Finite Group Databasebeta

← #28306299

Subgroups — #28306299

Order 13568.

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290500C211yesmaximal, characteristic
2#20290432C211yesmaximal, characteristic
3#20289591C214yesmaximal
4#20283076C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133272C2212yes—
7C2×#13133272C2212yes—
8#16811489—21no—
9#16811489—21no—
10#16811489—21no—
11C2×#13133208C412yes—
12#16810522—21no—
13#16810522—21no—
14#16810519—21no—
15#16810519—21no—
16#16810493C411yescharacteristic
17#16810488C2212yes—
18C42×Dic53C2211yescharacteristic
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133272—22no—
24#13133272—24no—
25#13133272Q814yes—
26#13133208—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38#13133124—22no—
39#13133124—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C42×C53C2×C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C22⋊C411yescharacteristic
51C22×Dic53C2×D411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C2×#88324—24no—
54C2×#88324—24no—
55C2×#88324—42no—
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—22no—
63C4⋊Dic53—44no—
64C4⋊Dic53—42no—
65C4⋊Dic53—42no—
66C4⋊Dic53—42no—
67#11719919—44no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53—41no—
71C4×Dic53—41no—
72C4×Dic53—41no—
73C4×Dic53C2×Q812yes—
74C4×Dic53#4112yes—
75C23×C53C2×C22⋊C411yescharacteristic
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53#10112yes—
86C2×C4×C53#10412yes—
87C2×C4×C53C4×Q812yes—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53#10012yes—
101C2×Dic53C22⋊Q812yes—
102C2×Dic53#12411yescharacteristic
103C2×Dic53C4⋊Q811yescharacteristic
104#88324—44no—
105#88324—44no—
106#9894—531nomaximal, Sylow
107C22×C53#39111yescharacteristic
108C22×C53C2×#10011yescharacteristic
109C22×C53C2×#10111yescharacteristic
110C22×C53C2×#10411yescharacteristic
111C22×C53#42111yescharacteristic
112C22×C53#42111yescharacteristic
113C22×C53#42111yescharacteristic
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—82no—
118Dic53—21no—
119Dic53—21no—
120Dic53—22no—
121Dic53—22no—
122Dic53—41no—
123Dic53—82no—
124#2859—531no—
125#1767—531no—
126#1745—531no—
127#1295—534no—
128C2×C53#152811yescharacteristic
129C2×C53#153011yescharacteristic
130C2×C53#153411yescharacteristic
131C2×C53#153611yescharacteristic
132C2×C53#198611yescharacteristic
133C2×C53#199511yescharacteristic
134C2×C53#199711yescharacteristic
135C2×#125—532no—
136C2×C4⋊C8—1061no—
137C2×#97—532no—
138C2×#97—532no—
139#389—1061no—
140#389—1061no—
141#389—1061no—
142#389—1061no—
143#384—531no—
144#384#615711yescharacteristic
145#382—532no—
146C43—531no—
147#336—1061no—
148#336—1061no—
149#336—1061no—
150C53#989411yescharacteristic, complemented, Sylow
151C22×C8—1062no—
152C22×C8—1062no—
153#125—2124no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4C2×#615712yes—
162C2×C42—531no—
163C2×C42—531no—
164C2×C42—531no—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42—531no—
168C2×C42C2×#615711yescharacteristic
169C4⋊C8—1064no—
170#97—534no—
171#97—1062no—
172#97—1064no—
173#95—1062no—
174#95—1062no—
175C22×C4—22no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—1062no—
181C22×C4C22⋊#615711yescharacteristic
182C22×C4C22⋊#615711yescharacteristic
183C22×C4C22×#615711yescharacteristic
184C2×C8—1064no—
185C2×C8—1064no—
186C2×C8—2122no—
187C2×C8—2122no—
188C2×C8—2122no—
189C2×C8—2122no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C4⋊C4—2124no—
197C4⋊C4—2124no—
198C42—22no—
199C42—532no—
200C42—532no—
201C42—1062no—
202C42—1062no—
203C42—2121no—
204C42—2121no—
205C42—2121no—
206C23C2×C22⋊#615711yescharacteristic
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—532no—
217C2×C4—531no—
218C2×C4—531no—
219C2×C4—532no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4#1313327812yes—
233C2×C4#1313328112yes—
234C2×C4Q8×#615712yes—
235C8—2124no—
236C8—2124no—
237C22(C2×#6157)⋊Q811yescharacteristic
238C22D53⋊#10011yescharacteristic
239C22C2×#1313327811yescharacteristic
240C22C2×#1313328111yescharacteristic
241C22#1681162111yescharacteristic
242C22#1681162211yescharacteristic
243C22#1681162211yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—82no—
248C4—1061no—
249C4—1061no—
250C4—1062no—
251C4—1062no—
252C4—4242no—
253C4—2121no—
254C2#2029017711yescharacteristic
255C2#2029018511yescharacteristic
256C2#2029019711yescharacteristic
257C2#2029020511yescharacteristic
258C2#2029089111yescharacteristic
259C2#2029090311yescharacteristic
260C2#2029090811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#9894
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
150. C53#9894semidirect

Maximal subgroups

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

subgroupindex
#202905002
#202904322
#202895912
#202830762
#989453