Finite Group Databasebeta

← #27281797

Subgroups — #27281797

Order 13568.

432 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283800C211yesmaximal, characteristic
2#20273432C212yesmaximal
3#20273059C211yesmaximal, characteristic
4#20272571C212yesmaximal
5C53×#2568C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—21no—
8#16810680C2212yes—
9#16810680C2212yes—
10#16810676—21no—
11#16810369—24no—
12#16810349—21no—
13#16810349—21no—
14C2×C4⋊Q8×C53C2211yescharacteristic
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#400—21no—
17C53×#400—21no—
18C53×#391—21no—
19C53×#391—21no—
20C53×#391C412yes—
21C53×#382C412yes—
22C4×C4⋊C4×C53C2211yescharacteristic
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322D414yes—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133128—22no—
30#13133128—22no—
31#13133128—22no—
32C4×#88322—22no—
33C22×Q8×C53—21no—
34C22×Q8×C53C2×C411yescharacteristic
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53—22no—
37C4×Q8×C53—22no—
38C4×Q8×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C42×C53C2×C412yes—
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49#95×C53—22no—
50#95×C53—22no—
51#95×C53—22no—
52#95×C53—22no—
53#95×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—44no—
58C2×Q8×C53—41no—
59C2×Q8×C53—41no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—42no—
80C42×C53C2×D412yes—
81C42×C53#4112yes—
82C2×#88322—24no—
83C2×#88322—42no—
84C23×C53C2×C22⋊C411yescharacteristic
85Q8×C53—44no—
86Q8×C53—82no—
87Q8×C53—82no—
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—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53#10112yes—
107C2×C4×C53C4×D412yes—
108C2×C4×C53#12011yescharacteristic
109C2×C4×C53#12112yes—
110C2×C4×C53#12311yescharacteristic
111#88322—44no—
112#88322—82no—
113#9456—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53#38611yescharacteristic
117C22×C53C2×#10111yescharacteristic
118C22×C53#41311yescharacteristic
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53—41no—
126C4×C53—82no—
127C4×C53—82no—
128C4×C53#45412yes—
129#2983—531no—
130#2568D5311yescharacteristic
131#1556—532no—
132#1467—531no—
133#1285—532no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53#188811yescharacteristic
137C2×C53#190111yescharacteristic
138C2×C53#190911yescharacteristic
139C2×C4⋊Q8C2×D5311yescharacteristic
140C2×C4×Q8C2×D5311yescharacteristic
141#446—1064no—
142#432—532no—
143#432—532no—
144#432—1061no—
145#431—1061no—
146#400—21no—
147#400—21no—
148#391—21no—
149#391—21no—
150#391Dic5312yes—
151#382Dic5312yes—
152C4×C4⋊C4C2×D5311yescharacteristic
153#332—1064no—
154#324—1061no—
155#324—1061no—
156C53#945611yescharacteristic, complemented, Sylow
157C22×Q8—21no—
158C22×Q8C2×Dic5311yescharacteristic
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×Dic5312yes—
166C2×C4⋊C4C2×Dic5311yescharacteristic
167C2×C4⋊C4C22⋊D5311yescharacteristic
168C2×C4⋊C4C22⋊D5311yescharacteristic
169C2×C42C2×Dic5312yes—
170C2×C42C22⋊D5311yescharacteristic
171C2×C42C22⋊D5311yescharacteristic
172C2×C42C22×D5311yescharacteristic
173C4⋊C8—534no—
174C4⋊C8—1062no—
175C4⋊C8—1062no—
176C4⋊C8—1062no—
177C22⋊C8—1062no—
178C22⋊C8—1062no—
179#97—1062no—
180#97—1062no—
181#97—1062no—
182C4×C8—1062no—
183#95—21no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—44no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4C22⋊Dic5311yescharacteristic
197C22×C4C22⋊Dic5311yescharacteristic
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22⋊Dic5311yescharacteristic
200C22×C4C22×Dic5311yescharacteristic
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203C2×C8—1064no—
204C2×C8—2122no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C42D4×D5312yes—
217C42#1171994712yes—
218C23C2×C22⋊Dic5311yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—82no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4#1313313612yes—
241C2×C4D4×Dic5312yes—
242C2×C4#1313318511yescharacteristic
243C2×C4#1313318912yes—
244C2×C4#1313319811yescharacteristic
245C8—4242no—
246C8—2124no—
247C22—21no—
248C22—21no—
249C22#1681050311yescharacteristic
250C22C2×#1313313611yescharacteristic
251C22#1681060711yescharacteristic
252C4—22no—
253C4—21no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#1681079512yes—
262C2—21no—
263C2—21no—
264C2#2027511911yescharacteristic
265C2#2027518411yescharacteristic
266C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2568C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2568D53
hypocenterC53#9456
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
156. C53#9456semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202734322
#202730592
#202725712
C53×#25682
#945653