Finite Group Databasebeta

← #31792684

Subgroups — #31792684

Order 17536.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578100C211yesmaximal, characteristic, complemented
2#22578060C211yesmaximal, characteristic, complemented
3#22577756C211yesmaximal, characteristic, complemented
4#22577745C211yesmaximal, characteristic, complemented
5#22577720C211yesmaximal, characteristic, complemented
6#22577712C211yesmaximal, characteristic, complemented
7#22577636C211yesmaximal, characteristic, complemented
8#22577630C211yesmaximal, characteristic, complemented
9#22577616C211yesmaximal, characteristic, complemented
10#22577540C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577399C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#449×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×#14555498C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984081—21no—
21#17984079C2211yescharacteristic
22#17984075—21no—
23#17984075C2211yescharacteristic
24#17984069—21no—
25#17984067—21no—
26#17984067C2211yescharacteristic
27#17984059—21no—
28#17984059—21no—
29#17984059C2211yescharacteristic
30#17984059C2211yescharacteristic
31Q8×Dic137C2211yescharacteristic
32D4×Dic137C2211yescharacteristic
33C4×C22⋊D137—21no—
34C4×C22⋊D137C2211yescharacteristic
35#17984048—21no—
36#17984047C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61Q8×D137—22no—
62Q8×D137—41no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137D411yescharacteristic
66C22×Dic137—21no—
67C2×C4×D137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2311yescharacteristic
70#14555498—41no—
71C2×#11764040—41no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C8×D137—41no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137D411yescharacteristic
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96#41×C137—22no—
97#41×C137—41no—
98C2×Q8×C137C2311yescharacteristic
99C2×D4×C137—21no—
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105SD16×C137—22no—
106D8×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—42no—
115C22⋊D137—22no—
116C22⋊D137—42no—
117C22⋊D137—42no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—81no—
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C4×D137—21no—
127C4×D137—21no—
128C4×D137—42no—
129C4×D137—81no—
130#11764040—21no—
131#11764040—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137C2×D411yescharacteristic
135Q8×C137—21no—
136Q8×C137#4112yes—
137D4×C137—21no—
138D4×C137—21no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—42no—
142D4×C137—81no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—42no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—21no—
164C2×D137#13512yes—
165C4×C137—21no—
166C4×C137—22no—
167C4×C137—41no—
168C4×C137C2×#4111yescharacteristic
169C4×C137H(5,𝔽2)11yescharacteristic
170Dic137—21no—
171Dic137—21no—
172Dic137—21no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177C2×C137—41no—
178C2×C137—81no—
179C2×C137—81no—
180C2×C137#54611yescharacteristic
181C2×C137C2×#13511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—22no—
184C137#330111yescharacteristic, complemented, Sylow
185#3301—1371nomaximal, Sylow
186#553—1371no—
187#548—1371no—
188#488—1371no—
189#485—1371no—
190#481—1371no—
191#479—1371no—
192#466—1371no—
193#465—1371no—
194#463—1371no—
195#449D13711yescharacteristic, complemented
196#448—1371no—
197C4×SD16—1371no—
198C8×D4—1371no—
199#416—1371no—
200C2×#103—1371no—
201C2×#41C2×D13711yescharacteristic
202C22×Q8—1371no—
203C2×SD16—1371no—
204C2×SD16C2×D13711yescharacteristic
205C2×D8C2×D13711yescharacteristic
206C22×C8—1371no—
207C22×C8—1371no—
208#126—2741no—
209#125—1371no—
210#124—2741no—
211#124—1371no—
212#123—2741no—
213C22⋊Q8—2741no—
214C22⋊Q8—1371no—
215#121—2741no—
216#121—2741no—
217#121—1371no—
218#121—1371no—
219#121C2×D13711yescharacteristic
220C4×Q8—1371no—
221C4×D4—2741no—
222C4×D4—2741no—
223C4×D4—1371no—
224C4×D4—1371no—
225#117—1371no—
226C2×C4⋊C4—1371no—
227#107—1371no—
228#106—1371no—
229#106—1371no—
230C4⋊C8—1371no—
231#103—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#103C2×D13711yescharacteristic
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102C2×D13711yescharacteristic
241C22⋊C8—1371no—
242C22⋊C8C2×D13711yescharacteristic
243C4×C8—1371no—
244#41—22no—
245#41—41no—
246C2×Q8—2741no—
247C2×Q8—2742no—
248C2×Q8—5481no—
249C2×Q8—1371no—
250C2×Q8C22×D13711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4C22×D13711yescharacteristic
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—1371no—
263C22×C4C22×D13711yescharacteristic
264SD16—22no—
265SD16—5481no—
266D8—41no—
267C2×C8—5481no—
268C2×C8—5481no—
269C2×C8—1371no—
270C2×C8—1371no—
271C2×C8C22×D13711yescharacteristic
272C2×C8C22×D13711yescharacteristic
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4C22×D13711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—1371no—
292C22⋊C4—1371no—
293C42—2741no—
294C42—2741no—
295C42—1371no—
296C23—21no—
297C23—21no—
298C23—1371no—
299C23D4×D13711yescharacteristic
300Q8—21no—
301Q8—2741no—
302Q8—2741no—
303Q8—5482no—
304Q8#1455550912yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—81no—
311D4—2742no—
312D4—5482no—
313D4—5482no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—2741no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—5482no—
326C2×C4—10961no—
327C2×C4—10961no—
328C2×C4—1371no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4D4×D13711yescharacteristic
332C2×C4C23×D13711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2741no—
336C8—2741no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2741no—
347C22—2741no—
348C22—1372no—
349C22C2×D4×D13711yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—2741no—
354C4—2741no—
355C4—2741no—
356C4—5481no—
357C4—5481no—
358C4—5481no—
359C4—5481no—
360C4C2×#1455550911yescharacteristic
361C4#1798413311yescharacteristic
362C2—41no—
363C2—81no—
364C2—81no—
365C2—2742no—
366C2#2257804011yescharacteristic
367C2#135×D13711yescharacteristic
368C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#449×C137C2
socleC22×C137C22×D4
hypercenter#449D137
hypocenterC137#3301
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22578100C2semidirect
2. #22578060C2semidirect
3. #22577756C2semidirect
4. #22577745C2semidirect
5. #22577720C2semidirect
6. #22577712C2semidirect
7. #22577636C2semidirect
8. #22577630C2semidirect
9. #22577616C2semidirect
10. #22577540C2semidirect
11. SD16×Dic137C2semidirect
12. #22577466C2semidirect
13. #22577399C2semidirect
14. #103×D137C2semidirect
15. #449×C137C2semidirect
184. C137#3301semidirect
195. #449D137semidirect

Maximal subgroups

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

subgroupindex
#225781002
#225780602
#225777562
#225777452
#225777202
#225777122
#225776362
#225776302
#225776162
#225775402
SD16×Dic1372
#225774662
#225773992
#103×D1372
#449×C1372
#3301137