Finite Group Databasebeta

← #31792695

Subgroups — #31792695

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#22578105C211yesmaximal, characteristic, complemented
2#22578064C211yesmaximal, characteristic, complemented
3#22577759C211yesmaximal, characteristic, complemented
4#22577742C211yesmaximal, characteristic, complemented
5#22577722C211yesmaximal, characteristic, complemented
6#22577708C211yesmaximal, characteristic, complemented
7#22577637C211yesmaximal, characteristic, complemented
8#22577629C211yesmaximal, characteristic, complemented
9#22577620C211yesmaximal, characteristic, complemented
10#22577547C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14C2×#17984012C211yesmaximal, characteristic, complemented
15#448×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984081—21no—
21#17984079C2211yescharacteristic
22#17984077—21no—
23#17984077C2211yescharacteristic
24#17984072—21no—
25#17984064—21no—
26#17984064C2211yescharacteristic
27#17984062—21no—
28#17984062C2211yescharacteristic
29#17984060—21no—
30#17984059C2211yescharacteristic
31Q8×Dic137C2211yescharacteristic
32D4×Dic137—21no—
33D4×Dic137C2211yescharacteristic
34C4×C22⋊D137—21no—
35C4×C22⋊D137C2211yescharacteristic
36#17984046C2211yescharacteristic
37C2×C4⋊Dic137C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984020C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984012C2211yescharacteristic
43#17984012C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C22×Q8×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555511—22no—
60#14555511—41no—
61C2×D548—21no—
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137D411yescharacteristic
66C22×Dic137—21no—
67C22×Dic137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2311yescharacteristic
70#14555495—41no—
71C2×#11764040—41no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C8×D137—41no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137—21no—
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137D411yescharacteristic
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96C2×Q8×C137—21no—
97C2×Q8×C137—22no—
98C2×Q8×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105SD16×C137—22no—
106SD16×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137C2×D411yescharacteristic
113D548—21no—
114D548—21no—
115D548—42no—
116C22⋊D137—22no—
117C22⋊D137—42no—
118C22⋊D137—81no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—81no—
123C2×Dic137—81no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—21no—
128C4×D137—21no—
129C4×D137—42no—
130#11764040—21no—
131#11764040—21no—
132C23×C137—21no—
133C23×C137C2×D411yescharacteristic
134Q8×C137—21no—
135Q8×C137—21no—
136Q8×C137—21no—
137Q8×C137—42no—
138Q8×C137#4112yes—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—42no—
142C2×C4×C137—21no—
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—41no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137#13512yes—
158C22×C137C22×D411yescharacteristic
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—21no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C4×C137—21no—
166C4×C137—22no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—21no—
172Dic137—21no—
173Dic137—21no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177C2×C137—22no—
178C2×C137—81no—
179C2×C137#54611yescharacteristic
180C2×C137C2×#13511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—41no—
183D137—81no—
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#449—1371no—
196#448D13711yescharacteristic, complemented
197C4×SD16—1371no—
198C8×D4—1371no—
199#416—1371no—
200C2×#103—1371no—
201C2×#41—1371no—
202C22×Q8C2×D13711yescharacteristic
203C2×SD16C2×D13711yescharacteristic
204C2×SD16C2×D13711yescharacteristic
205C2×D8—1371no—
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#103C2×D13711yescharacteristic
236#103C2×D13711yescharacteristic
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241C22⋊C8—1371no—
242C22⋊C8C2×D13711yescharacteristic
243C4×C8—1371no—
244#41—2742no—
245#41—5481no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D13711yescharacteristic
250C2×Q8C22×D13711yescharacteristic
251C2×D4—21no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—1371no—
263C22×C4C22×D13711yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—5481no—
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—2741no—
298C23—1371no—
299C23D4×D13711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1455550912yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—2742no—
309D4—2741no—
310D4—2741no—
311D4—5482no—
312D4—5482no—
313D4—10961no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
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—41no—
340C22—41no—
341C22—41no—
342C22—2741no—
343C22—2741no—
344C22—2741no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22#1798410912yes—
349C22C2×D4×D13711yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—2741no—
355C4—2741no—
356C4—2741no—
357C4—5481no—
358C4—5481no—
359C4—5481no—
360C4C2×#1455550911yescharacteristic
361C4#1798413311yescharacteristic
362C2—22no—
363C2—81no—
364C2—5481no—
365C2—10961no—
366C2#2257804011yescharacteristic
367C2C2×#1798410911yescharacteristic
368C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#448×C137C2
socleC22×C137C22×D4
hypercenter#448D137
hypocenterC137#3301
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #22578105C2semidirect
2. #22578064C2semidirect
3. #22577759C2semidirect
4. #22577742C2semidirect
5. #22577722C2semidirect
6. #22577708C2semidirect
7. #22577637C2semidirect
8. #22577629C2semidirect
9. #22577620C2semidirect
10. #22577547C2semidirect
11. SD16×Dic137C2semidirect
12. #22577466C2semidirect
13. #22577398C2semidirect
14. C2×#17984012C2semidirect
15. #448×C137C2semidirect
184. C137#3301semidirect
196. #448D137semidirect

Maximal subgroups

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

subgroupindex
#225781052
#225780642
#225777592
#225777422
#225777222
#225777082
#225776372
#225776292
#225776202
#225775472
SD16×Dic1372
#225774662
#225773982
C2×#179840122
#448×C1372
#3301137