Finite Group Databasebeta

← #27293758

Subgroups — #27293758

Order 13568.

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

normal subgroups only

This group has 358 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#20273039C211yesmaximal, characteristic
3#20272658C214yesmaximal
4C53×#1467C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6#16810676—21no—
7#16810562—21no—
8#16810562—21no—
9#16810530C412yes—
10#16810529—21no—
11#16810529—21no—
12#16810488C2212yes—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106—21no—
21#13133217—41no—
22#13133217—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C412yes—
26C2×C4×Dic53C2×C412yes—
27C4⋊#88322—41no—
28C22⋊#88322—22no—
29C4×#88322—41no—
30#13133124—22no—
31#13133124—22no—
32#13133124—21no—
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C4×Q8×C53—41no—
37C4×Q8×C53—41no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C4⋊C8×C53—41no—
45C22⋊C8×C53—22no—
46#97×C53—41no—
47C2×Q8×C53—22no—
48C2×Q8×C53—22no—
49C2×Q8×C53—41no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—22no—
59C2×Dic106—44no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×C8×C53—42no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C4⋊Dic53—42no—
73C4⋊Dic53—42no—
74C4⋊Dic53—42no—
75C4×Dic53—24no—
76C2×#88322—42no—
77C23×C53C2×C22⋊C411yescharacteristic
78Q8×C53—44no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53#9911yescharacteristic
94C2×C4×C53#9911yescharacteristic
95C2×C4×C53#10112yes—
96C2×C4×C53#13112yes—
97C2×Dic53—44no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102Dic106—82no—
103Dic106—82no—
104C8×C53—82no—
105#88322—82no—
106#11573—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53#40811yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53C2×#10111yescharacteristic
112C4×C53—21no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—41no—
116C4×C53—41no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53#35512yes—
120Dic53—82no—
121Dic53—82no—
122#2568—531no—
123#1467D5311yescharacteristic
124#1461—531no—
125#1324—534no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53#147211yescharacteristic
129C2×C53#212911yescharacteristic
130C2×C53C2×#35511yescharacteristic
131C2×C4⋊Q8—531no—
132C2×C4×Q8C2×D5311yescharacteristic
133#432—21no—
134#431—1061no—
135#400—1061no—
136#400—1061no—
137#391—532no—
138#391—1061no—
139#391—1061no—
140#382—532no—
141C4×C4⋊C4—531no—
142#324—21no—
143#324—532no—
144#324—1061no—
145#324C2×D5312yes—
146C53#1157311yescharacteristic, complemented, Sylow
147C22×Q8—1061no—
148C22×Q8C4×D5311yescharacteristic
149C4⋊Q8—2121no—
150C4⋊Q8—2121no—
151C4×Q8—41no—
152C4×Q8—41no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—1061no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4D21211yescharacteristic
157C2×C4⋊C4C22⋊D5311yescharacteristic
158C2×C4⋊C4C22×D5311yescharacteristic
159C2×C42—532no—
160C2×C42C4×D5311yescharacteristic
161C2×C42D21211yescharacteristic
162C2×C42C22⋊D5311yescharacteristic
163C4⋊C8—41no—
164C4⋊C8—2121no—
165C22⋊C8—22no—
166C22⋊C8—1062no—
167#97—41no—
168C4×C8—2121no—
169#95—1062no—
170#95—1062no—
171#95—1061no—
172#95—1062no—
173#95—1062no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—2124no—
178C2×Q8—2121no—
179C2×Q8—2121no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4#1171992111yescharacteristic
183C22×C4#1171992111yescharacteristic
184C22×C4#1171992111yescharacteristic
185C22×C4#1171992111yescharacteristic
186C22×C4C2×C4×D5311yescharacteristic
187C22×C4C2×D21211yescharacteristic
188C22×C4C2×C22⋊D5311yescharacteristic
189C2×C8—42no—
190C2×C8—2122no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—1064no—
204C23C2×#1171992111yescharacteristic
205Q8—44no—
206Q8—4242no—
207Q8—4242no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2124no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4#1313313111yescharacteristic
228C2×C4#1313313111yescharacteristic
229C2×C4#1313313512yes—
230C2×C4#1313322512yes—
231C8—82no—
232C8—4242no—
233C22—21no—
234C22—21no—
235C22#1681059111yescharacteristic
236C22C2×#1313313111yescharacteristic
237C22C2×#1313313511yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—4242no—
246C4—4242no—
247C4#1681041912yes—
248C2—21no—
249C2—21no—
250C2#2027308011yescharacteristic
251C2#2027638911yescharacteristic
252C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11573
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
146. C53#11573semidirect

Maximal subgroups

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

subgroupindex
#202799272
#202730392
#202726582
C53×#14672
#1157353