Finite Group Databasebeta

← #31792836

Subgroups — #31792836

Order 17536.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578088C211yesmaximal, characteristic
2Q8×C22⋊D137C211yesmaximal, characteristic
3#22577770C211yesmaximal, characteristic, complemented
4#22577743C211yesmaximal, characteristic, complemented
5#22577733C211yesmaximal, characteristic, complemented
6#22577712C211yesmaximal, characteristic, complemented
7#22577660C211yesmaximal, characteristic, complemented
8#22577644C211yesmaximal, characteristic, complemented
9#22577612C211yesmaximal, characteristic, complemented
10#22577560C211yesmaximal, characteristic, complemented
11#22577559C211yesmaximal, characteristic, complemented
12#22577507C211yesmaximal, characteristic, complemented
13#22577474C211yesmaximal, characteristic
14#22577410C211yesmaximal, characteristic, complemented
15C2×#103×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×Dic548C2211yescharacteristic
18C2×#14555500C2211yescharacteristic
19C2×#14555500C2211yescharacteristic
20C2×#14555493C2211yescharacteristic
21#17984086—21no—
22#17984086C2211yescharacteristic
23#17984083—21no—
24#17984079C2211yescharacteristic
25#17984078—21no—
26#17984071—21no—
27#17984068C2211yescharacteristic
28#17984067—21no—
29#17984067C2211yescharacteristic
30#17984067C2211yescharacteristic
31#17984064—21no—
32#17984064C2211yescharacteristic
33Q8×Dic137C2211yescharacteristic
34#17984053C2211yescharacteristic
35C4×C22⋊D137—21no—
36C4×C22⋊D137C2211yescharacteristic
37#17984047—21no—
38#17984046C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17983998C2211yescharacteristic
52C22×Q8×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#103×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61C2×Dic274C2311yescharacteristic
62Q8×D137—22no—
63Q8×D137—41no—
64C2×C22⋊D137D411yescharacteristic
65C2×C4×D137—21no—
66C2×C4×D137C2311yescharacteristic
67Dic548—41no—
68#14555500—22no—
69#14555500—41no—
70#14555493—41no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C4⋊Dic137—21no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489D411yescharacteristic
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×Q8×C137—21no—
98C2×Q8×C137—22no—
99C2×Q8×C137—41no—
100C2×Q8×C137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117C22⋊D137—22no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—21no—
126C4×D137—21no—
127C4×D137—42no—
128#11764040—21no—
129#11764040—21no—
130C23×C137C2×D411yescharacteristic
131Q8×C137—21no—
132Q8×C137—21no—
133Q8×C137—21no—
134Q8×C137—42no—
135Q8×C137#4112yes—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—42no—
143C2×C4×C137—42no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2411yescharacteristic
146C8×C137—21no—
147C8×C137—21no—
148C22×C137—21no—
149C22×C137—21no—
150C22×C137#13712yes—
151C22×C137C22×D411yescharacteristic
152C2×D137—21no—
153C2×D137—21no—
154C2×D137—21no—
155C4×C137—21no—
156C4×C137—22no—
157C4×C137—41no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137C2×#4111yescharacteristic
161C4×C137H(5,𝔽2)11yescharacteristic
162Dic137—21no—
163Dic137—21no—
164Dic137—21no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169C2×C137—22no—
170C2×C137#54611yescharacteristic
171C2×C137C2×#13711yescharacteristic
172C2×C137#57811yescharacteristic
173D137—41no—
174C137#331211yescharacteristic, complemented, Sylow
175#3312—1371nomaximal, Sylow
176#552—1371no—
177D4×Q8—1371no—
178#491—1371no—
179#485—1371no—
180#483—1371no—
181#479—1371no—
182#470—1371no—
183#467—1371no—
184#462—1371no—
185#451—1371no—
186#451—1371no—
187#441—1371no—
188#435—1371no—
189#419—1371no—
190C2×#103D13711yescharacteristic, complemented
191C22×Q8—1371no—
192C22×Q8C2×D13711yescharacteristic
193C2×Q16—1371no—
194C2×Q16—1371no—
195C2×Q16—1371no—
196C2×M16—1371no—
197C22×C8C2×D13711yescharacteristic
198C4⋊Q8—2741no—
199C4⋊Q8—1371no—
200#126—2741no—
201#125—2741no—
202#125—1371no—
203#123—2741no—
204C22⋊Q8—2741no—
205C22⋊Q8—2741no—
206C22⋊Q8—1371no—
207C22⋊Q8—1371no—
208C22⋊Q8—1371no—
209C22⋊Q8—1371no—
210C4×Q8—1371no—
211C4×Q8—1371no—
212C4×D4—2741no—
213C4×D4—1371no—
214#117—2741no—
215#117—1371no—
216C2×C4⋊C4C2×D13711yescharacteristic
217#107—1371no—
218#106—1371no—
219#106—1371no—
220C4⋊C8—1371no—
221#103—1371no—
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103C2×D13711yescharacteristic
228#103C2×D13711yescharacteristic
229#103C2×D13711yescharacteristic
230#103C2×D13711yescharacteristic
231C22⋊C8—1371no—
232C22⋊C8—1371no—
233#97—1371no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—2741no—
238C2×Q8—2742no—
239C2×Q8—5481no—
240C2×Q8—1371no—
241C2×Q8—1371no—
242C2×Q8C22×D13711yescharacteristic
243C2×Q8C22×D13711yescharacteristic
244C2×D4—1371no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2741no—
248C22×C4—1371no—
249C22×C4C22×D13711yescharacteristic
250Q16—2742no—
251Q16—5481no—
252Q16—5481no—
253M16—5481no—
254C2×C8—41no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8C22×D13711yescharacteristic
258C2×C8C22×D13711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—2741no—
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—1371no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4C22×D13711yescharacteristic
274C4⋊C4C22×D13711yescharacteristic
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—1371no—
281C22⋊C4—1371no—
282C42—2741no—
283C42—2741no—
284C42—2741no—
285C42—1371no—
286C23—1371no—
287C23D4×D13711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—2741no—
293Q8—2741no—
294Q8—2741no—
295Q8—2741no—
296Q8—5482no—
297Q8#1455551012yes—
298D4—2742no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—5482no—
314C2×C4—1371no—
315C2×C4—1371no—
316C2×C4—1371no—
317C2×C4D4×D13711yescharacteristic
318C2×C4C23×D13711yescharacteristic
319C8—21no—
320C8—21no—
321C8—2741no—
322C8—2741no—
323C22—21no—
324C22—21no—
325C22—2741no—
326C22—2741no—
327C22—2741no—
328C22#1798411712yes—
329C22C2×D4×D13711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—2741no—
336C4—2741no—
337C4—2741no—
338C4—5481no—
339C4—5481no—
340C4—5481no—
341C4—5481no—
342C4C2×#1455551011yescharacteristic
343C4#1798413311yescharacteristic
344C2—22no—
345C2—5481no—
346C2#2257804311yescharacteristic
347C2C2×#1798411711yescharacteristic
348C2#2257821611yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#103×C137C2
socleC22×C137C22×D4
prime radicalC2×C22⋊D137D4
hypercenterC2×#103D137
hypocenterC137#3312
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #22577770C2semidirect
4. #22577743C2semidirect
5. #22577733C2semidirect
6. #22577712C2semidirect
7. #22577660C2semidirect
8. #22577644C2semidirect
9. #22577612C2semidirect
10. #22577560C2semidirect
11. #22577559C2semidirect
12. #22577507C2semidirect
14. #22577410C2semidirect
15. C2×#103×C137C2semidirect
174. C137#3312semidirect
190. C2×#103D137semidirect

Maximal subgroups

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

subgroupindex
#225780882
Q8×C22⋊D1372
#225777702
#225777432
#225777332
#225777122
#225776602
#225776442
#225776122
#225775602
#225775592
#225775072
#225774742
#225774102
C2×#103×C1372
#3312137