Finite Group Databasebeta

← #27277580

Subgroups — #27277580

Order 13568.

450 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277199C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272572C212yesmaximal, complemented
4#20272570C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8#16810680C2212yes—
9#16810522—21no—
10#16810522—21no—
11#16810498—21no—
12#16810498—21no—
13C4×C4⋊Dic53C411yescharacteristic
14C4×C4⋊Dic53C411yescharacteristic
15C4×C4⋊Dic53C2211yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17#16810349—21no—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#432C2212yes—
21C53×#324—21no—
22C53×#323—21no—
23C23×Dic53—21no—
24#13133176—42no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—42no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C4⋊#88322—22no—
38C22⋊#88322—22no—
39#13133128—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45#117×C53—22no—
46#117×C53—22no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C4⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55#97×C53—22no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C22⋊C4×C53—22no—
78C22⋊C4×C53—22no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53#4112yes—
82C42×C53#4112yes—
83C22⋊Dic53—42no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4×Dic53—42no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53#10112yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—24no—
102C2×C4×C53—24no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#10012yes—
114C2×C4×C53#11711yescharacteristic
115C2×C4×C53#11711yescharacteristic
116C2×C4×C53#12312yes—
117C2×C4×C53#12312yes—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—48no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C8×C53—82no—
132#88322—82no—
133#8743—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53#38111yescharacteristic
142C22×C53C2×#10011yescharacteristic
143C22×C53C2×#10111yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—41no—
150C4×C53—41no—
151Dic53—42no—
152Dic53—42no—
153Dic53—81no—
154Dic53—81no—
155#2296—531no—
156#1464—531no—
157#1464D5311yescharacteristic
158#1285—532no—
159#1285—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53#182111yescharacteristic
164C2×C53#182711yescharacteristic
165C2×C53#182711yescharacteristic
166C2×#117—531no—
167C2×#117C2×D5311yescharacteristic
168C22×C42—531no—
169#432—532no—
170#432C2×D5312yes—
171#389—1061no—
172#389—1061no—
173#385—1061no—
174#385—1061no—
175C4×C4⋊C4—531no—
176C4×C4⋊C4—531no—
177C4×C4⋊C4—531no—
178C4×C4⋊C4—531no—
179#324—21no—
180#324—1061no—
181#323—21no—
182#323—1061no—
183C53#874311yescharacteristic, complemented, Sylow
184C23×C4—1061no—
185C23×C4C4×D5311yescharacteristic
186#117—22no—
187#117—22no—
188#117—2122no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4D21211yescharacteristic
195C2×C4⋊C4C22⋊D5311yescharacteristic
196C2×C22⋊C4—1061no—
197C2×C22⋊C4D21211yescharacteristic
198C2×C22⋊C4C22⋊D5311yescharacteristic
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—1061no—
204C2×C42—2122no—
205C2×C42C4×D5311yescharacteristic
206C2×C42C22×D5311yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—1062no—
209C22⋊C8—22no—
210C22⋊C8—1062no—
211#97—22no—
212#97—1062no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C24#1171992111yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4#1171992111yescharacteristic
230C22×C4#1171992111yescharacteristic
231C22×C4#1171992111yescharacteristic
232C22×C4C2×C4×D5311yescharacteristic
233C22×C4C2×D21211yescharacteristic
234C22×C4C2×C22⋊D5311yescharacteristic
235C2×C8—42no—
236C2×C8—2122no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—2122no—
247C42—21no—
248C42—21no—
249C42—2121no—
250C42—2121no—
251C42—2121no—
252C42—2121no—
253C42—2122no—
254C42#1171994712yes—
255C42#1171994712yes—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#1313313512yes—
261C23C2×#1171992111yescharacteristic
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2128no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4#1313313312yes—
288C2×C4#1313317311yescharacteristic
289C2×C4#1313317311yescharacteristic
290C2×C4#1313319812yes—
291C2×C4#1313320112yes—
292C8—82no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1681048411yescharacteristic
302C22C2×#1313313311yescharacteristic
303C22C2×#1313313511yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—4241no—
312C4—4241no—
313C4—2122no—
314C4—2122no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2027479311yescharacteristic
319C2#2027481611yescharacteristic
320C2#2027482011yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1464C2
socleC22×C53#381
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1464D53
hypocenterC53#8743
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20272572C2semidirect
4. #20272570C2semidirect
183. C53#8743semidirect

Maximal subgroups

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

subgroupindex
#202771992
#202730492
#202725722
#202725702
C53×#14642
#874353