Finite Group Databasebeta

← #21430244

Subgroups — #21430244

Order 7552.

498 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14273710C211yesmaximal, characteristic
2C2×#14273706C211yesmaximal, characteristic, complemented
3#16837342C211yesmaximal, characteristic, complemented
4#16837339C211yesmaximal, characteristic, complemented
5#16837327C212yesmaximal, complemented
6#16837323C211yesmaximal, characteristic, complemented
7#16837318C212yesmaximal
8#16837315C211yesmaximal, characteristic, complemented
9#16837300C212yesmaximal, complemented
10#16837293C212yesmaximal, complemented
11C4×C22⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×C22⋊D59C2211yescharacteristic
14#14273710—24no—
15#14273706—24no—
16C2×C4⋊Dic59C2211yescharacteristic
17C2×C4⋊Dic59C2212yes—
18C2×C4⋊Dic59C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×C22⋊Dic59C2212yes—
23C2×C22⋊Dic59C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C23×C4×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2212yes—
34C2×C42×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C22×C4×C59—42no—
40C22×C4×C59D414yes—
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2312yes—
45C2×C22⋊D59—22no—
46C2×C22⋊D59—42no—
47C22×Dic59C2311yescharacteristic
48C22×Dic59C2312yes—
49C22×Dic59C2311yescharacteristic
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C2×Dic118—22no—
53C2×Dic118—42no—
54C22⋊C4×C59—24no—
55C42×C59—42no—
56C22⋊Dic59—24no—
57C22⋊Dic59—22no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—24no—
61(C2×D59)⋊C4—42no—
62C4⋊Dic59—22no—
63C4⋊Dic59—24no—
64C4⋊Dic59—42no—
65Dic59⋊C4—24no—
66Dic59⋊C4—22no—
67Dic59⋊C4—22no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59C2×D412yes—
73C23×C59C2411yescharacteristic
74C22×D59—21no—
75C22×D59—22no—
76C22×D59—22no—
77C22×D59—21no—
78C22×D59—21no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—44no—
87C2×C4×C59C2×D412yes—
88C2×C4×C59C2×D412yes—
89C2×C4×C59#4114yes—
90C22⋊D59—44no—
91C2×Dic59—21no—
92C2×Dic59—22no—
93C2×Dic59—21no—
94C2×Dic59—21no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—21no—
104Dic118—44no—
105C22×C59—22no—
106C22×C59—22no—
107C22×C59—22no—
108C22×C59—42no—
109C22×C59—42no—
110C22×C59C22×D411yescharacteristic
111C22×C59C2×#4112yes—
112C22×C59H(5,𝔽2)12yes—
113C22×C592−1+411yescharacteristic
114C22×C592−1+411yescharacteristic
115C2×D59—41no—
116C2×D59—41no—
117C2×D59—41no—
118C2×D59—42no—
119C2×D59—42no—
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—24no—
123C4×C59—42no—
124Dic59—42no—
125Dic59—41no—
126Dic59—42no—
127Dic59—41no—
128Dic59—41no—
129#2663—591nomaximal, Sylow
130C2×C59—42no—
131C2×C59#53211yescharacteristic
132C2×C59#53411yescharacteristic
133C2×C59#53611yescharacteristic
134C2×C59#53912yes—
135C2×C59#54212yes—
136D59—81no—
137C2×#123—591no—
138C2×C22⋊Q8—591no—
139#400—591no—
140#399—591no—
141#397—592no—
142#395—591no—
143#394—592no—
144#393—591no—
145#389—592no—
146#388—592no—
147C4×C22⋊C4D5911yescharacteristic, complemented
148C59#266311yescharacteristic, complemented, Sylow
149C22×Q8—591no—
150C22×D4—591no—
151C23×C4C2×D5911yescharacteristic
152#123—1184no—
153C22⋊Q8—1184no—
154C2×C4⋊C4—592no—
155C2×C4⋊C4—591no—
156C2×C4⋊C4—591no—
157C2×C4⋊C4—591no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—592no—
160C2×C22⋊C4—592no—
161C2×C22⋊C4—591no—
162C2×C22⋊C4—591no—
163C2×C22⋊C4—591no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4C2×D5912yes—
167C2×C42C2×D5912yes—
168#95—592no—
169#95—592no—
170#95—592no—
171#95—592no—
172#95C2×D5911yescharacteristic
173#95C2×D5911yescharacteristic
174C24—591no—
175C24C22×D5911yescharacteristic
176C2×Q8—1182no—
177C2×Q8—2362no—
178C2×D4—1182no—
179C2×D4—2362no—
180C22×C4—42no—
181C22×C4—591no—
182C22×C4—591no—
183C22×C4—592no—
184C22×C4—592no—
185C22×C4—591no—
186C22×C4D23614yes—
187C22×C4C22×D5912yes—
188C22×C4C22×D5911yescharacteristic
189C22×C4C22×D5911yescharacteristic
190C22×C4C22×D5912yes—
191C4⋊C4—1184no—
192C4⋊C4—1182no—
193C4⋊C4—1182no—
194C4⋊C4—1182no—
195C4⋊C4—1184no—
196C4⋊C4—2362no—
197C22⋊C4—24no—
198C22⋊C4—1184no—
199C22⋊C4—1182no—
200C22⋊C4—1182no—
201C22⋊C4—1182no—
202C22⋊C4—1184no—
203C22⋊C4—2362no—
204C42—42no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—1181no—
210C23—1182no—
211C23—1182no—
212C23—1181no—
213C23—1181no—
214C23C2×D23612yes—
215C23C23×D5911yescharacteristic
216Q8—2364no—
217D4—2364no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—1181no—
227C2×C4—1181no—
228C2×C4—1182no—
229C2×C4—1182no—
230C2×C4—1181no—
231C2×C4—1182no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—1182no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1181no—
238C2×C4—1181no—
239C2×C4C2×D23612yes—
240C2×C4C2×D23612yes—
241C2×C4#1174568514yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—2361no—
248C22—2361no—
249C22—2361no—
250C22—2362no—
251C22—2362no—
252C22C22×D23611yescharacteristic
253C22C2×#1174568512yes—
254C22#1427377212yes—
255C22#1427377411yescharacteristic
256C22#1427377411yescharacteristic
257C4—22no—
258C4—22no—
259C4—24no—
260C4—42no—
261C4—2361no—
262C4—2362no—
263C4—2361no—
264C4—2362no—
265C4—2361no—
266C2—42no—
267C2—4721no—
268C2#1683795211yescharacteristic
269C2#1683796411yescharacteristic
270C2#1683797811yescharacteristic
271C2#1683799412yes—
272C2#1683801712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC4×C22⋊C4×C59C2
hypercenterC4×C22⋊C4D59
hypocenterC59#2663
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14273706C2semidirect
3. #16837342C2semidirect
4. #16837339C2semidirect
5. #16837327C2semidirect
6. #16837323C2semidirect
8. #16837315C2semidirect
9. #16837300C2semidirect
10. #16837293C2semidirect
11. C4×C22⋊C4×C59C2semidirect
147. C4×C22⋊C4D59semidirect
148. C59#2663semidirect