Finite Group Databasebeta

← #27273412

Subgroups — #27273412

Order 13568.

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

normal subgroups only

This group has 438 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#20276252C212yesmaximal, complemented
3#20276248C211yesmaximal, characteristic
4#20272540C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C2×#13133151—22no—
7C2×#13133148—21no—
8C2×#13133142—22no—
9C2×#13133138—21no—
10#16810562—21no—
11#16810535—22no—
12#16810529C2212yes—
13#16810503—21no—
14#16810498—22no—
15#16810493C2211yescharacteristic
16#16810378C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106D412yes—
20C23×Dic53—21no—
21C22×#88322—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53D412yes—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—21no—
28#13133151—44no—
29#13133148—42no—
30#13133142—44no—
31#13133138—42no—
32#13133124—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×D4×C53D411yescharacteristic
37C22×C8×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2311yescharacteristic
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53Q1612yes—
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—41no—
52C22×Dic53—82no—
53C22×Dic53—81no—
54C2×Dic106—22no—
55C2×Dic106—22no—
56C2×Dic106—42no—
57C2×Dic106—42no—
58C2×C8×C53—24no—
59C2×C8×C53—42no—
60C2×C8×C53—42no—
61C4⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C42×C53—41no—
65C22⋊Dic53—82no—
66C22⋊Dic53—81no—
67C4⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4⋊Dic53—42no—
72C4⋊Dic53—81no—
73C2×#88322—22no—
74C2×#88322—41no—
75C2×#88322—41no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—41no—
79C23×C53—42no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82D4×C53—42no—
83D4×C53—81no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53C2×Q1611yescharacteristic
91C2×C4×C53#13512yes—
92C2×C4×C53#13712yes—
93C2×C4×C53#13711yescharacteristic
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—84no—
101C2×Dic53—82no—
102C2×Dic53—81no—
103Dic106—44no—
104Dic106—82no—
105C8×C53—44no—
106#88322—42no—
107#7833—531nomaximal, Sylow
108C22×C53—42no—
109C22×C53—44no—
110C22×C53—82no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53#45111yescharacteristic
114C22×C53#45212yes—
115C22×C53#45312yes—
116C22×C53#45311yescharacteristic
117C22×C53#45811yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—81no—
123Dic53—82no—
124Dic53—82no—
125Dic53—161no—
126#2804—531no—
127#2095—532no—
128#2094—531no—
129#1272—532no—
130#1270D5311yescharacteristic
131C2×C53—82no—
132C2×C53#161411yescharacteristic
133C2×C53#161812yes—
134C2×C53#163011yescharacteristic
135C2×C53#163212yes—
136C2×C53#165511yescharacteristic
137C2×#107—1062no—
138C2×#106—1061no—
139C2×#103—1062no—
140C2×#102—1061no—
141#400—1061no—
142#393—1062no—
143#391—532no—
144#390C2×D5311yescharacteristic
145#386—1061no—
146#385—1062no—
147#384—531no—
148#336—531no—
149#336C2×D5312yes—
150C53#783311yescharacteristic, complemented, Sylow
151C22×Q8—532no—
152C22×D4D21211yescharacteristic
153C23×C4—1061no—
154C22×C8—22no—
155C22×C8—1061no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4D21211yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42C22×D5311yescharacteristic
165#107—2124no—
166#106—2122no—
167#103—2124no—
168#102—2122no—
169#95—1062no—
170#95—1061no—
171#95—1062no—
172#95—1062no—
173C24—21no—
174C2×Q8—1062no—
175C2×Q8—1062no—
176C2×Q8—2122no—
177C2×Q8—2122no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4Dic21212yes—
182C22×C4—21no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—4242no—
186C22×C4—4241no—
187C22×C4—2121no—
188C22×C4C2×D21211yescharacteristic
189C22×C4D4×D5312yes—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—1062no—
194C2×C8—2121no—
195C2×C8—2121no—
196C4⋊C4—41no—
197C4⋊C4—4241no—
198C4⋊C4—2121no—
199C4⋊C4—2122no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—4242no—
206C22⋊C4—4241no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1313318711yescharacteristic
214Q8—4242no—
215Q8—2124no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—4242no—
225C2×C4—4241no—
226C2×C4—4244no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4C2×Dic21211yescharacteristic
234C2×C4#1313323612yes—
235C2×C4#1313324612yes—
236C2×C4#1313325011yescharacteristic
237C8—44no—
238C8—2122no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1681078111yescharacteristic
245C22#1681078612yes—
246C22#1681079212yes—
247C22#1681079311yescharacteristic
248C22#1681081411yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—8481no—
255C4—4242no—
256C4—4242no—
257C2—82no—
258C2#2027374211yescharacteristic
259C2#2027376412yes—
260C2#2027383711yescharacteristic
261C2#2027385112yes—
262C2#2027400511yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7833
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20276252C2semidirect
4. #20272540C2semidirect
150. C53#7833semidirect

Maximal subgroups

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

subgroupindex
#202826612
#202762522
#202762482
#202725402
C53×#12702
#783353