Finite Group Databasebeta

← #21423017

Subgroups — #21423017

Order 7552.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#16838027C211yesmaximal, characteristic, complemented
2#16837553C211yesmaximal, characteristic, complemented
3#16837310C211yesmaximal, characteristic
4#16837140C211yesmaximal, characteristic, complemented
5#16837139C211yesmaximal, characteristic, complemented
6#16837125C211yesmaximal, characteristic, complemented
7C59×#450C211yesmaximal, characteristic, complemented
8C22×C22⋊D59—21no—
9C2×#11745670—21no—
10#14273707—21no—
11#14273706C2211yescharacteristic
12#14273705—21no—
13#14273705—21no—
14#14273704—21no—
15#14273703—21no—
16C2×C22⋊Dic59—21no—
17C2×(C2×D59)⋊C4C2211yescharacteristic
18C22⋊C4×D59—21no—
19#14273648—21no—
20C22⋊(C59⋊C8)C2211yescharacteristic
21C59⋊#95C2211yescharacteristic
22C22×D4×C59C2211yescharacteristic
23C2×SD16×C59—21no—
24C22⋊Q8×C59C2211yescharacteristic
25#102×C59—21no—
26C22⋊C8×C59C2211yescharacteristic
27C24×C59—21no—
28C23×D59D411yescharacteristic
29C2×Q8×C59D411yescharacteristic, complemented
30C2×D4×C59—21no—
31C2×D4×C59—21no—
32C2×D4×C59—42no—
33C22×C4×C59C2311yescharacteristic
34C2×C22⋊D59—42no—
35C2×C22⋊D59—42no—
36C2×C22⋊D59—42no—
37C22×Dic59—21no—
38C22×Dic59D411yescharacteristic
39C2×C4×D59—21no—
40C2×C4×D59—21no—
41C2×Dic118—21no—
42C2×Dic118D411yescharacteristic, complemented
43SD16×C59—42no—
44C2×C8×C59—21no—
45C4⋊C4×C59—21no—
46C4⋊C4×C59D411yescharacteristic, complemented
47C22⋊C4×C59—21no—
48C22⋊Dic59—21no—
49C22⋊Dic59—42no—
50#11745670—42no—
51(C2×D59)⋊C4—21no—
52(C2×D59)⋊C4—21no—
53(C2×D59)⋊C4—21no—
54(C2×D59)⋊C4—21no—
55C4⋊Dic59—21no—
56C4⋊Dic59D411yescharacteristic, complemented
57Dic59⋊C4—21no—
58Dic59⋊C4—21no—
59Dic59⋊C4—21no—
60C2×C59⋊C8—21no—
61C23×C59—22no—
62C23×C59—41no—
63C23×C59—41no—
64C23×C59—41no—
65C23×C59—41no—
66C23×C59—41no—
67C23×C59—41no—
68C23×C59C2×D411yescharacteristic
69C22×D59—21no—
70C22×D59—21no—
71C22×D59—41no—
72C22×D59—41no—
73C22×D59SD1612yescomplemented
74Q8×C59—41no—
75D4×C59—22no—
76D4×C59—22no—
77D4×C59—41no—
78D4×C59—41no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—41no—
82C2×C4×C59C2×D411yescharacteristic
83C2×C4×C59C2×D411yescharacteristic
84C22⋊D59—48no—
85C2×Dic59—22no—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—21no—
89C2×Dic59—41no—
90C2×Dic59—41no—
91C2×Dic59—41no—
92C4×D59—81no—
93C4×D59—81no—
94Dic118—41no—
95Dic118—81no—
96C8×C59—41no—
97C59⋊C8—41no—
98C22×C59—21no—
99C22×C59—21no—
100C22×C59—42no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59—81no—
104C22×C59—81no—
105C22×C59—81no—
106C22×C59—81no—
107C22×C59#12011yescharacteristic
108C22×C59C2×SD1611yescharacteristic
109C22×C59#13711yescharacteristic
110C2×D59—22no—
111C2×D59—41no—
112C2×D59—41no—
113C2×D59—42no—
114C2×D59—81no—
115C4×C59—21no—
116C4×C59—21no—
117C4×C59—41no—
118C4×C59—41no—
119Dic59—42no—
120Dic59—41no—
121Dic59—41no—
122Dic59—41no—
123#1596—591nomaximal, Sylow
124C2×C59—21no—
125C2×C59—21no—
126C2×C59—82no—
127C2×C59#44811yescharacteristic
128C2×C59#45311yescharacteristic
129C2×C59B(4,𝔽2)11yescharacteristic
130D59—42no—
131#543—591no—
132#450—591no—
133#450D5911yescharacteristic, complemented
134#392—591no—
135#328—591no—
136#328—591no—
137#323—591no—
138C59#159611yescharacteristic, complemented, Sylow
139C22×D4—1181no—
140C22×D4C2×D5911yescharacteristic
141C2×SD16—21no—
142C2×SD16—1181no—
143C22⋊Q8—591no—
144C22⋊Q8—1181no—
145C22⋊Q8—1181no—
146C22⋊Q8—1181no—
147C22⋊Q8—1181no—
148C22⋊Q8—1181no—
149C22⋊Q8C2×D5911yescharacteristic
150C2×C22⋊C4—591no—
151C2×C22⋊C4—1181no—
152C2×C22⋊C4—1181no—
153#102—21no—
154#102—1181no—
155C22⋊C8—591no—
156C22⋊C8C2×D5911yescharacteristic
157#95—591no—
158C24—21no—
159C24—591no—
160C2×Q8—591no—
161C2×Q8—1181no—
162C2×Q8C22⋊D5911yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—2362no—
167C2×D4—2362no—
168C2×D4—2362no—
169C22×C4—591no—
170C22×C4—1181no—
171C22×C4—1181no—
172C22×C4—1181no—
173C22×C4C22×D5911yescharacteristic
174SD16—42no—
175SD16—2362no—
176C2×C8—21no—
177C2×C8—1181no—
178C4⋊C4—21no—
179C4⋊C4—591no—
180C4⋊C4—1181no—
181C4⋊C4—1181no—
182C4⋊C4—1181no—
183C4⋊C4—1181no—
184C4⋊C4C22⋊D5911yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—1181no—
187C22⋊C4—1181no—
188C22⋊C4—1181no—
189C22⋊C4—1181no—
190C22⋊C4—1181no—
191C22⋊C4—2362no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—592no—
200C23—1181no—
201C23—1181no—
202C23—2361no—
203C23—2361no—
204C23D4×D5911yescharacteristic
205Q8—41no—
206Q8—4721no—
207Q8—2361no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—2368no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—1182no—
217C2×C4—1181no—
218C2×C4—1181no—
219C2×C4—1181no—
220C2×C4—4721no—
221C2×C4—4721no—
222C2×C4—2361no—
223C2×C4—2361no—
224C2×C4—2361no—
225C2×C4D4×D5911yescharacteristic
226C2×C4C2×C22⋊D5911yescharacteristic
227C8—41no—
228C8—2361no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—1182no—
239C22—4721no—
240C22—2362no—
241C22—2361no—
242C22—2361no—
243C22#1427369511yescharacteristic
244C22SD16×D5911yescharacteristic
245C22#1427375611yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—2362no—
251C4—2361no—
252C4—2361no—
253C4—2361no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—2362no—
258C2#1683753911yescharacteristic
259C2#1683756811yescharacteristic
260C2#1683758811yescharacteristic
subgroupG/H
centerC22#14273695
commutatorC22×C4×C59C23
FrattiniC22×C4C22×D59
FittingC59×#450C2
socleC22×C59#120
hypercenter#450D59
hypocenterC59#1596
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16838027C2semidirect
2. #16837553C2semidirect
4. #16837140C2semidirect
5. #16837139C2semidirect
6. #16837125C2semidirect
7. C59×#450C2semidirect
29. C2×Q8×C59D4semidirect
42. C2×Dic118D4semidirect
46. C4⋊C4×C59D4semidirect
56. C4⋊Dic59D4semidirect
73. C22×D59SD16semidirect
133. #450D59semidirect
138. C59#1596semidirect
162. C2×Q8C22⋊D59semidirect
184. C4⋊C4C22⋊D59semidirect