Finite Group Databasebeta

← #27273409

Subgroups — #27273409

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#20276254C212yesmaximal, complemented
3#20272540C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2094C211yesmaximal, characteristic
6C2×#13133150—22no—
7C2×#13133140—22no—
8#16810562—21no—
9#16810535—22no—
10#16810529C2212yes—
11#16810503—21no—
12#16810498—22no—
13#16810378C2212yes—
14C2×#106×C53—21no—
15C2×#102×C53—21no—
16C53×#390C2211yescharacteristic
17C53×#384C2211yescharacteristic
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#13133150—44no—
27#13133140—44no—
28#13133124—22no—
29#13133124—21no—
30#13133124—22no—
31#13133124—22no—
32C22×D4×C53D411yescharacteristic
33C22×C8×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C22⋊C4×C53—22no—
38C2×C42×C53C2311yescharacteristic
39#106×C53—42no—
40#102×C53—42no—
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53Q1612yes—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—82no—
53C22×Dic53—81no—
54C2×Dic106—22no—
55C2×Dic106—22no—
56C2×Dic106—42no—
57C2×Dic106—42no—
58C2×C8×C53—22no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68C42×C53—41no—
69C22⋊Dic53—82no—
70C22⋊Dic53—81no—
71C4⋊Dic53—42no—
72C4⋊Dic53—81no—
73C2×#88322—24no—
74C2×#88322—42no—
75C2×#88322—42no—
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×C53—82no—
91C2×C4×C53—81no—
92C2×C4×C53C2×Q1611yescharacteristic
93C2×C4×C53#13512yes—
94C2×C4×C53#13712yes—
95C2×C4×C53#13711yescharacteristic
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—84no—
103Dic106—44no—
104Dic106—82no—
105C8×C53—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#45312yes—
116C22×C53#45311yescharacteristic
117C22×C53#45811yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—81no—
123C4×C53—161no—
124Dic53—82no—
125Dic53—82no—
126#2804—531no—
127#2095—532no—
128#2094D5311yescharacteristic
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—21no—
139C2×#103—1062no—
140C2×#102—21no—
141#400—1061no—
142#393—1062no—
143#391—532no—
144#390C2×D5311yescharacteristic
145#386—1061no—
146#385—1062no—
147#384C2×D5311yescharacteristic
148#336—532no—
149#336C2×D5311yescharacteristic
150C53#783311yescharacteristic, complemented, Sylow
151C22×Q8—532no—
152C22×D4C22⋊D5311yescharacteristic
153C23×C4—1061no—
154C22×C8—21no—
155C22×C8—1062no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4C22⋊D5311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42C22×D5311yescharacteristic
165#107—2124no—
166#106—42no—
167#103—2124no—
168#102—42no—
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×D4#1171992512yes—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4—4242no—
187C22×C4—4241no—
188C22×C4D4×D5312yes—
189C22×C4C2×C22⋊D5311yescharacteristic
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—1064no—
194C2×C8—2122no—
195C2×C8—2122no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—4241no—
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#1313318511yescharacteristic
214Q8—4242no—
215Q8—2124no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—4244no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4C2×#1171992511yescharacteristic
234C2×C4#1313323812yes—
235C2×C4#1313324612yes—
236C2×C4#1313324711yescharacteristic
237C8—42no—
238C8—2124no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1681077811yescharacteristic
245C22#1681078212yes—
246C22#1681078912yes—
247C22#1681079011yescharacteristic
248C22#1681081211yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—161no—
255C4—4242no—
256C4—4242no—
257C2—82no—
258C2#2027373911yescharacteristic
259C2#2027376012yes—
260C2#2027383411yescharacteristic
261C2#2027384712yes—
262C2#2027400311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2094C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2094D53
hypocenterC53#7833
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
2. #20276254C2semidirect
3. #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
#202762542
#202725402
#202725362
C53×#20942
#783353