Finite Group Databasebeta

← #27294739

Subgroups — #27294739

Order 13568.

378 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279927C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272741C214yesmaximal
4C4×#101×C53C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6#16810680—21no—
7#16810562C412yes—
8#16810562C2212yes—
9#16810530—21no—
10#16810529—21no—
11#16810529—21no—
12#16810488—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C2×#101×C53—21no—
18C4×M16×C53—21no—
19C53×#343C2212yes—
20C22×Dic106—21no—
21#13133217—41no—
22#13133217—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C412yes—
26C2×C4×Dic53—21no—
27C4⋊#88322—41no—
28C22⋊#88322—22no—
29#13133128—41no—
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124C2×C412yes—
35C22×Q8×C53D411yescharacteristic
36C2×M16×C53—22no—
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45#101×C53—24no—
46#97×C53—41no—
47C4×C8×C53—41no—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—44no—
61C2×Dic106—41no—
62C2×Dic106—41no—
63M16×C53—24no—
64M16×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C4⋊Dic53—42no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C4×Dic53—42no—
79C2×#88322—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—44no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53#9911yescharacteristic
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53#10412yes—
99C2×C4×C53#13712yes—
100C2×Dic53—44no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105Dic106—82no—
106Dic106—82no—
107C8×C53—44no—
108#88322—82no—
109#11794—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53C2×#9911yescharacteristic
113C22×C53#41911yescharacteristic
114C22×C53C2×#10411yescharacteristic
115C4×C53—21no—
116C4×C53—22no—
117C4×C53—22no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53—41no—
121C4×C53—41no—
122C4×C53#35512yes—
123Dic53—82no—
124Dic53—82no—
125#2568—531no—
126C4×#101D5311yescharacteristic
127#1467—531no—
128#1353—534no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53#150811yescharacteristic
132C2×C53#212311yescharacteristic
133C2×C53C2×#35511yescharacteristic
134C2×C4⋊Q8—531no—
135C2×C4×Q8C2×D5311yescharacteristic
136#432—1061no—
137C2×#101—21no—
138C4×M16—21no—
139#400—532no—
140#400—532no—
141#391—1061no—
142#391—1061no—
143#391—1061no—
144#382—1061no—
145C4×C4⋊C4—531no—
146#343C2×D5312yes—
147#324—532no—
148#324—1061no—
149C53#1179411yescharacteristic, complemented, Sylow
150C22×Q8—1061no—
151C22×Q8C22⋊D5311yescharacteristic
152C2×M16—22no—
153C4⋊Q8—2121no—
154C4⋊Q8—2121no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4C4×D5311yescharacteristic
161C2×C4⋊C4D21211yescharacteristic
162C2×C4⋊C4C22×D5311yescharacteristic
163C2×C42—1061no—
164C2×C42C4×D5311yescharacteristic
165C2×C42D21211yescharacteristic
166C2×C42C22⋊D5311yescharacteristic
167C4⋊C8—2121no—
168#101—24no—
169C22⋊C8—1062no—
170#97—41no—
171#97—2121no—
172C4×C8—41no—
173#95—532no—
174#95—1062no—
175#95—1062no—
176#95—1062no—
177#95—1062no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—2124no—
182C2×Q8—2121no—
183C2×Q8—2121no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4#1171992111yescharacteristic
187C22×C4#1171992111yescharacteristic
188C22×C4#1171992111yescharacteristic
189C22×C4#1171992111yescharacteristic
190C22×C4C2×C4×D5311yescharacteristic
191C22×C4C2×D21211yescharacteristic
192C22×C4C2×C22⋊D5311yescharacteristic
193M16—24no—
194M16—42no—
195C2×C8—42no—
196C2×C8—2122no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—2122no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—2122no—
210C23C2×#1171992111yescharacteristic
211Q8—44no—
212Q8—4242no—
213Q8—4242no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2124no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4#1313313111yescharacteristic
234C2×C4#1313313111yescharacteristic
235C2×C4#1313314512yes—
236C2×C4#1313324712yes—
237C8—44no—
238C8—4242no—
239C22—21no—
240C22—21no—
241C22C2×#1313313111yescharacteristic
242C22#1681063111yescharacteristic
243C22C2×#1313314511yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—4242no—
252C4—4242no—
253C4#1681041912yes—
254C2—21no—
255C2—21no—
256C2#2027323411yescharacteristic
257C2#2027637011yescharacteristic
258C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#101D53
hypocenterC53#11794
LCSG ▷ C22×C4×C53 ▷ C23×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
149. C53#11794semidirect

Maximal subgroups

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

subgroupindex
#202799272
#202730592
#202727412
C4×#101×C532
#1179453