Finite Group Databasebeta

← #28308276

Subgroups — #28308276

Order 13568.

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

normal subgroups only

This group has 424 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#20289978C212yesmaximal, complemented
2#20289657C212yesmaximal, complemented
3#20289643C212yesmaximal
4#20277428C211yesmaximal, characteristic
5#16811643—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×Q8×Dic53C2211yescharacteristic
9C2×D4×Dic53C411yescharacteristic
10#16810530C411yescharacteristic
11#16810526C2211yescharacteristic
12#16810516—22no—
13#16810498—22no—
14#16810493C2211yescharacteristic
15#16810469C411yescharacteristic
16C2×#124×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C23×Dic53—21no—
21Q8×Dic53—42no—
22D4×Dic53—42no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53D412yes—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—22no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32#13133124—22no—
33#13133124—21no—
34#13133124—21no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53C2×C411yescharacteristic
37#124×C53—22no—
38#124×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—21no—
42C2×Q8×C53—21no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53M1612yes—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53—42no—
56C22×Dic53—82no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D412yes—
60C2×#88324—44no—
61C22⋊C4×C53—28no—
62C42×C53—21no—
63C42×C53M1612yes—
64C22⋊Dic53—42no—
65C22⋊Dic53—82no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—82no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—41no—
74C23×C53—22no—
75C23×C53—42no—
76C23×C53—42no—
77C23×C53—41no—
78C23×C53—41no—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82D4×C53—42no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C2×M1611yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—42no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—84no—
105C2×Dic53#9912yes—
106#88324—84no—
107#11534—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—44no—
111C22×C53—42no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53#40711yescharacteristic
116C22×C53C2×#9911yescharacteristic
117C22×C53#41311yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122Dic53—22no—
123Dic53—42no—
124Dic53—41no—
125Dic53—81no—
126Dic53—81no—
127#2326—531no—
128#1467—532no—
129#1325—532no—
130#1318—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#146511yescharacteristic
135C2×C53#212811yescharacteristic
136C2×C53#213311yescharacteristic
137C2×#124#615711yescharacteristic
138C2×C4×Q8—531no—
139C2×C4×D4—531no—
140#432—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#1153411yescharacteristic, complemented, Sylow
150C22×Q8C2×#615711yescharacteristic
151C22×D4C2×#615711yescharacteristic
152C23×C4—1061no—
153#124—22no—
154#124—22no—
155C4×Q8—2122no—
156C4×D4—2122no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1062no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42—532no—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42C2×#615711yescharacteristic
168C4⋊C8—2122no—
169C22⋊C8—1064no—
170#97—2122no—
171#95—1061no—
172#95—1061no—
173#95—1062no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1171994012yes—
183C22×C4—21no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—532no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—4242no—
190C22×C4—2122no—
191C22×C4C22⋊#615712yes—
192C22×C4C22×#615711yescharacteristic
193C2×C8—2124no—
194C4⋊C4—4242no—
195C4⋊C4—2121no—
196C4⋊C4—2122no—
197C4⋊C4—2121no—
198C22⋊C4—28no—
199C22⋊C4—4242no—
200C22⋊C4—2122no—
201C42—21no—
202C42—1062no—
203C42—1062no—
204C42—1062no—
205C42—2121no—
206C42#1171994012yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—532no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—4244no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4—2122no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4C2×#1171994011yescharacteristic
238C2×C4C2×#1171994011yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#1681159311yescharacteristic
248C22D53⋊#9911yescharacteristic
249C22#1681161011yescharacteristic
250C4—21no—
251C4—21no—
252C4—42no—
253C4—42no—
254C4—1062no—
255C4—4241no—
256C4—4241no—
257C4—2122no—
258C4—2121no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2028997111yescharacteristic
263C2#2029099511yescharacteristic
264C2#2029101111yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#124#6157
hypocenterC53#11534
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20289978C2semidirect
2. #20289657C2semidirect
149. C53#11534semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202896572
#202896432
#202774282
#1153453