Finite Group Databasebeta

← #28304346

Subgroups — #28304346

Order 13568.

498 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 116 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#20291608C211yesmaximal, characteristic
2#20290019C212yesmaximal
3#20289958C211yesmaximal, characteristic
4#20289539C212yesmaximal
5#20277433C211yesmaximal, characteristic
6Q8×#88324—24no—
7#16811639—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10#16811473—24no—
11#16811468—22no—
12C2×#13133215C411yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×#13133182C2211yescharacteristic
15#16810529C411yescharacteristic
16#16810510—22no—
17#16810505—22no—
18#16810488C411yescharacteristic
19#16810469C2211yescharacteristic
20C53×#391C411yescharacteristic
21C4⋊#88324—24no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26C4×#88324—24no—
27C4×#88324—22no—
28C4×#88324D414yes—
29C22×Dic106C2×C411yescharacteristic
30#13133215—22no—
31#13133215C814yescomplemented
32Q8×Dic53—22no—
33Q8×Dic53—22no—
34#13133182—22no—
35#13133182—22no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×#11719919—22no—
39C2×#11719919—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D412yes—
43C2×C4×Dic53C2311yescharacteristic
44#13133124—22no—
45#13133124—22no—
46#13133124—21no—
47#13133124—21no—
48C22×Q8×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53C2×C812yes—
55C2×Q8×C53M1612yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C412yes—
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—21no—
68C2×Dic106C2×C812yes—
69C2×#88324—24no—
70C2×#88324—42no—
71C4⋊C4×C53—22no—
72C42×C53—22no—
73C4⋊Dic53—22no—
74C4⋊Dic53—24no—
75#11719919—28no—
76#11719919—41no—
77#11719919—41no—
78C4×Dic53—21no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53C2×C812yes—
83C4×Dic53C2×D412yes—
84C4×Dic53#4112yes—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—24no—
87Q8×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C22⋊C814yes—
96C2×C4×C53C4×D412yes—
97C2×C4×C53C22×C811yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53#13112yes—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53#12011yescharacteristic
111C2×Dic53#12112yes—
112C2×Dic53#12311yescharacteristic
113Dic106—24no—
114Dic106—42no—
115#88324—44no—
116#88324—82no—
117#9100—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#38611yescharacteristic
121C22×C53C2×C22⋊C811yescharacteristic
122C22×C53#40811yescharacteristic
123C4×C53—24no—
124C4×C53—81no—
125C4×C53—81no—
126C4×C53C8×D412yes—
127C4×C53#43512yes—
128Dic53—22no—
129Dic53—21no—
130Dic53—21no—
131Dic53—42no—
132Dic53—42no—
133Dic53—42no—
134#2982—531no—
135#2327—531no—
136#1479—532no—
137#1461—531no—
138#1274—532no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#186911yescharacteristic
142C2×C53#190911yescharacteristic
143C2×C53#191211yescharacteristic
144C2×C4⋊Q8—531no—
145C2×C4×Q8—531no—
146C2×C4×Q8—531no—
147C8×Q8—1064no—
148#431—532no—
149#431—532no—
150#431—1062no—
151#391—531no—
152#391#615711yescharacteristic
153#387—1062no—
154#387—1062no—
155#382—531no—
156#376—531no—
157Q8⋊C8—1064no—
158#324—1062no—
159C53#910011yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×Q8C2×#615711yescharacteristic
162C4⋊Q8—534no—
163C4⋊Q8—1062no—
164C4×Q8—1062no—
165C4×Q8—1062no—
166C4×Q8—1062no—
167C4×Q8—1062no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4C2×#615711yescharacteristic
173C2×C42—531no—
174C2×C42—531no—
175C2×C42—532no—
176C2×C42—1061no—
177C2×C42C2×#615711yescharacteristic
178C4⋊C8—1062no—
179C4⋊C8—1062no—
180C4⋊C8—1064no—
181C22⋊C8—1062no—
182C22⋊C8—1062no—
183C4×C8—534no—
184C4×C8—1062no—
185C4×C8—1064no—
186#95—22no—
187#95—1061no—
188#95—1061no—
189#95—1062no—
190#95—1062no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—532no—
194C2×Q8—1061no—
195C2×Q8—1062no—
196C2×Q8—1062no—
197C2×Q8D53⋊C812yes—
198C2×Q8#1171993612yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4C22⋊#615712yes—
207C22×C4C22×#615711yescharacteristic
208C2×C8—1064no—
209C2×C8—2122no—
210C4⋊C4—22no—
211C4⋊C4—1064no—
212C4⋊C4—1068no—
213C4⋊C4—1062no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C42—22no—
217C42—532no—
218C42—532no—
219C42—532no—
220C42—1062no—
221C42—1062no—
222C42—1061no—
223C42—2122no—
224C23C2×C22⋊#615711yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—1064no—
228Q8—2122no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—531no—
237C2×C4—532no—
238C2×C4—531no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2122no—
249C2×C4#1313327414yes—
250C2×C4D4×#615712yes—
251C2×C4C2×D53⋊C811yescharacteristic
252C2×C4C2×#1171993611yescharacteristic
253C2×C4#1313330412yes—
254C8—4242no—
255C8—2124no—
256C22—21no—
257C22—21no—
258C22#1681157111yescharacteristic
259C22C2×#1313327411yescharacteristic
260C22#1681159611yescharacteristic
261C4—24no—
262C4—81no—
263C4—81no—
264C4—1062no—
265C4—1061no—
266C4—1061no—
267C4—2122no—
268C4—2122no—
269C4—2122no—
270C4#1681165012yes—
271C4#1681165212yes—
272C2—21no—
273C2—21no—
274C2#2029069011yescharacteristic
275C2#2029075711yescharacteristic
276C2#2029076311yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#9100
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
31. #13133215C8semidirect
159. C53#9100semidirect

Maximal subgroups

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

subgroupindex
#202916082
#202900192
#202899582
#202895392
#202774332
#910053