Finite Group Databasebeta

← #27295258

Subgroups — #27295258

Order 13568.

388 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279920C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272744C214yesmaximal
4C53×#1801C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6#16810676—21no—
7#16810543C412yes—
8#16810543C2212yes—
9#16810530—21no—
10#16810521—21no—
11#16810521—21no—
12#16810492—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#423—21no—
18C2×#101×C53—21no—
19C53×#343C2212yes—
20C22×Dic106C2×C412yes—
21#13133217—41no—
22#13133217—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×C4×Dic53—21no—
29C4⋊#88322—41no—
30C22⋊#88322—22no—
31C4×#88322—41no—
32#13133124—22no—
33#13133124—22no—
34#13133124C2×C412yes—
35C22×Q8×C53D411yescharacteristic
36C2×M16×C53—22no—
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C4⋊C8×C53—41no—
47#101×C53—42no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53SD1612yes—
51C2×Q8×C53Q1612yes—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C2×Dic106—42no—
62C2×Dic106—82no—
63M16×C53—42no—
64M16×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C4⋊Dic53—24no—
76C4⋊Dic53—42no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79#11719919—82no—
80#11719919—82no—
81C4×Dic53—42no—
82C2×#88322—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84Q8×C53—42no—
85Q8×C53—42no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53#9911yescharacteristic
100C2×C4×C53#9911yescharacteristic
101C2×C4×C53#10312yes—
102C2×C4×C53#10312yes—
103C2×C4×C53C2×SD1611yescharacteristic
104C2×C4×C53C2×Q1611yescharacteristic
105C2×Dic53—24no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111Dic106—84no—
112C8×C53—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—41no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53#35612yes—
129Dic53—44no—
130Dic53—82no—
131#2567—531no—
132#1801D5311yescharacteristic
133#1461—531no—
134#1354—534no—
135C2×C53—21no—
136C2×C53—21no—
137C2×C53#152011yescharacteristic
138C2×C53#212211yescharacteristic
139C2×C53C2×#35611yescharacteristic
140C2×C4⋊Q8—531no—
141C2×C4×Q8C2×D5311yescharacteristic
142#431—1061no—
143#423—21no—
144C2×#101—21no—
145#395—532no—
146#395—532no—
147#391—1061no—
148#389—1061no—
149#389—1061no—
150#384—1061no—
151C4×C4⋊C4—531no—
152#343C2×D5312yes—
153#324—532no—
154#324—1061no—
155C53#1190811yescharacteristic, complemented, Sylow
156C22×Q8—532no—
157C22×Q8C22⋊D5311yescharacteristic
158C2×M16—22no—
159C4⋊Q8—2121no—
160C4⋊Q8—2121no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4—1061no—
168C2×C4⋊C4C4×D5311yescharacteristic
169C2×C4⋊C4D21211yescharacteristic
170C2×C4⋊C4C22×D5311yescharacteristic
171C2×C42—1061no—
172C2×C42C4×D5311yescharacteristic
173C2×C42D21211yescharacteristic
174C2×C42C22⋊D5311yescharacteristic
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—2121no—
178#101—42no—
179C22⋊C8—1062no—
180C4×C8—2121no—
181#95—532no—
182#95—1062no—
183#95—1062no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—4242no—
187C2×Q8—2122no—
188C2×Q8#1171992312yes—
189C2×Q8#1171992512yes—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4#1171992111yescharacteristic
193C22×C4#1171992111yescharacteristic
194C22×C4#1171992111yescharacteristic
195C22×C4#1171992111yescharacteristic
196C22×C4C2×C4×D5311yescharacteristic
197C22×C4C2×D21211yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199M16—42no—
200M16—42no—
201C2×C8—42no—
202C2×C8—2122no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—1064no—
207C4⋊C4—4242no—
208C4⋊C4—4242no—
209C4⋊C4—2122no—
210C4⋊C4—2121no—
211C4⋊C4—2121no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—2122no—
219C23C2×#1171992111yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—4244no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—1064no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4#1313313111yescharacteristic
243C2×C4#1313313111yescharacteristic
244C2×C4#1313314012yes—
245C2×C4#1313314012yes—
246C2×C4C2×#1171992311yescharacteristic
247C2×C4C2×#1171992511yescharacteristic
248C8—82no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22C2×#1313313111yescharacteristic
253C22C2×#1313314011yescharacteristic
254C22#1681064111yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—4242no—
264C4—2124no—
265C4#1681042112yes—
266C2—21no—
267C2—21no—
268C2#2027328611yescharacteristic
269C2#2027636611yescharacteristic
270C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1801C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1801D53
hypocenterC53#11908
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

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
#202799202
#202730392
#202727442
C53×#18012
#1190853