Finite Group Databasebeta

← #21424083

Subgroups — #21424083

Order 7552.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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×(C22⋊D59)⋊C4C211yesmaximal, characteristic
2C2×#14273641C212yesmaximal
3#16837260C212yesmaximal, complemented
4#16837169C211yesmaximal, characteristic, complemented
5C59×#342C211yesmaximal, characteristic, complemented
6C22×C22⋊D59C2211yescharacteristic
7C23×Dic59C2211yescharacteristic
8C22×C4×D59C411yescharacteristic, complemented
9(C22⋊D59)⋊C4—22no—
10(C22⋊D59)⋊C4—22no—
11C2×Dic59⋊C4—22no—
12C2×Dic59⋊C4C411yescharacteristic, complemented
13C2×C22⋊Dic59C2212yes—
14C2×(C2×D59)⋊C4C411yescharacteristic, complemented
15C2×C4×Dic59C411yescharacteristic, complemented
16#14273641—24no—
17C59⋊#95—22no—
18C2×C22⋊C4×C59C2212yes—
19C2×C22⋊C4×C59C2211yescharacteristic
20C24×C59C2311yescharacteristic
21C23×D59C2×C411yescharacteristic
22C22×C4×C59—22no—
23C22×C4×C59C2×C411yescharacteristic
24C22×C4×C59C2×C411yescharacteristic
25C2×C22⋊D59—22no—
26C2×C22⋊D59—22no—
27C2×C22⋊D59D412yes—
28C2×C22⋊D59Q812yes—
29C22×Dic59—22no—
30C22×Dic59—42no—
31C22×Dic59C2×C411yescharacteristic
32C22×Dic59C2×C411yescharacteristic
33C22×Dic59C2×C411yescharacteristic
34C22×Dic59D412yes—
35C22×Dic59D412yes—
36C2×C4×D59—21no—
37C2×C4×D59—21no—
38C2×C4×D59—42no—
39C22⋊C4×C59—22no—
40C22⋊C4×C59—22no—
41C22⋊C4×C59—22no—
42C22⋊Dic59—22no—
43C22⋊Dic59—22no—
44(C2×D59)⋊C4—22no—
45Dic59⋊C4—22no—
46Dic59⋊C4—44no—
47C4×Dic59—21no—
48C4×Dic59—21no—
49C23×C59—41no—
50C23×C59—41no—
51C23×C59C22×C411yescharacteristic
52C23×C59C2×D412yes—
53C23×C59C2×D411yescharacteristic
54C23×C59C2×Q811yescharacteristic
55C23×C59#4112yes—
56C22×D59—21no—
57C22×D59—21no—
58C22×D59—41no—
59C22×D59—41no—
60C22×D59C4⋊C412yescomplemented
61C2×C4×C59—21no—
62C2×C4×C59—21no—
63C2×C4×C59—41no—
64C2×C4×C59—41no—
65C2×C4×C59—42no—
66C2×C4×C59—42no—
67C2×C4×C59—42no—
68C22⋊D59—24no—
69C22⋊D59—24no—
70C2×Dic59—21no—
71C2×Dic59—21no—
72C2×Dic59—24no—
73C2×Dic59—22no—
74C2×Dic59—41no—
75C2×Dic59—42no—
76C2×Dic59—42no—
77C2×Dic59—42no—
78C2×Dic59—42no—
79C2×Dic59C22⋊C411yescharacteristic
80C2×Dic59C22⋊C412yes—
81C2×Dic59C22⋊C411yescharacteristic
82C2×Dic59C4⋊C412yescomplemented
83C4×D59—42no—
84C4×D59—42no—
85C22×C59—22no—
86C22×C59—21no—
87C22×C59—22no—
88C22×C59—21no—
89C22×C59—41no—
90C22×C59—41no—
91C22×C59—42no—
92C22×C59C2×C22⋊C411yescharacteristic
93C22×C59C2×C4⋊C411yescharacteristic
94C22×C59#11711yescharacteristic
95C22×C59#12112yes—
96C22×C59C22⋊Q812yes—
97C2×D59—22no—
98C2×D59—42no—
99C2×D59—41no—
100C2×D59—41no—
101C2×D59—42no—
102C4×C59—41no—
103C4×C59—41no—
104C4×C59—82no—
105Dic59—22no—
106Dic59—21no—
107Dic59—22no—
108Dic59—82no—
109Dic59#9912yescomplemented
110#1799—591nomaximal, Sylow
111C2×C59—22no—
112C2×C59—21no—
113C2×C59—21no—
114C2×C59—22no—
115C2×C59#38011yescharacteristic
116C2×C59C2×#9911yescharacteristic
117C2×C59#41011yescharacteristic
118D59—42no—
119C2×C4×D4—591no—
120C2×#99—592no—
121#380—592no—
122#342—591no—
123#342D5911yescharacteristic, complemented
124C59#179911yescharacteristic, complemented, Sylow
125C22×D4—591no—
126C23×C4—591no—
127C23×C4—591no—
128C4×D4—1182no—
129C4×D4—1182no—
130C2×C4⋊C4—591no—
131C2×C4⋊C4—1182no—
132C2×C22⋊C4—591no—
133C2×C22⋊C4—592no—
134C2×C22⋊C4C2×D5912yes—
135C2×C22⋊C4C2×D5911yescharacteristic
136C2×C42—591no—
137#99—1184no—
138#95—1182no—
139C24—591no—
140C24C22×D5911yescharacteristic
141C2×D4—592no—
142C2×D4—592no—
143C2×D4—1182no—
144C2×D4—1182no—
145C22×C4—22no—
146C22×C4—591no—
147C22×C4—591no—
148C22×C4—592no—
149C22×C4—591no—
150C22×C4—592no—
151C22×C4—1181no—
152C22×C4—1181no—
153C22×C4—1182no—
154C22×C4—2362no—
155C22×C4—2362no—
156C22×C4C4×D5911yescharacteristic
157C22×C4C4×D5911yescharacteristic
158C4⋊C4—1182no—
159C4⋊C4—2364no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—1182no—
164C22⋊C4—1182no—
165C22⋊C4—1182no—
166C42—1181no—
167C42—1181no—
168C23—41no—
169C23—41no—
170C23—592no—
171C23—1181no—
172C23—1181no—
173C23—2361no—
174C23—2361no—
175C23C2×C4×D5911yescharacteristic
176C23#1174568512yes—
177C23D4×D5912yes—
178C23D4×D5911yescharacteristic
179C23Q8×D5911yescharacteristic
180D4—1184no—
181D4—1184no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—591no—
190C2×C4—591no—
191C2×C4—592no—
192C2×C4—592no—
193C2×C4—1184no—
194C2×C4—1182no—
195C2×C4—1181no—
196C2×C4—1181no—
197C2×C4—2362no—
198C2×C4—2362no—
199C2×C4—2362no—
200C2×C4—2361no—
201C2×C4—2362no—
202C2×C4—2362no—
203C2×C4—2362no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—1182no—
212C22—2362no—
213C22—2362no—
214C22—2361no—
215C22—2361no—
216C22C22⋊C4×D5911yescharacteristic
217C22C4⋊C4×D5911yescharacteristic
218C22#1427368411yescharacteristic
219C22#1427370012yes—
220C22#1427370512yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—592no—
225C4—1182no—
226C4—1181no—
227C4—1182no—
228C4—4722no—
229C2—22no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—2362no—
234C2#1683726111yescharacteristic
235C2#99×D5911yescharacteristic
236C2#1683737411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D59
commutatorC23×C59C22×C4
FrattiniC24C22×D59
FittingC59×#342C2
socleC22×C59C2×C22⋊C4
hypercenter#342D59
hypocenterC59#1799
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #16837260C2semidirect
4. #16837169C2semidirect
5. C59×#342C2semidirect
8. C22×C4×D59C4semidirect
12. C2×Dic59⋊C4C4semidirect
14. C2×(C2×D59)⋊C4C4semidirect
15. C2×C4×Dic59C4semidirect
60. C22×D59C4⋊C4semidirect
82. C2×Dic59C4⋊C4semidirect
109. Dic59#99semidirect
123. #342D59semidirect
124. C59#1799semidirect