Finite Group Databasebeta

← #28309413

Subgroups — #28309413

Order 13568.

358 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 74 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#20291078C211yesmaximal, characteristic, complemented
2#20290637C211yesmaximal, characteristic, complemented
3#20289702C214yesmaximal, complemented
4#20285295C211yesmaximal, characteristic
5#16811567—21no—
6#16811544—21no—
7#16811518C2212yes—
8#16811511—21no—
9#16811497—21no—
10#16811497C2212yes—
11#16811174—21no—
12#16811000C411yescharacteristic, complemented
13#16810971—21no—
14#16810860—21no—
15#16810860C2212yes—
16#16810705—21no—
17#16810705C412yescomplemented
18C4×C8×D53C2211yescharacteristic
19C53×#500C411yescharacteristic, complemented
20C2×C4⋊#6157—22no—
21C2×#11719910—22no—
22(C2×#6157)⋊C4—41no—
23(C2×#6157)⋊C4—41no—
24C2×#11719949—41no—
25C2×D424—21no—
26C2×D424C2×C412yes—
27C2×C8×D53D411yescharacteristic
28C2×C8×D53C2311yescharacteristic
29#13133214—21no—
30#13133206—41no—
31#13133192—22no—
32#13133192—41no—
33#13133179—22no—
34#13133179—41no—
35C42×D53D411yescharacteristic
36#13133137—22no—
37#13133137—22no—
38C4×#88322D411yescharacteristic
39C8×Dic53C2×C411yescharacteristic
40C8×Dic53D411yescharacteristic
41C4⋊Q8×C53—21no—
42#107×C53—21no—
43#107×C53C2×C412yes—
44C4×C8×C53C2×C411yescharacteristic
45C22×#6157—42no—
46C2×Q8×C53—41no—
47#11719949—84no—
48C2×D212—22no—
49C2×D212—42no—
50C2×D212—41no—
51C2×D212—41no—
52C2×C4×D53—42no—
53C2×C4×D53D811yescharacteristic
54C2×C4×D53D811yescharacteristic
55C2×C4×D53C2×D411yescharacteristic
56C4⋊#6157—42no—
57C4⋊#6157—42no—
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C22×C411yescharacteristic
60C4⋊C4×C53—22no—
61C4⋊C4×C53—41no—
62C42×C53C22⋊C411yescharacteristic
63#11719921—42no—
64#11719921—42no—
65C4×Dic53C22⋊C411yescharacteristic
66C4×Dic53SD1611yescharacteristic
67C4×Dic53SD1611yescharacteristic
68C2×#88322C22⋊C411yescharacteristic
69C2×#88322C2×D411yescharacteristic
70D424—42no—
71D424—42no—
72C8×D53—41no—
73C8×D53—41no—
74#11719910—42no—
75C22×D53—42no—
76C22×D53C2×D811yescharacteristic
77C2×#6157—82no—
78C2×#6157—82no—
79C2×#6157—82no—
80Q8×C53—82no—
81C2×C4×C53—42no—
82C2×C4×C53#10211yescharacteristic
83C2×C4×C53#10211yescharacteristic
84C2×C4×C53C2×C22⋊C411yescharacteristic
85C2×Dic53#10211yescharacteristic
86C2×Dic53#10211yescharacteristic
87C2×Dic53C2×SD1611yescharacteristic
88D212—42no—
89D212—42no—
90D212—82no—
91D212—82no—
92D212—84no—
93C4×D53—22no—
94C4×D53—21no—
95C4×D53—21no—
96C4×D53—41no—
97C4×D53—84no—
98C8×C53—21no—
99C8×C53—21no—
100#88322—21no—
101#88322—21no—
102#11858—531nomaximal, Sylow
103C22×C53C2×#10211yescharacteristic
104C2×D53—21no—
105C2×D53—21no—
106C2×D53—82no—
107C2×D53—82no—
108C2×D53—82no—
109C2×D53#50912yes—
110#6157—162no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—82no—
114C4×C53#41011yescharacteristic
115C4×C53#42111yescharacteristic
116Dic53—21no—
117Dic53—21no—
118Dic53—21no—
119#3169—531no—
120#2163—531no—
121#1835—531no—
122#1347—534no—
123C2×C53#151711yescharacteristic
124C2×C53#214011yescharacteristic
125C2×C53#214511yescharacteristic
126D53—22no—
127D53—162no—
128#532—1061no—
129#500#615711yescharacteristic, complemented
130#493—531no—
131#486—1061no—
132#466—532no—
133#466—1061no—
134C4×D8—532no—
135C4×D8—1061no—
136C2×C4×C8—531no—
137#382—1061no—
138C4⋊C16—1061no—
139C22⋊C16—532no—
140#346—1061no—
141#340—532no—
142#340—1061no—
143C53#1185811yescharacteristic, complemented, Sylow
144C2×#41—2121no—
145C2×D8—532no—
146C2×D8—1061no—
147C22×C8—531no—
148C22×C8—531no—
149C4⋊Q8—21no—
150#127—1061no—
151#124—2121no—
152#121—1062no—
153#121—2121no—
154C4×D4—1062no—
155C4×D4—2121no—
156C2×C4⋊C4—1062no—
157C2×C42—531no—
158C2×C16—1062no—
159#107—21no—
160#107C2×#615712yes—
161#102—1062no—
162#102—1062no—
163C4×C8—531no—
164C4×C8—531no—
165C4×C8—531no—
166C4×C8C2×#615711yescharacteristic
167#95—2121no—
168#95—2121no—
169#41—4244no—
170C2×Q8—41no—
171C2×D4—1062no—
172C2×D4—2122no—
173C2×D4—2121no—
174C2×D4—2121no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—2122no—
179C22×C4—2122no—
180D8—2122no—
181D8—2122no—
182C2×C8—531no—
183C2×C8—531no—
184C2×C8—2121no—
185C2×C8—2121no—
186C2×C8C22⋊#615711yescharacteristic
187C2×C8C22×#615711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—2122no—
191C4⋊C4—2122no—
192C22⋊C4—2122no—
193C22⋊C4—2122no—
194C42—531no—
195C42—531no—
196C42—531no—
197C42C22⋊#615711yescharacteristic
198C16—2122no—
199C23—531no—
200C23—2122no—
201Q8—82no—
202D4—4242no—
203D4—4242no—
204D4—4244no—
205D4—2122no—
206D4—2122no—
207C2×C4—42no—
208C2×C4—531no—
209C2×C4—531no—
210C2×C4—531no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—4244no—
215C2×C4—4242no—
216C2×C4—4242no—
217C2×C4—4242no—
218C2×C4—2121no—
219C2×C4#1313327911yescharacteristic
220C2×C4#1313327911yescharacteristic
221C2×C4C2×C22⋊#615711yescharacteristic
222C8—21no—
223C8—21no—
224C8—1061no—
225C8—1061no—
226C22—532no—
227C22—1061no—
228C22—1061no—
229C22—4242no—
230C22—4242no—
231C22—4242no—
232C22C2×#1313327911yescharacteristic
233C4—21no—
234C4—21no—
235C4—82no—
236C4—8482no—
237C4—1061no—
238C4—1061no—
239C4—1061no—
240C4#1681160411yescharacteristic
241C4#1681162211yescharacteristic
242C2—8482no—
243C2—1062no—
244C2#2029014611yescharacteristic
245C2#2029102411yescharacteristic
246C2#2029103711yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#500C4
socleC22×C53C2×#102
prime radical#20285295C2
cosocleC2×C8×D53C23
hypercenter#500#6157
hypocenterC53#11858
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #20291078C2semidirect
2. #20290637C2semidirect
3. #20289702C2semidirect
12. #16811000C4semidirect
17. #16810705C4semidirect
19. C53×#500C4semidirect
129. #500#6157semidirect
143. C53#11858semidirect

Maximal subgroups

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

subgroupindex
#202910782
#202906372
#202897022
#202852952
#1185853