Finite Group Databasebeta

← #21435906

Subgroups — #21435906

Order 7552.

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

normal subgroups only

This group has 494 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#16838055C211yesmaximal, characteristic, complemented
2D4×D236C211yesmaximal, characteristic, complemented
3#16837750C211yesmaximal, characteristic, complemented
4#16837717C212yesmaximal, complemented
5#16837639C212yesmaximal, complemented
6#16837598C211yesmaximal, characteristic, complemented
7#16837537C212yesmaximal, complemented
8C4×#11745669C211yesmaximal, characteristic, complemented
9#16837467C211yesmaximal, characteristic, complemented
10#102×D59C212yesmaximal, complemented
11C59×#474C211yesmaximal, characteristic, complemented
12C2×#11745689—21no—
13C22×D236—21no—
14C2×D4×D59C2212yes—
15C2×D472C2212yescomplemented
16C2×#11745669C2211yescharacteristic
17C2×C8×D59C2212yescomplemented
18#14273724C2211yescharacteristic
19#14273712—22no—
20#14273703—21no—
21#14273702C2212yes—
22#14273702C2212yescomplemented
23#14273701—21no—
24#14273697—22no—
25C4×D236C2211yescharacteristic
26C4×D236C2211yescharacteristic, complemented
27C4⋊C4×D59C2212yescomplemented
28#14273660C2211yescharacteristic, complemented
29#14273649C2212yescomplemented
30#14273648C2211yescharacteristic
31#14273647C2211yescharacteristic, complemented
32#14273647C2212yescomplemented
33#14273647C2212yescomplemented
34(C2×D59)⋊C8C2212yescomplemented
35C4×C59⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C59C2211yescharacteristic, complemented
37C4×D4×C59C2211yescharacteristic
38#107×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—
46#11745689—44no—
47D4×D59—24no—
48D4×D59—42no—
49C2×D236—22no—
50C2×D236—22no—
51C2×D236—21no—
52C2×D236—21no—
53C2×D236—44no—
54C2×D236D411yescharacteristic
55C2×D236C2312yes—
56C2×D236C2311yescharacteristic
57C2×C4×D59—22no—
58C2×C4×D59—21no—
59C2×C4×D59C2312yes—
60C2×C8×C59C2312yes—
61C4⋊C4×C59—21no—
62C4⋊C4×C59C2312yes—
63C4⋊C4×C59C2311yescharacteristic
64C22⋊C4×C59—21no—
65C42×C59C2311yescharacteristic
66#11745669—22no—
67(C2×D59)⋊C4—21no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4D412yes—
71C4⋊Dic59D411yescharacteristic
72C4⋊Dic59C2311yescharacteristic
73Dic59⋊C4—22no—
74C2×C59⋊C8C2311yescharacteristic
75C2×C59⋊C8C2311yescharacteristic
76D472—42no—
77C8×D59—42no—
78C23×C59—21no—
79C22×D59—21no—
80C22×D59—22no—
81C22×D59—44no—
82C22×D59—44no—
83C22×D59—82no—
84C22×D59C2×D412yes—
85Q8×C59—42no—
86D4×C59—21no—
87D4×C59#4112yescomplemented
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—42no—
91C2×C4×C59C2×D411yescharacteristic
92C2×C4×C59C2×D411yescharacteristic
93C2×C4×C59C2411yescharacteristic
94C22⋊D59—44no—
95C2×Dic59—21no—
96C2×Dic59C2×D412yes—
97D236—21no—
98D236—22no—
99D236—21no—
100D236—22no—
101D236—44no—
102D236—44no—
103D236—42no—
104D236—81no—
105D236D814yescomplemented
106C4×D59—22no—
107C4×D59—22no—
108C4×D59—44no—
109C4×D59—42no—
110C8×C59—22no—
111C59⋊C8—21no—
112C59⋊C8—21no—
113C22×C59—22no—
114C22×C59—41no—
115C22×C59—41no—
116C22×C59C22×D411yescharacteristic
117C2×D59—22no—
118C2×D59—22no—
119C2×D59—42no—
120C2×D59—41no—
121C2×D59—41no—
122C2×D59—42no—
123C2×D59—42no—
124C2×D59—41no—
125C2×D59—88no—
126C2×D59C2×D814yes—
127C4×C59—21no—
128C4×C59—42no—
129C4×C59—41no—
130C4×C59C2×D812yes—
131C4×C59C2×#4111yescharacteristic
132C4×C59H(5,𝔽2)11yescharacteristic
133Dic59—22no—
134Dic59—41no—
135#3294—591nomaximal, Sylow
136C2×C59—42no—
137C2×C59#54611yescharacteristic
138C2×C59C22×D811yescharacteristic
139C2×C59#57711yescharacteristic
140D59—24no—
141D59—81no—
142D59—82no—
143#547—591no—
144D42—591no—
145#486—591no—
146#480—592no—
147#474D5911yescharacteristic, complemented
148#466—592no—
149#459—591no—
150#447—592no—
151C4×D8—591no—
152C8×D4—591no—
153C2×#102—592no—
154C59#329411yescharacteristic, complemented, Sylow
155C2×#41—1181no—
156C22×D4—592no—
157C22×D4—1181no—
158C2×D8—591no—
159C2×D8—592no—
160C22×C8—592no—
161C4⋊Q8C2×D5911yescharacteristic, complemented
162#127—591no—
163#123—1182no—
164C22⋊Q8—1181no—
165#121—592no—
166#121—592no—
167#121—1181no—
168#120—1182no—
169C4×D4—591no—
170C4×D4—591no—
171C4×D4C2×D5911yescharacteristic
172C2×C4⋊C4—592no—
173#107—591no—
174#107C2×D5912yescomplemented
175C4⋊C8C2×D5911yescharacteristic, complemented
176#102—591no—
177#102—592no—
178#102—592no—
179#102—591no—
180#102—592no—
181#102C2×D5912yes—
182C22⋊C8—592no—
183C4×C8—591no—
184C24—1182no—
185#41—2364no—
186C2×Q8—21no—
187C2×D4—591no—
188C2×D4—592no—
189C2×D4—591no—
190C2×D4—1182no—
191C2×D4—1182no—
192C2×D4—1182no—
193C2×D4—1184no—
194C2×D4—1181no—
195C2×D4—1181no—
196C2×D4—2364no—
197C2×D4—2362no—
198C2×D4C22×D5911yescharacteristic
199C22×C4—21no—
200C22×C4—592no—
201C22×C4—1182no—
202C22×C4—1181no—
203D8—1182no—
204D8—2362no—
205C2×C8—591no—
206C2×C8—591no—
207C2×C8—2362no—
208C2×C8C22×D5912yes—
209C4⋊C4—21no—
210C4⋊C4—591no—
211C4⋊C4—591no—
212C4⋊C4—1182no—
213C4⋊C4C22×D5912yes—
214C4⋊C4C22×D5911yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—592no—
217C22⋊C4—1182no—
218C22⋊C4—1182no—
219C22⋊C4—1181no—
220C42C22×D5911yescharacteristic
221C23—21no—
222C23—592no—
223C23—1181no—
224C23—1182no—
225C23—4722no—
226C23—2364no—
227C23—2364no—
228Q8—42no—
229D4—21no—
230D4—594no—
231D4—1181no—
232D4—1182no—
233D4—1181no—
234D4—1182no—
235D4—4721no—
236D4—2364no—
237D4—2364no—
238D4—2364no—
239D4—2362no—
240D4#1174568912yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—592no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1181no—
248C2×C4—2364no—
249C2×C4—2362no—
250C2×C4D4×D5911yescharacteristic
251C2×C4D4×D5911yescharacteristic
252C2×C4C23×D5911yescharacteristic
253C8—22no—
254C8—1181no—
255C8—1181no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—594no—
260C22—1182no—
261C22—1182no—
262C22—4728no—
263C22—2361no—
264C22—2362no—
265C22—2362no—
266C22—2361no—
267C22—2362no—
268C22—2361no—
269C22C2×D4×D5911yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—1182no—
274C4—2361no—
275C4D8×D5912yes—
276C4C2×#1174568911yescharacteristic
277C4#1427377311yescharacteristic
278C2—42no—
279C2—1184no—
280C2—4722no—
281C2—4721no—
282C2#1683804311yescharacteristic
283C2C2×D8×D5911yescharacteristic
284C2#1683821111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#474C2
socleC22×C59C22×D4
hypercenter#474D59
hypocenterC59#3294
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16838055C2semidirect
2. D4×D236C2semidirect
3. #16837750C2semidirect
4. #16837717C2semidirect
5. #16837639C2semidirect
6. #16837598C2semidirect
7. #16837537C2semidirect
8. C4×#11745669C2semidirect
9. #16837467C2semidirect
10. #102×D59C2semidirect
11. C59×#474C2semidirect
15. C2×D472C22semidirect
17. C2×C8×D59C22semidirect
22. #14273702C22semidirect
26. C4×D236C22semidirect
27. C4⋊C4×D59C22semidirect
28. #14273660C22semidirect
29. #14273649C22semidirect
31. #14273647C22semidirect
32. #14273647C22semidirect
33. #14273647C22semidirect
34. (C2×D59)⋊C8C22semidirect
35. C4×C59⋊C8C22semidirect
36. C4⋊Q8×C59C22semidirect
38. #107×C59C22semidirect
39. C4⋊C8×C59C22semidirect
87. D4×C59#41semidirect
105. D236D8semidirect
147. #474D59semidirect
154. C59#3294semidirect
161. C4⋊Q8C2×D59semidirect
174. #107C2×D59semidirect
175. C4⋊C8C2×D59semidirect
240. D4#11745689semidirect