Finite Group Databasebeta

← #21434811

Subgroups — #21434811

Order 7552.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16837956C211yesmaximal, characteristic, complemented
2#16837949C211yesmaximal, characteristic
3#16837806C211yesmaximal, characteristic, complemented
4#16837773C211yesmaximal, characteristic, complemented
5#16837751C211yesmaximal, characteristic, complemented
6#16837747C211yesmaximal, characteristic, complemented
7#16837651C211yesmaximal, characteristic, complemented
8#16837649C211yesmaximal, characteristic, complemented
9#16837647C211yesmaximal, characteristic, complemented
10#16837645C211yesmaximal, characteristic, complemented
11#16837504C211yesmaximal, characteristic, complemented
12#16837502C211yesmaximal, characteristic, complemented
13#16837500C211yesmaximal, characteristic, complemented
14#16837354C211yesmaximal, characteristic, complemented
15C59×#440C211yesmaximal, characteristic, complemented
16C2×#11745685—21no—
17C2×#11745660C2211yescharacteristic
18C2×#11745670C2211yescharacteristic
19C2×#11745671C2211yescharacteristic
20C2×#11745659C2211yescharacteristic
21C2×#11745659C2211yescharacteristic
22#14273726C2211yescharacteristic
23#14273726C2211yescharacteristic
24#14273725C2211yescharacteristic
25#14273723C2211yescharacteristic
26#14273717—21no—
27#14273713—21no—
28#14273707C2211yescharacteristic
29#14273705—21no—
30#14273703C2211yescharacteristic
31#14273702C2211yescharacteristic
32#14273700—21no—
33#14273698C2211yescharacteristic
34C4×Dic118—21no—
35C4×Dic118C2211yescharacteristic
36C4×D236C2211yescharacteristic
37C4×C22⋊D59—21no—
38#14273684—21no—
39C42×D59C2211yescharacteristic
40#14273661C2211yescharacteristic
41#14273660C2211yescharacteristic
42#14273660C2211yescharacteristic
43#14273652C2211yescharacteristic
44#14273651C2211yescharacteristic
45#14273650C2211yescharacteristic
46#14273649C2211yescharacteristic
47#14273648C2211yescharacteristic
48#14273647C2211yescharacteristic
49#14273638C2211yescharacteristic
50C8×Dic59C2211yescharacteristic
51C8×Dic59C2211yescharacteristic
52C2×SD16×C59C2211yescharacteristic
53C4×Q8×C59C2211yescharacteristic
54C4×D4×C59C2211yescharacteristic
55#107×C59C2211yescharacteristic
56#103×C59C2211yescharacteristic
57#102×C59C2211yescharacteristic
58#97×C59C2211yescharacteristic
59C2×Q8×C59C2311yescharacteristic
60C2×D4×C59C2311yescharacteristic
61C22×C4×C59—21no—
62C2×C22⋊D59—21no—
63C2×C22⋊D59—21no—
64C2×C22⋊D59—21no—
65#11745685—44no—
66C2×D236—21no—
67C2×D236C2311yescharacteristic
68C2×C4×D59—21no—
69C2×C4×D59D411yescharacteristic
70C2×C4×D59D411yescharacteristic
71C2×C4×D59C2311yescharacteristic
72C2×Dic118—21no—
73C2×Dic118—21no—
74C2×Dic118C2311yescharacteristic
75SD16×C59—22no—
76C2×C8×C59C2311yescharacteristic
77C2×C8×C59C2311yescharacteristic
78C4⋊C4×C59—21no—
79C4⋊C4×C59C2311yescharacteristic
80C4⋊C4×C59C2311yescharacteristic
81C22⋊C4×C59—21no—
82C42×C59—21no—
83C42×C59C2311yescharacteristic
84C22⋊Dic59—21no—
85#11745671—22no—
86#11745670—22no—
87(C2×D59)⋊C4—21no—
88(C2×D59)⋊C4—21no—
89(C2×D59)⋊C4—21no—
90(C2×D59)⋊C4—21no—
91C4⋊Dic59—21no—
92C4⋊Dic59C2311yescharacteristic
93C4⋊Dic59C2311yescharacteristic
94Dic59⋊C4—21no—
95Dic59⋊C4—21no—
96Dic59⋊C4—21no—
97Dic59⋊C4—21no—
98Dic59⋊C4—21no—
99Dic59⋊C4—21no—
100C4×Dic59—21no—
101C4×Dic59D411yescharacteristic
102C4×Dic59D411yescharacteristic
103C4×Dic59C2311yescharacteristic
104C2×C59⋊C8C2311yescharacteristic
105C2×C59⋊C8C2311yescharacteristic
106#11745660—22no—
107#11745659—22no—
108#11745659—22no—
109C23×C59—21no—
110C22×D59—21no—
111C22×D59C2×D411yescharacteristic
112Q8×C59—21no—
113Q8×C59—21no—
114D4×C59—21no—
115D4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—42no—
120C2×C4×C59C2×D411yescharacteristic
121C2×C4×C59C2×D411yescharacteristic
122C2×C4×C59C2411yescharacteristic
123C22⋊D59—42no—
124C22⋊D59—42no—
125C22⋊D59—42no—
126C2×Dic59—21no—
127C2×Dic59—21no—
128C2×Dic59—21no—
129C2×Dic59C2×D411yescharacteristic
130C2×Dic59C2×D411yescharacteristic
131C2×Dic59C2×D411yescharacteristic
132D236—21no—
133D236—21no—
134D236—42no—
135C4×D59—21no—
136C4×D59—21no—
137C4×D59—22no—
138C4×D59—22no—
139C4×D59—42no—
140Dic118—21no—
141Dic118—21no—
142Dic118—42no—
143Dic118—42no—
144C8×C59—21no—
145C8×C59#4112yescomplemented
146C59⋊C8—21no—
147C59⋊C8#4112yescomplemented
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59C22×D411yescharacteristic
152C2×D59—21no—
153C2×D59—21no—
154C2×D59—21no—
155C2×D59—41no—
156C2×D59—41no—
157C2×D59—41no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59C2×#4111yescharacteristic
164C4×C59C2×#4111yescharacteristic
165Dic59—21no—
166Dic59—41no—
167Dic59—41no—
168Dic59—41no—
169Dic59AGL(1,ℤ/8)12yes—
170Dic59#13712yes—
171#3182—591nomaximal, Sylow
172C2×C59—81no—
173C2×C59#53211yescharacteristic
174C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C59C2×#13711yescharacteristic
176D59—41no—
177D59—81no—
178#533—591no—
179#532—591no—
180#500—591no—
181#492—591no—
182#487—591no—
183#486—591no—
184#468—591no—
185#468—591no—
186#468—591no—
187#468—591no—
188#440—591no—
189#440—591no—
190#440—591no—
191#440D5911yescharacteristic, complemented
192C4×M16—591no—
193C59#318211yescharacteristic, complemented, Sylow
194C2×#41—1181no—
195C2×SD16—591no—
196C2×SD16—591no—
197C2×SD16—591no—
198C2×SD16C2×D5911yescharacteristic
199C2×M16—591no—
200C2×M16—591no—
201C4⋊Q8—591no—
202C4⋊Q8—591no—
203C4⋊Q8—591no—
204#127—591no—
205#125—1181no—
206#124—1181no—
207C22⋊Q8—591no—
208C22⋊Q8—591no—
209C22⋊Q8—1181no—
210#121—591no—
211#121—591no—
212#121—1181no—
213C4×Q8—591no—
214C4×Q8—1181no—
215C4×Q8C2×D5911yescharacteristic
216C4×D4—591no—
217C4×D4—1181no—
218C4×D4C2×D5911yescharacteristic
219#117—1181no—
220C2×C42—591no—
221#107—591no—
222#107—591no—
223#107—591no—
224#107C2×D5911yescharacteristic
225#103—591no—
226#103—591no—
227#103—591no—
228#103C2×D5911yescharacteristic
229#102—591no—
230#102—591no—
231#102—591no—
232#102C2×D5911yescharacteristic
233#97—591no—
234#97C2×D5911yescharacteristic
235C4×C8—591no—
236C4×C8—591no—
237#41—2364no—
238C2×Q8—591no—
239C2×Q8—1181no—
240C2×Q8—1181no—
241C2×Q8C22×D5911yescharacteristic
242C2×D4—591no—
243C2×D4—1181no—
244C2×D4—1181no—
245C2×D4—1181no—
246C2×D4—1181no—
247C2×D4C22×D5911yescharacteristic
248C22×C4—21no—
249C22×C4—591no—
250C22×C4—591no—
251C22×C4—591no—
252C22×C4—1181no—
253SD16—22no—
254SD16—1182no—
255SD16—1182no—
256SD16—1182no—
257M16—1182no—
258M16—1182no—
259C2×C8—591no—
260C2×C8—591no—
261C2×C8C22×D5911yescharacteristic
262C2×C8C22×D5911yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—591no—
265C4⋊C4—591no—
266C4⋊C4—1181no—
267C4⋊C4—1181no—
268C4⋊C4—1181no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—1181no—
273C4⋊C4C22×D5911yescharacteristic
274C4⋊C4C22×D5911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—1181no—
277C22⋊C4—1181no—
278C22⋊C4—1181no—
279C22⋊C4—1181no—
280C22⋊C4—1181no—
281C42—21no—
282C42—591no—
283C42—591no—
284C42—591no—
285C42—1181no—
286C42C22×D5911yescharacteristic
287C23—21no—
288C23—591no—
289C23—1181no—
290Q8—21no—
291Q8—21no—
292Q8—1181no—
293Q8—1181no—
294Q8—2362no—
295Q8—2362no—
296D4—21no—
297D4—21no—
298D4—1181no—
299D4—1181no—
300D4—2362no—
301D4—2362no—
302D4—2362no—
303D4—2362no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—591no—
309C2×C4—591no—
310C2×C4—591no—
311C2×C4—1182no—
312C2×C4—1182no—
313C2×C4—1181no—
314C2×C4—1181no—
315C2×C4—1181no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—2362no—
319C2×C4D4×D5911yescharacteristic
320C2×C4D4×D5911yescharacteristic
321C2×C4C23×D5911yescharacteristic
322C8—21no—
323C8—592no—
324C8—1181no—
325C8#1174568512yescomplemented
326C22—41no—
327C22—41no—
328C22—41no—
329C22—1181no—
330C22—1181no—
331C22—1181no—
332C22—2361no—
333C22—2361no—
334C22—2361no—
335C22C2×D4×D5911yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—592no—
342C4—592no—
343C4—1181no—
344C4—2361no—
345C4—2361no—
346C4—2361no—
347C4#41×D5911yescharacteristic
348C4C2×#1174568511yescharacteristic
349C2—81no—
350C2—4721no—
351C2—2361no—
352C2#1683794911yescharacteristic
353C2AGL(1,ℤ/8)×D5911yescharacteristic
354C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#440C2
socleC22×C59C22×D4
hypercenter#440D59
hypocenterC59#3182
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #16837956C2semidirect
3. #16837806C2semidirect
4. #16837773C2semidirect
5. #16837751C2semidirect
6. #16837747C2semidirect
7. #16837651C2semidirect
8. #16837649C2semidirect
9. #16837647C2semidirect
10. #16837645C2semidirect
11. #16837504C2semidirect
12. #16837502C2semidirect
13. #16837500C2semidirect
14. #16837354C2semidirect
15. C59×#440C2semidirect
145. C8×C59#41semidirect
147. C59⋊C8#41semidirect
191. #440D59semidirect
193. C59#3182semidirect
325. C8#11745685semidirect