Finite Group Databasebeta

← #27294796

Subgroups — #27294796

Order 13568.

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

normal subgroups only

This group has 388 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#20274798C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272721C214yesmaximal, complemented
4C53×#2621C211yesmaximal, characteristic
5#16810680—21no—
6#16810587—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C22⋊Q8×C53—21no—
13C2×#117×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398C2212yes—
16C53×#397—21no—
17C53×#397C412yes—
18C53×#391—21no—
19C53×#382C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—41no—
24C22⋊#88322—41no—
25#13133128—41no—
26C22×Q8×C53—21no—
27C23×C4×C53D411yescharacteristic
28C4⋊Q8×C53—41no—
29C4⋊Q8×C53—41no—
30C22⋊Q8×C53—44no—
31#117×C53—41no—
32#117×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44#95×C53—21no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C2×Q8×C53—42no—
50C2×Q8×C53—42no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—44no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—44no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78#11719917—42no—
79#11719917—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—82no—
89Q8×C53—82no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53AGL(1,ℤ/8)12yes—
113#88322—82no—
114#88322—82no—
115#11807—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—81no—
123C22×C53C2×#9911yescharacteristic
124C22×C53#41811yescharacteristic
125C22×C53#42111yescharacteristic
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—82no—
134C4×C53—82no—
135#2621D5311yescharacteristic
136#1822—531no—
137#1464—531no—
138#1346—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—81no—
142C2×C53#151011yescharacteristic
143C2×C53#212311yescharacteristic
144C2×C53#213511yescharacteristic
145C2×C4⋊Q8Dic5311yescharacteristic
146C2×C22⋊Q8—21no—
147C2×#117C2×D5311yescharacteristic
148#432—1061no—
149#407—1061no—
150#398—21no—
151#398C2×D5312yes—
152#397—21no—
153#397Dic5312yes—
154#391—21no—
155#382Dic5311yescharacteristic
156#343—532no—
157#343—1061no—
158#324—532no—
159#323—1061no—
160C53#1180711yescharacteristic, complemented, Sylow
161C22×Q8—21no—
162C23×C4C22⋊D5311yescharacteristic
163C2×M16—1062no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C22⋊Q8—44no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×Dic5312yes—
173C2×C4⋊C4C22⋊D5311yescharacteristic
174C2×C4⋊C4C22×D5311yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4C2×Dic5311yescharacteristic
177C2×C22⋊C4C22⋊D5311yescharacteristic
178C2×C42C2×Dic5311yescharacteristic
179C2×C42C22⋊D5311yescharacteristic
180C4⋊C8—2121no—
181C22⋊C8—1062no—
182C22⋊C8—2121no—
183C22⋊C8—2121no—
184#97—2121no—
185#95—22no—
186#95—21no—
187#95—22no—
188#95C2×Dic5312yes—
189C24C22⋊Dic5311yescharacteristic
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4C22⋊Dic5311yescharacteristic
201C22×C4C22⋊Dic5311yescharacteristic
202C22×C4C22⋊Dic5311yescharacteristic
203C22×C4C22×Dic5311yescharacteristic
204C22×C4C2×C22⋊D5311yescharacteristic
205C22×C4C2×C22⋊D5311yescharacteristic
206M16—2122no—
207M16—2122no—
208C2×C8—2122no—
209C2×C8—2122no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—44no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×C22⋊Dic5311yescharacteristic
229Q8—82no—
230Q8—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4#1313313211yescharacteristic
252C2×C4#1313313211yescharacteristic
253C2×C4#1313324312yes—
254C8—4242no—
255C8—4242no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#1313313211yescharacteristic
264C22#1681062811yescharacteristic
265C22#1681064011yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—82no—
274C4—82no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#2027324411yescharacteristic
279C2#2027636911yescharacteristic
280C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2621C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2621D53
hypocenterC53#11807
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
160. C53#11807semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730492
#202727212
C53×#26212
#1180753