Finite Group Databasebeta

← #27295259

Subgroups — #27295259

Order 13568.

388 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274711C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272745C214yesmaximal
4C53×#2567C211yesmaximal, characteristic
5#16810676—21no—
6#16810647—21no—
7C2×#13133136—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#395C412yes—
14C53×#395C2212yes—
15C53×#391—21no—
16C53×#389—21no—
17C53×#389—21no—
18C53×#384—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133136—42no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27C22×Q8×C53C2×C412yes—
28C22×Q8×C53D411yescharacteristic
29C4⋊Q8×C53—41no—
30C4⋊Q8×C53—41no—
31C4×Q8×C53—41no—
32C4×Q8×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—42no—
51C2×Q8×C53—82no—
52C2×Q8×C53SD1612yes—
53C2×Q8×C53Q1612yes—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C4⋊C4×C53—24no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—82no—
71C4⋊C4×C53—82no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—42no—
79#11719917—42no—
80#11719917—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84Q8×C53—42no—
85Q8×C53—42no—
86Q8×C53—84no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—24no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#10312yes—
109C2×C4×C53#10312yes—
110C2×C4×C53C2×SD1611yescharacteristic
111C2×C4×C53C2×Q1611yescharacteristic
112#88322—82no—
113#88322—82no—
114#11908—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53C2×#9911yescharacteristic
118C22×C53C2×#10311yescharacteristic
119C22×C53#42111yescharacteristic
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—21no—
125C4×C53—44no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—82no—
130C4×C53#35612yes—
131#2567D5311yescharacteristic
132#1801—531no—
133#1461—531no—
134#1354—534no—
135C2×C53—21no—
136C2×C53—21no—
137C2×C53#152011yescharacteristic
138C2×C53#212211yescharacteristic
139C2×C53C2×#35611yescharacteristic
140C2×C4⋊Q8Dic5311yescharacteristic
141C2×C4×Q8C2×D5311yescharacteristic
142#431—1061no—
143#423—1061no—
144C2×#101—1061no—
145#395Dic5312yes—
146#395C2×D5312yes—
147#391—21no—
148#389—21no—
149#389—21no—
150#384—21no—
151C4×C4⋊C4Dic5311yescharacteristic
152#343—532no—
153#324—532no—
154#324—1061no—
155C53#1190811yescharacteristic, complemented, Sylow
156C22×Q8C2×Dic5312yes—
157C22×Q8C22⋊D5311yescharacteristic
158C2×M16—1062no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×Dic5311yescharacteristic
169C2×C4⋊C4C22⋊D5311yescharacteristic
170C2×C4⋊C4C22×D5311yescharacteristic
171C2×C42—21no—
172C2×C42C2×Dic5311yescharacteristic
173C2×C42C22⋊D5311yescharacteristic
174C2×C42C22⋊D5311yescharacteristic
175C4⋊C8—2121no—
176C4⋊C8—2121no—
177C4⋊C8—2121no—
178#101—2122no—
179C22⋊C8—1062no—
180C4×C8—2121no—
181#95—22no—
182#95—22no—
183#95C2×Dic5312yes—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8#1171992412yes—
189C2×Q8#1171992512yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4C22⋊Dic5311yescharacteristic
193C22×C4C22⋊Dic5311yescharacteristic
194C22×C4C22⋊Dic5311yescharacteristic
195C22×C4C22⋊Dic5311yescharacteristic
196C22×C4C22×Dic5311yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199M16—2122no—
200M16—2122no—
201C2×C8—2122no—
202C2×C8—2122no—
203C4⋊C4—24no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23C2×C22⋊Dic5311yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—84no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4#1313313211yescharacteristic
243C2×C4#1313313211yescharacteristic
244C2×C4#1313314112yes—
245C2×C4#1313314112yes—
246C2×C4C2×#1171992411yescharacteristic
247C2×C4C2×#1171992511yescharacteristic
248C8—4242no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22C2×#1313313211yescharacteristic
253C22C2×#1313314111yescharacteristic
254C22#1681064011yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#1681042212yes—
266C2—21no—
267C2—21no—
268C2#2027328711yescharacteristic
269C2#2027636511yescharacteristic
270C2C2×#1681042211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2567C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2567D53
hypocenterC53#11908
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
155. C53#11908semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202727452
C53×#25672
#1190853