Finite Group Databasebeta

← #28303085

Subgroups — #28303085

Order 13568.

486 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290595C212yesmaximal, complemented
2#20289972C212yesmaximal
3#20289587C212yesmaximal, complemented
4C2×#16810688C211yesmaximal, characteristic
5C2×#13133288—22no—
6#16811594C2214yes—
7#16811499—22no—
8#16811468—22no—
9#16811467—22no—
10C22×C8×D53C411yescharacteristic
11C22×C8×D53C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16810688—22no—
14#16810688—22no—
15C2×#13133129C411yescharacteristic
16C2×#13133129C2211yescharacteristic
17C2×C4×#88322C411yescharacteristic
18C2×C4⋊C8×C53C411yescharacteristic
19C2×#11719936—24no—
20#13133288—44no—
21#13133274—24no—
22C22⋊#88324—24no—
23C22×D212Q811yescharacteristic
24C22×C4×D53C2×C411yescharacteristic
25C22×C4×D53C2311yescharacteristic
26C22×#88322C2×C411yescharacteristic
27C22×#88322C2×C411yescharacteristic
28C2×C8×D53—21no—
29C2×C8×D53—21no—
30C2×C8×D53—42no—
31C2×C8×D53—42no—
32C2×C8×D53D412yes—
33C2×C8×D53D412yes—
34C4×D212—22no—
35C4×D212—22no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719921D411yescharacteristic
38C2×#11719921Q811yescharacteristic
39#13133129—22no—
40#13133129—22no—
41C4×#88322—21no—
42C4×#88322—21no—
43C22×C8×C53C2×C411yescharacteristic
44C22×C8×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C4⋊C8×C53—22no—
47C23×D53C4⋊C411yescharacteristic
48C23×D53C2×Q811yescharacteristic
49C22×C4×C53C4⋊C411yescharacteristic
50C22×C4×C53C4⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53C4⋊C411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C2×D212—22no—
55C2×D212—22no—
56C2×D212—22no—
57C2×C4×D53—21no—
58C2×C4×D53—21no—
59C2×C4×D53—41no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×C4×D53C2×D412yes—
64C2×C4×D53#4112yes—
65C2×#88324—44no—
66#11719936—44no—
67#11719936—44no—
68C2×C8×C53—21no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53C22⋊C412yes—
72C42×C53—21no—
73C42×C53—21no—
74#11719921—22no—
75#11719921—22no—
76C4⋊Dic53—22no—
77C2×#88322—22no—
78C2×#88322—21no—
79C2×#88322—41no—
80C2×#88322C22⋊C411yescharacteristic
81C2×#88322C22⋊C411yescharacteristic
82C8×D53—24no—
83C8×D53—42no—
84C8×D53—42no—
85C8×D53—42no—
86C23×C53C2×C4⋊C411yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53#10812yescomplemented
96C22×D53C22⋊Q812yes—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53#10812yescomplemented
104C2×C4×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53#11711yescharacteristic
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53#12112yes—
110D212—28no—
111C4×D53—22no—
112C4×D53—22no—
113C4×D53—42no—
114C4×D53—42no—
115C4×D53—41no—
116C4×D53—41no—
117#88324—84no—
118C8×C53—22no—
119C8×C53—41no—
120#88322—21no—
121#88322—41no—
122#88322#9912yes—
123#8611—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53#38011yescharacteristic
127C22×C53C2×#10811yescharacteristic
128C22×C53C2×#10811yescharacteristic
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—22no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—42no—
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53C2×#9911yescharacteristic
143C4×C53#41011yescharacteristic
144Dic53—22no—
145Dic53—41no—
146C2×C8×D4—531no—
147#1810—532no—
148#1465—532no—
149#1293—532no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53#179911yescharacteristic
153C2×C53#181011yescharacteristic
154C2×C53#181311yescharacteristic
155D53—42no—
156D53—42no—
157C23×C8—531no—
158C23×C8—531no—
159C2×C4×D4—531no—
160C8×D4—1062no—
161C8×D4—1062no—
162C2×#108—1062no—
163C2×C4⋊C8#615711yescharacteristic
164#407—534no—
165C2×C22⋊C8—531no—
166C2×C22⋊C8—531no—
167C2×C4×C8—531no—
168#341—1062no—
169#324—1062no—
170#323—1062no—
171C53#861111yescharacteristic, complemented, Sylow
172C22×D4—531no—
173C23×C4—531no—
174C23×C4—531no—
175C2×M16—1064no—
176C22×C8—531no—
177C22×C8—532no—
178C22×C8—532no—
179C22×C8—531no—
180C22×C8—1061no—
181C22×C8—1061no—
182C22×C8—2122no—
183C22×C8—2122no—
184C22×C8C2×#615711yescharacteristic
185C22×C8C2×#615711yescharacteristic
186C4×D4—1062no—
187C4×D4—1062no—
188C2×C4⋊C4—531no—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—531no—
191C2×C42C2×#615711yescharacteristic
192#108—2124no—
193C4⋊C8—22no—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196C22⋊C8—1064no—
197C22⋊C8—1064no—
198C4×C8—1061no—
199C4×C8—1061no—
200C24—531no—
201C24—531no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—1062no—
205C22×C4—531no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—531no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—2121no—
212C22×C4—2121no—
213C22×C4—2121no—
214C22×C4—2121no—
215C22×C4C4⋊#615711yescharacteristic
216C22×C4C4⋊#615711yescharacteristic
217C22×C4C22×#615711yescharacteristic
218M16—2124no—
219M16—2124no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—531no—
224C2×C8—531no—
225C2×C8—1064no—
226C2×C8—1062no—
227C2×C8—1061no—
228C2×C8—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C2×C8—2124no—
232C2×C8—2121no—
233C2×C8C22⋊#615712yes—
234C4⋊C4—1062no—
235C22⋊C4—1062no—
236C22⋊C4—1062no—
237C42—21no—
238C42—21no—
239C23—532no—
240C23—532no—
241C23—1061no—
242C23—1061no—
243C23—1061no—
244C23—1061no—
245C23—2121no—
246C23—2121no—
247C23—2121no—
248C23—2121no—
249C23C2×C4⋊#615711yescharacteristic
250D4—1068no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—532no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1061no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313328812yescomplemented
268C2×C4C2×C22⋊#615711yescharacteristic
269C2×C4#1313329611yescharacteristic
270C8—22no—
271C8—41no—
272C8—532no—
273C8—1061no—
274C8—4244no—
275C8—2121no—
276C22—21no—
277C22—21no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—1062no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22—2121no—
287C22—2121no—
288C22#1681156511yescharacteristic
289C22D53⋊#10811yescharacteristic
290C22C2×#1313328811yescharacteristic
291C4—21no—
292C4—22no—
293C4—41no—
294C4—1062no—
295C4—2121no—
296C4D53⋊#9911yescharacteristic
297C4#1681160411yescharacteristic
298C2—21no—
299C2—21no—
300C2—2122no—
301C2—2122no—
302C2#2029057811yescharacteristic
303C2#2029059511yescharacteristic
304C2#2029060011yescharacteristic
subgroupG/H
centerC22#16811565
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊C8×C53C4
socleC22×C53#380
prime radicalC22×D212Q8
cosocleC22×C4×D53C23
hypercenterC2×C4⋊C8#6157
hypocenterC53#8611
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20290595C2semidirect
3. #20289587C2semidirect
95. C22×D53#108semidirect
103. C2×C4×C53#108semidirect
171. C53#8611semidirect
267. C2×C4#13133288semidirect

Maximal subgroups

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

subgroupindex
#202905952
#202899722
#202895872
C2×#168106882
#861153