Finite Group Databasebeta

← #21429667

Subgroups — #21429667

Order 7552.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14273684C211yesmaximal, characteristic
2#16837333C211yesmaximal, characteristic, complemented
3#16837328C211yesmaximal, characteristic
4#16837321C211yesmaximal, characteristic, complemented
5#16837313C211yesmaximal, characteristic
6#16837293C212yesmaximal
7#16837286C212yesmaximal, complemented
8#16837283C212yesmaximal, complemented
9#16837273C212yesmaximal
10#16837270C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C59C211yesmaximal, characteristic, complemented
12C22×C4×D59C2211yescharacteristic
13#14273684—24no—
14C2×Dic59⋊C4C2212yes—
15C2×Dic59⋊C4C2212yes—
16C2×Dic59⋊C4C2211yescharacteristic
17C2×Dic59⋊C4C2211yescharacteristic
18C2×(C2×D59)⋊C4C2212yes—
19C2×(C2×D59)⋊C4C2212yes—
20C2×(C2×D59)⋊C4C2211yescharacteristic
21C2×(C2×D59)⋊C4C2211yescharacteristic
22C2×C4×Dic59C2211yescharacteristic
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2211yescharacteristic
26C59⋊#95C2211yescharacteristic
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C22×Q8×C59C2211yescharacteristic
32C4×Q8×C59—24no—
33C2×C4⋊C4×C59C2212yes—
34C2×C4⋊C4×C59C2211yescharacteristic
35C2×C42×C59C2212yes—
36C2×C42×C59C2211yescharacteristic
37C23×D59C2311yescharacteristic
38C2×Q8×C59—22no—
39C2×Q8×C59—24no—
40C22×C4×C59C2312yes—
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2312yes—
47C22×Dic59C2311yescharacteristic
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2311yescharacteristic
50C2×C4×D59—22no—
51C2×C4×D59—22no—
52C2×C4×D59—22no—
53C4⋊C4×C59—24no—
54C4⋊C4×C59—22no—
55C42×C59—24no—
56C42×C59D414yes—
57(C2×D59)⋊C4—24no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—24no—
61Dic59⋊C4—22no—
62Dic59⋊C4—22no—
63Dic59⋊C4—24no—
64Dic59⋊C4—24no—
65C4×Dic59—22no—
66C23×C59C2411yescharacteristic
67C22×D59—21no—
68C22×D59—22no—
69C22×D59—21no—
70C22×D59—22no—
71C22×D59#4112yes—
72Q8×C59—28no—
73C2×C4×C59—22no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59C2×D412yes—
83C2×C4×C59C2×D412yes—
84C2×C4×C59C2×D412yes—
85C2×C4×C59#4114yes—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—21no—
89C2×Dic59—21no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59#4112yes—
99C4×D59—28no—
100C22×C59C22×D411yescharacteristic
101C22×C59C2×#4111yescharacteristic
102C22×C59C2×#4111yescharacteristic
103C22×C59C2×#4112yes—
104C22×C592−1+412yes—
105C2×D59—22no—
106C2×D59—24no—
107C2×D59—41no—
108C2×D59—41no—
109C2×D59—42no—
110C4×C59—22no—
111C4×C59—22no—
112C4×C59—24no—
113C4×C59—22no—
114C4×C59—42no—
115Dic59—22no—
116Dic59—41no—
117Dic59—41no—
118Dic59—42no—
119Dic59—42no—
120#2599—591nomaximal, Sylow
121C2×C59#52511yescharacteristic
122C2×C59#52511yescharacteristic
123C2×C59#53611yescharacteristic
124C2×C59#55212yes—
125C2×C59#55612yes—
126D59—42no—
127C2×C4×Q8D5911yescharacteristic, complemented
128C2×#117—591no—
129#398—591no—
130#397—591no—
131#395—591no—
132#393—591no—
133#388—592no—
134#387—592no—
135#387—592no—
136#385—592no—
137#383—591no—
138C59#259911yescharacteristic, complemented, Sylow
139C22×Q8C2×D5911yescharacteristic
140C23×C4—591no—
141C4×Q8—24no—
142#117—1184no—
143C2×C4⋊C4—591no—
144C2×C4⋊C4—591no—
145C2×C4⋊C4—592no—
146C2×C4⋊C4—592no—
147C2×C4⋊C4C2×D5912yes—
148C2×C4⋊C4C2×D5911yescharacteristic
149C2×C22⋊C4—591no—
150C2×C22⋊C4—591no—
151C2×C22⋊C4—592no—
152C2×C22⋊C4—592no—
153C2×C42—591no—
154C2×C42C2×D5912yes—
155C2×C42C2×D5911yescharacteristic
156#95—592no—
157#95—592no—
158#95—591no—
159#95—591no—
160#95—592no—
161#95—592no—
162#95—592no—
163#95—592no—
164C24—591no—
165C2×Q8—24no—
166C2×Q8—22no—
167C22×C4—591no—
168C22×C4—592no—
169C22×C4—592no—
170C22×C4—591no—
171C22×C4—591no—
172C22×C4—1182no—
173C22×C4—1182no—
174C22×C4—1182no—
175C22×C4C22×D5912yes—
176C22×C4C22×D5912yes—
177C22×C4C22×D5911yescharacteristic
178C22×C4C22×D5911yescharacteristic
179C22×C4C22×D5911yescharacteristic
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—1184no—
183C4⋊C4—1184no—
184C4⋊C4—1182no—
185C4⋊C4—1182no—
186C22⋊C4—1184no—
187C22⋊C4—1184no—
188C22⋊C4—1182no—
189C22⋊C4—1182no—
190C42—24no—
191C42—1182no—
192C42C22⋊D5914yes—
193C23—592no—
194C23—1181no—
195C23—1181no—
196C23—1182no—
197C23—1182no—
198C23C23×D5911yescharacteristic
199Q8—28no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—592no—
210C2×C4—1188no—
211C2×C4—1181no—
212C2×C4—1181no—
213C2×C4—1181no—
214C2×C4—1181no—
215C2×C4—1182no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4—1182no—
220C2×C4—1182no—
221C2×C4—1182no—
222C2×C4—1182no—
223C2×C4#1174568514yes—
224C2×C4C2×C22⋊D5912yes—
225C2×C4C2×C22⋊D5912yes—
226C2×C4C2×C22⋊D5912yes—
227C22—1182no—
228C22—1184no—
229C22—2361no—
230C22—2361no—
231C22—2362no—
232C22C22×C22⋊D5911yescharacteristic
233C22#41×D5911yescharacteristic
234C22#41×D5911yescharacteristic
235C22C2×#1174568512yes—
236C22#1427377512yes—
237C4—22no—
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—1182no—
243C4—2361no—
244C4—2361no—
245C4—2362no—
246C4—2362no—
247C2—2362no—
248C2#1683790911yescharacteristic
249C2#1683791311yescharacteristic
250C2#1683797611yescharacteristic
251C2#1683809312yes—
252C2#1683812412yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×C4×Q8×C59C2
hypercenterC2×C4×Q8D59
hypocenterC59#2599
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #16837333C2semidirect
4. #16837321C2semidirect
7. #16837286C2semidirect
8. #16837283C2semidirect
10. #16837270C2semidirect
11. C2×C4×Q8×C59C2semidirect
127. C2×C4×Q8D59semidirect
138. C59#2599semidirect