Finite Group Databasebeta

← #21424511

Subgroups — #21424511

Order 7552.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#16838168C211yesmaximal, characteristic, complemented
2#16837913C211yesmaximal, characteristic
3#16837384C211yesmaximal, characteristic, complemented
4#16837366C211yesmaximal, characteristic, complemented
5#16837174C211yesmaximal, characteristic, complemented
6#16837173C211yesmaximal, characteristic, complemented
7C59×#407C211yesmaximal, characteristic, complemented
8C2×#11745685C2211yescharacteristic
9#14273735—22no—
10#14273733—22no—
11C2×#11745664C2211yescharacteristic
12C2×#11745664C2211yescharacteristic
13M16×D59—22no—
14#14273710C412yes—
15#14273706—21no—
16#14273701—21no—
17#14273696C412yes—
18C4×C22⋊D59—22no—
19#14273686C2211yescharacteristic
20#14273645—21no—
21#14273643—21no—
22C22⋊(C59⋊C8)—22no—
23C23×C4×C59C2211yescharacteristic
24C2×M16×C59C2211yescharacteristic
25C2×M16×C59C2211yescharacteristic
26C22⋊C8×C59—22no—
27C24×C59C2×C411yescharacteristic
28C22×C4×C59—21no—
29C22×C4×C59—21no—
30C22×C4×C59—21no—
31C22×C4×C59—21no—
32C22×C4×C59—21no—
33C22×C4×C59—21no—
34C22×C4×C59C2×C411yescharacteristic
35C22×C4×C59C2311yescharacteristic
36C2×C22⋊D59—22no—
37C2×C22⋊D59C2×C412yes—
38#11745685—21no—
39#11745685—21no—
40#11745685—21no—
41#11745685—21no—
42C2×D236D411yescharacteristic
43C2×C4×D59D412yes—
44C2×Dic118D411yescharacteristic
45M16×C59—21no—
46M16×C59—21no—
47M16×C59—21no—
48M16×C59—21no—
49C2×C8×C59—21no—
50C2×C8×C59D412yescomplemented
51C22⋊Dic59—22no—
52C22⋊Dic59C2×C412yes—
53(C2×D59)⋊C4—22no—
54C4⋊Dic59—21no—
55Dic59⋊C4—22no—
56C4×Dic59—21no—
57#11745664—21no—
58#11745664—21no—
59#11745664—21no—
60#11745664—21no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8D412yescomplemented
63#11745659—24no—
64C8×D59—24no—
65C23×C59—21no—
66C23×C59—21no—
67C23×C59—21no—
68C23×C59—42no—
69C23×C59C22×C411yescharacteristic
70C22×D59C22⋊C412yes—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2×D411yescharacteristic
81C2×C4×C59C2×D411yescharacteristic
82C2×C4×C59#4111yescharacteristic
83C2×C4×C59#4111yescharacteristic
84C22⋊D59—22no—
85C22⋊D59—22no—
86C22⋊D59—44no—
87C2×Dic59—22no—
88C2×Dic59C22⋊C412yes—
89D236—21no—
90D236—21no—
91C4×D59—22no—
92C4×D59—22no—
93Dic118—21no—
94Dic118—21no—
95C8×C59—22no—
96C8×C59—41no—
97C59⋊C8—22no—
98C59⋊C8—41no—
99C22×C59—22no—
100C22×C59—22no—
101C22×C59—22no—
102C22×C59—41no—
103C22×C59—41no—
104C22×C59—41no—
105C22×C59—41no—
106C22×C59—41no—
107C22×C59C2×C22⋊C411yescharacteristic
108C22×C59C4×D411yescharacteristic
109C22×C59C4×D411yescharacteristic
110C2×D59—22no—
111C2×D59—42no—
112C4×C59—41no—
113C4×C59#12011yescharacteristic
114C4×C59#12111yescharacteristic
115C4×C59#12111yescharacteristic
116C4×C59#12311yescharacteristic
117Dic59—22no—
118Dic59—42no—
119#1891—591nomaximal, Sylow
120C2×C59—21no—
121C2×C59—21no—
122C2×C59—21no—
123C2×C59—42no—
124C2×C59#38611yescharacteristic
125D59—42no—
126#568—591no—
127#525—591no—
128#413—591no—
129#407—591no—
130#407D5911yescharacteristic, complemented
131#344—591no—
132#344—591no—
133C59#189111yescharacteristic, complemented, Sylow
134C2×#41—591no—
135C23×C4C2×D5911yescharacteristic
136#131—1182no—
137#131—1182no—
138C2×M16—591no—
139C2×M16—591no—
140C2×M16—1182no—
141C2×M16C2×D5911yescharacteristic
142C2×M16C2×D5911yescharacteristic
143#123—592no—
144C22⋊Q8—1181no—
145#121—1181no—
146#120—592no—
147C4×D4—1182no—
148#117—591no—
149#101—1181no—
150#100—1181no—
151C22⋊C8—22no—
152C22⋊C8—1182no—
153C24C4×D5911yescharacteristic
154#41—1181no—
155#41—1181no—
156#41—1181no—
157#41—1181no—
158C2×Q8—591no—
159C2×D4—592no—
160C2×D4—591no—
161C2×D4—1182no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—592no—
169C22×C4C4×D5911yescharacteristic
170C22×C4C22×D5911yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—1184no—
176M16—1181no—
177M16—1181no—
178M16—1181no—
179M16—1181no—
180C2×C8—21no—
181C2×C8—592no—
182C2×C8—1184no—
183C2×C8—1181no—
184C2×C8C22⋊D5912yescomplemented
185C4⋊C4—1182no—
186C4⋊C4—1181no—
187C22⋊C4—592no—
188C22⋊C4—1182no—
189C22⋊C4—1182no—
190C42—1181no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—592no—
196C23C2×C4×D5911yescharacteristic
197Q8—1181no—
198Q8—1181no—
199D4—1182no—
200D4—1182no—
201D4—1181no—
202D4—1181no—
203D4—2364no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—592no—
212C2×C4—1182no—
213C2×C4—1182no—
214C2×C4—1182no—
215C2×C4#1174568511yescharacteristic
216C2×C4D4×D5911yescharacteristic
217C2×C4D4×D5911yescharacteristic
218C2×C4D4×D5911yescharacteristic
219C2×C4#1174568711yescharacteristic
220C2×C4C2×C22⋊D5911yescharacteristic
221C8—22no—
222C8—41no—
223C8—1182no—
224C8—2361no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—1182no—
234C22—2362no—
235C22C22⋊C4×D5911yescharacteristic
236C22(C22⋊D59)⋊C411yescharacteristic
237C22C4×C22⋊D5911yescharacteristic
238C4—41no—
239C4—1182no—
240C4—2362no—
241C4#1427369511yescharacteristic
242C4#1427369811yescharacteristic
243C4#1427370011yescharacteristic
244C4#1427370911yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—2362no—
250C2#1683727811yescharacteristic
subgroupG/H
centerC4#14273695
commutatorC23×C59C22×C4
FrattiniC22×C4C22×D59
FittingC59×#407C2
socleC2×C59#386
hypercenter#407D59
hypocenterC59#1891
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16838168C2semidirect
3. #16837384C2semidirect
4. #16837366C2semidirect
5. #16837174C2semidirect
6. #16837173C2semidirect
7. C59×#407C2semidirect
50. C2×C8×C59D4semidirect
62. C2×C59⋊C8D4semidirect
130. #407D59semidirect
133. C59#1891semidirect
184. C2×C8C22⋊D59semidirect