Finite Group Databasebeta

← #28308087

Subgroups — #28308087

Order 13568.

440 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 86 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#20289957C212yesmaximal, complemented
2#20289658C212yesmaximal, complemented
3#20289643C212yesmaximal
4#20277427C211yesmaximal, characteristic
5#16811640—22no—
6#16811469—22no—
7#16811468C2214yes—
8C2×#13133204C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10#16810528C411yescharacteristic
11#16810526C2211yescharacteristic
12#16810511—22no—
13#16810496—22no—
14#16810492C2211yescharacteristic
15#16810469C411yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20C22×Dic106C2×C411yescharacteristic
21#13133204—22no—
22#13133204C814yescomplemented
23C4×Dic106—42no—
24C2×C4⋊Dic53C2311yescharacteristic
25C2×#11719919—22no—
26C2×#11719919D412yes—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32#13133124—21no—
33#13133124—21no—
34#13133124—22no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53—21no—
37C4×D4×C53—42no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53C2×C812yes—
46C22×C4×C53—21no—
47C22×C4×C53—42no—
48C22×C4×C53—82no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106C2×C812yes—
59C2×Dic106M1612yes—
60C2×#88324—44no—
61C4⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C42×C53—22no—
64C22⋊Dic53—28no—
65C22⋊Dic53—82no—
66C4⋊Dic53—41no—
67#11719919—42no—
68#11719919—82no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—21no—
72C4×Dic53—41no—
73C4×Dic53C2×C812yes—
74C23×C53—22no—
75C23×C53—42no—
76C23×C53—42no—
77C23×C53—41no—
78C23×C53—41no—
79C23×C53C2×C22⋊C411yescharacteristic
80D4×C53—42no—
81D4×C53—42no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—84no—
90C2×C4×C53#9912yes—
91C2×C4×C53C22×C811yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53C22⋊C814yes—
104Dic106—42no—
105Dic106—42no—
106#88324—84no—
107#11480—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—44no—
111C22×C53—42no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53C2×C22⋊C811yescharacteristic
116C22×C53C2×#9911yescharacteristic
117C22×C53#41311yescharacteristic
118C4×C53—22no—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53—81no—
123Dic53—24no—
124Dic53—42no—
125Dic53—42no—
126Dic53—81no—
127#2326—531no—
128#1461—532no—
129#1325—532no—
130#1318—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#145911yescharacteristic
135C2×C53#212811yescharacteristic
136C2×C53#213311yescharacteristic
137C2×#124—531no—
138C2×C4×Q8—531no—
139C2×C4×D4#615711yescharacteristic
140#431—1062no—
141#391—531no—
142#390—531no—
143#388—1062no—
144#385—1062no—
145#384—531no—
146#376—531no—
147#324—534no—
148#324—1062no—
149C53#1148011yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×D4C2×#615711yescharacteristic
152C23×C4—21no—
153#124—534no—
154#124—1062no—
155C4×Q8—2122no—
156C4×D4—42no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1062no—
164C2×C42—531no—
165C2×C42—532no—
166C2×C42—531no—
167C2×C42C2×#615711yescharacteristic
168C4⋊C8—2122no—
169C22⋊C8—1064no—
170C4×C8—2122no—
171#95—1062no—
172#95—1061no—
173#95—1061no—
174C24—21no—
175C2×Q8—532no—
176C2×Q8—532no—
177C2×Q8—2121no—
178C2×Q8—2121no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4D53⋊C812yes—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—82no—
186C22×C4—532no—
187C22×C4—532no—
188C22×C4—1061no—
189C22×C4—1061no—
190C22×C4C22⋊#615711yescharacteristic
191C22×C4C22⋊#615711yescharacteristic
192C22×C4C22×#615711yescharacteristic
193C2×C8—2124no—
194C4⋊C4—41no—
195C4⋊C4—4242no—
196C4⋊C4—2122no—
197C4⋊C4—2121no—
198C22⋊C4—42no—
199C22⋊C4—1068no—
200C22⋊C4—4242no—
201C42—22no—
202C42—532no—
203C42—1062no—
204C42—1062no—
205C42—1061no—
206C42—2121no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—2122no—
214Q8—2122no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—84no—
225C2×C4—534no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4#1313327512yes—
237C2×C4C2×D53⋊C811yescharacteristic
238C2×C4D53⋊M1611yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22D53⋊(C22⋊C8)11yescharacteristic
248C22C2×#1313327511yescharacteristic
249C22#1681160911yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—41no—
254C4—81no—
255C4—1064no—
256C4—4241no—
257C4—2122no—
258C4—2122no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2028995011yescharacteristic
263C2#2029099411yescharacteristic
264C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#11480
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20289957C2semidirect
2. #20289658C2semidirect
22. #13133204C8semidirect
149. C53#11480semidirect

Maximal subgroups

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

subgroupindex
#202899572
#202896582
#202896432
#202774272
#1148053