Finite Group Databasebeta

← #27300102

Subgroups — #27300102

Order 13568.

432 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277433C211yesmaximal, characteristic
2C2×#16810421C212yesmaximal
3#20274709C212yesmaximal
4#20272842C211yesmaximal, characteristic
5C53×#1394C211yesmaximal, characteristic
6C2×#13133215C2211yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133182C411yescharacteristic
9#16810646—22no—
10C2×#13133136C2212yes—
11#16810529C411yescharacteristic
12#16810510—22no—
13#16810505—22no—
14#16810488C411yescharacteristic
15#16810469C411yescharacteristic
16#16810421—24no—
17#16810391—22no—
18C2×#101×C53C2212yes—
19C53×#391C2211yescharacteristic
20C22×Dic106C2×C411yescharacteristic
21C2×#11719917—22no—
22#13133215—22no—
23#13133215D412yes—
24#13133215Q812yes—
25Q8×Dic53—42no—
26#13133182—42no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×#11719919—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35#13133146—44no—
36#13133136—22no—
37#13133136—22no—
38#13133124—21no—
39#13133124—22no—
40#13133124—22no—
41#13133124—21no—
42C22×Q8×C53C2311yescharacteristic
43C2×M16×C53—22no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46#101×C53—22no—
47#101×C53—22no—
48#95×C53—22no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C2×D411yescharacteristic
52C2×Q8×C53C2×Q811yescharacteristic
53C2×Q8×C53#4112yes—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53C22⋊C412yes—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C2×Dic106—21no—
64C2×Dic106—41no—
65C2×Dic106—41no—
66C2×Dic106C4⋊C412yes—
67M16×C53—42no—
68M16×C53—42no—
69C2×C8×C53—42no—
70C4⋊C4×C53—41no—
71C42×C53—41no—
72C4⋊Dic53—41no—
73C4⋊Dic53—42no—
74#11719919—24no—
75#11719919—24no—
76#11719919—42no—
77C4×Dic53—21no—
78C4×Dic53—24no—
79C4×Dic53—22no—
80C4×Dic53—42no—
81C4×Dic53C4⋊C412yes—
82#11719917—42no—
83#11719917—42no—
84C2×#88322—42no—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—41no—
87Q8×C53—41no—
88Q8×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53C2×C4⋊C411yescharacteristic
99C2×C4×C53#11711yescharacteristic
100C2×C4×C53#12112yes—
101C2×C4×C53C22⋊Q812yes—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53#9912yes—
114Dic106—42no—
115Dic106—42no—
116C8×C53—82no—
117#88322—82no—
118#12804—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#38011yescharacteristic
122C22×C53C2×#9911yescharacteristic
123C22×C53#41011yescharacteristic
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—81no—
129C4×C53—81no—
130Dic53—22no—
131Dic53—42no—
132Dic53—42no—
133Dic53—41no—
134Dic53—42no—
135#2327—531no—
136C2×#356—532no—
137#1801—532no—
138#1394—531no—
139#1394D5311yescharacteristic
140C2×C53—21no—
141C2×C53—21no—
142C2×C53#179911yescharacteristic
143C2×C53#213311yescharacteristic
144C2×C53#213511yescharacteristic
145C2×C4⋊Q8—531no—
146C2×C4×Q8—531no—
147C2×C4×Q8—531no—
148#423—1062no—
149C2×#101—532no—
150C2×#101C2×D5312yes—
151#391—531no—
152#391C2×D5311yescharacteristic
153#387—1062no—
154#387—1062no—
155#382—531no—
156#376—531no—
157#356—1064no—
158#343—1062no—
159C53#1280411yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×Q8C22×D5311yescharacteristic
162C2×M16—22no—
163C2×M16—1062no—
164C4⋊Q8—532no—
165C4⋊Q8—532no—
166C4⋊Q8—1062no—
167C4×Q8—2122no—
168C4×Q8—2122no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1061no—
173C2×C4⋊C4C4×D5311yescharacteristic
174C2×C42—531no—
175C2×C42—531no—
176C2×C42—532no—
177C2×C42—1061no—
178C2×C42C4×D5311yescharacteristic
179C4⋊C8—2124no—
180#101—22no—
181#101—22no—
182#101—1062no—
183#101—1062no—
184#95—22no—
185#95—1061no—
186#95—1061no—
187#95—1062no—
188#95—1062no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—532no—
192C2×Q8—1061no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C2×Q8#1171994512yes—
196C2×Q8D4×D5311yescharacteristic
197C2×Q8Q8×D5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4D4×D5312yes—
207M16—42no—
208M16—42no—
209M16—2122no—
210M16—2122no—
211C2×C8—42no—
212C2×C8—2122no—
213C4⋊C4—41no—
214C4⋊C4—1064no—
215C4⋊C4—1064no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C4⋊C4—2122no—
219C42—41no—
220C42—532no—
221C42—1064no—
222C42—1062no—
223C42—1061no—
224C42—2122no—
225C23C22⋊C4×D5311yescharacteristic
226Q8—42no—
227Q8—41no—
228Q8—41no—
229Q8—2122no—
230Q8—2122no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—532no—
241C2×C4—1061no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—2121no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2122no—
251C2×C4—2121no—
252C2×C4C4⋊C4×D5311yescharacteristic
253C2×C4#1313317411yescharacteristic
254C2×C4#1313319012yes—
255C2×C4#1313319512yes—
256C8—82no—
257C8—4242no—
258C22—21no—
259C22—21no—
260C22#1681048211yescharacteristic
261C22#99×D5311yescharacteristic
262C22#1681059511yescharacteristic
263C4—22no—
264C4—21no—
265C4—21no—
266C4—22no—
267C4—81no—
268C4—81no—
269C4—1062no—
270C4—2122no—
271C4—2122no—
272C4—2121no—
273C4—2122no—
274C2—21no—
275C2—21no—
276C2#2027470211yescharacteristic
277C2#2027640111yescharacteristic
278C2#2027640811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#1394D53
hypocenterC53#12804
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
159. C53#12804semidirect

Maximal subgroups

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

subgroupindex
#202774332
C2×#168104212
#202747092
#202728422
C53×#13942
#1280453