Finite Group Databasebeta

← #21433354

Subgroups — #21433354

Order 7552.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#11745685C211yesmaximal, characteristic
2C2×#14273758C212yesmaximal, complemented
3#16837562C214yesmaximal, complemented
4#16837557C212yesmaximal, complemented
5#16837552C212yesmaximal, complemented
6#16837401C212yesmaximal, complemented
7#16837366C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
9C2×#11745685—21no—
10C2×#11745685—21no—
11C2×#11745685—28no—
12C2×#11745685C2212yes—
13C2×#11745685C2212yes—
14C22×Dic118C2211yescharacteristic
15C22×D236C2211yescharacteristic
16C22×C22⋊D59—21no—
17C22×C4×D59—21no—
18#14273758—28no—
19C2×#11745672C2214yescomplemented
20C2×#11745671C2214yescomplemented
21C2×#11745664C2212yescomplemented
22#14273650C2214yescomplemented
23#14273647C2214yescomplemented
24C22⋊(C59⋊C8)C2212yes—
25C22⋊(C59⋊C8)C2212yescomplemented
26C22×Q8×C59C2211yescharacteristic, complemented
27C23×C4×C59C2211yescharacteristic
28C22⋊Q8×C59—21no—
29C22⋊Q8×C59—21no—
30C22⋊Q8×C59C2212yes—
31C22⋊Q8×C59C2212yescomplemented
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59C2211yescharacteristic, complemented
34C2×C22⋊C4×C59—21no—
35C24×C59D411yescharacteristic
36C23×D59—21no—
37C2×Q8×C59—21no—
38C2×Q8×C59—24no—
39C2×Q8×C59C2312yes—
40C22×C4×C59—24no—
41C22×C4×C59—21no—
42C22×C4×C59—21no—
43C22×C4×C59D411yescharacteristic
44C22×C4×C59D411yescharacteristic
45C22×C4×C59D411yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2311yescharacteristic
48C22×C4×C59C2311yescharacteristic
49C2×C22⋊D59—28no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C22×Dic59—21no—
53#11745685—28no—
54#11745685—28no—
55#11745685—24no—
56#11745685—24no—
57#11745685—24no—
58#11745685D418yes—
59C2×D236—21no—
60C2×D236—24no—
61C2×D236C2312yes—
62C2×C4×D59—28no—
63C2×C4×D59—22no—
64C2×C4×D59—22no—
65C2×Dic118—21no—
66C2×Dic118—24no—
67C2×Dic118C2312yes—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—21no—
71C4⋊C4×C59C2312yes—
72C22⋊C4×C59—22no—
73C22⋊C4×C59—22no—
74#11745672—28no—
75#11745671—28no—
76#11745664—24no—
77C2×C59⋊C8C2314yes—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—41no—
81C23×C59C2×D411yescharacteristic
82C23×C59C2×D411yescharacteristic
83C23×C59C2×D411yescharacteristic
84C22×D59—24no—
85C22×D59—22no—
86C22×D59—42no—
87C22×D59—42no—
88Q8×C59—24no—
89Q8×C59—42no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59C2×D414yes—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—28no—
105C22⋊D59—28no—
106C2×Dic59—24no—
107C2×Dic59—22no—
108D236—24no—
109D236—22no—
110D236C2×D414yes—
111C4×D59—28no—
112C4×D59—28no—
113Dic118—24no—
114Dic118—22no—
115Dic118C2×D414yes—
116C59⋊C8—24no—
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—44no—
125C22×C59#12012yes—
126C22×C59#13712yescomplemented
127C22×C59C22×D411yescharacteristic
128C2×D59—24no—
129C2×D59—48no—
130C2×D59—42no—
131C2×D59—42no—
132C4×C59—21no—
133C4×C59—21no—
134C4×C59—42no—
135C4×C59—42no—
136C4×C59#12012yes—
137C4×C59C22×D412yes—
138Dic59—24no—
139#3007—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59—22no—
142C2×C59—41no—
143C2×C59C2×#12011yescharacteristic
144C2×C59C2×#13711yescharacteristic
145C2×C59#57511yescharacteristic
146D59—44no—
147C22×#41—591no—
148C2×#137—592no—
149C2×C22⋊Q8D5911yescharacteristic, complemented
150#452—594no—
151#451—592no—
152#450—592no—
153#417—592no—
154#407—591no—
155C59#300711yescharacteristic, complemented, Sylow
156C2×#41—592no—
157C2×#41—592no—
158C2×#41—1188no—
159C2×#41—1181no—
160C2×#41—1181no—
161C22×Q8—591no—
162C22×Q8C2×D5911yescharacteristic, complemented
163C22×D4—591no—
164C22×D4—1181no—
165C23×C4—1181no—
166C23×C4C2×D5911yescharacteristic
167#137—1188no—
168C2×Q16—594no—
169C2×SD16—594no—
170C2×M16—592no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D5912yes—
174C22⋊Q8C2×D5912yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D5911yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—594no—
179#102—594no—
180C22⋊C8—592no—
181C22⋊C8—592no—
182C24—1181no—
183C24C22⋊D5911yescharacteristic
184#41—598no—
185#41—1184no—
186#41—1184no—
187#41—1188no—
188#41—1188no—
189#41—1184no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—592no—
193C2×Q8—1184no—
194C2×Q8—1181no—
195C2×Q8C22×D5912yes—
196C2×D4—592no—
197C2×D4—1184no—
198C2×D4—1188no—
199C2×D4—1182no—
200C2×D4—1182no—
201C2×D4—1181no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1188no—
206C22×C4—1181no—
207C22×C4—1182no—
208C22×C4—1182no—
209C22×C4C22⋊D5911yescharacteristic
210C22×C4C22⋊D5911yescharacteristic
211C22×C4C22⋊D5911yescharacteristic
212C22×C4C22×D5911yescharacteristic
213C22×C4C22×D5911yescharacteristic
214C22×C4C22×D5911yescharacteristic
215Q16—1188no—
216SD16—1188no—
217M16—1184no—
218C2×C8—594no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D5912yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1184no—
229C23—1182no—
230C23—2362no—
231C23—2362no—
232C23C2×C22⋊D5911yescharacteristic
233C23C2×C22⋊D5911yescharacteristic
234C23C2×C22⋊D5911yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—594no—
238Q8—1184no—
239Q8—1182no—
240D4—594no—
241D4—1184no—
242D4—1188no—
243D4—1188no—
244D4—1182no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1188no—
255C2×C4—1184no—
256C2×C4—1182no—
257C2×C4—1188no—
258C2×C4D4×D5914yes—
259C2×C4C2×C22⋊D5911yescharacteristic
260C2×C4C2×C22⋊D5911yescharacteristic
261C2×C4C2×C22⋊D5911yescharacteristic
262C2×C4C23×D5911yescharacteristic
263C8—1184no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—1184no—
273C22—2368no—
274C22—2362no—
275C22—2362no—
276C22#1427369512yes—
277C22#1427375812yescomplemented
278C22C22×C22⋊D5911yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1184no—
284C4#1427369512yes—
285C4C2×D4×D5912yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—2364no—
290C2C2×#1427369511yescharacteristic
291C2C2×#1427375811yescharacteristic
292C2#1683820011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C22⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C22⋊Q8D59
hypocenterC59#3007
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14273758C2semidirect
3. #16837562C2semidirect
4. #16837557C2semidirect
5. #16837552C2semidirect
6. #16837401C2semidirect
7. #16837366C2semidirect
8. C2×C22⋊Q8×C59C2semidirect
19. C2×#11745672C22semidirect
20. C2×#11745671C22semidirect
21. C2×#11745664C22semidirect
22. #14273650C22semidirect
23. #14273647C22semidirect
25. C22⋊(C59⋊C8)C22semidirect
26. C22×Q8×C59C22semidirect
31. C22⋊Q8×C59C22semidirect
33. C2×C4⋊C4×C59C22semidirect
126. C22×C59#137semidirect
149. C2×C22⋊Q8D59semidirect
155. C59#3007semidirect
162. C22×Q8C2×D59semidirect
174. C22⋊Q8C2×D59semidirect
176. C2×C4⋊C4C2×D59semidirect
277. C22#14273758semidirect