Finite Group Databasebeta

← #21435917

Subgroups — #21435917

Order 7552.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16838113C211yesmaximal, characteristic, complemented
2D4×D236C211yesmaximal, characteristic
3#16837750C211yesmaximal, characteristic, complemented
4#16837722C212yesmaximal, complemented
5#16837632C212yesmaximal, complemented
6#16837603C211yesmaximal, characteristic, complemented
7#16837554C212yesmaximal, complemented
8C4×#11745670C211yesmaximal, characteristic, complemented
9#16837467C211yesmaximal, characteristic, complemented
10#102×D59C212yesmaximal, complemented
11C59×#476C211yesmaximal, characteristic, complemented
12C22×D236—21no—
13C2×D4×D59C2212yes—
14C2×#11745660C2212yescomplemented
15C2×#11745670C2211yescharacteristic
16C2×C8×D59C2212yescomplemented
17#14273724C2211yescharacteristic
18#14273718—21no—
19#14273707C2212yescomplemented
20#14273703—22no—
21#14273702C2212yes—
22#14273701—21no—
23#14273697—22no—
24C4×Dic118C2211yescharacteristic, complemented
25C4×D236C2211yescharacteristic
26#14273689—21no—
27C4⋊C4×D59C2212yescomplemented
28#14273658C2211yescharacteristic, complemented
29#14273650C2211yescharacteristic, complemented
30#14273649C2212yescomplemented
31#14273648C2211yescharacteristic
32#14273647C2212yescomplemented
33#14273647C2212yescomplemented
34(C2×D59)⋊C8C2212yescomplemented
35C4×C59⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C59C2211yescharacteristic, complemented
37C4×D4×C59C2211yescharacteristic
38#106×C59C2212yescomplemented
39C4⋊C8×C59C2211yescharacteristic, complemented
40#102×C59C2212yes—
41C23×D59—22no—
42C2×Q8×C59—21no—
43C2×D4×C59C2311yescharacteristic
44C22×C4×C59—21no—
45C2×C22⋊D59—22no—
46D4×D59—24no—
47D4×D59—42no—
48C2×D236—22no—
49C2×D236—21no—
50C2×D236—21no—
51C2×D236—44no—
52C2×D236D411yescharacteristic
53C2×D236C2312yes—
54C2×C4×D59—22no—
55C2×C4×D59C2312yes—
56C2×Dic118C2311yescharacteristic
57C2×C8×C59C2312yes—
58C4⋊C4×C59—21no—
59C4⋊C4×C59C2312yes—
60C4⋊C4×C59C2311yescharacteristic
61C22⋊C4×C59—21no—
62C42×C59C2311yescharacteristic
63#11745670—22no—
64(C2×D59)⋊C4—22no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4D412yes—
67C4⋊Dic59—22no—
68C4⋊Dic59D411yescharacteristic
69C4⋊Dic59C2311yescharacteristic
70Dic59⋊C4—21no—
71Dic59⋊C4—22no—
72C4×Dic59—21no—
73C2×C59⋊C8C2311yescharacteristic
74C2×C59⋊C8C2311yescharacteristic
75#11745660—42no—
76C8×D59—42no—
77C23×C59—21no—
78C22×D59—22no—
79C22×D59—44no—
80C22×D59—44no—
81C22×D59—82no—
82C22×D59C2×D412yes—
83Q8×C59—81no—
84D4×C59—21no—
85D4×C59#4112yes—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—42no—
89C2×C4×C59C2×D411yescharacteristic
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2411yescharacteristic
92C22⋊D59—44no—
93C2×Dic59—21no—
94C2×Dic59—21no—
95C2×Dic59C2×D412yes—
96D236—22no—
97D236—22no—
98D236—44no—
99D236—42no—
100D236—81no—
101D236SD1614yescomplemented
102C4×D59—22no—
103C4×D59—22no—
104C4×D59—82no—
105Dic118—21no—
106Dic118—21no—
107C8×C59—22no—
108C59⋊C8—21no—
109C59⋊C8—21no—
110C22×C59—22no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59C22×D411yescharacteristic
114C2×D59—22no—
115C2×D59—22no—
116C2×D59—42no—
117C2×D59—42no—
118C2×D59—42no—
119C2×D59—88no—
120C2×D59C2×SD1614yes—
121C4×C59—21no—
122C4×C59—41no—
123C4×C59—42no—
124C4×C59C2×SD1612yes—
125C4×C59C2×#4111yescharacteristic
126C4×C59H(5,𝔽2)11yescharacteristic
127Dic59—22no—
128Dic59—41no—
129Dic59—41no—
130#3295—591nomaximal, Sylow
131C2×C59—42no—
132C2×C59#54611yescharacteristic
133C2×C59C22×SD1611yescharacteristic
134C2×C59#57611yescharacteristic
135D59—24no—
136D59—82no—
137#554—591no—
138D42—591no—
139#486—591no—
140#481—592no—
141#476D5911yescharacteristic, complemented
142#465—592no—
143#460—591no—
144#450—592no—
145C4×SD16—591no—
146C8×D4—591no—
147C2×#102—592no—
148C59#329511yescharacteristic, complemented, Sylow
149C22×D4—592no—
150C22×D4—1181no—
151C2×SD16—591no—
152C2×SD16—592no—
153C22×C8—592no—
154C4⋊Q8C2×D5911yescharacteristic, complemented
155#127—591no—
156#125—1181no—
157C22⋊Q8—592no—
158C22⋊Q8—1182no—
159#121—592no—
160#121—1181no—
161#120—1182no—
162C4×Q8—591no—
163C4×D4—591no—
164C4×D4—1181no—
165C4×D4C2×D5911yescharacteristic
166C2×C4⋊C4—592no—
167#106—591no—
168#106C2×D5912yescomplemented
169C4⋊C8C2×D5911yescharacteristic, complemented
170#103—591no—
171#102—591no—
172#102—592no—
173#102—592no—
174#102—592no—
175#102C2×D5912yes—
176C22⋊C8—592no—
177C4×C8—591no—
178C24—1182no—
179C2×Q8—21no—
180C2×Q8—591no—
181C2×D4—591no—
182C2×D4—592no—
183C2×D4—1182no—
184C2×D4—1182no—
185C2×D4—1184no—
186C2×D4—1181no—
187C2×D4—1181no—
188C2×D4—2364no—
189C2×D4—2362no—
190C2×D4C22×D5911yescharacteristic
191C22×C4—21no—
192C22×C4—592no—
193C22×C4—1182no—
194SD16—1182no—
195SD16—2362no—
196C2×C8—591no—
197C2×C8—591no—
198C2×C8—2362no—
199C2×C8C22×D5912yes—
200C4⋊C4—21no—
201C4⋊C4—591no—
202C4⋊C4—591no—
203C4⋊C4—1182no—
204C4⋊C4—1182no—
205C4⋊C4—1181no—
206C4⋊C4C22×D5912yes—
207C4⋊C4C22×D5911yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—592no—
210C22⋊C4—1182no—
211C22⋊C4—1182no—
212C42—1181no—
213C42C22×D5911yescharacteristic
214C23—21no—
215C23—592no—
216C23—1182no—
217C23—4722no—
218C23—2364no—
219C23—2364no—
220Q8—81no—
221Q8—1181no—
222Q8—1181no—
223D4—21no—
224D4—594no—
225D4—1182no—
226D4—1182no—
227D4—4721no—
228D4—2364no—
229D4—2364no—
230D4—2362no—
231D4#1174568912yes—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—592no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1181no—
239C2×C4—1181no—
240C2×C4—4722no—
241C2×C4D4×D5911yescharacteristic
242C2×C4D4×D5911yescharacteristic
243C2×C4C23×D5911yescharacteristic
244C8—22no—
245C8—1181no—
246C8—1181no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—594no—
251C22—1182no—
252C22—1182no—
253C22—4728no—
254C22—2362no—
255C22—2362no—
256C22—2362no—
257C22C2×D4×D5911yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—1182no—
262C4—2361no—
263C4—2361no—
264C4SD16×D5912yes—
265C4C2×#1174568911yescharacteristic
266C4#1427377311yescharacteristic
267C2—42no—
268C2—1184no—
269C2—4722no—
270C2#1683804311yescharacteristic
271C2C2×SD16×D5911yescharacteristic
272C2#1683820611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#476C2
socleC22×C59C22×D4
hypercenter#476D59
hypocenterC59#3295
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #16838113C2semidirect
3. #16837750C2semidirect
4. #16837722C2semidirect
5. #16837632C2semidirect
6. #16837603C2semidirect
7. #16837554C2semidirect
8. C4×#11745670C2semidirect
9. #16837467C2semidirect
10. #102×D59C2semidirect
11. C59×#476C2semidirect
14. C2×#11745660C22semidirect
16. C2×C8×D59C22semidirect
19. #14273707C22semidirect
24. C4×Dic118C22semidirect
27. C4⋊C4×D59C22semidirect
28. #14273658C22semidirect
29. #14273650C22semidirect
30. #14273649C22semidirect
32. #14273647C22semidirect
33. #14273647C22semidirect
34. (C2×D59)⋊C8C22semidirect
35. C4×C59⋊C8C22semidirect
36. C4⋊Q8×C59C22semidirect
38. #106×C59C22semidirect
39. C4⋊C8×C59C22semidirect
101. D236SD16semidirect
141. #476D59semidirect
148. C59#3295semidirect
154. C4⋊Q8C2×D59semidirect
168. #106C2×D59semidirect
169. C4⋊C8C2×D59semidirect