Finite Group Databasebeta

← #27272906

Subgroups — #27272906

Order 13568.

480 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 83 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#20282661C211yesmaximal, characteristic
2#20275853C212yesmaximal, complemented
3#20272540C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2013C211yesmaximal, characteristic
6C2×#13133140—22no—
7#16810562—21no—
8#16810535—22no—
9#16810529C2212yes—
10#16810503—21no—
11#16810498—22no—
12#16810385—22no—
13#16810378C2212yes—
14C22×SD16×C53—21no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C2×#103×C53—21no—
17C53×#390C2211yescharacteristic
18C53×#336C2211yescharacteristic
19C22×Dic106D412yes—
20C23×Dic53—21no—
21C22×#88322—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—21no—
26#13133140—44no—
27#13133124—22no—
28#13133124—21no—
29#13133124—22no—
30#13133124—22no—
31C22×Q8×C53—21no—
32C22×D4×C53D411yescharacteristic
33C2×SD16×C53—42no—
34C2×SD16×C53—42no—
35C2×SD16×C53—42no—
36C22×C8×C53—21no—
37C4⋊Q8×C53—42no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C42×C53C2311yescharacteristic
42#103×C53—42no—
43C24×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×D4×C53—21no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53SD1612yes—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53—81no—
61C22×Dic53—82no—
62C2×Dic106—22no—
63C2×Dic106—22no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66SD16×C53—48no—
67C2×C8×C53—22no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C42×C53—41no—
76C22⋊Dic53—82no—
77C22⋊Dic53—81no—
78C4⋊Dic53—42no—
79C4⋊Dic53—81no—
80C2×#88322—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—41no—
86C23×C53—42no—
87C23×C53—41no—
88C23×C53#12011yescharacteristic
89Q8×C53—42no—
90Q8×C53—42no—
91Q8×C53—81no—
92Q8×C53—81no—
93D4×C53—42no—
94D4×C53—81no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—81no—
103C2×C4×C53—81no—
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53#13512yes—
106C2×C4×C53#13712yes—
107C2×C4×C53#13711yescharacteristic
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—84no—
115Dic106—44no—
116Dic106—82no—
117C8×C53—42no—
118#88322—82no—
119#7755—531nomaximal, Sylow
120C22×C53—42no—
121C22×C53—44no—
122C22×C53—82no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53#44811yescharacteristic
126C22×C53#45212yes—
127C22×C53#45311yescharacteristic
128C22×C53#45512yes—
129C22×C53B(4,𝔽2)11yescharacteristic
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—82no—
135C4×C53—81no—
136Dic53—82no—
137Dic53—82no—
138#2804—531no—
139#2024—532no—
140#2013D5311yescharacteristic
141#1272—532no—
142#1270—531no—
143C2×C53—82no—
144C2×C53#159611yescharacteristic
145C2×C53#160612yes—
146C2×C53#163011yescharacteristic
147C2×C53#164312yes—
148C2×C53#165911yescharacteristic
149C22×SD16—21no—
150C2×C4⋊Q8C2×D5311yescharacteristic
151C2×#103—21no—
152C2×#103—1062no—
153#400—1061no—
154#393—1062no—
155#391—532no—
156#390C2×D5311yescharacteristic
157#386—1061no—
158#385—1062no—
159#340—1062no—
160#336—532no—
161#336C2×D5311yescharacteristic
162C53#775511yescharacteristic, complemented, Sylow
163C22×Q8—21no—
164C22×Q8—532no—
165C22×D4C22⋊D5311yescharacteristic
166C23×C4—1061no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C22×C8—21no—
171C22×C8—1062no—
172C4⋊Q8—42no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4C22⋊D5311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4—1061no—
180C2×C42C22×D5311yescharacteristic
181#103—42no—
182#103—2124no—
183#95—1062no—
184#95—1061no—
185#95—1062no—
186#95—1062no—
187C24—21no—
188C2×Q8—22no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—1062no—
195C2×Q8—1062no—
196C2×Q8—2122no—
197C2×Q8—2122no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4#1171992312yes—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4—4241no—
207C22×C4—4242no—
208C22×C4D4×D5312yes—
209C22×C4C2×C22⋊D5311yescharacteristic
210SD16—48no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—2122no—
215C2×C8—2122no—
216C2×C8—2122no—
217C4⋊C4—24no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—4241no—
221C4⋊C4—2122no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—4242no—
225C22⋊C4—4241no—
226C42—41no—
227C23—22no—
228C23—42no—
229C23—41no—
230C23—42no—
231C23—41no—
232C23#1313318511yescharacteristic
233Q8—42no—
234Q8—42no—
235Q8—81no—
236Q8—81no—
237Q8—4242no—
238Q8—2124no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—4244no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4C2×#1171992311yescharacteristic
258C2×C4#1313323812yes—
259C2×C4#1313324612yes—
260C2×C4#1313324711yescharacteristic
261C8—42no—
262C8—4242no—
263C22—44no—
264C22—42no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22#1681076111yescharacteristic
269C22#1681078212yes—
270C22#1681079011yescharacteristic
271C22#1681079912yes—
272C22#1681080911yescharacteristic
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—81no—
278C4—82no—
279C4—4242no—
280C4—4242no—
281C2—82no—
282C2#2027363711yescharacteristic
283C2#2027369212yes—
284C2#2027383411yescharacteristic
285C2#2027392012yes—
286C2#2027401911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2013C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2013D53
hypocenterC53#7755
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #20275853C2semidirect
3. #20272540C2semidirect
162. C53#7755semidirect

Maximal subgroups

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

subgroupindex
#202826612
#202758532
#202725402
#202725362
C53×#20132
#775553