Finite Group Databasebeta

← #31790732

Subgroups — #31790732

Order 17536.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊Q8×D137C211yesmaximal, characteristic
2C2×#17984046C211yesmaximal, characteristic
3#22577747C212yesmaximal, complemented
4#22577743C212yesmaximal, complemented
5#22577739C211yesmaximal, characteristic, complemented
6#22577736C211yesmaximal, characteristic, complemented
7#22577733C211yesmaximal, characteristic, complemented
8#22577442C212yesmaximal, complemented
9#22577410C212yesmaximal, complemented
10#22577365C211yesmaximal, characteristic
11#483×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137C2211yescharacteristic
13C22×C4×D137C2211yescharacteristic
14C2×#14555493C2212yes—
15#17984068C2211yescharacteristic
16#17984067C2211yescharacteristic
17#17984066—21no—
18#17984065—21no—
19#17984064C2211yescharacteristic
20#17984046—21no—
21#17984046—21no—
22#17984046C2212yes—
23#17984046C2212yes—
24C2×C4⋊Dic137C2211yescharacteristic
25C4⋊C4×D137—21no—
26C4⋊C4×D137C2211yescharacteristic
27C2×#14555487—21no—
28C22⋊C4×D137—21no—
29C2×C4×Dic137—21no—
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984019C2212yescomplemented
33#17984013C2212yescomplemented
34#17984012C2212yescomplemented
35#17984011C2212yescomplemented
36C22⋊#11764040C2211yescharacteristic
37#17984000C2212yes—
38C22⋊Q8×C137C2211yescharacteristic
39C2×C4⋊C4×C137C2211yescharacteristic
40#106×C137C2212yescomplemented
41#103×C137C2212yescomplemented
42C22⋊C8×C137C2211yescharacteristic
43C23×D137D411yescharacteristic
44C2×Dic274—21no—
45C2×Dic274C2311yescharacteristic
46Q8×D137—24no—
47C22×Dic137—21no—
48C22×Dic137D411yescharacteristic
49C2×C4×D137—24no—
50C2×C4×D137—21no—
51C2×C4×D137—21no—
52C2×C4×D137D411yescharacteristic
53C2×C4×D137D411yescharacteristic
54C2×C4×D137C2311yescharacteristic
55C2×C4×D137C2311yescharacteristic
56#14555493—24no—
57C2×#11764040C2312yes—
58C4⋊Dic137—21no—
59C4⋊Dic137—21no—
60C4⋊Dic137C2312yes—
61C4⋊Dic137C2311yescharacteristic
62#14555489—21no—
63#14555489—21no—
64#14555489—21no—
65C22⋊Dic137—21no—
66#14555487—21no—
67#14555487—21no—
68#14555487—22no—
69#14555487—22no—
70C4×Dic137—22no—
71C4×Dic137—22no—
72C2×Q8×C137C2311yescharacteristic
73C22×C4×C137—21no—
74C22×C4×C137C2311yescharacteristic
75C2×C8×C137C2312yes—
76C4⋊C4×C137—21no—
77C4⋊C4×C137—21no—
78C4⋊C4×C137C2312yes—
79C4⋊C4×C137C2311yescharacteristic
80C22⋊C4×C137—21no—
81C22×D137—22no—
82C22×D137—22no—
83C22×D137—41no—
84C22×D137C2×D411yescharacteristic
85C22×D137C2×D411yescharacteristic
86Dic274—22no—
87Dic274—42no—
88C2×Dic137—22no—
89C2×Dic137—22no—
90C2×Dic137—21no—
91C2×Dic137—21no—
92C2×Dic137—42no—
93C2×Dic137C2×D411yescharacteristic
94C2×Dic137C2×D411yescharacteristic
95C4×D137—22no—
96C4×D137—22no—
97C4×D137—22no—
98C4×D137—42no—
99C4×D137—42no—
100C4×D137#4114yes—
101#11764040—22no—
102C23×C137C2×D411yescharacteristic
103Q8×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—21no—
106C2×C4×C137—21no—
107C2×C4×C137—22no—
108C2×C4×C137—42no—
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2411yescharacteristic
111C8×C137—22no—
112C22×C137—21no—
113C22×C137—21no—
114C22×C137#13712yes—
115C22×C137C22×D411yescharacteristic
116C2×D137—21no—
117C2×D137—21no—
118C2×D137—21no—
119C2×D137—21no—
120C2×D137—21no—
121C2×D137—44no—
122C2×D137#12312yes—
123C4×C137—21no—
124C4×C137—41no—
125C4×C137—42no—
126C4×C137—41no—
127C4×C137C2×#4112yes—
128Dic137—21no—
129Dic137—41no—
130Dic137—41no—
131Dic137—42no—
132Dic137#12312yes—
133C2×C137—22no—
134C2×C137C2×#12311yescharacteristic
135C2×C137C2×#13711yescharacteristic
136C2×C137#57511yescharacteristic
137D137—22no—
138D137—41no—
139C137#309511yescharacteristic, complemented, Sylow
140#3095—1371nomaximal, Sylow
141C2×C22⋊Q8—1371no—
142C2×#117—1371no—
143#485—1372no—
144#485—1372no—
145#484—1371no—
146#484—1371no—
147#483—1371no—
148#483D13711yescharacteristic, complemented
149#428—1372no—
150#419—1372no—
151#407—1371no—
152C22×Q8—1371no—
153C23×C4—1371no—
154C2×M16—1372no—
155C22⋊Q8—2741no—
156C22⋊Q8—2741no—
157C22⋊Q8—1371no—
158C22⋊Q8—1371no—
159C22⋊Q8—1371no—
160C22⋊Q8C2×D13711yescharacteristic
161#117—2741no—
162#117—2741no—
163#117—1372no—
164#117—1372no—
165C2×C4⋊C4—2741no—
166C2×C4⋊C4—1371no—
167C2×C4⋊C4—1371no—
168C2×C4⋊C4C2×D13711yescharacteristic
169C2×C22⋊C4—2741no—
170C2×C22⋊C4—2741no—
171C2×C42—2741no—
172#107—1372no—
173#107—1372no—
174#106—1372no—
175#106C2×D13712yescomplemented
176#103—1372no—
177#103—1372no—
178#103—1372no—
179#103C2×D13712yescomplemented
180C22⋊C8—1371no—
181C22⋊C8—1372no—
182C22⋊C8C2×D13711yescharacteristic
183C24—1371no—
184C2×Q8—2741no—
185C2×Q8—2744no—
186C2×Q8—1371no—
187C2×Q8C22×D13711yescharacteristic
188C22×C4—21no—
189C22×C4—2744no—
190C22×C4—2741no—
191C22×C4—2741no—
192C22×C4—2741no—
193C22×C4—1371no—
194C22×C4—1371no—
195C22×C4—1371no—
196C22×C4—1371no—
197C22×C4—1371no—
198C22×C4C22×D13711yescharacteristic
199M16—2744no—
200C2×C8—1372no—
201C2×C8C22×D13712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—2741no—
205C4⋊C4—2741no—
206C4⋊C4—2741no—
207C4⋊C4—2741no—
208C4⋊C4—2741no—
209C4⋊C4—1372no—
210C4⋊C4—1371no—
211C4⋊C4C22×D13712yes—
212C4⋊C4C22×D13711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2742no—
215C22⋊C4—2742no—
216C22⋊C4—2741no—
217C22⋊C4—2741no—
218C22⋊C4—2741no—
219C42—2742no—
220C42—2742no—
221C23—2742no—
222C23—2742no—
223C23—5481no—
224C23—1371no—
225C23—1371no—
226C23D4×D13711yescharacteristic
227Q8—22no—
228Q8—2742no—
229Q8—5482no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2742no—
236C2×C4—2742no—
237C2×C4—2742no—
238C2×C4—2742no—
239C2×C4—2742no—
240C2×C4—2741no—
241C2×C4—2741no—
242C2×C4—5482no—
243C2×C4—5482no—
244C2×C4—5482no—
245C2×C4—1371no—
246C2×C4—1371no—
247C2×C4—1374no—
248C2×C4D4×D13711yescharacteristic
249C2×C4C23×D13711yescharacteristic
250C8—22no—
251C8—2742no—
252C22—21no—
253C22—21no—
254C22—2741no—
255C22—2741no—
256C22—2741no—
257C22—2741no—
258C22—2741no—
259C22—5484no—
260C22—1372no—
261C22#1798411712yes—
262C22C2×D4×D13711yescharacteristic
263C4—21no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—2741no—
268C4—5482no—
269C4—5481no—
270C4—5481no—
271C4—1372no—
272C4#41×D13712yes—
273C2—22no—
274C2—2742no—
275C2—5481no—
276C2#123×D13711yescharacteristic
277C2C2×#1798411711yescharacteristic
278C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
prime radicalC23×D137D4
hypercenter#483D137
hypocenterC137#3095
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #22577747C2semidirect
4. #22577743C2semidirect
5. #22577739C2semidirect
6. #22577736C2semidirect
7. #22577733C2semidirect
8. #22577442C2semidirect
9. #22577410C2semidirect
11. #483×C137C2semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984019C22semidirect
33. #17984013C22semidirect
34. #17984012C22semidirect
35. #17984011C22semidirect
40. #106×C137C22semidirect
41. #103×C137C22semidirect
139. C137#3095semidirect
148. #483D137semidirect
175. #106C2×D137semidirect
179. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1372
C2×#179840462
#225777472
#225777432
#225777392
#225777362
#225777332
#225774422
#225774102
#225773652
#483×C1372
#3095137