Finite Group Databasebeta

← #27273410

Subgroups — #27273410

Order 13568.

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

normal subgroups only

This group has 438 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#20276253C212yesmaximal, complemented
2#20276248C211yesmaximal, characteristic
3#20272541C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2804C211yesmaximal, characteristic
6C2×#13133150—22no—
7C2×#13133148—21no—
8C2×#13133141—22no—
9C2×#13133138—21no—
10#16810493C2211yescharacteristic
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C53×#400—21no—
14C53×#393—22no—
15C53×#391C2212yes—
16C53×#390C2211yescharacteristic
17C53×#386—21no—
18C53×#385—22no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23#13133150—44no—
24#13133148—42no—
25#13133141—44no—
26#13133138—42no—
27C22×Q8×C53D412yes—
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C42×C53C2311yescharacteristic
37#95×C53—22no—
38#95×C53—21no—
39#95×C53—22no—
40#95×C53—22no—
41C24×C53—21no—
42C2×Q8×C53—22no—
43C2×Q8×C53—22no—
44C2×Q8×C53—42no—
45C2×Q8×C53—42no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53Q1612yes—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—82no—
54C22×C4×C53—81no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D412yes—
57C22×Dic53—41no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—81no—
61C22⋊C4×C53—42no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—82no—
64C22⋊C4×C53—81no—
65C42×C53—41no—
66C4⋊Dic53—41no—
67C4⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—42no—
70C2×#88322—22no—
71C2×#88322—24no—
72C2×#88322—41no—
73C2×#88322—42no—
74C2×#88322—42no—
75C2×#88322—41no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—41no—
79C23×C53—42no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82Q8×C53—44no—
83Q8×C53—82no—
84D4×C53—42no—
85D4×C53—81no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—84no—
99C2×C4×C53C2×Q1611yescharacteristic
100C2×C4×C53#13512yes—
101C2×C4×C53#13712yes—
102C2×C4×C53#13711yescharacteristic
103C2×Dic53—81no—
104C2×Dic53—82no—
105#88322—42no—
106#88322—44no—
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#45311yescharacteristic
116C22×C53#45312yes—
117C22×C53#45811yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—82no—
123C4×C53—81no—
124C4×C53—82no—
125Dic53—161no—
126#2804D5311yescharacteristic
127#2095—532no—
128#2094—531no—
129#1272—532no—
130#1270—531no—
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—21no—
142#393—22no—
143#391C2×D5312yes—
144#390C2×D5311yescharacteristic
145#386—21no—
146#385—22no—
147#384—531no—
148#336—531no—
149#336—532no—
150C53#783311yescharacteristic, complemented, Sylow
151C22×Q8C22⋊D5312yes—
152C22×D4C22⋊D5311yescharacteristic
153C23×C4—21no—
154C22×C8—1061no—
155C22×C8—1062no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4C22⋊D5312yes—
160C2×C4⋊C4C22⋊D5311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C22×D5311yescharacteristic
165#107—2124no—
166#106—2122no—
167#103—2124no—
168#102—2122no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173C24—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4#1171992512yes—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—2121no—
188C22×C4C2×C22⋊D5311yescharacteristic
189C22×C4C2×C22⋊D5312yes—
190C2×C8—1062no—
191C2×C8—1064no—
192C2×C8—2121no—
193C2×C8—2122no—
194C2×C8—2122no—
195C2×C8—2121no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—2121no—
200C4⋊C4—2122no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1313318611yescharacteristic
214Q8—44no—
215Q8—82no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—4241no—
232C2×C4—4242no—
233C2×C4C2×#1171992511yescharacteristic
234C2×C4#1313323712yes—
235C2×C4#1313324712yes—
236C2×C4#1313324811yescharacteristic
237C8—2122no—
238C8—2124no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1681077911yescharacteristic
245C22#1681078512yes—
246C22#1681079111yescharacteristic
247C22#1681079112yes—
248C22#1681081311yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—8481no—
257C2—82no—
258C2#2027374011yescharacteristic
259C2#2027376312yes—
260C2#2027383611yescharacteristic
261C2#2027384912yes—
262C2#2027400411yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2804C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2804D53
hypocenterC53#7833
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
1. #20276253C2semidirect
3. #20272541C2semidirect
150. C53#7833semidirect

Maximal subgroups

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

subgroupindex
#202762532
#202762482
#202725412
#202725362
C53×#28042
#783353