Finite Group Databasebeta

← #27294933

Subgroups — #27294933

Order 13568.

286 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 79 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#20276527C211yesmaximal, characteristic, complemented
2#20274864C211yesmaximal, characteristic, complemented
3#20272709C211yesmaximal, characteristic, complemented
4#20272706C211yesmaximal, characteristic
5#20272703C211yesmaximal, characteristic, complemented
6#20272700C211yesmaximal, characteristic
7C53×#3183C211yesmaximal, characteristic
8#16810444—21no—
9#16810405C2211yescharacteristic
10#16810405C2211yescharacteristic
11#16810401—21no—
12#16810394—21no—
13#16810383—21no—
14#16810381C2211yescharacteristic
15#16810380C2211yescharacteristic
16C53×#529—21no—
17C53×#499C411yescharacteristic, complemented
18C53×#495C411yescharacteristic, complemented
19C53×#488—21no—
20C53×#470C2211yescharacteristic
21C53×#469C2211yescharacteristic
22C53×#468—21no—
23C53×#442C411yescharacteristic, complemented
24C53×#441C411yescharacteristic, complemented
25C53×#440—21no—
26C4×M16×C53C2211yescharacteristic
27C2×#11719917—21no—
28#13133176—21no—
29#13133157—21no—
30#13133157—21no—
31C2×Q16×C53C2×C411yescharacteristic
32C2×SD16×C53—21no—
33C2×D8×C53C2×C411yescharacteristic
34C2×M16×C53D411yescharacteristic
35C2×M16×C53C2311yescharacteristic
36#126×C53—41no—
37#125×C53—21no—
38#124×C53—21no—
39#123×C53—41no—
40C22⋊Q8×C53—21no—
41#121×C53—21no—
42C4×Q8×C53—21no—
43C4×D4×C53—21no—
44C4×D4×C53—41no—
45#117×C53—41no—
46C2×C42×C53D411yescharacteristic
47#107×C53—21no—
48#106×C53C2×C411yescharacteristic
49#106×C53C2×C411yescharacteristic
50#103×C53—21no—
51#103×C53—21no—
52#102×C53—21no—
53#102×C53—21no—
54#97×C53C2×C411yescharacteristic
55#97×C53D411yescharacteristic
56C4×C8×C53C2×C411yescharacteristic
57C4×C8×C53D411yescharacteristic
58C2×Q8×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64Q16×C53—41no—
65SD16×C53—42no—
66D8×C53—41no—
67M16×C53—41no—
68M16×C53—41no—
69C2×C8×C53C22⋊C411yescharacteristic
70C2×C8×C53C22⋊C411yescharacteristic
71C2×C8×C53C22×C411yescharacteristic
72C2×C8×C53C2×D411yescharacteristic
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C42×C53—21no—
85C42×C53—41no—
86C42×C53C22⋊C411yescharacteristic
87C42×C53C22⋊C411yescharacteristic
88C22⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4×Dic53—41no—
91#11719917—41no—
92#11719917—41no—
93C2×#88322—41no—
94#11719908—41no—
95#11719908—41no—
96C23×C53—41no—
97C23×C53AGL(1,ℤ/8)11yescharacteristic
98Q8×C53—41no—
99Q8×C53—41no—
100D4×C53—41no—
101D4×C53—41no—
102D4×C53—82no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—81no—
111C2×C4×C53—82no—
112C2×C4×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
114C2×Dic53—81no—
115C8×C53—21no—
116C8×C53—21no—
117C8×C53—21no—
118C8×C53—21no—
119#88322—81no—
120#11836—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—41no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53—81no—
126C22×C53#41811yescharacteristic
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53—81no—
131C4×C53—81no—
132C4×C53—81no—
133C4×C53#41011yescharacteristic
134C4×C53#42111yescharacteristic
135Dic53—161no—
136#3183D5311yescharacteristic
137#2165—531no—
138#1841—531no—
139#1342—531no—
140#1341—531no—
141#1340—531no—
142#1339—531no—
143C2×C53—21no—
144C2×C53—41no—
145C2×C53—161no—
146C2×C53#151411yescharacteristic
147#529—21no—
148#499Dic5311yescharacteristic, complemented
149#495Dic5311yescharacteristic, complemented
150#488—21no—
151#470C2×D5311yescharacteristic
152#469C2×D5311yescharacteristic
153#468—21no—
154#442Dic5311yescharacteristic, complemented
155#441Dic5311yescharacteristic, complemented
156#440—21no—
157C4×M16C2×D5311yescharacteristic
158#364—1061no—
159#349—531no—
160#349—531no—
161#347—1061no—
162#344—1061no—
163#339—1061no—
164#338—531no—
165H(3,ℤ/4)—531no—
166C53#1183611yescharacteristic, complemented, Sylow
167C2×Q16C2×Dic5311yescharacteristic
168C2×SD16—21no—
169C2×D8C2×Dic5311yescharacteristic
170C2×M16—1061no—
171C2×M16C22⋊D5311yescharacteristic
172C2×M16C22×D5311yescharacteristic
173#126—41no—
174#125—21no—
175#124—21no—
176#123—41no—
177C22⋊Q8—21no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—41no—
182#117—41no—
183#117—1061no—
184C2×C42C22⋊D5311yescharacteristic
185M32—1061no—
186M32—1061no—
187#107—21no—
188#106C2×Dic5311yescharacteristic
189#106C2×Dic5311yescharacteristic
190#103—21no—
191#103—21no—
192#102—21no—
193#102—21no—
194#97C2×Dic5311yescharacteristic
195#97C22⋊D5311yescharacteristic
196C4×C8C2×Dic5311yescharacteristic
197C4×C8C22⋊D5311yescharacteristic
198C2×Q8—21no—
199C2×D4—21no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4C2×C22⋊D5311yescharacteristic
204Q16—41no—
205SD16—42no—
206D8—41no—
207M16—41no—
208M16—41no—
209M16—2121no—
210M16—2121no—
211C2×C8—2121no—
212C2×C8C22⋊Dic5311yescharacteristic
213C2×C8C22⋊Dic5311yescharacteristic
214C2×C8C22×Dic5311yescharacteristic
215C2×C8C2×C22⋊D5311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—2121no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—2121no—
229C42—21no—
230C42—41no—
231C42—2121no—
232C42C22⋊Dic5311yescharacteristic
233C42C22⋊Dic5311yescharacteristic
234C16—2121no—
235C16—2121no—
236C23—41no—
237C23#1313324311yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—41no—
241D4—41no—
242D4—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—4241no—
253C2×C4C2×C22⋊Dic5311yescharacteristic
254C2×C4#1313324311yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—4241no—
260C22—21no—
261C22—41no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#1681062811yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—8481no—
273C4#1681059711yescharacteristic
274C4#1681064011yescharacteristic
275C2—21no—
276C2—41no—
277C2—161no—
278C2#2027326211yescharacteristic
subgroupG/H
centerC2#20273262
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#3183C2
socleC2×C53#1514
prime radicalC53×#469C22
cosocleC2×M16×C53C23
hypercenter#3183D53
hypocenterC53#11836
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3183

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20274864C2semidirect
3. #20272709C2semidirect
5. #20272703C2semidirect
17. C53×#499C4semidirect
18. C53×#495C4semidirect
23. C53×#442C4semidirect
24. C53×#441C4semidirect
148. #499Dic53semidirect
149. #495Dic53semidirect
154. #442Dic53semidirect
155. #441Dic53semidirect
166. C53#11836semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202748642
#202727092
#202727062
#202727032
#202727002
C53×#31832
#1183653