Finite Group Databasebeta

← #31792647

Subgroups — #31792647

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D137C211yesmaximal, characteristic, complemented
2#22578053C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577732C211yesmaximal, characteristic, complemented
6#22577686C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577629C211yesmaximal, characteristic, complemented
9#22577606C211yesmaximal, characteristic, complemented
10#22577540C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13C2×#17984012C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#448×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×Q8×D137C2211yescharacteristic
18C2×#14555498C2211yescharacteristic
19C22×#11764040C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic
21#17984087C2211yescharacteristic
22#17984086—21no—
23#17984086C2211yescharacteristic
24#17984070—21no—
25#17984069—21no—
26#17984067—21no—
27#17984067C2211yescharacteristic
28#17984065—21no—
29#17984064—21no—
30#17984064C2211yescharacteristic
31#17984059C2211yescharacteristic
32Q8×Dic137C2211yescharacteristic
33D4×Dic137C2211yescharacteristic
34C4×C22⋊D137—21no—
35C4×C22⋊D137C2211yescharacteristic
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984020C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984012C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984009C2211yescharacteristic
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#14555509—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×Dic274C2311yescharacteristic
63Q8×D137—22no—
64Q8×D137—41no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137D411yescharacteristic
67C22×Dic137—21no—
68C22×Dic137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137C2311yescharacteristic
72#14555498—41no—
73C2×#11764040—41no—
74C2×#11764040C2311yescharacteristic
75C2×#11764040C2311yescharacteristic
76C8×D137—41no—
77C4⋊Dic137—21no—
78C4⋊Dic137—21no—
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137D411yescharacteristic
93#14555487—21no—
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×Q8×C137—21no—
98C2×Q8×C137—22no—
99C2×Q8×C137—41no—
100C2×Q8×C137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×D4×C137—21no—
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137—21no—
105C22×C4×C137C2311yescharacteristic
106SD16×C137—22no—
107SD16×C137—41no—
108C2×C8×C137C2311yescharacteristic
109C2×C8×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115Dic274—42no—
116Dic274—42no—
117C22⋊D137—42no—
118C22⋊D137SD1614yescomplemented
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—42no—
124C2×Dic137—42no—
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C4×D137—21no—
129C4×D137—21no—
130C4×D137—42no—
131C4×D137—42no—
132#11764040—21no—
133#11764040—21no—
134C23×C137—21no—
135C23×C137C2×D411yescharacteristic
136Q8×C137—21no—
137Q8×C137—21no—
138Q8×C137—21no—
139Q8×C137—42no—
140Q8×C137#4112yescomplemented
141D4×C137—21no—
142D4×C137—21no—
143D4×C137—42no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—42no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2411yescharacteristic
152C8×C137—21no—
153C8×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137C2×SD1612yes—
160C22×C137C22×D411yescharacteristic
161C2×D137—21no—
162C2×D137—21no—
163C2×D137C2×SD1612yes—
164C4×C137—21no—
165C4×C137—22no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137C2×#4111yescharacteristic
169C4×C137H(5,𝔽2)11yescharacteristic
170Dic137—21no—
171Dic137—21no—
172Dic137—41no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137C2×SD1612yes—
177C2×C137—22no—
178C2×C137—81no—
179C2×C137#54611yescharacteristic
180C2×C137C22×SD1611yescharacteristic
181C2×C137#57811yescharacteristic
182D137—22no—
183C137#329811yescharacteristic, complemented, Sylow
184#3298—1371nomaximal, Sylow
185D4×Q8—1371no—
186#547—1371no—
187#487—1371no—
188#483—1371no—
189#483—1371no—
190#475—1371no—
191#465—1371no—
192#465—1371no—
193#461—1371no—
194#448—1371no—
195#448D13711yescharacteristic, complemented
196C4×SD16—1371no—
197C8×D4—1371no—
198C2×#103—1371no—
199C2×#103—1371no—
200C2×#41—2741no—
201C22×Q8—1371no—
202C22×Q8C2×D13711yescharacteristic
203C2×SD16—1371no—
204C2×SD16C2×D13711yescharacteristic
205C2×SD16C2×D13711yescharacteristic
206C22×C8—1371no—
207C22×C8—1371no—
208C4⋊Q8—2741no—
209C4⋊Q8—1371no—
210C4⋊Q8—1371no—
211#123—2741no—
212#123—2741no—
213C22⋊Q8—2741no—
214C22⋊Q8—2741no—
215C22⋊Q8—2741no—
216C22⋊Q8—1371no—
217C22⋊Q8—1371no—
218#121—1371no—
219#121C2×D13711yescharacteristic
220C4×Q8—1371no—
221C4×D4—2741no—
222C4×D4—1371no—
223C4×D4—1371no—
224C2×C4⋊C4—1371no—
225C2×C4⋊C4—1371no—
226#106—1371no—
227#106—1371no—
228#106—1371no—
229C4⋊C8—1371no—
230#103—1371no—
231#103—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#103—1371no—
237#103C2×D13711yescharacteristic
238#103C2×D13711yescharacteristic
239#102—1371no—
240C22⋊C8—1371no—
241C22⋊C8C2×D13711yescharacteristic
242C4×C8—1371no—
243#41—5484no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—2741no—
248C2×Q8—2741no—
249C2×Q8—2742no—
250C2×Q8—5481no—
251C2×Q8—1371no—
252C2×Q8C22×D13711yescharacteristic
253C2×Q8C22×D13711yescharacteristic
254C2×D4—21no—
255C2×D4—2741no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—2741no—
263C22×C4—1371no—
264C22×C4C22×D13711yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—5481no—
268C2×C8—5481no—
269C2×C8—5481no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8C22×D13711yescharacteristic
273C2×C8C22×D13711yescharacteristic
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—2741no—
280C4⋊C4—2741no—
281C4⋊C4—2741no—
282C4⋊C4—1371no—
283C4⋊C4—1371no—
284C4⋊C4—1371no—
285C4⋊C4—1371no—
286C4⋊C4C22×D13711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—2741no—
295C42—1371no—
296C23—21no—
297C23—1371no—
298C23D4×D13711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—2741no—
304Q8—2741no—
305Q8—5482no—
306Q8—5482no—
307Q8#1455550912yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—5482no—
312D4—1374no—
313C2×C4—21no—
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—5482no—
327C2×C4—5482no—
328C2×C4—5482no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—2741no—
344C22—2741no—
345C22—1372no—
346C22SD16×D13712yes—
347C22C2×D4×D13711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—2741no—
353C4—2741no—
354C4—5481no—
355C4—5481no—
356C4—5481no—
357C4—5481no—
358C4—1372no—
359C4C2×#1455550911yescharacteristic
360C4#1798413311yescharacteristic
361C2—22no—
362C2—81no—
363C2—2742no—
364C2#2257804011yescharacteristic
365C2C2×SD16×D13711yescharacteristic
366C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#448×C137C2
socleC22×C137C22×D4
prime radical#22578053C2
hypercenter#448D137
hypocenterC137#3298
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. Q8×C22⋊D137C2semidirect
3. #22577752C2semidirect
4. #22577734C2semidirect
5. #22577732C2semidirect
6. #22577686C2semidirect
7. #22577630C2semidirect
8. #22577629C2semidirect
9. #22577606C2semidirect
10. #22577540C2semidirect
11. SD16×Dic137C2semidirect
12. #22577466C2semidirect
13. C2×#17984012C2semidirect
14. #103×D137C2semidirect
15. #448×C137C2semidirect
118. C22⋊D137SD16semidirect
140. Q8×C137#41semidirect
183. C137#3298semidirect
195. #448D137semidirect
307. Q8#14555509semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1372
#225780532
#225777522
#225777342
#225777322
#225776862
#225776302
#225776292
#225776062
#225775402
SD16×Dic1372
#225774662
C2×#179840122
#103×D1372
#448×C1372
#3298137