Finite Group Databasebeta

← #27303771

Subgroups — #27303771

Order 13568.

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

normal subgroups only

This group has 484 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×#2804C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810663—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810496—22no—
11#16810389—22no—
12#16810383—21no—
13C53×#400—21no—
14C53×#393—22no—
15C53×#391—21no—
16C53×#390C2211yescharacteristic
17C53×#386C2212yes—
18C53×#385—22no—
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—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53D412yes—
42C2×C42×C53D411yescharacteristic
43#95×C53—22no—
44#95×C53—22no—
45#95×C53—21no—
46#95×C53—22no—
47C24×C53C2×D412yes—
48C2×Q8×C53—42no—
49C2×Q8×C53—81no—
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—21no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—82no—
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—81no—
68C4⋊C4×C53—81no—
69C22⋊C4×C53—44no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—82no—
72C42×C53—41no—
73C22⋊Dic53—24no—
74C22⋊Dic53—42no—
75C22⋊Dic53—42no—
76C22⋊Dic53—41no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79#11719919—84no—
80C4×Dic53—41no—
81#11719917—22no—
82#11719917—41no—
83C2×#88322—41no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53#12011yescharacteristic
89C23×C53C22⋊Q812yes—
90C23×C53C4⋊Q812yes—
91Q8×C53—84no—
92D4×C53—41no—
93D4×C53—42no—
94D4×C53—41no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—44no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—82no—
110C2×C4×C53C22⋊Q811yescharacteristic
111C2×C4×C53C4⋊Q811yescharacteristic
112C2×Dic53—44no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—82no—
116C2×Dic53—82no—
117C2×Dic53—82no—
118C2×Dic53—82no—
119#88322—42no—
120#13463—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—82no—
130C22×C53#39511yescharacteristic
131C22×C53B(4,𝔽2)11yescharacteristic
132C22×C53#45811yescharacteristic
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—42no—
136C4×C53—82no—
137C4×C53—81no—
138C4×C53—82no—
139Dic53—84no—
140Dic53—81no—
141Dic53—81no—
142#2804D5311yescharacteristic
143#1867—532no—
144#1856—531no—
145#1393—531no—
146#1390—532no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53—42no—
150C2×C53—42no—
151C2×C53#206211yescharacteristic
152C2×C53#219811yescharacteristic
153C2×C53#219911yescharacteristic
154#518—531no—
155#428—1061no—
156C2×#100—531no—
157C2×#99—532no—
158#400—21no—
159#393—22no—
160#391—21no—
161#390C2×D5311yescharacteristic
162#386C2×D5312yes—
163#385—22no—
164#385—1062no—
165#342—1062no—
166#339—1061no—
167C53#1346311yescharacteristic, complemented, Sylow
168C22×Q8—21no—
169C22×D4C22×D5311yescharacteristic
170C23×C4C22⋊D5312yes—
171C2×M16—1061no—
172C4×D4—2121no—
173C4×D4—2121no—
174#117—1061no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—2122no—
178C2×C4⋊C4C22⋊D5311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—1062no—
182C2×C22⋊C4—1062no—
183C2×C22⋊C4C22⋊D5312yes—
184C2×C42C22⋊D5311yescharacteristic
185#107—2121no—
186#106—2121no—
187#100—532no—
188#100—1061no—
189#99—534no—
190#99—1062no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—2122no—
196C24C2×C22⋊D5312yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D5311yescharacteristic
203C2×D4Q8×D5312yes—
204C2×D4#1171994911yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—2122no—
214C22×C4C2×C22⋊D5311yescharacteristic
215M16—1062no—
216M16—2121no—
217C2×C8—2121no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—1061no—
222C4⋊C4—1061no—
223C4⋊C4—4244no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2121no—
231C42—41no—
232C42—2121no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1313318611yescharacteristic
238C23#1313319312yes—
239C23#1313321512yes—
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—4242no—
260C2×C4—4242no—
261C2×C4—4242no—
262C2×C4—4242no—
263C2×C4—2124no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4#1313319311yescharacteristic
267C2×C4#1313321511yescharacteristic
268C8—2122no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1681054211yescharacteristic
279C22#1681081011yescharacteristic
280C22#1681081311yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—82no—
287C4—4244no—
288C4—4241no—
289C4—4241no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2027606711yescharacteristic
295C2#2027671111yescharacteristic
296C2#2027671611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, prime radical, cosocleC22×D4×C53C23
FrattiniC22×D4C22×D53
FittingC53×#2804C2
socleC22×C53B(4,𝔽2)
hypercenter#2804D53
hypocenterC53#13463
LCSG ▷ C22×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2804

Complements

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

NH ≅ G/Ntype
29. #13133134Q8semidirect
31. #13133132Q8semidirect
167. C53#13463semidirect

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×#28042
#1346353