Finite Group Databasebeta

← #31790989

Subgroups — #31790989

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578006C211yesmaximal, characteristic, complemented
2#22577886C211yesmaximal, characteristic, complemented
3#22577722C211yesmaximal, characteristic, complemented
4#22577714C211yesmaximal, characteristic, complemented
5#22577708C211yesmaximal, characteristic, complemented
6#22577705C211yesmaximal, characteristic, complemented
7#22577601C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577515C211yesmaximal, characteristic, complemented
11#22577506C211yesmaximal, characteristic, complemented
12SD16×Dic137C211yesmaximal, characteristic, complemented
13#22577483C211yesmaximal, characteristic, complemented
14#22577461C211yesmaximal, characteristic, complemented
15#449×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18#17984084C2211yescharacteristic
19#17984079C2211yescharacteristic
20#17984072—21no—
21#17984070—21no—
22#17984062C2211yescharacteristic
23#17984060—21no—
24#17984059—21no—
25Q8×Dic137C2211yescharacteristic
26Q8×Dic137C2211yescharacteristic
27D4×Dic137—21no—
28D4×Dic137—21no—
29D4×Dic137C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32#17984050C2211yescharacteristic
33#17984047—21no—
34#17984047C2211yescharacteristic
35C2×C4⋊Dic137—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37#17984021C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984020C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984012C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17983998C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×D548C2311yescharacteristic
60C2×C22⋊D137—21no—
61C2×C22⋊D137—21no—
62C2×C22⋊D137—21no—
63C22×Dic137—21no—
64C22×Dic137—21no—
65C22×Dic137—21no—
66C22×Dic137—21no—
67C2×C4×D137—21no—
68#14555497—22no—
69#14555495—22no—
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C4⋊Dic137—21no—
73C4⋊Dic137—21no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137D411yescharacteristic
77C4⋊Dic137D411yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137D411yescharacteristic
86C22⋊Dic137D411yescharacteristic
87#14555487—21no—
88#14555487—21no—
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C4×Dic137C2311yescharacteristic
93#41×C137—22no—
94#41×C137—22no—
95C2×Q8×C137C2311yescharacteristic
96C2×D4×C137—21no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137—21no—
101C22×C4×C137C2311yescharacteristic
102SD16×C137—22no—
103D8×C137—22no—
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108C22×D137—21no—
109D548—21no—
110D548—21no—
111C22⋊D137—81no—
112C22⋊D137—81no—
113C22⋊D137—81no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—81no—
118C2×Dic137—81no—
119C2×Dic137—81no—
120C2×Dic137—81no—
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—81no—
126#11764040—21no—
127#11764040—21no—
128C23×C137—21no—
129C23×C137—21no—
130C23×C137C2×D411yescharacteristic
131Q8×C137—21no—
132Q8×C137#4112yes—
133D4×C137—24no—
134D4×C137—21no—
135D4×C137—21no—
136D4×C137—21no—
137D4×C137—81no—
138D4×C137#4112yes—
139C2×C4×C137—24no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2411yescharacteristic
146C8×C137—21no—
147C8×C137—21no—
148C22×C137—21no—
149C22×C137—22no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137C22×D411yescharacteristic
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—41no—
161C4×C137—21no—
162C4×C137—22no—
163C4×C137—41no—
164C4×C137C2×#4111yescharacteristic
165C4×C137C2×#4111yescharacteristic
166Dic137—21no—
167Dic137—21no—
168Dic137—21no—
169Dic137—21no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173C2×C137—41no—
174C2×C137—42no—
175C2×C137—81no—
176C2×C137#52511yescharacteristic
177C2×C137#57611yescharacteristic
178C2×C137#57711yescharacteristic
179D137—81no—
180C137#311911yescharacteristic, complemented, Sylow
181#3119—1371nomaximal, Sylow
182#540—1371no—
183#520—1371no—
184#481—1371no—
185#480—1371no—
186#479—1371no—
187#478—1371no—
188#460—1371no—
189#459—1371no—
190#449—1371no—
191#449D13711yescharacteristic, complemented
192#442—1371no—
193#440—1371no—
194C4×SD16—1371no—
195C4×D8—1371no—
196#433—1371no—
197C2×#41C2×D13711yescharacteristic
198C2×SD16—1371no—
199C2×SD16C2×D13711yescharacteristic
200C2×D8—1371no—
201C2×D8C2×D13711yescharacteristic
202#127—1371no—
203#125—1371no—
204#123—2741no—
205#123—2741no—
206#121—2741no—
207#121—2741no—
208#121—1371no—
209#121C2×D13711yescharacteristic
210C4×Q8—1371no—
211C4×Q8—1371no—
212C4×D4—2741no—
213C4×D4—2741no—
214C4×D4—1371no—
215C4×D4—1371no—
216C4×D4—1371no—
217C4×D4—1371no—
218#117—2741no—
219#117—1371no—
220C2×C4⋊C4—2741no—
221C2×C4⋊C4—1371no—
222#107—1371no—
223#107—1371no—
224#106—1371no—
225#106—1371no—
226C4⋊C8—1371no—
227C4⋊C8—1371no—
228#103—1371no—
229#103C2×D13711yescharacteristic
230#102—1371no—
231#102—1371no—
232#102—1371no—
233#102—1371no—
234#102—1371no—
235#102C2×D13711yescharacteristic
236C22⋊C8—1371no—
237C22⋊C8C2×D13711yescharacteristic
238#97—1371no—
239C4×C8—1371no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D13711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—2741no—
246C2×D4—2741no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4C22×D13711yescharacteristic
250C2×D4C22×D13711yescharacteristic
251C22×C4—21no—
252C22×C4—2741no—
253C22×C4—2741no—
254C22×C4—2741no—
255C22×C4—2741no—
256C22×C4—2741no—
257C22×C4C22×D13711yescharacteristic
258SD16—22no—
259SD16—2742no—
260D8—22no—
261D8—2742no—
262C2×C8—1371no—
263C2×C8—1371no—
264C2×C8C22×D13711yescharacteristic
265C2×C8C22×D13711yescharacteristic
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—1371no—
284C22⋊C4—1371no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C42—1371no—
289C23—21no—
290C23—21no—
291C23—2741no—
292C23D4×D13711yescharacteristic
293Q8—21no—
294Q8#1455550912yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—2741no—
301D4—2741no—
302D4—10961no—
303D4—10961no—
304D4—10961no—
305D4#1455550912yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—10961no—
315C2×C4—10961no—
316C2×C4—10961no—
317C2×C4—10961no—
318C2×C4—10961no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4—1371no—
323C2×C4D4×D13711yescharacteristic
324C2×C4C23×D13711yescharacteristic
325C8—21no—
326C8—21no—
327C8—2741no—
328C8—2741no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—5481no—
339C22—5481no—
340C22—5481no—
341C22C2×D4×D13711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—2741no—
346C4—2741no—
347C4—2741no—
348C4—2741no—
349C4—5481no—
350C4—5481no—
351C4—5481no—
352C4C2×#1455550911yescharacteristic
353C4C2×#1455550911yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—10961no—
358C2#2257790911yescharacteristic
359C2#2257820711yescharacteristic
360C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#449×C137C2
socleC22×C137C22×D4
hypercenter#449D137
hypocenterC137#3119
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22578006C2semidirect
2. #22577886C2semidirect
3. #22577722C2semidirect
4. #22577714C2semidirect
5. #22577708C2semidirect
6. #22577705C2semidirect
7. #22577601C2semidirect
8. #22577596C2semidirect
9. #22577548C2semidirect
10. #22577515C2semidirect
11. #22577506C2semidirect
12. SD16×Dic137C2semidirect
13. #22577483C2semidirect
14. #22577461C2semidirect
15. #449×C137C2semidirect
180. C137#3119semidirect
191. #449D137semidirect

Maximal subgroups

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

subgroupindex
#225780062
#225778862
#225777222
#225777142
#225777082
#225777052
#225776012
#225775962
#225775482
#225775152
#225775062
SD16×Dic1372
#225774832
#225774612
#449×C1372
#3119137