Finite Group Databasebeta

← #27303759

Subgroups — #27303759

Order 13568.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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#20274995C212yesmaximal
2#20274924C211yesmaximal, characteristic
3#20272840C211yesmaximal, characteristic
4#20272833C212yesmaximal
5C53×#2683C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810663—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810496—22no—
11#16810389—22no—
12#16810383—21no—
13C2×C22⋊Q8×C53—22no—
14C53×#396—22no—
15C53×#395—21no—
16C53×#390C2211yescharacteristic
17C53×#389—21no—
18C53×#386C2212yes—
19C2×#11719917—21no—
20D4×Dic53—41no—
21D4×Dic53—41no—
22#13133176—21no—
23C2×#11719919—42no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26#13133150—41no—
27#13133148—41no—
28#13133134—21no—
29#13133134Q812yescomplemented
30#13133132—22no—
31#13133132Q814yescomplemented
32#13133124—42no—
33C22×Q8×C53—21no—
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53D412yes—
36C22⋊Q8×C53—48no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53D412yes—
45C2×C42×C53D411yescharacteristic
46#95×C53—21no—
47#95×C53—22no—
48C24×C53C2×D412yes—
49C2×Q8×C53—44no—
50C2×Q8×C53—81no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53C2×Q811yescharacteristic
54C2×D4×C53C2×Q812yes—
55C2×D4×C53#4111yescharacteristic
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—44no—
61C22×C4×C53—42no—
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—42no—
66C4⋊C4×C53—24no—
67C4⋊C4×C53—44no—
68C4⋊C4×C53—44no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—81no—
71C22⋊C4×C53—44no—
72C22⋊C4×C53—44no—
73C22⋊C4×C53—42no—
74C42×C53—41no—
75C22⋊Dic53—24no—
76C22⋊Dic53—42no—
77C22⋊Dic53—42no—
78C22⋊Dic53—41no—
79C4⋊Dic53—21no—
80C4⋊Dic53—21no—
81#11719919—84no—
82C4×Dic53—41no—
83#11719917—22no—
84#11719917—41no—
85C2×#88322—41no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53#12011yescharacteristic
91C23×C53C22⋊Q812yes—
92C23×C53C4⋊Q812yes—
93Q8×C53—84no—
94D4×C53—41no—
95D4×C53—42no—
96D4×C53—41no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—24no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—44no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—82no—
112C2×C4×C53C22⋊Q811yescharacteristic
113C2×C4×C53C4⋊Q811yescharacteristic
114C2×Dic53—44no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—82no—
118C2×Dic53—82no—
119C2×Dic53—82no—
120C2×Dic53—82no—
121#88322—42no—
122#13461—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—82no—
132C22×C53#39511yescharacteristic
133C22×C53B(4,𝔽2)11yescharacteristic
134C22×C53#45811yescharacteristic
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—42no—
138C4×C53—44no—
139C4×C53—82no—
140C4×C53—81no—
141Dic53—84no—
142Dic53—81no—
143Dic53—81no—
144#2683D5311yescharacteristic
145#1867—532no—
146#1856—531no—
147#1393—531no—
148#1390—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—42no—
152C2×C53—42no—
153C2×C53#206211yescharacteristic
154C2×C53#219611yescharacteristic
155C2×C53#220111yescharacteristic
156C2×C22⋊Q8—22no—
157#518—531no—
158#428—1061no—
159C2×#100—531no—
160C2×#99—532no—
161#396—22no—
162#395—21no—
163#390C2×D5311yescharacteristic
164#389—21no—
165#386C2×D5312yes—
166#385—1062no—
167#342—1062no—
168#339—1061no—
169C53#1346111yescharacteristic, complemented, Sylow
170C22×Q8—21no—
171C22×D4C22×D5311yescharacteristic
172C23×C4C22⋊D5312yes—
173C2×M16—1061no—
174C22⋊Q8—48no—
175C4×D4—2121no—
176C4×D4—2121no—
177#117—1061no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—2122no—
183C2×C4⋊C4C22⋊D5311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—1062no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4C22⋊D5312yes—
189C2×C42C22⋊D5311yescharacteristic
190#107—2121no—
191#106—2121no—
192#100—532no—
193#100—1061no—
194#99—534no—
195#99—1062no—
196#95—21no—
197#95—22no—
198#95—2122no—
199C24C2×C22⋊D5312yes—
200C2×Q8—44no—
201C2×Q8—81no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4Q8×D5311yescharacteristic
205C2×D4Q8×D5312yes—
206C2×D4#1171994911yescharacteristic
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—2122no—
216C22×C4C2×C22⋊D5311yescharacteristic
217M16—1062no—
218M16—2121no—
219C2×C8—2121no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—81no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—4244no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—1064no—
232C22⋊C4—2122no—
233C22⋊C4—2122no—
234C22⋊C4—2121no—
235C42—41no—
236C42—2121no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1313318611yescharacteristic
242C23#1313319312yes—
243C23#1313321512yes—
244Q8—84no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—4242no—
264C2×C4—4242no—
265C2×C4—4242no—
266C2×C4—4242no—
267C2×C4—2124no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4#1313319311yescharacteristic
271C2×C4#1313321511yescharacteristic
272C8—2122no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1681054211yescharacteristic
283C22#1681081011yescharacteristic
284C22#1681081311yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—44no—
289C4—82no—
290C4—81no—
291C4—4244no—
292C4—4241no—
293C4—4241no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2027606711yescharacteristic
299C2#2027669911yescharacteristic
300C2#2027672811yescharacteristic
subgroupG/H
centerC22#16810810
commutator, prime radical, cosocleC22×D4×C53C23
FrattiniC22×D4C22×D53
FittingC53×#2683C2
socleC22×C53B(4,𝔽2)
hypercenter#2683D53
hypocenterC53#13461
LCSG ▷ C22×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
29. #13133134Q8semidirect
31. #13133132Q8semidirect
169. C53#13461semidirect

Maximal subgroups

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

subgroupindex
#202749952
#202749242
#202728402
#202728332
C53×#26832
#1346153