Finite Group Databasebeta

← #27293716

Subgroups — #27293716

Order 13568.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279707C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272648C214yesmaximal, complemented
4C53×#1459C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133204C412yes—
7#16810587—21no—
8#16810548—21no—
9#16810548—21no—
10#16810528C2212yes—
11#16810483—21no—
12#16810483—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C2×C22⋊C8×C53—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C22×Dic106C2×C412yes—
21C2×#11719917—41no—
22#13133217—41no—
23#13133217—41no—
24#13133204—44no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53C2×C412yes—
31C22⋊#88322—22no—
32C22⋊#88322—41no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C22×C8×C53—41no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C22⋊C8×C53—22no—
47C22⋊C8×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C22⋊C411yescharacteristic
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C2×Dic106—24no—
67C2×Dic106—42no—
68C2×Dic106—42no—
69C2×C8×C53—42no—
70C2×C8×C53—82no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76C22⋊Dic53—44no—
77C22⋊Dic53—44no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4×Dic53—42no—
83#11719917—82no—
84C2×#88322—42no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#10012yes—
95C23×C53C2×C22⋊C411yescharacteristic
96D4×C53—44no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53#13112yes—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119Dic106—84no—
120C8×C53—82no—
121#88322—82no—
122#11564—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—81no—
134C22×C53—81no—
135C22×C53#40811yescharacteristic
136C22×C53C2×#9911yescharacteristic
137C22×C53C2×#10011yescharacteristic
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53#35412yes—
144Dic53—82no—
145Dic53—82no—
146#2547—531no—
147#1465—531no—
148#1459D5311yescharacteristic
149#1319—534no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—42no—
153C2×C53—81no—
154C2×C53#147111yescharacteristic
155C2×C53C2×#35411yescharacteristic
156C2×C53#213511yescharacteristic
157C2×C4⋊Q8—531no—
158C2×#124—532no—
159C2×C4×D4C2×D5311yescharacteristic
160#407—1061no—
161C2×C22⋊C8—21no—
162#397—1061no—
163#397—1061no—
164#391—532no—
165#380—1061no—
166#380—1061no—
167C4×C4⋊C4—531no—
168#324—532no—
169#324C2×D5312yes—
170#323—21no—
171#323—1061no—
172C53#1156411yescharacteristic, complemented, Sylow
173C22×Q8—532no—
174C22×D4C4×D5311yescharacteristic
175C23×C4D21211yescharacteristic
176C23×C4C22⋊D5311yescharacteristic
177C2×M16—2121no—
178C22×C8—41no—
179C4⋊Q8—2121no—
180C4⋊Q8—2121no—
181#124—2124no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4C22×D5311yescharacteristic
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4D21211yescharacteristic
191C2×C22⋊C4C22⋊D5311yescharacteristic
192C2×C42—532no—
193C2×C42C4×D5311yescharacteristic
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—1062no—
197C22⋊C8—2121no—
198#95—1062no—
199#95—1062no—
200#95—1061no—
201C24#1171992111yescharacteristic
202C24#1171992111yescharacteristic
203C2×Q8—1064no—
204C2×Q8—2122no—
205C2×Q8—2122no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4#1171992111yescharacteristic
218C22×C4#1171992111yescharacteristic
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4C2×D21211yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222M16—4242no—
223C2×C8—42no—
224C2×C8—82no—
225C2×C8—2122no—
226C4⋊C4—41no—
227C4⋊C4—2122no—
228C4⋊C4—2122no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—2124no—
234C22⋊C4—2124no—
235C42—21no—
236C42—21no—
237C42—2122no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1313313312yes—
248C23C2×#1171992111yescharacteristic
249Q8—4244no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4#1313313111yescharacteristic
271C2×C4#1313313111yescharacteristic
272C2×C4#1313322312yes—
273C8—82no—
274C8—4242no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1681058911yescharacteristic
288C22C2×#1313313111yescharacteristic
289C22C2×#1313313311yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—4242no—
296C4—4242no—
297C4#1681041712yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2027307411yescharacteristic
303C2C2×#1681041711yescharacteristic
304C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1459C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1459D53
hypocenterC53#11564
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272648C2semidirect
172. C53#11564semidirect

Maximal subgroups

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

subgroupindex
#202797072
#202730532
#202726482
C53×#14592
#1156453