Finite Group Databasebeta

← #21429901

Subgroups — #21429901

Order 7552.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14273706C211yesmaximal, characteristic, complemented
2#16837338C212yesmaximal, complemented
3#16837325C214yesmaximal, complemented
4#16837314C212yesmaximal, complemented
5#16837275C212yesmaximal, complemented
6#16837262C211yesmaximal, characteristic, complemented
7#16837258C212yesmaximal
8C22×C22⋊C4×C59C211yesmaximal, characteristic
9C22×Dic118C2211yescharacteristic, complemented
10#14273706—24no—
11C2×C4⋊Dic59C2211yescharacteristic, complemented
12C2×C4⋊Dic59C2212yescomplemented
13C2×Dic59⋊C4C2212yescomplemented
14C2×Dic59⋊C4C2214yescomplemented
15C2×C22⋊Dic59C2212yes—
16C2×C22⋊Dic59C2212yes—
17C2×C22⋊Dic59C2214yes—
18C2×C22⋊Dic59C2214yes—
19C59⋊#95C2212yescomplemented
20C59⋊#95C2214yescomplemented
21C25×C59C2211yescharacteristic
22C23×C4×C59C2211yescharacteristic
23C23×C4×C59C2211yescharacteristic
24C2×C22⋊C4×C59—24no—
25C2×C22⋊C4×C59C2212yes—
26C2×C22⋊C4×C59C2212yes—
27C24×C59—22no—
28C24×C59—22no—
29C24×C59—42no—
30C24×C59—42no—
31C24×C59Q814yescomplemented
32C24×C59C2312yes—
33C24×C59C2311yescharacteristic
34C22×C4×C59—22no—
35C22×C4×C59—42no—
36C22×C4×C59—42no—
37C22×C4×C59D414yes—
38C22×C4×C59C2312yes—
39C22×C4×C59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×Dic59C2314yes—
42C22×Dic59C2312yes—
43C22×Dic59C2312yes—
44C2×Dic118—22no—
45C2×Dic118—42no—
46C22⋊C4×C59—24no—
47C22⋊C4×C59—24no—
48C22⋊Dic59—24no—
49C22⋊Dic59—24no—
50C22⋊Dic59—44no—
51C22⋊Dic59—44no—
52C4⋊Dic59—22no—
53C4⋊Dic59—42no—
54Dic59⋊C4—24no—
55Dic59⋊C4—44no—
56C23×C59—22no—
57C23×C59—22no—
58C23×C59—28no—
59C23×C59—22no—
60C23×C59—22no—
61C23×C59—24no—
62C23×C59—24no—
63C23×C59—48no—
64C23×C59—44no—
65C23×C59—44no—
66C23×C59—48no—
67C23×C59C2×D412yes—
68C23×C59C2×Q814yes—
69C23×C59C2×Q812yes—
70C23×C59#4112yes—
71C23×C59C2411yescharacteristic
72C2×C4×C59—22no—
73C2×C4×C59—22no—
74C2×C4×C59—22no—
75C2×C4×C59—24no—
76C2×C4×C59—44no—
77C2×C4×C59—44no—
78C2×C4×C59C2×D412yes—
79C2×C4×C59C2×D412yes—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—24no—
83C2×Dic59—24no—
84C2×Dic59—24no—
85C2×Dic59—24no—
86C2×Dic59—24no—
87Dic118—44no—
88C22×C59—22no—
89C22×C59—22no—
90C22×C59—24no—
91C22×C59—24no—
92C22×C59—24no—
93C22×C59—28no—
94C22×C59—416no—
95C22×C59—42no—
96C22×C59—42no—
97C22×C59—44no—
98C22×C59C22⋊Q814yes—
99C22×C59C22×D411yescharacteristic
100C22×C59C22×Q811yescharacteristic
101C22×C59C2×#4111yescharacteristic
102C22×C59H(5,𝔽2)12yes—
103C22×C59H(5,𝔽2)12yes—
104C4×C59—22no—
105C4×C59—22no—
106C4×C59—42no—
107Dic59—42no—
108Dic59—42no—
109Dic59—44no—
110#2626—591nomaximal, Sylow
111C2×C59—24no—
112C2×C59—44no—
113C2×C59C2×C22⋊Q811yescharacteristic
114C2×C59#53411yescharacteristic
115C2×C59#53511yescharacteristic
116C2×C59#53812yes—
117C2×C59#54312yes—
118C2×C22⋊Q8—591no—
119C22×C22⋊C4D5911yescharacteristic
120#399—592no—
121#396—594no—
122#393—592no—
123#385—592no—
124#380—591no—
125#379—592no—
126C59#262611yescharacteristic, complemented, Sylow
127C25C2×D5911yescharacteristic
128C22×Q8—591no—
129C23×C4C2×D5911yescharacteristic
130C23×C4C2×D5911yescharacteristic
131C22⋊Q8—1184no—
132C2×C4⋊C4—592no—
133C2×C4⋊C4—591no—
134C2×C4⋊C4—592no—
135C2×C4⋊C4—594no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—592no—
138C2×C22⋊C4—592no—
139C2×C22⋊C4—594no—
140C2×C22⋊C4—594no—
141C2×C22⋊C4C2×D5912yes—
142C2×C22⋊C4C2×D5912yes—
143#95—592no—
144#95—594no—
145C24—22no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24Dic11814yescomplemented
150C24C22×D5912yes—
151C24C22×D5911yescharacteristic
152C2×Q8—1182no—
153C2×Q8—2362no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—594no—
158C22×C4—592no—
159C22×C4—592no—
160C22×C4C22⋊D5914yes—
161C22×C4C22×D5912yes—
162C22×C4C22×D5911yescharacteristic
163C22×C4C22×D5911yescharacteristic
164C4⋊C4—1184no—
165C4⋊C4—1182no—
166C4⋊C4—2364no—
167C4⋊C4—2362no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—1184no—
171C22⋊C4—1184no—
172C22⋊C4—2364no—
173C22⋊C4—2364no—
174C23—22no—
175C23—22no—
176C23—28no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—48no—
182C23—44no—
183C23—44no—
184C23—48no—
185C23C2×Dic11814yes—
186C23C2×Dic11812yes—
187C23#1174568512yes—
188C23C2×C22⋊D5912yes—
189C23C23×D5911yescharacteristic
190Q8—2364no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—1182no—
198C2×C4—1182no—
199C2×C4—1184no—
200C2×C4—1184no—
201C2×C4—1184no—
202C2×C4—1184no—
203C2×C4—1184no—
204C2×C4C2×C22⋊D5912yes—
205C2×C4C2×C22⋊D5912yes—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—24no—
211C22—28no—
212C22—416no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22#1427370614yes—
217C22C22×C22⋊D5911yescharacteristic
218C22C22×Dic11811yescharacteristic
219C22C2×#1174568511yescharacteristic
220C22#1427377212yes—
221C22#1427377212yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—2362no—
226C4—2362no—
227C4—2364no—
228C2—24no—
229C2—44no—
230C2C2×#1427370611yescharacteristic
231C2#1683796111yescharacteristic
232C2#1683796911yescharacteristic
233C2#1683799012yes—
234C2#1683802812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC22×C22⋊C4×C59C2
hypercenterC22×C22⋊C4D59
hypocenterC59#2626
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14273706C2semidirect
2. #16837338C2semidirect
3. #16837325C2semidirect
4. #16837314C2semidirect
5. #16837275C2semidirect
6. #16837262C2semidirect
9. C22×Dic118C22semidirect
11. C2×C4⋊Dic59C22semidirect
12. C2×C4⋊Dic59C22semidirect
13. C2×Dic59⋊C4C22semidirect
14. C2×Dic59⋊C4C22semidirect
19. C59⋊#95C22semidirect
20. C59⋊#95C22semidirect
31. C24×C59Q8semidirect
126. C59#2626semidirect
149. C24Dic118semidirect