Finite Group Databasebeta

← #28298513

Subgroups — #28298513

Order 13568.

290 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 52 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#20290445C211yesmaximal, characteristic
2#20289967C211yesmaximal, characteristic
3#20274791C211yesmaximal, characteristic
4#16811644—21no—
5C2×C22⋊#88324—21no—
6C2×#13133272C411yescharacteristic, complemented
7C2×C4×#88324C411yescharacteristic, complemented
8#16811489—21no—
9#16811469C412yescomplemented
10#16811466—21no—
11C2×#13133173C2211yescharacteristic
12#16810484—21no—
13#16810390C411yescharacteristic
14#16810389—21no—
15C53×#342C411yescharacteristic
16C22×#88324—21no—
17C22×#88324—21no—
18C4⋊#88324—41no—
19C22⋊#88324—22no—
20C22⋊#88324—22no—
21C22⋊#88324—22no—
22#13133272—41no—
23#13133272—41no—
24C4×#88324—41no—
25C23×Dic53D411yescharacteristic
26#13133173—41no—
27#13133173—41no—
28C2×#11719919C2×C411yescharacteristic
29C2×#11719919Q811yescharacteristic
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53D411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34#13133124—41no—
35#13133124—41no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53C2×C411yescharacteristic
38C24×C53C22⋊C411yescharacteristic
39C22×C4×C53—41no—
40C22×C4×C53C4211yescharacteristic
41C22×C4×C53C4⋊C411yescharacteristic
42C22×Dic53—21no—
43C22×Dic53—21no—
44C22×Dic53—21no—
45C22×Dic53—21no—
46C22×Dic53—41no—
47C22×Dic53—41no—
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C4⋊C411yescharacteristic
51C22×Dic53C4⋊C411yescharacteristic
52C2×#88324—22no—
53C2×#88324—22no—
54C2×#88324—41no—
55C2×#88324—41no—
56C2×#88324—41no—
57C2×#88324—41no—
58C2×#88324—42no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊Dic53—41no—
63C22⋊Dic53—41no—
64C22⋊Dic53—41no—
65#11719919—41no—
66#11719919—41no—
67C4×Dic53—21no—
68C4×Dic53—21no—
69C4×Dic53—21no—
70C4×Dic53—21no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—41no—
74C23×C53—41no—
75C23×C53#9511yescharacteristic
76C23×C53#10411yescharacteristic
77C23×C53#10411yescharacteristic
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—81no—
83C2×C4×C53—81no—
84C2×C4×C53—81no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—81no—
99C2×Dic53—81no—
100C2×Dic53—81no—
101C2×Dic53#9911yescharacteristic
102C2×Dic53#9911yescharacteristic
103#88324—42no—
104#88324—42no—
105#88324—82no—
106#7214—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—81no—
116C22×C53H(3,ℤ/4)11yescharacteristic
117C22×C53#33911yescharacteristic
118C22×C53#34211yescharacteristic
119C4×C53—41no—
120C4×C53—41no—
121C4×C53—161no—
122Dic53—21no—
123Dic53—21no—
124Dic53—41no—
125Dic53—41no—
126Dic53—41no—
127Dic53—161no—
128#1820—531no—
129#1750—531no—
130#1464—531no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—41no—
134C2×C53—41no—
135C2×C53#128411yescharacteristic
136C2×C53#139111yescharacteristic
137C2×C53#139511yescharacteristic
138C2×#117—531no—
139#432—1061no—
140C2×C22⋊C8—1061no—
141C2×#97—531no—
142C2×C4×C8—531no—
143#381—1061no—
144#342—531no—
145#342—1061no—
146#342#615711yescharacteristic
147#336—1061no—
148#324—532no—
149#323—1061no—
150C53#721411yescharacteristic, complemented, Sylow
151C23×C4—531no—
152C22×C8—1061no—
153C22×C8—1061no—
154#117—2121no—
155#117—2121no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—531no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—531no—
160C2×C22⋊C4—1061no—
161C2×C22⋊C4C2×#615711yescharacteristic
162C2×C42—531no—
163C2×C42—531no—
164C4⋊C8—2121no—
165C22⋊C8—1062no—
166C22⋊C8—1062no—
167C22⋊C8—1062no—
168#97—2121no—
169#97—2121no—
170C4×C8—2121no—
171#95—2121no—
172#95—2121no—
173C24C22⋊#615711yescharacteristic
174C22×C4—41no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—1061no—
180C22×C4—1061no—
181C22×C4—1061no—
182C22×C4—1061no—
183C22×C4—2121no—
184C22×C4—2121no—
185C22×C4C4×#615711yescharacteristic
186C22×C4C4⋊#615711yescharacteristic
187C2×C8—1062no—
188C2×C8—1062no—
189C2×C8—2122no—
190C2×C8—2121no—
191C2×C8—2121no—
192C2×C8—2121no—
193C2×C8—2121no—
194C4⋊C4—2121no—
195C4⋊C4—2121no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—2121no—
200C22⋊C4—2121no—
201C22⋊C4—2121no—
202C42—1061no—
203C42—1061no—
204C42—1061no—
205C42—1061no—
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23(C2×#6157)⋊C411yescharacteristic
211C23#1313328111yescharacteristic
212C23#1313328211yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—531no—
221C2×C4—531no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—4241no—
230C2×C4—4241no—
231C2×C4—4241no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2121no—
238C8—4242no—
239C8—2122no—
240C8—2122no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—81no—
250C22#1681149011yescharacteristic
251C22#1681149311yescharacteristic
252C22#1681150011yescharacteristic
253C4—41no—
254C4—41no—
255C4—161no—
256C4—8481no—
257C4—1061no—
258C4—1061no—
259C4—2121no—
260C4—2121no—
261C4—2121no—
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2#2028956211yescharacteristic
267C2#2028980211yescharacteristic
268C2#2028981411yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C22⋊C4C2×#6157
FittingC53×#342C4
socleC22×C53#342
prime radicalC24×C53C22⋊C4
cosocleC2×#13133173C22
hypercenter#342#6157
hypocenterC53#7214
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
6. C2×#13133272C4semidirect
7. C2×C4×#88324C4semidirect
9. #16811469C4semidirect
150. C53#7214semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202904452
#202899672
#202747912
#721453