Finite Group Databasebeta

← #27276255

Subgroups — #27276255

Order 13568.

658 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 143 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#20275656C211yesmaximal, characteristic
2#20273415C212yesmaximal
3#20273330C212yesmaximal
4#20273059C211yesmaximal, characteristic
5C53×#3502C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×#13133128C2211yescharacteristic
9#16810378—22no—
10#16810369—22no—
11#16810369—22no—
12#16810365—24no—
13#16810349—22no—
14Q82×C53—22no—
15C53×#557—22no—
16C53×#557—24no—
17C53×#544—22no—
18C53×#544C412yes—
19C53×#533—22no—
20C53×#533C2212yes—
21C53×#519C412yes—
22C2×C4×Q8×C53C2211yescharacteristic
23C22×#88322—21no—
24C22×#88322—21no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28#13133128—22no—
29#13133128—22no—
30C22×Q8×C53C2×C411yescharacteristic
31C4⋊Q8×C53—22no—
32C4⋊Q8×C53—22no—
33C4⋊Q8×C53—28no—
34C4⋊Q8×C53—24no—
35C4⋊Q8×C53D412yes—
36C4⋊Q8×C53D412yes—
37#125×C53—24no—
38#125×C53—28no—
39#125×C53—24no—
40#125×C53D414yes—
41C22⋊Q8×C53—24no—
42C22⋊Q8×C53—22no—
43C22⋊Q8×C53—22no—
44C4×Q8×C53—24no—
45C4×Q8×C53—24no—
46C4×Q8×C53—24no—
47C4×Q8×C53—24no—
48C4×Q8×C53—22no—
49C4×Q8×C53—22no—
50C4×Q8×C53—22no—
51C4×Q8×C53—22no—
52#117×C53—24no—
53#117×C53C2×C414yes—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D412yes—
56C2×C42×C53D412yes—
57C2×C42×C53C2311yescharacteristic
58C2×Q8×C53—24no—
59C2×Q8×C53—24no—
60C2×Q8×C53—22no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C4⋊C4×C53—24no—
69C4⋊C4×C53—24no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—28no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53C22⋊C414yes—
79C4⋊C4×C53C22⋊C414yes—
80C22⋊C4×C53—24no—
81C42×C53—28no—
82C42×C53—24no—
83C42×C53—22no—
84C42×C53—22no—
85C42×C53C2×D412yes—
86C42×C53C2×D412yes—
87C2×#88322—22no—
88C2×#88322—41no—
89C2×#88322—41no—
90C2×#88322—42no—
91C2×#88322—42no—
92C2×#88322—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—28no—
95Q8×C53—42no—
96Q8×C53—42no—
97Q8×C53—82no—
98Q8×C53—84no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—24no—
106C2×C4×C53—24no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53#10114yes—
110C2×C4×C53C2×C22⋊C412yes—
111C2×C4×C53#12012yes—
112C2×C4×C53#12011yescharacteristic
113C2×C4×C53#12011yescharacteristic
114#88322—42no—
115#88322—82no—
116#88322—81no—
117#8460—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#37911yescharacteristic
121C22×C53C2×#10112yes—
122C4×C53—21no—
123C4×C53—22no—
124C4×C53—24no—
125C4×C53—22no—
126C4×C53—44no—
127C4×C53—42no—
128C4×C53—44no—
129C4×C53#45411yescharacteristic
130C4×C53#45611yescharacteristic
131#3502D5311yescharacteristic
132#1985—531no—
133#1552—532no—
134#1531—532no—
135#1467—531no—
136C2×C53—22no—
137C2×C53#178411yescharacteristic
138C2×C53#179311yescharacteristic
139C2×C53#179411yescharacteristic
140Q82—22no—
141#557—22no—
142#557—24no—
143#544—22no—
144#544Dic5312yes—
145#533—22no—
146#533C2×D5312yes—
147#519Dic5312yes—
148C2×C4×Q8C2×D5311yescharacteristic
149#432—532no—
150C2×C4⋊C8—531no—
151C2×#97—531no—
152#336—1062no—
153#332—1062no—
154#332—1062no—
155#330—1064no—
156#324—1062no—
157C53#846011yescharacteristic, complemented, Sylow
158C22×Q8C2×Dic5311yescharacteristic
159C22×C8—1061no—
160C22×C8—1061no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—28no—
164C4⋊Q8—24no—
165C4⋊Q8C22⋊D5312yes—
166C4⋊Q8C22⋊D5312yes—
167#125—24no—
168#125—28no—
169#125—24no—
170#125C22⋊D5314yes—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182#117—24no—
183#117C2×Dic5314yes—
184C2×C4⋊C4C2×Dic5311yescharacteristic
185C2×C4⋊C4C22⋊D5312yes—
186C2×C42C22⋊D5312yes—
187C2×C42C22×D5311yescharacteristic
188C4⋊C8—1062no—
189C4⋊C8—1062no—
190C22⋊C8—1062no—
191#97—1062no—
192#97—1062no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C22×C4C22⋊Dic5312yes—
199C22×C4C22⋊Dic5312yes—
200C22×C4C22×Dic5311yescharacteristic
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203C2×C8—1062no—
204C2×C8—2122no—
205C2×C8—2122no—
206C2×C8—2121no—
207C2×C8—2121no—
208C2×C8—2121no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—28no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4C22⋊Dic5314yes—
220C4⋊C4C22⋊Dic5314yes—
221C22⋊C4—24no—
222C42—28no—
223C42—24no—
224C42—22no—
225C42—22no—
226C42C2×C22⋊D5312yes—
227C42C2×C22⋊D5312yes—
228C23C2×C22⋊Dic5311yescharacteristic
229Q8—28no—
230Q8—42no—
231Q8—42no—
232Q8—82no—
233Q8—84no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4#1313313614yes—
245C2×C4C2×C22⋊Dic5312yes—
246C2×C4#1313318611yescharacteristic
247C2×C4#1313318612yes—
248C2×C4#1313318611yescharacteristic
249C8—4242no—
250C8—4241no—
251C8—2122no—
252C22—21no—
253C22—21no—
254C22#1681047911yescharacteristic
255C22C2×#1313313612yes—
256C4—24no—
257C4—22no—
258C4—21no—
259C4—22no—
260C4—44no—
261C4—42no—
262C4—44no—
263C4#1681079611yescharacteristic
264C4#1681080611yescharacteristic
265C2—22no—
266C2#2027463911yescharacteristic
267C2#2027467411yescharacteristic
268C2#2027467911yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3502C2
socleC22×C53#379
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#3502D53
hypocenterC53#8460
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
157. C53#8460semidirect

Maximal subgroups

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

subgroupindex
#202756562
#202734152
#202733302
#202730592
C53×#35022
#846053