Finite Group Databasebeta

← #28312296

Subgroups — #28312296

Order 13568.

404 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 84 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#20290581C211yesmaximal, characteristic
2C4×#13133278C211yesmaximal, characteristic
3#20289812C214yesmaximal
4#20279930C211yesmaximal, characteristic
5#16811626—21no—
6C2×#13133278C2212yes—
7C2×#13133278C2212yes—
8C4×#11719940—21no—
9#16811503—21no—
10#16811503—21no—
11C2×#13133217C411yescharacteristic
12C2×C4×Dic106C2211yescharacteristic
13#16810563—21no—
14#16810563—21no—
15#16810529C2212yes—
16#16810528—21no—
17#16810528—21no—
18#16810489C412yes—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C2×#11719940—22no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133278—22no—
25#13133278—24no—
26#13133278Q814yescomplemented
27#13133272—41no—
28C4×#88324—41no—
29C22×Dic106—21no—
30C22×Dic106C2311yescharacteristic
31#13133217—41no—
32#13133217—41no—
33C4×Dic106—41no—
34C4×Dic106—41no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C412yes—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919D411yescharacteristic
39C2×#11719919D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42#13133124—22no—
43#13133124—22no—
44#13133124—22no—
45#13133124—22no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C412yes—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—22no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106—44no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×Dic106C2×Q812yes—
65C2×Dic106#4112yes—
66#11719940—24no—
67#11719940—42no—
68#11719940—42no—
69#11719940—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—42no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—42no—
76C4⋊Dic53—41no—
77C4⋊Dic53—42no—
78C4⋊Dic53—42no—
79#11719919—41no—
80#11719919—41no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C23×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53#9911yescharacteristic
95C2×C4×C53#9911yescharacteristic
96C2×C4×C53C4×Q812yes—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—44no—
106C2×Dic53—42no—
107C2×Dic53C22⋊Q812yes—
108C2×Dic53#12411yescharacteristic
109C2×Dic53C4⋊Q811yescharacteristic
110Dic106—42no—
111Dic106—42no—
112Dic106—82no—
113Dic106—82no—
114#88324—44no—
115#88324—82no—
116#13301—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53#39111yescharacteristic
120C22×C53C2×#9911yescharacteristic
121C22×C53#41011yescharacteristic
122C4×C53—21no—
123C4×C53—22no—
124C4×C53—41no—
125C4×C53—82no—
126C4×C53#35512yes—
127Dic53—22no—
128Dic53—21no—
129Dic53—21no—
130Dic53—41no—
131Dic53—41no—
132Dic53—82no—
133#2568—531no—
134#1801—531no—
135C4×#101—531no—
136#1394—534no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53#199611yescharacteristic
140C2×C53C2×#35511yescharacteristic
141C2×C53#213511yescharacteristic
142C2×C4⋊Q8—531no—
143C2×C4×Q8—531no—
144#423—1061no—
145C2×#101—532no—
146C2×#101—532no—
147C4×M16—1061no—
148#400—1061no—
149#400—1061no—
150#391—532no—
151#391—1061no—
152#391—1061no—
153#382—532no—
154C4×C4⋊C4#615711yescharacteristic
155#343—1061no—
156#343—1061no—
157C53#1330111yescharacteristic, complemented, Sylow
158C22×Q8—531no—
159C22×Q8—1061no—
160C2×M16—1062no—
161C2×M16—1062no—
162C4⋊Q8—2121no—
163C4⋊Q8—2121no—
164C4×Q8—2121no—
165C4×Q8—2121no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—531no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1061no—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42C2×#615712yes—
175C2×C42C2×#615711yescharacteristic
176C4⋊C8—2121no—
177C4⋊C8—2121no—
178#101—534no—
179#101—1062no—
180#101—1064no—
181#97—2121no—
182C4×C8—2121no—
183#95—21no—
184#95—1062no—
185#95—1062no—
186#95—1062no—
187#95—1062no—
188C2×Q8—532no—
189C2×Q8—532no—
190C2×Q8—2121no—
191C2×Q8—2121no—
192C2×Q8—2124no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C22×C4—22no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—1062no—
201C22×C4C22⋊#615711yescharacteristic
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22×#615711yescharacteristic
204M16—1064no—
205M16—2122no—
206M16—2122no—
207M16—2122no—
208C2×C8—2122no—
209C2×C8—2122no—
210C4⋊C4—42no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2122no—
215C4⋊C4—2122no—
216C42—21no—
217C42—21no—
218C42—42no—
219C42—1061no—
220C42—1061no—
221C42—1061no—
222C42—1061no—
223C23C2×C22⋊#615711yescharacteristic
224Q8—4242no—
225Q8—4242no—
226Q8—2122no—
227Q8—2122no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—532no—
237C2×C4—531no—
238C2×C4—531no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—2124no—
246C2×C4—2122no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4#1313327611yescharacteristic
250C2×C4#1313327611yescharacteristic
251C2×C4Q8×#615712yes—
252C8—4242no—
253C8—2124no—
254C22—21no—
255C22—21no—
256C22(C2×#6157)⋊Q811yescharacteristic
257C22C2×#1313327611yescharacteristic
258C22#1681160311yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—1061no—
264C4—1062no—
265C4—1061no—
266C4—4242no—
267C4—2121no—
268C4—2121no—
269C4#1681153212yes—
270C2—21no—
271C2—21no—
272C2#2029090511yescharacteristic
273C2C2×#1681153211yescharacteristic
274C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#13301
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
26. #13133278Q8semidirect
157. C53#13301semidirect

Maximal subgroups

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

subgroupindex
#202905812
C4×#131332782
#202898122
#202799302
#1330153