Finite Group Databasebeta

← #31793379

Subgroups — #31793379

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578117C211yesmaximal, characteristic, complemented
2#22578053C211yesmaximal, characteristic, complemented
3#22577800C211yesmaximal, characteristic, complemented
4#22577725C211yesmaximal, characteristic, complemented
5#22577724C211yesmaximal, characteristic, complemented
6#22577720C211yesmaximal, characteristic, complemented
7#22577719C211yesmaximal, characteristic, complemented
8#22577691C211yesmaximal, characteristic, complemented
9#22577681C211yesmaximal, characteristic, complemented
10#22577653C211yesmaximal, characteristic, complemented
11D8×Dic137C211yesmaximal, characteristic, complemented
12#22577479C211yesmaximal, characteristic, complemented
13#22577445C211yesmaximal, characteristic, complemented
14#22577442C211yesmaximal, characteristic, complemented
15#469×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×#14555493C2211yescharacteristic
18#17984087C2211yescharacteristic
19#17984087C2211yescharacteristic
20#17984075—21no—
21#17984074—21no—
22#17984070—21no—
23#17984069—21no—
24#17984065—21no—
25#17984060—21no—
26#17984059C2211yescharacteristic
27#17984059C2211yescharacteristic
28D4×Dic137C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30C4×C22⋊D137C2211yescharacteristic
31#17984047C2211yescharacteristic
32#17984046C2211yescharacteristic
33#17984044—21no—
34C2×C4⋊Dic137C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36#17984022C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984020C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984017C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×D8×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic
54#121×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65C22×Dic137—21no—
66C22×Dic137—21no—
67C22×Dic137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2311yescharacteristic
70#14555493—22no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489D411yescharacteristic
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137D411yescharacteristic
88#14555487—21no—
89#14555487D411yescharacteristic
90C4×Dic137—21no—
91C4×Dic137—21no—
92C4×Dic137C2311yescharacteristic
93C2×D4×C137—21no—
94C2×D4×C137—21no—
95C2×D4×C137C2311yescharacteristic
96C2×D4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98D8×C137—22no—
99M16×C137—22no—
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22×D137C2×D411yescharacteristic
107Dic274—42no—
108Dic274—81no—
109C22⋊D137—22no—
110C22⋊D137—42no—
111C22⋊D137—81no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—42no—
117C2×Dic137—42no—
118C2×Dic137—42no—
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C4×D137—21no—
123C4×D137—21no—
124C4×D137—42no—
125#11764040—21no—
126#11764040—21no—
127C23×C137—21no—
128C23×C137—21no—
129C23×C137C2×D411yescharacteristic
130D4×C137—21no—
131D4×C137—21no—
132D4×C137—21no—
133D4×C137—21no—
134D4×C137—42no—
135D4×C137—81no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2411yescharacteristic
142C8×C137—21no—
143C8×C137#4112yescomplemented
144C22×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—41no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137C22×D411yescharacteristic
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137C2×#4111yescharacteristic
161C4×C1372−1+411yescharacteristic
162Dic137—21no—
163Dic137—21no—
164Dic137—41no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168Dic137AGL(1,ℤ/8)12yes—
169C2×C137—41no—
170C2×C137—81no—
171C2×C137—81no—
172C2×C137#54711yescharacteristic
173C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C137#57711yescharacteristic
175D137—41no—
176C137#335411yescharacteristic, complemented, Sylow
177#3354—1371nomaximal, Sylow
178#555—1371no—
179#547—1371no—
180#498—1371no—
181#482—1371no—
182#482—1371no—
183#481—1371no—
184#481—1371no—
185#476—1371no—
186#474—1371no—
187#469—1371no—
188#469D13711yescharacteristic, complemented
189C4×D8—1371no—
190#436—1371no—
191#428—1371no—
192#428—1371no—
193C2×#41—2741no—
194C2×D8C2×D13711yescharacteristic
195C2×M16—1371no—
196C2×M16C2×D13711yescharacteristic
197C4⋊Q8—1371no—
198C4⋊Q8—1371no—
199#124—2741no—
200#124—2741no—
201#123—2741no—
202#123—2741no—
203C22⋊Q8—2741no—
204#121—2741no—
205#121—1371no—
206#121—1371no—
207#121C2×D13711yescharacteristic
208#121C2×D13711yescharacteristic
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212#117—2741no—
213#117—1371no—
214#117—1371no—
215C2×C4⋊C4—1371no—
216C2×C4⋊C4—1371no—
217#107—1371no—
218#107—1371no—
219#107—1371no—
220#106—1371no—
221#106—1371no—
222#106—1371no—
223#106—1371no—
224#106—1371no—
225#106C2×D13711yescharacteristic
226C4⋊C8—1371no—
227#102—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231#102C2×D13711yescharacteristic
232#102C2×D13711yescharacteristic
233C22⋊C8—1371no—
234C22⋊C8—1371no—
235C4×C8—1371no—
236#41—5484no—
237C2×Q8—2741no—
238C2×Q8—2741no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—2741no—
242C2×D4—2741no—
243C2×D4—1371no—
244C2×D4C22×D13711yescharacteristic
245C2×D4C22×D13711yescharacteristic
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—1371no—
251C22×C4C22×D13711yescharacteristic
252D8—22no—
253M16—22no—
254M16—2742no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8C22×D13711yescharacteristic
258C2×C8C22×D13711yescharacteristic
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C4⋊C4—1371no—
268C4⋊C4C22×D13711yescharacteristic
269C4⋊C4C22×D13711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—1371no—
279C22⋊C4—1371no—
280C42—2741no—
281C42—2741no—
282C42—1371no—
283C23—21no—
284C23—21no—
285C23—1371no—
286C23D4×D13711yescharacteristic
287Q8—5482no—
288Q8—10961no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—2742no—
296D4—5482no—
297D4—10961no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—5482no—
309C2×C4—5482no—
310C2×C4—5482no—
311C2×C4—5482no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1371no—
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—2741no—
319C8—2741no—
320C8#1455550912yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—2741no—
331C22—2741no—
332C22—2741no—
333C22C2×D4×D13711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—2741no—
338C4—2741no—
339C4—5481no—
340C4—5481no—
341C4—5481no—
342C4—5481no—
343C4—1372no—
344C4C2×#1455550911yescharacteristic
345C4#1798413511yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—5481no—
350C2#2257805311yescharacteristic
351C2AGL(1,ℤ/8)×D13711yescharacteristic
352C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3354
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22578117C2semidirect
2. #22578053C2semidirect
3. #22577800C2semidirect
4. #22577725C2semidirect
5. #22577724C2semidirect
6. #22577720C2semidirect
7. #22577719C2semidirect
8. #22577691C2semidirect
9. #22577681C2semidirect
10. #22577653C2semidirect
11. D8×Dic137C2semidirect
12. #22577479C2semidirect
13. #22577445C2semidirect
14. #22577442C2semidirect
15. #469×C137C2semidirect
143. C8×C137#41semidirect
176. C137#3354semidirect
188. #469D137semidirect
320. C8#14555509semidirect

Maximal subgroups

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

subgroupindex
#225781172
#225780532
#225778002
#225777252
#225777242
#225777202
#225777192
#225776912
#225776812
#225776532
D8×Dic1372
#225774792
#225774452
#225774422
#469×C1372
#3354137