Finite Group Databasebeta

← #27293571

Subgroups — #27293571

Order 13568.

412 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 77 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#20277282C211yesmaximal, characteristic
2#20273057C212yesmaximal
3#20272661C211yesmaximal, characteristic
4#20272648C212yesmaximal
5C53×#1325C211yesmaximal, characteristic
6C2×#13133217C411yescharacteristic
7C2×#13133182C411yescharacteristic
8C2×#13133182C2211yescharacteristic
9#16810677—22no—
10#16810528C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810519—22no—
13C4⋊C4×Dic53—22no—
14C42×Dic53C411yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#384C2211yescharacteristic
18C53×#324C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×Dic106C2×C411yescharacteristic
21#13133217—22no—
22#13133217—22no—
23#13133182—41no—
24#13133182—41no—
25#13133182—41no—
26#13133182—41no—
27C2×C4⋊Dic53—22no—
28C2×#11719919—21no—
29C2×#11719919D412yes—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D412yes—
36#13133146—42no—
37C22⋊#88322—22no—
38#13133127—42no—
39#13133124—22no—
40#13133124—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C22⋊C8×C53—22no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53C22⋊C412yes—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C2×Dic106—21no—
56C2×Dic106—21no—
57C2×Dic106—21no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106M1612yes—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—44no—
66C4⋊C4×C53—41no—
67C42×C53—21no—
68C42×C53M1612yes—
69C4⋊Dic53—28no—
70#11719919—42no—
71#11719919—42no—
72C4×Dic53—24no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—42no—
78C4×Dic53—41no—
79C4×Dic53—41no—
80C4×Dic53—41no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53C2×M1611yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53#9911yescharacteristic
107C2×Dic53#9911yescharacteristic
108Dic106—42no—
109Dic106—42no—
110Dic106—42no—
111Dic106—42no—
112C8×C53—82no—
113#88322—82no—
114#11532—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C4×C53—24no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—42no—
124C4×C53—81no—
125Dic53—21no—
126Dic53—42no—
127Dic53—41no—
128Dic53—42no—
129Dic53—42no—
130Dic53#35612yes—
131C4×C4⋊Q8—531no—
132#1467—532no—
133#1325—531no—
134#1325D5311yescharacteristic
135#1319—532no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53#146511yescharacteristic
139C2×C53#212611yescharacteristic
140C2×C53C2×#35611yescharacteristic
141C2×C4⋊Q8—531no—
142C2×C4×Q8—531no—
143C2×C4×Q8—531no—
144#432—1062no—
145#391—531no—
146#391—531no—
147#389—1062no—
148#384C2×D5311yescharacteristic
149C4×C4⋊C4—1062no—
150C43—531no—
151#324—532no—
152#324—1062no—
153#324C2×D5312yes—
154C53#1153211yescharacteristic, complemented, Sylow
155C22×Q8—531no—
156C22×Q8—531no—
157C4⋊Q8—1062no—
158C4⋊Q8—1062no—
159C4×Q8—2121no—
160C4×Q8—2121no—
161C4×Q8—2121no—
162C4×Q8—2121no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4C4×D5311yescharacteristic
168C2×C4⋊C4C22×D5311yescharacteristic
169C2×C42—531no—
170C2×C42—532no—
171C2×C42—531no—
172C2×C42—1061no—
173C2×C42—1061no—
174C2×C42—1061no—
175C2×C42C4×D5311yescharacteristic
176C4⋊C8—2122no—
177C22⋊C8—22no—
178C22⋊C8—1062no—
179#97—2122no—
180#95—1062no—
181#95—1062no—
182C2×Q8—532no—
183C2×Q8—1061no—
184C2×Q8—1061no—
185C2×Q8—1061no—
186C2×Q8—2121no—
187C2×Q8—2121no—
188C2×Q8—2121no—
189C2×Q8—2121no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—531no—
193C22×C4—532no—
194C22×C4—531no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4C2×C4×D5311yescharacteristic
198C22×C4D4×D5312yes—
199C2×C8—42no—
200C2×C8—2122no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—44no—
204C4⋊C4—1068no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C42—21no—
208C42—1062no—
209C42—1062no—
210C42—1061no—
211C42—1061no—
212C42—1064no—
213C42—2121no—
214C42—2122no—
215C42—2121no—
216C42—2121no—
217C42#1171991212yes—
218C23C22⋊C4×D5311yescharacteristic
219Q8—2122no—
220Q8—2122no—
221Q8—2122no—
222Q8—2122no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—531no—
233C2×C4—531no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1061no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4M16×D5311yescharacteristic
247C2×C4C2×#1171991211yescharacteristic
248C8—82no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22#1681058511yescharacteristic
253C22#99×D5311yescharacteristic
254C22#1681060611yescharacteristic
255C4—24no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—81no—
260C4—532no—
261C4—1061no—
262C4—2122no—
263C4—2121no—
264C4—2122no—
265C4—2122no—
266C2—21no—
267C2—21no—
268C2#2027305211yescharacteristic
269C2#2027637711yescharacteristic
270C2#356×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11532
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
154. C53#11532semidirect

Maximal subgroups

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

subgroupindex
#202772822
#202730572
#202726612
#202726482
C53×#13252
#1153253