Finite Group Databasebeta

← #27272884

Subgroups — #27272884

Order 13568.

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

normal subgroups only

This group has 492 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#20276043C212yesmaximal, complemented
2#20275797C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2724C211yesmaximal, characteristic
6C22×#11719923—21no—
7C2×#13133217C2211yescharacteristic
8C2×#13133150—22no—
9C2×#13133140—21no—
10#16810385—22no—
11#16810378C2212yes—
12#16810378C2211yescharacteristic
13C2×#123×C53—21no—
14C53×#398—21no—
15C53×#397—22no—
16C53×#390C2211yescharacteristic
17C53×#389C2212yes—
18C53×#385—22no—
19C22×Dic106—21no—
20C2×#11719923—42no—
21C2×#11719923—42no—
22C2×#11719923—42no—
23C22×#88322—21no—
24C22×#88322—22no—
25#13133217—42no—
26C2×C4⋊Dic53—22no—
27#13133150—44no—
28#13133140—42no—
29C22×D4×C53D411yescharacteristic
30C23×C4×C53—21no—
31#123×C53—42no—
32#123×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C4⋊C4×C53D412yes—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53C2311yescharacteristic
42#95×C53—22no—
43#95×C53—21no—
44#95×C53—22no—
45C24×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53D812yes—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—81no—
55C22×C4×C53—81no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—41no—
59C2×Dic106—22no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—81no—
72C22⋊C4×C53—44no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C42×C53—41no—
77#11719923—48no—
78C4⋊Dic53—24no—
79C4⋊Dic53—42no—
80C2×#88322—22no—
81C2×#88322—24no—
82C2×#88322—42no—
83C2×#88322—41no—
84C2×#88322—41no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—41no—
89C23×C53—42no—
90C23×C53—41no—
91C23×C53#12011yescharacteristic
92D4×C53—42no—
93D4×C53—81no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—84no—
107C2×C4×C53C2×D811yescharacteristic
108C2×C4×C53#13512yes—
109C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
110C2×C4×C53#13712yes—
111C2×Dic53—42no—
112C2×Dic53—81no—
113C2×Dic53—81no—
114Dic106—42no—
115Dic106—42no—
116Dic106—81no—
117Dic106—81no—
118#88322—44no—
119#88322—42no—
120#7751—531nomaximal, Sylow
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—82no—
124C22×C53—81no—
125C22×C53—81no—
126C22×C53#44711yescharacteristic
127C22×C53#45212yes—
128C22×C53#45311yescharacteristic
129C22×C53#45612yes—
130C22×C53B(4,𝔽2)11yescharacteristic
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—82no—
136C4×C53—81no—
137C4×C53—82no—
138Dic53—82no—
139#2724D5311yescharacteristic
140#2057—532no—
141#2013—531no—
142#1271—532no—
143#1270—531no—
144C2×C53—82no—
145C2×C53#159511yescharacteristic
146C2×C53#160312yes—
147C2×C53#163011yescharacteristic
148C2×C53#164712yes—
149C2×C53#166011yescharacteristic
150C22×SD16—1061no—
151C2×C4⋊Q8—531no—
152C2×#123—21no—
153C2×#107—1062no—
154C2×#103—1061no—
155#398—21no—
156#397—22no—
157#390C2×D5311yescharacteristic
158#389C2×D5312yes—
159#385—22no—
160#340—1062no—
161#336—532no—
162#336—531no—
163C53#775111yescharacteristic, complemented, Sylow
164C22×Q8—1061no—
165C22×D4C22⋊D5311yescharacteristic
166C23×C4—21no—
167C2×SD16—2122no—
168C2×SD16—2122no—
169C2×SD16—2122no—
170C22×C8—1061no—
171C22×C8—1062no—
172C4⋊Q8—2122no—
173#123—42no—
174#123—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—1062no—
178C2×C4⋊C4C22⋊D5312yes—
179C2×C4⋊C4C22⋊D5312yes—
180C2×C4⋊C4C22⋊D5311yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C42C22×D5311yescharacteristic
185#107—2124no—
186#103—2122no—
187#95—22no—
188#95—21no—
189#95—22no—
190C24—21no—
191C2×Q8—1062no—
192C2×Q8—2121no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#1171992212yes—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—2121no—
208C22×C4C2×C22⋊D5311yescharacteristic
209C22×C4C2×C22⋊D5312yes—
210SD16—2128no—
211C2×C8—1062no—
212C2×C8—1064no—
213C2×C8—2122no—
214C2×C8—2121no—
215C2×C8—2121no—
216C2×C8—2122no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—1064no—
225C4⋊C4—2122no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#1313318611yescharacteristic
237Q8—4241no—
238Q8—4241no—
239Q8—2122no—
240Q8—2122no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—84no—
256C2×C4—4241no—
257C2×C4—4241no—
258C2×C4—2122no—
259C2×C4C2×#1171992211yescharacteristic
260C2×C4#1313323712yes—
261C2×C4#1313324311yescharacteristic
262C2×C4#1313324712yes—
263C8—2124no—
264C8—2122no—
265C22—44no—
266C22—42no—
267C22—82no—
268C22—81no—
269C22—81no—
270C22#1681075711yescharacteristic
271C22#1681078512yes—
272C22#1681079111yescharacteristic
273C22#1681080612yes—
274C22#1681081011yescharacteristic
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—82no—
280C4—81no—
281C4—82no—
282C4—4242no—
283C2—82no—
284C2#2027363311yescharacteristic
285C2#2027367812yes—
286C2#2027383611yescharacteristic
287C2#2027395012yes—
288C2#2027402611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2724C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2724D53
hypocenterC53#7751
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #20276043C2semidirect
3. #20272538C2semidirect
163. C53#7751semidirect

Maximal subgroups

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

subgroupindex
#202760432
#202757972
#202725382
#202725362
C53×#27242
#775153