Finite Group Databasebeta

← #21427706

Subgroups — #21427706

Order 7552.

914 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 173 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, complemented
2C2×#14273710C211yesmaximal, characteristic, complemented
3C2×#14273701C212yesmaximal, complemented
4#16837323C211yesmaximal, characteristic, complemented
5#16837320C211yesmaximal, characteristic, complemented
6#16837315C211yesmaximal, characteristic, complemented
7#16837312C211yesmaximal, characteristic, complemented
8#16837281C212yesmaximal, complemented
9#16837280C212yesmaximal, complemented
10#16837276C212yesmaximal, complemented
11C2×#95×C59C211yesmaximal, characteristic, complemented
12C2×#11745685—24no—
13C2×#11745685—28no—
14C22×Dic118C2211yescharacteristic
15C22×D236C2211yescharacteristic
16C22×D236C2212yes—
17C22×C22⋊D59C2211yescharacteristic
18C22×C22⋊D59C2212yes—
19C22×C4×D59C2212yes—
20#14273710—24no—
21#14273701—28no—
22C2×C4⋊Dic59C2212yes—
23C2×Dic59⋊C4C2211yescharacteristic
24C2×Dic59⋊C4C2211yescharacteristic
25C2×C22⋊Dic59C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×(C2×D59)⋊C4C2211yescharacteristic
30C2×(C2×D59)⋊C4C2212yes—
31C2×(C2×D59)⋊C4C2212yes—
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C23×C4×C59C2212yes—
35C23×C4×C59C2211yescharacteristic
36#95×C59C2212yes—
37#95×C59C2211yescharacteristic
38#95×C59C2211yescharacteristic
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C23×D59C2312yes—
42C22×C4×C59—24no—
43C22×C4×C59—24no—
44C22×C4×C59—24no—
45C22×C4×C59—42no—
46C22×C4×C59D414yescomplemented
47C22×C4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2311yescharacteristic
50C22×C4×C59C2312yes—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—24no—
53C2×C22⋊D59—24no—
54C2×C22⋊D59—22no—
55C2×C22⋊D59D418yescomplemented
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2311yescharacteristic
58C22×Dic59C2311yescharacteristic
59C22×Dic59C2312yes—
60#11745685—232no—
61C2×D236—24no—
62C2×D236—22no—
63C2×D236—24no—
64C2×D236—24no—
65C2×D236—42no—
66C2×C4×D59—24no—
67C2×C4×D59—24no—
68C2×C4×D59—24no—
69C2×Dic118—22no—
70C2×Dic118—24no—
71C22⋊Dic59—22no—
72(C2×D59)⋊C4—24no—
73(C2×D59)⋊C4—22no—
74(C2×D59)⋊C4—22no—
75(C2×D59)⋊C4—24no—
76(C2×D59)⋊C4—24no—
77(C2×D59)⋊C4—42no—
78C4⋊Dic59—24no—
79Dic59⋊C4—22no—
80Dic59⋊C4—22no—
81C23×C59—22no—
82C23×C59—21no—
83C23×C59—21no—
84C23×C59C2×D412yes—
85C23×C59C2×D414yes—
86C23×C59C2411yescharacteristic
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59—22no—
91C22×D59—22no—
92C22×D59—22no—
93C22×D59—21no—
94C22×D59—22no—
95C22×D59—22no—
96C22×D59—21no—
97C22×D59—21no—
98C22×D59C2×D414yes—
99C2×C4×C59—28no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—48no—
108C2×C4×C59C2×D412yes—
109C2×C4×C59C2×D412yes—
110C2×C4×C59#4114yescomplemented
111C22⋊D59—216no—
112C22⋊D59—44no—
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59—21no—
116C2×Dic59—21no—
117C2×Dic59—21no—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—21no—
121C2×Dic59—21no—
122C2×Dic59C2×D414yes—
123D236—28no—
124D236—48no—
125C4×D59—216no—
126Dic118—28no—
127C22×C59—22no—
128C22×C59—22no—
129C22×C59—24no—
130C22×C59—24no—
131C22×C59#12014yes—
132C22×C59C22×D412yes—
133C22×C59C22×D411yescharacteristic
134C22×C59C2×#4112yes—
135C22×C59H(5,𝔽2)11yescharacteristic
136C22×C592−1+411yescharacteristic
137C2×D59—24no—
138C2×D59—24no—
139C2×D59—24no—
140C2×D59—42no—
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—42no—
144C2×D59—41no—
145C2×D59—41no—
146C2×D59—41no—
147C2×D59—42no—
148C2×D59—42no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—24no—
152C4×C59—42no—
153Dic59—24no—
154Dic59—41no—
155Dic59—41no—
156Dic59—41no—
157#2430—591nomaximal, Sylow
158C2×C59—24no—
159C2×C59C2×#12011yescharacteristic
160C2×C59#53211yescharacteristic
161C2×C59#53711yescharacteristic
162C2×C59#54612yes—
163C2×C59#54712yes—
164D59—44no—
165D59—81no—
166C22×#41—591no—
167C2×#123—591no—
168C2×#121—592no—
169#395—591no—
170#394—591no—
171#393—591no—
172#392—591no—
173#386—592no—
174#386—592no—
175#385—592no—
176C2×#95D5911yescharacteristic, complemented
177C59#243011yescharacteristic, complemented, Sylow
178C2×#41—1188no—
179C2×#41—1184no—
180C22×Q8—591no—
181C22×D4—592no—
182C22×D4—591no—
183C22×D4—591no—
184C22×D4—592no—
185C23×C4—592no—
186C23×C4C2×D5912yes—
187C23×C4C2×D5911yescharacteristic
188#123—1184no—
189#121—1188no—
190C2×C4⋊C4—592no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—591no—
193C2×C22⋊C4—591no—
194C2×C22⋊C4—592no—
195C2×C22⋊C4—592no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—591no—
198C2×C22⋊C4—592no—
199C2×C22⋊C4—592no—
200#95—592no—
201#95—592no—
202#95C2×D5912yes—
203#95C2×D5911yescharacteristic
204#95C2×D5911yescharacteristic
205C24—592no—
206C24—591no—
207C24C22×D5911yescharacteristic
208#41—11832no—
209C2×Q8—1184no—
210C2×Q8—1182no—
211C2×D4—598no—
212C2×D4—1184no—
213C2×D4—1184no—
214C2×D4—1184no—
215C2×D4—1184no—
216C2×D4—1182no—
217C2×D4—1184no—
218C2×D4—1182no—
219C2×D4—1184no—
220C2×D4—2362no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—592no—
226C22×C4—591no—
227C22×C4—591no—
228C22×C4—591no—
229C22×C4—1184no—
230C22×C4—1184no—
231C22×C4—1184no—
232C22×C4D23614yescomplemented
233C22×C4C22×D5912yes—
234C22×C4C22×D5911yescharacteristic
235C22×C4C22×D5911yescharacteristic
236C22×C4C22×D5912yes—
237C4⋊C4—1184no—
238C4⋊C4—1182no—
239C4⋊C4—1182no—
240C22⋊C4—1184no—
241C22⋊C4—1184no—
242C22⋊C4—1182no—
243C22⋊C4—1182no—
244C22⋊C4—1182no—
245C22⋊C4—1184no—
246C22⋊C4—2362no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—594no—
251C23—1182no—
252C23—1182no—
253C23—1182no—
254C23—1182no—
255C23—1182no—
256C23—1182no—
257C23—1182no—
258C23—1181no—
259C23—1182no—
260C23—1181no—
261C23—1181no—
262C23C2×D23612yes—
263C23D4×D5914yes—
264C23C23×D5911yescharacteristic
265Q8—1188no—
266D4—1188no—
267D4—11816no—
268D4—2368no—
269D4—2364no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—594no—
280C2×C4—11816no—
281C2×C4—1182no—
282C2×C4—1182no—
283C2×C4—1182no—
284C2×C4—1181no—
285C2×C4—1181no—
286C2×C4—1181no—
287C2×C4—1182no—
288C2×C4—1181no—
289C2×C4—1181no—
290C2×C4C2×D23612yes—
291C2×C4C2×D23612yes—
292C2×C4#1174568514yescomplemented
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—1184no—
298C22—1184no—
299C22—1184no—
300C22—2361no—
301C22—2361no—
302C22—2361no—
303C22—2362no—
304C22—2362no—
305C22—2362no—
306C22—2362no—
307C22—2362no—
308C22—2362no—
309C22#1427369714yes—
310C22C2×D4×D5912yes—
311C22C22×D23611yescharacteristic
312C22C2×#1174568512yes—
313C22#1427377211yescharacteristic
314C22#1427377511yescharacteristic
315C4—22no—
316C4—22no—
317C4—24no—
318C4—42no—
319C4—1184no—
320C4—2361no—
321C4—2361no—
322C4—2361no—
323C2—24no—
324C2—4721no—
325C2—2364no—
326C2C2×#1427369711yescharacteristic
327C2#1683795211yescharacteristic
328C2#1683798311yescharacteristic
329C2#1683804212yes—
330C2#1683805312yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#95×C59C2
hypercenterC2×#95D59
hypocenterC59#2430
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×#11745685C2semidirect
2. C2×#14273710C2semidirect
3. C2×#14273701C2semidirect
4. #16837323C2semidirect
5. #16837320C2semidirect
6. #16837315C2semidirect
7. #16837312C2semidirect
8. #16837281C2semidirect
9. #16837280C2semidirect
10. #16837276C2semidirect
11. C2×#95×C59C2semidirect
46. C22×C4×C59D4semidirect
55. C2×C22⋊D59D4semidirect
110. C2×C4×C59#41semidirect
176. C2×#95D59semidirect
177. C59#2430semidirect
232. C22×C4D236semidirect
292. C2×C4#11745685semidirect