Finite Group Databasebeta

← #27287334

Subgroups — #27287334

Order 13568.

594 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 75 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#20280385C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal
3#20272623C211yesmaximal, characteristic, complemented
4#20272567C212yesmaximal, complemented
5C53×#1782C211yesmaximal, characteristic
6C2×#13133144—22no—
7C2×#13133144—22no—
8#16810587C2212yes—
9#16810546—22no—
10#16810520—22no—
11#16810506—22no—
12C4×C22⋊Dic53C2212yes—
13C22×#41×C53C2211yescharacteristic
14C2×#99×C53—22no—
15C53×#381C2211yescharacteristic
16C53×#381C2211yescharacteristic
17C2×#11719917—22no—
18C2×C4⋊Dic53—22no—
19C2×#11719919—22no—
20C2×#11719919—22no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53D412yes—
23C2×C4×Dic53D412yes—
24#13133144—44no—
25#13133144—44no—
26C22⋊#88322—22no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30C2×#41×C53—22no—
31C2×#41×C53—22no—
32C2×#41×C53—44no—
33C22×Q8×C53Q811yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53Q811yescharacteristic
36C23×C4×C53—21no—
37C23×C4×C53C2311yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53D412yes—
40#99×C53—44no—
41#95×C53—22no—
42#95×C53—22no—
43C24×C53—21no—
44C24×C53C2×D411yescharacteristic
45#41×C53—24no—
46#41×C53—24no—
47#41×C53—42no—
48#41×C53—42no—
49#41×C53—44no—
50#41×C53—44no—
51C2×Q8×C53—22no—
52C2×Q8×C53—42no—
53C2×D4×C53—24no—
54C2×D4×C53—22no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—41no—
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×Q811yescharacteristic
68C22×C4×C53#4112yes—
69C22×C4×C53#4111yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C22⋊Dic53—44no—
75C4⋊Dic53—44no—
76#11719919—44no—
77#11719919—82no—
78C4×Dic53—42no—
79C4×Dic53—42no—
80#11719917—42no—
81#11719917—42no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53—82no—
89C23×C53#12011yescharacteristic
90C23×C53C22⋊Q811yescharacteristic
91C23×C53#12411yescharacteristic
92Q8×C53—24no—
93Q8×C53—42no—
94D4×C53—24no—
95D4×C53—42no—
96D4×C53—44no—
97D4×C53—44no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—44no—
105C2×C4×C53—44no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—82no—
112C2×C4×C53—81no—
113C2×C4×C53C22⋊Q812yes—
114C2×C4×C53#12412yes—
115C2×Dic53—24no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121#88322—82no—
122#10388—531nomaximal, Sylow
123C22×C53—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—42no—
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53#39311yescharacteristic
134C22×C53B(4,𝔽2)12yes—
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—82no—
139C4×C53—161no—
140Dic53—44no—
141Dic53—82no—
142#2608—531no—
143#1789—532no—
144#1782D5311yescharacteristic
145#1306—531no—
146#1284—532no—
147C2×C53—22no—
148C2×C53—41no—
149C2×C53—41no—
150C2×C53—82no—
151C2×C53#202911yescharacteristic
152C2×C53#203112yes—
153C22×#41C2×D5311yescharacteristic
154C2×#104—1062no—
155C2×#104—1062no—
156C2×#99—22no—
157#407—532no—
158#396—1062no—
159#389—1062no—
160#387—1062no—
161#381C2×D5311yescharacteristic
162#381C2×D5311yescharacteristic
163C4×C22⋊C4—532no—
164C53#1038811yescharacteristic, complemented, Sylow
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—44no—
168C22×Q8Dic10611yescharacteristic
169C22×D4—21no—
170C22×D4Dic10611yescharacteristic
171C23×C4—21no—
172C23×C4C22×D5311yescharacteristic
173C2×M16—1062no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4—1062no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4C22⋊D5312yes—
180C2×C42—532no—
181C2×C42—532no—
182#104—2124no—
183#104—2124no—
184#99—44no—
185C22⋊C8—1062no—
186C22⋊C8—1062no—
187#95—22no—
188#95—22no—
189#95—1062no—
190#95—1062no—
191C24—21no—
192C24C2×C22⋊D5311yescharacteristic
193#41—24no—
194#41—24no—
195#41—42no—
196#41—42no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4C2×Dic10611yescharacteristic
217C22×C4#1171994511yescharacteristic
218C22×C4D4×D5312yes—
219C22×C4#1171994712yes—
220M16—2122no—
221M16—2122no—
222C2×C8—2122no—
223C4⋊C4—4242no—
224C4⋊C4—2124no—
225C4⋊C4—2124no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—2124no—
229C42—2122no—
230C42—2122no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23#1313318511yescharacteristic
238C23#1313319611yescharacteristic
239C23#1313320411yescharacteristic
240Q8—24no—
241Q8—42no—
242D4—24no—
243D4—42no—
244D4—44no—
245D4—44no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—1064no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4#1313319412yes—
268C2×C4#1313320312yes—
269C8—4242no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22#1681053511yescharacteristic
281C22#1681080912yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—82no—
286C4—161no—
287C4—4242no—
288C4—2124no—
289C2—22no—
290C2—41no—
291C2—41no—
292C2—82no—
293C2#2027587911yescharacteristic
294C2#2027588912yes—
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1782C2
socleC22×C53#393
prime radicalC22×#41×C53C22
hypercenter#1782D53
hypocenterC53#10388
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #20280385C2semidirect
3. #20272623C2semidirect
4. #20272567C2semidirect
164. C53#10388semidirect

Maximal subgroups

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

subgroupindex
#202803852
#202746592
#202726232
#202725672
C53×#17822
#1038853