Finite Group Databasebeta

← #31793390

Subgroups — #31793390

Order 17536.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578105C211yesmaximal, characteristic, complemented
2#22578100C211yesmaximal, characteristic, complemented
3#22577803C211yesmaximal, characteristic, complemented
4#22577726C211yesmaximal, characteristic, complemented
5#22577724C211yesmaximal, characteristic, complemented
6#22577715C211yesmaximal, characteristic, complemented
7#22577714C211yesmaximal, characteristic, complemented
8#22577705C211yesmaximal, characteristic, complemented
9#22577701C211yesmaximal, 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×#14555493C2211yescharacteristic
17#17984081—21no—
18#17984081—21no—
19#17984079C2211yescharacteristic
20#17984079C2211yescharacteristic
21#17984072—21no—
22#17984069—21no—
23#17984060—21no—
24#17984059—21no—
25#17984059C2211yescharacteristic
26#17984059C2211yescharacteristic
27D4×Dic137—21no—
28D4×Dic137C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30C4×C22⋊D137C2211yescharacteristic
31#17984048—21no—
32#17984047C2211yescharacteristic
33#17984046C2211yescharacteristic
34C2×C4⋊Dic137C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36#17984022C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984020C2211yescharacteristic
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
59C2×C22⋊D137—21no—
60C2×C22⋊D137—21no—
61C2×C22⋊D137D411yescharacteristic
62C22×Dic137—21no—
63C22×Dic137—21no—
64C22×Dic137—21no—
65C2×C4×D137—21no—
66C2×C4×D137C2311yescharacteristic
67#14555493—22no—
68C2×#11764040C2311yescharacteristic
69C2×#11764040C2311yescharacteristic
70C4⋊Dic137—21no—
71C4⋊Dic137—21no—
72C4⋊Dic137—21no—
73C4⋊Dic137C2311yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489D411yescharacteristic
81C22⋊Dic137—21no—
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137D411yescharacteristic
87#14555487—21no—
88#14555487D411yescharacteristic
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C2×D4×C137—21no—
93C2×D4×C137—21no—
94C2×D4×C137C2311yescharacteristic
95C2×D4×C137C2311yescharacteristic
96C22×C4×C137C2311yescharacteristic
97D8×C137—22no—
98M16×C137—22no—
99C2×C8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22×D137C2×D411yescharacteristic
106C22⋊D137—41no—
107C22⋊D137—42no—
108C22⋊D137—81no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—81no—
114C2×Dic137—81no—
115C2×Dic137—81no—
116C2×Dic137C2×D411yescharacteristic
117C2×Dic137C2×D411yescharacteristic
118C2×Dic137C2×D411yescharacteristic
119C4×D137—21no—
120C4×D137—21no—
121C4×D137—81no—
122#11764040—21no—
123#11764040—21no—
124C23×C137—21no—
125C23×C137—21no—
126C23×C137C2×D411yescharacteristic
127D4×C137—21no—
128D4×C137—21no—
129D4×C137—21no—
130D4×C137—21no—
131D4×C137—42no—
132D4×C137—81no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2411yescharacteristic
139C8×C137—21no—
140C8×C137#4112yes—
141C22×C137—21no—
142C22×C137—21no—
143C22×C137—21no—
144C22×C137—41no—
145C22×C137—41no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137C22×D411yescharacteristic
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—21no—
154C4×C137—21no—
155C4×C137—41no—
156C4×C137—41no—
157C4×C137C2×#4111yescharacteristic
158C4×C1372−1+411yescharacteristic
159Dic137—21no—
160Dic137—21no—
161Dic137—21no—
162Dic137—41no—
163Dic137—41no—
164Dic137—41no—
165Dic137—41no—
166C2×C137—41no—
167C2×C137—81no—
168C2×C137—81no—
169C2×C137#54711yescharacteristic
170C2×C137#57511yescharacteristic
171C2×C137#57611yescharacteristic
172D137—41no—
173C137#335511yescharacteristic, complemented, Sylow
174#3355—1371nomaximal, Sylow
175#553—1371no—
176#553—1371no—
177#499—1371no—
178#482—1371no—
179#482—1371no—
180#480—1371no—
181#480—1371no—
182#478—1371no—
183#478—1371no—
184#469—1371no—
185#469D13711yescharacteristic, complemented
186C4×D8—1371no—
187#436—1371no—
188#428—1371no—
189#428—1371no—
190C2×D8C2×D13711yescharacteristic
191C2×M16—1371no—
192C2×M16C2×D13711yescharacteristic
193#126—2741no—
194#126—2741no—
195#125—1371no—
196#125—1371no—
197#123—2741no—
198#123—2741no—
199#121—2741no—
200#121—2741no—
201#121—1371no—
202#121—1371no—
203#121C2×D13711yescharacteristic
204#121C2×D13711yescharacteristic
205C4×D4—2741no—
206C4×D4—2741no—
207C4×D4—1371no—
208C4×D4—1371no—
209C4×D4—1371no—
210#117—1371no—
211#117—1371no—
212C2×C4⋊C4—1371no—
213C2×C4⋊C4—1371no—
214#107—1371no—
215#107—1371no—
216#107—1371no—
217#107—1371no—
218#107—1371no—
219#106—1371no—
220#106—1371no—
221#106—1371no—
222#106C2×D13711yescharacteristic
223C4⋊C8—1371no—
224#102—1371no—
225#102—1371no—
226#102—1371no—
227#102—1371no—
228#102C2×D13711yescharacteristic
229#102C2×D13711yescharacteristic
230C22⋊C8—1371no—
231C22⋊C8—1371no—
232C4×C8—1371no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—2741no—
236C2×D4—2741no—
237C2×D4—1371no—
238C2×D4C22×D13711yescharacteristic
239C2×D4C22×D13711yescharacteristic
240C22×C4—2741no—
241C22×C4—2741no—
242C22×C4—2741no—
243C22×C4—2741no—
244C22×C4—1371no—
245C22×C4C22×D13711yescharacteristic
246D8—22no—
247M16—22no—
248M16—2742no—
249C2×C8—1371no—
250C2×C8—1371no—
251C2×C8C22×D13711yescharacteristic
252C2×C8C22×D13711yescharacteristic
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—2741no—
256C4⋊C4—2741no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—1371no—
260C4⋊C4—1371no—
261C4⋊C4—1371no—
262C4⋊C4—1371no—
263C4⋊C4—1371no—
264C4⋊C4C22×D13711yescharacteristic
265C4⋊C4C22×D13711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—2741no—
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—1371no—
275C22⋊C4—1371no—
276C42—2741no—
277C42—2741no—
278C42—1371no—
279C23—21no—
280C23—21no—
281C23—1371no—
282C23D4×D13711yescharacteristic
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—81no—
289D4—5481no—
290D4—5482no—
291D4—10961no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—10961no—
303C2×C4—10961no—
304C2×C4—10961no—
305C2×C4—10961no—
306C2×C4—1371no—
307C2×C4—1371no—
308C2×C4—1371no—
309C2×C4D4×D13711yescharacteristic
310C2×C4C23×D13711yescharacteristic
311C8—21no—
312C8—2741no—
313C8—2741no—
314C8#1455550912yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—2741no—
325C22—2741no—
326C22—2741no—
327C22C2×D4×D13711yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—2741no—
332C4—2741no—
333C4—2741no—
334C4—5481no—
335C4—5481no—
336C4—5481no—
337C4—5481no—
338C4C2×#1455550911yescharacteristic
339C4#1798413511yescharacteristic
340C2—41no—
341C2—81no—
342C2—81no—
343C2—5481no—
344C2#2257805311yescharacteristic
345C2#2257819911yescharacteristic
346C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3355
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. #22578105C2semidirect
2. #22578100C2semidirect
3. #22577803C2semidirect
4. #22577726C2semidirect
5. #22577724C2semidirect
6. #22577715C2semidirect
7. #22577714C2semidirect
8. #22577705C2semidirect
9. #22577701C2semidirect
10. #22577653C2semidirect
11. D8×Dic137C2semidirect
12. #22577479C2semidirect
13. #22577445C2semidirect
14. #22577442C2semidirect
15. #469×C137C2semidirect
173. C137#3355semidirect
185. #469D137semidirect

Maximal subgroups

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

subgroupindex
#225781052
#225781002
#225778032
#225777262
#225777242
#225777152
#225777142
#225777052
#225777012
#225776532
D8×Dic1372
#225774792
#225774452
#225774422
#469×C1372
#3355137