019C2212yescomplemented
32#17984013C2212yescomplemented
33#17984013C2212yescomplemented
34#17984011C2212yescomplemented
35#17984011C2212yescomplemented
36#17984000C2212yes—
37#17984000C2212yes—
38C2×M16×C137C2211yescharacteristic
39#117×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#107×C137C2212yescomplemented
42#106×C137C2212yescomplemented
43C23×D137D411yescharacteristic
44C2×Dic274—21no—
45C2×Dic274—22no—
46C2×Dic274—22no—
47C2×Dic274C2312yes—
48C22×Dic137—21no—
49C22×Dic137D411yescharacteristic
50C2×C4×D137—21no—
51C2×C4×D137—21no—
52C2×C4×D137—22no—
53C2×C4×D137—22no—
54C2×C4×D137D411yescharacteristic
55C2×C4×D137D411yescharacteristic
56C2×C4×D137C2311yescharacteristic
57C2×C4×D137C2311yescharacteristic
58#14555494—22no—
59C2×#11764040C2312yes—
60C4⋊Dic137—22no—
61C4⋊Dic137—22no—
62C4⋊Dic137C2311yescharacteristic
63C4⋊Dic137C2311yescharacteristic
64#14555489—21no—
65#14555489—21no—
66C22⋊Dic137—21no—
67#14555487—21no—
68#14555487—21no—
69#14555487—22no—
70#14555487—22no—
71C4×Dic137—21no—
72C4×Dic137—21no—
73C4×Dic137—21no—
74C22×C4×C137—21no—
75C22×C4×C137C2311yescharacteristic
76M16×C137—22no—
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2311yescharacteristic
80C4⋊C4×C137C2311yescharacteristic
81C4⋊C4×C137C2312yes—
82C22⋊C4×C137—21no—
83C42×C137—21no—
84C22×D137—22no—
85C22×D137—22no—
86C22×D137—41no—
87C22×D137C2×D411yescharacteristic
88C22×D137C2×D411yescharacteristic
89Dic274—22no—
90Dic274—22no—
91Dic274—42no—
92C2×Dic137—22no—
93C2×Dic137—21no—
94C2×Dic137—21no—
95C2×Dic137—22no—
96C2×Dic137—42no—
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C4×D137—22no—
100C4×D137—22no—
101C4×D137—21no—
102C4×D137—21no—
103C4×D137—21no—
104C4×D137—21no—
105C4×D137—42no—
106C4×D137—42no—
107#11764040—22no—
108C23×C137C2×D411yescharacteristic
109C2×C4×C137—21no—
110C2×C4×C137—21no—
111C2×C4×C137—22no—
112C2×C4×C137—42no—
113C2×C4×C137C2×D411yescharacteristic
114C2×C4×C137#4112yes—
115C2×C4×C137#4112yes—
116C2×C4×C137C2411yescharacteristic
117C8×C137—22no—
118C22×C137—21no—
119C22×C137—21no—
120C22×C137#12312yes—
121C22×C137C22×D411yescharacteristic
122C2×D137—21no—
123C2×D137—21no—
124C2×D137—21no—
125C2×D137—21no—
126C2×D137—21no—
127C2×D137—44no—
128C2×D137#13712yes—
129C4×C137—41no—
130C4×C137—42no—
131C4×C137—41no—
132C4×C137#12312yes—
133C4×C137C2×#4111yescharacteristic
134C4×C137C2×#4111yescharacteristic
135Dic137—21no—
136Dic137—21no—
137Dic137—42no—
138Dic137—41no—
139Dic137—41no—
140C2×C137—22no—
141C2×C137C2×#12311yescharacteristic
142C2×C137C2×#13711yescharacteristic
143C2×C137#57511yescharacteristic
144D137—22no—
145D137—41no—
146C137#309511yescharacteristic, complemented, Sylow
147#3095—1371nomaximal, Sylow
148C2×C22⋊Q8—1371no—
149C2×#117—1371no—
150#485—1372no—
151#485—1372no—
152#484—1372no—
153#483—1372no—
154#428—1371no—
155#428D13711yescharacteristic, complemented
156#419—1371no—
157#419—1371no—
158#407—1371no—
159C22×Q8—1371no—
160C23×C4—1371no—
161C2×M16—1371no—
162C2×M16C2×D13711yescharacteristic
163C22⋊Q8—2741no—
164C22⋊Q8—2741no—
165C22⋊Q8—1372no—
166C22⋊Q8—1372no—
167#117—2741no—
168#117—2741no—
169#117—1371no—
170#117—1371no—
171#117—1371no—
172#117C2×D13711yescharacteristic
173C2×C4⋊C4—2741no—
174C2×C4⋊C4—1371no—
175C2×C4⋊C4—1371no—
176C2×C4⋊C4C2×D13711yescharacteristic
177C2×C22⋊C4—2741no—
178C2×C22⋊C4—2741no—
179C2×C42—2741no—
180#107—1372no—
181#107C2×D13712yescomplemented
182#106—1372no—
183#106C2×D13712yescomplemented
184#103—1372no—
185#103—1372no—
186#103—1372no—
187#103—1372no—
188C22⋊C8—1372no—
189C22⋊C8—1372no—
190C24—1371no—
191C2×Q8—2741no—
192C2×Q8—2742no—
193C2×Q8—2742no—
194C2×Q8—1372no—
195C22×C4—21no—
196C22×C4—2741no—
197C22×C4—2741no—
198C22×C4—2742no—
199C22×C4—2742no—
200C22×C4—2741no—
201C22×C4—1371no—
202C22×C4—1371no—
203C22×C4—1371no—
204C22×C4—1371no—
205C22×C4—1371no—
206C22×C4C22×D13711yescharacteristic
207M16—22no—
208M16—2742no—
209C2×C8—1372no—
210C2×C8C22×D13712yes—
211C4⋊C4—21no—
212C4⋊C4—2741no—
213C4⋊C4—2741no—
214C4⋊C4—2742no—
215C4⋊C4—2742no—
216C4⋊C4—1371no—
217C4⋊C4—1371no—
218C4⋊C4C22×D13711yescharacteristic
219C4⋊C4C22×D13711yescharacteristic
220C4⋊C4C22×D13712yes—
221C22⋊C4—21no—
222C22⋊C4—2741no—
223C22⋊C4—2741no—
224C22⋊C4—2742no—
225C22⋊C4—2742no—
226C22⋊C4—2741no—
227C42—21no—
228C42—2741no—
229C42—2741no—
230C42—2741no—
231C23—2742no—
232C23—2742no—
233C23—5481no—
234C23—1371no—
235C23—1371no—
236C23D4×D13711yescharacteristic
237Q8—2742no—
238Q8—2742no—
239Q8—5482no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2742no—
245C2×C4—2742no—
246C2×C4—2742no—
247C2×C4—2741no—
248C2×C4—2741no—
249C2×C4—2741no—
250C2×C4—2741no—
251C2×C4—2741no—
252C2×C4—2741no—
253C2×C4—2742no—
254C2×C4—5482no—
255C2×C4—5482no—
256C2×C4—5482no—
257C2×C4—1371no—
258C2×C4—1371no—
259C2×C4D4×D13711yescharacteristic
260C2×C4#1455551012yes—
261C2×C4#1455551112yes—
262C2×C4C23×D13711yescharacteristic
263C8—22no—
264C8—2742no—
265C22—21no—
266C22—21no—
267C22—2741no—
268C22—2741no—
269C22—2741no—
270C22—2741no—
271C22—2741no—
272C22—5484no—
273C22—1372no—
274C22#1798407312yes—
275C22C2×D4×D13711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2741no—
280C4—2741no—
281C4—5481no—
282C4—5481no—
283C4—5482no—
284C4#1798407312yes—
285C4C2×#1455551011yescharacteristic
286C4C2×#1455551111yescharacteristic
287C2—22no—
288C2—2742no—
289C2—5481no—
290C2C2×#1798407311yescharacteristic
291C2#137×D13711yescharacteristic
292C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radicalC23×D137D4
hypercenter#428D137
hypocenterC137#3095
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22577747C2semidirect
4. #22577746C2semidirect
5. #22577740C2semidirect
6. #22577735C2semidirect
7. #22577443C2semidirect
8. #22577413C2semidirect
9. #22577411C2semidirect
11. #428×C137C2semidirect
30. #17984021C22semidirect
31. #17984019C22semidirect
32. #17984013C22semidirect
33. #17984013C22semidirect
34. #17984011C22semidirect
35. #17984011C22semidirect
41. #107×C137C22semidirect
42. #106×C137C22semidirect
146. C137#3095semidirect
155. #428D137semidirect
181. #107C2×D137semidirect
183. #106C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840682
#117×D1372
#225777472
#225777462
#225777402
#225777352
#225774432
#225774132
#225774112
#225773662
#428×C1372
#3095137