Finite Group Databasebeta

← #27302779

Subgroups — #27302779

Order 13568.

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

normal subgroups only

This group has 420 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#20279920C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20272841C214yesmaximal
4C4×#101×C53C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6#16810647—21no—
7C2×#13133136C2212yes—
8#16810543—21no—
9#16810543—21no—
10#16810530C2212yes—
11#16810521—21no—
12#16810521—21no—
13#16810492C412yes—
14C4×C4⋊Dic53C411yescharacteristic
15#16810392—21no—
16C2×C4×Q8×C53C2211yescharacteristic
17C2×#101×C53C2212yes—
18C4×M16×C53—21no—
19C53×#343—21no—
20C22×Dic106—21no—
21C2×#11719917—22no—
22#13133217—41no—
23#13133217—41no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×C4×Dic53C2×C412yes—
30C4⋊#88322—41no—
31C4⋊#88322—41no—
32#13133136—24no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53C2311yescharacteristic
37C2×M16×C53—22no—
38C4×Q8×C53—41no—
39C4×Q8×C53—41no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46#101×C53—22no—
47#101×C53Q814yescomplemented
48#97×C53—41no—
49C4×C8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C2×Q812yes—
53C2×Q8×C53#4112yes—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C2×Dic106—42no—
64C2×Dic106—82no—
65M16×C53—24no—
66M16×C53—42no—
67C2×C8×C53—42no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C4⋊Dic53—24no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53—42no—
81#11719919—82no—
82#11719919—82no—
83C4×Dic53—42no—
84#11719917—42no—
85#11719917—42no—
86C2×#88322—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—42no—
89Q8×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53C4×Q812yes—
106C2×C4×C53C22⋊Q812yes—
107C2×C4×C53#12411yescharacteristic
108C2×C4×C53C4⋊Q811yescharacteristic
109C2×Dic53—24no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115Dic106—84no—
116C8×C53—44no—
117#88322—82no—
118#13302—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#39111yescharacteristic
122C22×C53C2×#9911yescharacteristic
123C22×C53#41011yescharacteristic
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53#35612yes—
133Dic53—44no—
134Dic53—82no—
135#2567—531no—
136#1801—531no—
137C4×#101D5311yescharacteristic
138#1394—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#199611yescharacteristic
142C2×C53C2×#35611yescharacteristic
143C2×C53#213411yescharacteristic
144C2×C4⋊Q8—531no—
145C2×C4×Q8C2×D5311yescharacteristic
146#423—1061no—
147C2×#101—532no—
148C2×#101C2×D5312yes—
149C4×M16—21no—
150#395—1061no—
151#395—1061no—
152#391—532no—
153#389—1061no—
154#389—1061no—
155#384—532no—
156C4×C4⋊C4—531no—
157#343—21no—
158#343—1061no—
159C53#1330211yescharacteristic, complemented, Sylow
160C22×Q8—1061no—
161C22×Q8C22×D5311yescharacteristic
162C2×M16—22no—
163C2×M16—1062no—
164C4⋊Q8—2121no—
165C4⋊Q8—2121no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1061no—
173C2×C4⋊C4C4×D5311yescharacteristic
174C2×C4⋊C4D21211yescharacteristic
175C2×C4⋊C4C22⋊D5311yescharacteristic
176C2×C42—532no—
177C2×C42C4×D5311yescharacteristic
178C2×C42D21211yescharacteristic
179C2×C42C22⋊D5311yescharacteristic
180C4⋊C8—2121no—
181C4⋊C8—2121no—
182#101—22no—
183#101—1064no—
184#101Dic10614yescomplemented
185#97—41no—
186C4×C8—41no—
187#95—1062no—
188#95—1062no—
189#95—1061no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—4242no—
193C2×Q8—2122no—
194C2×Q8C2×Dic10612yes—
195C2×Q8#1171994512yes—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4#1171992111yescharacteristic
199C22×C4#1171992111yescharacteristic
200C22×C4#1171992111yescharacteristic
201C22×C4#1171992111yescharacteristic
202C22×C4C2×C4×D5311yescharacteristic
203C22×C4C2×D21211yescharacteristic
204C22×C4C2×C22⋊D5311yescharacteristic
205M16—24no—
206M16—42no—
207M16—2122no—
208M16—2122no—
209C2×C8—42no—
210C2×C8—2122no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—1064no—
215C4⋊C4—4242no—
216C4⋊C4—4242no—
217C4⋊C4—2122no—
218C4⋊C4—2121no—
219C4⋊C4—2121no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2122no—
227C23C2×#1171992111yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—4244no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—1064no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4#1313313111yescharacteristic
251C2×C4#1313313111yescharacteristic
252C2×C4C4×Dic10612yes—
253C2×C4#1313319612yes—
254C2×C4#1313320511yescharacteristic
255C2×C4#1313321711yescharacteristic
256C8—44no—
257C8—4242no—
258C22—21no—
259C22—21no—
260C22#1681052911yescharacteristic
261C22C2×#1313313111yescharacteristic
262C22#1681059611yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—4242no—
272C4—2124no—
273C4#1681042112yes—
274C2—21no—
275C2—21no—
276C2#2027570011yescharacteristic
277C2C2×#1681042111yescharacteristic
278C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC4×#101D53
hypocenterC53#13302
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
47. #101×C53Q8semidirect
159. C53#13302semidirect
184. #101Dic106semidirect

Maximal subgroups

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

subgroupindex
#202799202
#202747112
#202728412
C4×#101×C532
#1330253