Finite Group Databasebeta

← #27293717

Subgroups — #27293717

Order 13568.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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#20273053C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272649C214yesmaximal, complemented
4C53×#2547C211yesmaximal, characteristic
5#16810587—21no—
6C2×C22⋊#88322—21no—
7#16810349C2212yes—
8#16810349C2212yes—
9#16810347—21no—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×#124×C53C412yes—
13C2×C4×D4×C53C2211yescharacteristic
14C53×#397—21no—
15C53×#397—21no—
16C53×#391C2212yes—
17C53×#380—21no—
18C53×#380—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—41no—
21C22×#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24C22⋊#88322—41no—
25C22⋊#88322—41no—
26C22×Q8×C53C2×C412yes—
27C22×D4×C53C2×C411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32#124×C53—44no—
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C42×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45#95×C53—21no—
46#95×C53—22no—
47#95×C53—22no—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C22⋊C411yescharacteristic
50C2×Q8×C53—24no—
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—41no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—44no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—42no—
81#11719917—82no—
82C2×#88322—42no—
83C2×#88322—42no—
84C2×#88322—82no—
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
96Q8×C53—84no—
97D4×C53—44no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53—81no—
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53#13112yes—
120#88322—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—82no—
144C4×C53—82no—
145C4×C53#35412yes—
146#2547D5311yescharacteristic
147#1465—531no—
148#1459—531no—
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⋊Q8Dic5311yescharacteristic
158C2×#124Dic5312yes—
159C2×C4×D4C2×D5311yescharacteristic
160#407—1061no—
161C2×C22⋊C8—1061no—
162#397—21no—
163#397—21no—
164#391C2×D5312yes—
165#380—21no—
166#380—21no—
167C4×C4⋊C4Dic5311yescharacteristic
168#324—532no—
169#324—532no—
170#323—1061no—
171#323—1061no—
172C53#1156411yescharacteristic, complemented, Sylow
173C22×Q8C2×Dic5312yes—
174C22×D4C2×Dic5311yescharacteristic
175C23×C4C22⋊D5311yescharacteristic
176C23×C4C22⋊D5311yescharacteristic
177C2×M16—2121no—
178C22×C8—2121no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181#124—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22×D5311yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊D5311yescharacteristic
191C2×C22⋊C4C22⋊D5311yescharacteristic
192C2×C42C2×Dic5311yescharacteristic
193C2×C42C2×Dic5312yes—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196C22⋊C8—2121no—
197C22⋊C8—2121no—
198#95—22no—
199#95—22no—
200#95—21no—
201C24C22⋊Dic5311yescharacteristic
202C24C22⋊Dic5311yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22⋊Dic5311yescharacteristic
219C22×C4C22×Dic5311yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222M16—4242no—
223C2×C8—4242no—
224C2×C8—2122no—
225C2×C8—2122no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1313313412yes—
248C23C2×C22⋊Dic5311yescharacteristic
249Q8—84no—
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—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#1313313211yescharacteristic
271C2×C4#1313313211yescharacteristic
272C2×C4#1313322412yes—
273C8—4242no—
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#1681059011yescharacteristic
288C22C2×#1313313211yescharacteristic
289C22C2×#1313313411yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#1681041812yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2027307511yescharacteristic
303C2C2×#1681041811yescharacteristic
304C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2547C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2547D53
hypocenterC53#11564
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #20272649C2semidirect
172. C53#11564semidirect

Maximal subgroups

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

subgroupindex
#202730532
#202730332
#202726492
C53×#25472
#1156453