Finite Group Databasebeta

← #31790736

Subgroups — #31790736

Order 17536.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D137C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic
3#22577745C211yesmaximal, characteristic, complemented
4#22577744C211yesmaximal, characteristic, complemented
5#22577741C211yesmaximal, characteristic, complemented
6#22577740C211yesmaximal, characteristic, complemented
7#22577737C211yesmaximal, characteristic, complemented
8#22577735C211yesmaximal, characteristic, complemented
9#22577732C211yesmaximal, characteristic, complemented
10#22577444C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577410C211yesmaximal, characteristic, complemented
13#22577410C211yesmaximal, characteristic, complemented
14#22577365C211yesmaximal, characteristic
15#485×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20#17984068C2211yescharacteristic
21#17984067C2211yescharacteristic
22#17984066—21no—
23#17984065—21no—
24#17984064C2211yescharacteristic
25#17984047C2211yescharacteristic
26#17984046C2211yescharacteristic
27#17984046C2211yescharacteristic
28#17984045—21no—
29#17984044—21no—
30C4⋊C4×D137—21no—
31C4⋊C4×D137C2211yescharacteristic
32C4⋊C4×D137C2211yescharacteristic
33C4⋊C4×D137C2211yescharacteristic
34C22⋊C4×D137—21no—
35C22⋊C4×D137—21no—
36C42×D137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984019C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984011C2211yescharacteristic, complemented
48#17984011C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic
52C22⋊Q8×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137D411yescharacteristic
60C2×Dic274—21no—
61C2×Dic274C2311yescharacteristic
62Q8×D137—22no—
63Q8×D137—22no—
64C22×Dic137D411yescharacteristic
65C2×C4×D137—22no—
66C2×C4×D137—22no—
67C2×C4×D137—21no—
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137D411yescharacteristic
72C2×C4×D137D411yescharacteristic
73C2×C4×D137C2311yescharacteristic
74C2×C4×D137C2311yescharacteristic
75#14555493—22no—
76#14555493—22no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×Q8×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C22×D137—22no—
109C22×D137—22no—
110C22×D137—41no—
111C22×D137C2×D411yescharacteristic
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115Dic274—42no—
116C2×Dic137—22no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C4×D137—22no—
124C4×D137—21no—
125C4×D137—21no—
126C4×D137—22no—
127C4×D137—42no—
128C4×D137—42no—
129C4×D137—42no—
130C4×D137—42no—
131C4×D137#4112yes—
132C4×D137#4112yes—
133#11764040—21no—
134#11764040—21no—
135C23×C137C2×D411yescharacteristic
136Q8×C137—21no—
137Q8×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2411yescharacteristic
146C8×C137—21no—
147C8×C137—21no—
148C22×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137C22×D411yescharacteristic
152C2×D137—21no—
153C2×D137—21no—
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—44no—
157C2×D137#12312yes—
158C2×D137#13712yes—
159C4×C137—21no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137C2×#4111yescharacteristic
165C4×C137C2×#4111yescharacteristic
166Dic137—21no—
167Dic137—41no—
168Dic137—41no—
169Dic137—41no—
170Dic137—41no—
171Dic137#12312yes—
172C2×C137—41no—
173C2×C137C2×#12311yescharacteristic
174C2×C137C2×#13711yescharacteristic
175C2×C137#57511yescharacteristic
176D137—22no—
177D137—22no—
178C137#309511yescharacteristic, complemented, Sylow
179#3095—1371nomaximal, Sylow
180C2×C22⋊Q8—1371no—
181C2×#117—1371no—
182#485—1371no—
183#485—1371no—
184#485—1371no—
185#485D13711yescharacteristic, complemented
186#484—1371no—
187#484—1371no—
188#483—1371no—
189#483—1371no—
190#428—1371no—
191#428—1371no—
192#419—1371no—
193#419—1371no—
194#407—1371no—
195C22×Q8—1371no—
196C23×C4—1371no—
197C2×M16—1371no—
198C2×M16—1371no—
199C22⋊Q8—2741no—
200C22⋊Q8—2741no—
201C22⋊Q8—1371no—
202C22⋊Q8—1371no—
203C22⋊Q8—1371no—
204C22⋊Q8C2×D13711yescharacteristic
205#117—2741no—
206#117—2741no—
207#117—1371no—
208#117—1371no—
209#117—1371no—
210#117C2×D13711yescharacteristic
211C2×C4⋊C4—2741no—
212C2×C4⋊C4—1371no—
213C2×C4⋊C4—1371no—
214C2×C4⋊C4—1371no—
215C2×C22⋊C4—2741no—
216C2×C22⋊C4—2741no—
217C2×C42—2741no—
218#107—1371no—
219#107—1371no—
220#107—1371no—
221#107C2×D13711yescharacteristic, complemented
222#106—1371no—
223#106—1371no—
224#106—1371no—
225#106C2×D13711yescharacteristic, complemented
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103—1371no—
232#103C2×D13711yescharacteristic, complemented
233#103C2×D13711yescharacteristic, complemented
234C22⋊C8—1371no—
235C22⋊C8—1371no—
236C22⋊C8—1371no—
237C22⋊C8C2×D13711yescharacteristic
238C24—1371no—
239C2×Q8—2741no—
240C2×Q8—2742no—
241C2×Q8—2742no—
242C2×Q8—1371no—
243C2×Q8C22×D13711yescharacteristic
244C22×C4—2742no—
245C22×C4—2742no—
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—1371no—
251C22×C4—1371no—
252C22×C4—1371no—
253C22×C4—1371no—
254C22×C4—1371no—
255C22×C4C22×D13711yescharacteristic
256M16—2742no—
257M16—2742no—
258C2×C8—1371no—
259C2×C8—1371no—
260C2×C8C22×D13711yescharacteristic
261C2×C8C22×D13711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—2741no—
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—1371no—
270C4⋊C4—1371no—
271C4⋊C4—1371no—
272C4⋊C4C22×D13711yescharacteristic
273C4⋊C4C22×D13711yescharacteristic
274C4⋊C4C22×D13711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C42—21no—
284C42—2741no—
285C42—2741no—
286C42—2741no—
287C23—2742no—
288C23—2742no—
289C23—5481no—
290C23—1371no—
291C23—1371no—
292C23D4×D13711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2741no—
296Q8—2741no—
297Q8—5482no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—2742no—
305C2×C4—2742no—
306C2×C4—2742no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—5482no—
314C2×C4—5482no—
315C2×C4—5482no—
316C2×C4—5482no—
317C2×C4—1371no—
318C2×C4—1371no—
319C2×C4—1372no—
320C2×C4—1372no—
321C2×C4D4×D13711yescharacteristic
322C2×C4C23×D13711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2741no—
326C8—2741no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—2741no—
331C22—2741no—
332C22—2741no—
333C22—2741no—
334C22—5484no—
335C22—1372no—
336C22—1372no—
337C22C2×D4×D13711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2741no—
344C4—5481no—
345C4—5481no—
346C4—5481no—
347C4—5481no—
348C4—1372no—
349C4#41×D13711yescharacteristic
350C4#41×D13711yescharacteristic
351C2—41no—
352C2—2742no—
353C2—2742no—
354C2#123×D13711yescharacteristic
355C2#137×D13711yescharacteristic
356C2#2257820011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#485×C137C2
socleC22×C137C22×D4
prime radicalC23×D137D4
hypercenter#485D137
hypocenterC137#3095
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #22577745C2semidirect
4. #22577744C2semidirect
5. #22577741C2semidirect
6. #22577740C2semidirect
7. #22577737C2semidirect
8. #22577735C2semidirect
9. #22577732C2semidirect
10. #22577444C2semidirect
11. #22577442C2semidirect
12. #22577410C2semidirect
13. #22577410C2semidirect
15. #485×C137C2semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984012C22semidirect
46. #17984012C22semidirect
47. #17984011C22semidirect
48. #17984011C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
178. C137#3095semidirect
185. #485D137semidirect
221. #107C2×D137semidirect
225. #106C2×D137semidirect
232. #103C2×D137semidirect
233. #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
#117×D1372
#225777452
#225777442
#225777412
#225777402
#225777372
#225777352
#225777322
#225774442
#225774422
#225774102
#225773652
#485×C1372
#3095137