Finite Group Databasebeta

← #21435223

Subgroups — #21435223

Order 7552.

404 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 123 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#16837949C211yesmaximal, characteristic
2C4⋊Q8×D59C211yesmaximal, characteristic, complemented
3#16837799C212yesmaximal, complemented
4#16837747C212yesmaximal, complemented
5#16837722C212yesmaximal, complemented
6#16837719C212yesmaximal, complemented
7#16837693C211yesmaximal, characteristic, complemented
8#16837692C211yesmaximal, characteristic, complemented
9#16837689C211yesmaximal, characteristic, complemented
10#16837454C211yesmaximal, characteristic, complemented
11C59×#476C211yesmaximal, characteristic, complemented
12C2×#11745685—21no—
13C2×Q8×D59—21no—
14#14273727C2211yescharacteristic
15#14273726—21no—
16#14273726C2212yes—
17#14273726C2211yescharacteristic
18#14273725—21no—
19#14273723C2211yescharacteristic
20#14273713—21no—
21#14273702C2211yescharacteristic
22#14273700—21no—
23#14273698C2211yescharacteristic
24C4×D236C2211yescharacteristic
25C4×C22⋊D59—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59C2212yes—
28C42×D59C2211yescharacteristic
29#14273659C2212yes—
30#14273658C2212yes—
31#14273658C2212yes—
32C4⋊(C59⋊C8)C2211yescharacteristic
33#14273649C2212yes—
34#14273648C2212yes—
35#14273647C2212yes—
36(C2×D59)⋊C8C2212yes—
37C8×Dic59C2212yes—
38C4⋊Q8×C59C2211yescharacteristic
39C4×D4×C59C2211yescharacteristic
40#106×C59C2212yes—
41C4⋊C8×C59C2211yescharacteristic
42#102×C59C2212yes—
43C2×Q8×C59—21no—
44C2×D4×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C2×C22⋊D59—21no—
47C2×C22⋊D59—21no—
48C2×C22⋊D59—21no—
49Q8×D59—44no—
50#11745685—44no—
51C2×D236—21no—
52C2×D236C2311yescharacteristic
53C2×C4×D59—22no—
54C2×C4×D59—21no—
55C2×C4×D59D411yescharacteristic
56C2×C4×D59D411yescharacteristic
57C2×C4×D59C2311yescharacteristic
58C2×Dic118—21no—
59C2×Dic118—21no—
60C2×Dic118—22no—
61C2×C8×C59C2312yes—
62C4⋊C4×C59—21no—
63C4⋊C4×C59C2312yes—
64C4⋊C4×C59C2311yescharacteristic
65C22⋊C4×C59—21no—
66C42×C59C2311yescharacteristic
67C22⋊Dic59—21no—
68(C2×D59)⋊C4—21no—
69(C2×D59)⋊C4—21no—
70C4⋊Dic59—21no—
71C4⋊Dic59C2312yes—
72C4⋊Dic59C2311yescharacteristic
73Dic59⋊C4—22no—
74Dic59⋊C4—22no—
75Dic59⋊C4—21no—
76C4×Dic59D411yescharacteristic
77C4×Dic59D411yescharacteristic
78C4×Dic59C2311yescharacteristic
79C2×C59⋊C8C2312yes—
80C23×C59—21no—
81C22×D59—21no—
82C22×D59C2×D411yescharacteristic
83Q8×C59—42no—
84D4×C59—21no—
85D4×C59—21no—
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—42no—
93C22⋊D59—42no—
94C22⋊D59—42no—
95C2×Dic59—22no—
96C2×Dic59—21no—
97C2×Dic59C2×D411yescharacteristic
98C2×Dic59C2×D411yescharacteristic
99C2×Dic59C2×D411yescharacteristic
100D236—21no—
101D236—21no—
102D236—42no—
103C4×D59—21no—
104C4×D59—21no—
105C4×D59—22no—
106C4×D59—44no—
107C4×D59—42no—
108C4×D59SD1614yes—
109Dic118—42no—
110Dic118—42no—
111Dic118—44no—
112C8×C59—22no—
113C59⋊C8—22no—
114C22×C59—41no—
115C22×C59—41no—
116C22×C59—41no—
117C22×C59C22×D411yescharacteristic
118C2×D59—21no—
119C2×D59—21no—
120C2×D59—41no—
121C2×D59—41no—
122C2×D59—41no—
123C2×D59C2×SD1612yes—
124C4×C59—21no—
125C4×C59—41no—
126C4×C59—42no—
127C4×C59C2×SD1612yes—
128C4×C592−1+411yescharacteristic
129C4×C592−1+411yescharacteristic
130Dic59—21no—
131Dic59—41no—
132Dic59—42no—
133Dic59C2×SD1612yes—
134Dic59AGL(1,ℤ/8)12yes—
135#3227—591nomaximal, Sylow
136C2×C59—81no—
137C2×C59#53611yescharacteristic
138C2×C59C22×SD1611yescharacteristic
139C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
140D59—22no—
141D59—81no—
142#532—591no—
143C2×C4⋊Q8—591no—
144#498—592no—
145#486—592no—
146#481—592no—
147#481—592no—
148#476—591no—
149#476—591no—
150#476—591no—
151#476D5911yescharacteristic, complemented
152#431—591no—
153C59#322711yescharacteristic, complemented, Sylow
154C2×#41—1181no—
155C22×Q8—1181no—
156C4⋊Q8—592no—
157C4⋊Q8—591no—
158C4⋊Q8—591no—
159C4⋊Q8—1181no—
160C4⋊Q8—1181no—
161C4⋊Q8C2×D5911yescharacteristic
162#127—591no—
163#124—1181no—
164#121—591no—
165#121—591no—
166#121—1181no—
167C4×D4—591no—
168C4×D4—1181no—
169C4×D4C2×D5911yescharacteristic
170C2×C4⋊C4—592no—
171C2×C4⋊C4—1181no—
172C2×C42—591no—
173#106—592no—
174#106—592no—
175#106—592no—
176#106C2×D5912yes—
177C4⋊C8—591no—
178C4⋊C8C2×D5911yescharacteristic
179#102—592no—
180#102—592no—
181#102—592no—
182#102C2×D5912yes—
183C22⋊C8—592no—
184C4×C8—592no—
185#41—2364no—
186C2×Q8—21no—
187C2×Q8—1181no—
188C2×Q8—1182no—
189C2×Q8—1181no—
190C2×Q8—2364no—
191C2×D4—591no—
192C2×D4—1181no—
193C2×D4—1181no—
194C2×D4—1181no—
195C2×D4—1181no—
196C2×D4C22×D5911yescharacteristic
197C22×C4—21no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—1182no—
202C22×C4—1181no—
203C2×C8—592no—
204C2×C8C22×D5912yes—
205C4⋊C4—21no—
206C4⋊C4—592no—
207C4⋊C4—591no—
208C4⋊C4—1182no—
209C4⋊C4—1182no—
210C4⋊C4—1181no—
211C4⋊C4—1181no—
212C4⋊C4C22×D5912yes—
213C4⋊C4C22×D5911yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1181no—
216C22⋊C4—1181no—
217C22⋊C4—1181no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42C22×D5911yescharacteristic
222C23—21no—
223C23—591no—
224C23—1181no—
225Q8—42no—
226Q8—2362no—
227Q8—2364no—
228Q8—2362no—
229D4—21no—
230D4—21no—
231D4—1181no—
232D4—1181no—
233D4—2362no—
234D4—2362no—
235D4—2362no—
236D4—2362no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—594no—
243C2×C4—591no—
244C2×C4—1182no—
245C2×C4—1181no—
246C2×C4—1181no—
247C2×C4—1181no—
248C2×C4—1182no—
249C2×C4—2364no—
250C2×C4—2362no—
251C2×C4D4×D5911yescharacteristic
252C2×C4D4×D5911yescharacteristic
253C2×C4C23×D5911yescharacteristic
254C8—22no—
255C8—1182no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—592no—
260C22—1181no—
261C22—1181no—
262C22—2361no—
263C22—2361no—
264C22—2361no—
265C22C2×D4×D5911yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—592no—
270C4—592no—
271C4—1181no—
272C4—2361no—
273C4—2362no—
274C4SD16×D5912yes—
275C4#1427377411yescharacteristic
276C4#1427377511yescharacteristic
277C2—81no—
278C2—1182no—
279C2—4721no—
280C2#1683797511yescharacteristic
281C2C2×SD16×D5911yescharacteristic
282C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#476C2
socleC22×C59C22×D4
hypercenter#476D59
hypocenterC59#3227
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
2. C4⋊Q8×D59C2semidirect
3. #16837799C2semidirect
4. #16837747C2semidirect
5. #16837722C2semidirect
6. #16837719C2semidirect
7. #16837693C2semidirect
8. #16837692C2semidirect
9. #16837689C2semidirect
10. #16837454C2semidirect
11. C59×#476C2semidirect
151. #476D59semidirect
153. C59#3227semidirect