Finite Group Databasebeta

← #31792834

Subgroups — #31792834

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
1D4×Dic274C211yesmaximal, characteristic
2#22577773C211yesmaximal, characteristic, complemented
3#22577744C211yesmaximal, characteristic, complemented
4#22577733C211yesmaximal, characteristic, complemented
5#22577707C211yesmaximal, characteristic, complemented
6#22577659C211yesmaximal, characteristic, complemented
7#22577642C211yesmaximal, characteristic, complemented
8#22577613C211yesmaximal, characteristic, complemented
9#22577558C211yesmaximal, characteristic, complemented
10#22577558C211yesmaximal, characteristic, complemented
11#22577508C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic
13#22577411C211yesmaximal, characteristic, complemented
14C2×#17984011C211yesmaximal, characteristic, complemented
15#552×C137C211yesmaximal, characteristic
16C22×Dic274C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C2×#14555500C2211yescharacteristic
19C2×#14555500C2211yescharacteristic
20C2×#14555500C2211yescharacteristic
21C2×#14555494C2211yescharacteristic
22C22×#11764040C2211yescharacteristic
23#17984088C2211yescharacteristic
24#17984087—21no—
25#17984067C2211yescharacteristic
26#17984067C2211yescharacteristic
27#17984065—21no—
28#17984065—21no—
29C4×Dic274C2211yescharacteristic
30D4×Dic137—21no—
31#17984021C2211yescharacteristic
32#17984019C2211yescharacteristic
33#17984019C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic
35#17984012C2211yescharacteristic
36#17984011C2211yescharacteristic
37#17984011C2211yescharacteristic
38#17984011C2211yescharacteristic
39#17984011C2211yescharacteristic
40#17984011C2211yescharacteristic
41#17984011C2211yescharacteristic
42#17984011C2211yescharacteristic
43#17984011C2211yescharacteristic
44#17984011C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic
47#17983999C2211yescharacteristic
48#126×C137—21no—
49#125×C137—21no—
50#125×C137C2211yescharacteristic
51#123×C137—21no—
52C22⋊Q8×C137C2211yescharacteristic
53C22⋊Q8×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56#117×C137—21no—
57#117×C137C2211yescharacteristic
58C2×C4⋊C4×C137C2211yescharacteristic
59C2×Dic274—22no—
60C2×Dic274—22no—
61C2×Dic274—21no—
62C2×Dic274—21no—
63C2×Dic274—41no—
64C2×Dic274—41no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68C22×Dic137—21no—
69C22×Dic137—21no—
70#14555500—22no—
71#14555500—41no—
72#14555500—41no—
73#14555494—41no—
74C2×#11764040—41no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C4⋊Dic137—21no—
80C4⋊Dic137—21no—
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C4×Dic137—21no—
88C2×Q8×C137C2311yescharacteristic
89C2×D4×C137D411yescharacteristic
90C22×C4×C137—21no—
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C4⋊C4×C137—21no—
94C4⋊C4×C137—21no—
95C4⋊C4×C137—21no—
96C4⋊C4×C137—21no—
97C4⋊C4×C137—21no—
98C4⋊C4×C137—21no—
99C4⋊C4×C137D411yescharacteristic
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22⋊C4×C137D411yescharacteristic
107C22⋊C4×C137D411yescharacteristic
108C42×C137—21no—
109C42×C137—21no—
110C42×C137C2311yescharacteristic
111Dic274—21no—
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117Dic274—42no—
118Dic274#4112yes—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—42no—
124C2×Dic137—42no—
125#11764040—21no—
126#11764040—21no—
127#11764040—21no—
128#11764040—21no—
129C23×C137C2×D411yescharacteristic
130C23×C137C2×D411yescharacteristic
131Q8×C137—21no—
132Q8×C137—21no—
133D4×C137—22no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
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×C137C2×D411yescharacteristic
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C22×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137#13712yes—
154C22×C137C22×D411yescharacteristic
155C4×C137—21no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137C2×#4111yescharacteristic
164C4×C137H(5,𝔽2)11yescharacteristic
165Dic137—22no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169C2×C137—22no—
170C2×C137—41no—
171C2×C137#54611yescharacteristic
172C2×C137C2×#13711yescharacteristic
173C2×C137#57811yescharacteristic
174C137#331211yescharacteristic, complemented, Sylow
175#3312—1371nomaximal, Sylow
176#552D13711yescharacteristic
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×#103—1371no—
191C22×Q8—1371no—
192C22×Q8—1371no—
193C2×Q16—1371no—
194C2×Q16—1371no—
195C2×Q16—1371no—
196C2×M16—1371no—
197C22×C8—1371no—
198C4⋊Q8—2741no—
199C4⋊Q8—1371no—
200#126—21no—
201#125—21no—
202#125C2×D13711yescharacteristic
203#123—21no—
204C22⋊Q8—2741no—
205C22⋊Q8—2741no—
206C22⋊Q8—1371no—
207C22⋊Q8—1371no—
208C22⋊Q8C2×D13711yescharacteristic
209C22⋊Q8C2×D13711yescharacteristic
210C4×Q8—1371no—
211C4×Q8C2×D13711yescharacteristic
212C4×D4—2741no—
213C4×D4C2×D13711yescharacteristic
214#117—21no—
215#117C2×D13711yescharacteristic
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#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231C22⋊C8—1371no—
232C22⋊C8—1371no—
233#97—1371no—
234C2×Q8—2742no—
235C2×Q8—2742no—
236C2×Q8—2741no—
237C2×Q8—2741no—
238C2×Q8—5481no—
239C2×Q8—5481no—
240C2×Q8—1371no—
241C2×Q8—1371no—
242C2×Q8—1371no—
243C2×Q8C22×D13711yescharacteristic
244C2×D4C22⋊D13711yescharacteristic
245C22×C4—21no—
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4C22×D13711yescharacteristic
249C22×C4C22×D13711yescharacteristic
250Q16—2742no—
251Q16—5481no—
252Q16—5481no—
253M16—5481no—
254C2×C8—5481no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8—1371no—
258C2×C8—1371no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—1371no—
271C4⋊C4C22⋊D13711yescharacteristic
272C4⋊C4C22×D13711yescharacteristic
273C4⋊C4C22×D13711yescharacteristic
274C4⋊C4C22×D13711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4C22⋊D13711yescharacteristic
281C22⋊C4C22⋊D13711yescharacteristic
282C42—21no—
283C42—21no—
284C42—2741no—
285C42C22×D13711yescharacteristic
286C23C2×C22⋊D13711yescharacteristic
287C23C2×C22⋊D13711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2741no—
291Q8—2741no—
292Q8—2741no—
293Q8—2741no—
294Q8—2741no—
295Q8—5482no—
296Q8—5482no—
297Q8—1372no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—5482no—
313C2×C4—5482no—
314C2×C4C2×C22⋊D13711yescharacteristic
315C2×C4C2×C22⋊D13711yescharacteristic
316C2×C4C2×C22⋊D13711yescharacteristic
317C2×C4C2×C22⋊D13711yescharacteristic
318C2×C4C23×D13711yescharacteristic
319C8—2741no—
320C8—2741no—
321C8—2741no—
322C8—2741no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1798411812yes—
329C22C22×C22⋊D13711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2742no—
339C4—5481no—
340C4—5481no—
341C4—5481no—
342C4#41×D13711yescharacteristic
343C4#1798413411yescharacteristic
344C2—22no—
345C2—41no—
346C2#2257804111yescharacteristic
347C2C2×#1798411811yescharacteristic
348C2#2257821811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#552×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×C137D4
hypercenter#552D137
hypocenterC137#3312
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #22577773C2semidirect
3. #22577744C2semidirect
4. #22577733C2semidirect
5. #22577707C2semidirect
6. #22577659C2semidirect
7. #22577642C2semidirect
8. #22577613C2semidirect
9. #22577558C2semidirect
10. #22577558C2semidirect
11. #22577508C2semidirect
13. #22577411C2semidirect
14. C2×#17984011C2semidirect
174. C137#3312semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2742
#225777732
#225777442
#225777332
#225777072
#225776592
#225776422
#225776132
#225775582
#225775082
#225774752
#225774112
C2×#179840112
#552×C1372
#3312137