Finite Group Databasebeta

← #28310641

Subgroups — #28310641

Order 13568.

464 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#16811533C212yesmaximal
2#20290468C212yesmaximal
3#20289812C212yesmaximal
4#20277433C211yesmaximal, characteristic
5C2×#13133278C2214yes—
6C4×#11719940—22no—
7#16811533C418yescomplemented
8#16811503—22no—
9C2×#13133215C2211yescharacteristic
10C2×Q8×Dic53C411yescharacteristic
11C2×#13133182C2211yescharacteristic
12#16810529C2211yescharacteristic
13#16810510—22no—
14#16810505—22no—
15#16810488C411yescharacteristic
16#16810469C411yescharacteristic
17C53×#391C411yescharacteristic
18C2×#11719940—24no—
19#13133278—24no—
20#13133278C2×C418yes—
21#13133272—42no—
22C4×#88324—42no—
23C22×Dic106C2311yescharacteristic
24#13133215—22no—
25#13133215C2×C414yes—
26Q8×Dic53—42no—
27#13133182—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—22no—
31C2×#11719919D412yes—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D412yes—
36#13133124—22no—
37#13133124—21no—
38#13133124—21no—
39#13133124—22no—
40C22×Q8×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43#95×C53—22no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C4212yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D412yes—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106C22×C412yes—
60C2×Dic106#4112yes—
61#11719940—28no—
62#11719940—44no—
63C2×#88324—44no—
64C4⋊C4×C53—41no—
65C42×C53—41no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68#11719919—28no—
69#11719919—42no—
70C4×Dic53—21no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—42no—
75C4×Dic53C4212yes—
76C23×C53C2×C22⋊C411yescharacteristic
77Q8×C53—42no—
78Q8×C53—42no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53C2×C4211yescharacteristic
90C2×C4×C53#11711yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53—41no—
101C2×Dic53#9912yes—
102C2×Dic53C4×D414yes—
103Dic106—42no—
104Dic106—42no—
105#88324—48no—
106#12631—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53C4×C22⋊C411yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53#41011yescharacteristic
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53—81no—
117Dic53—22no—
118Dic53—24no—
119Dic53—42no—
120Dic53—41no—
121Dic53—81no—
122#2327—531no—
123C2×#356—532no—
124C4×#101—532no—
125#1394—532no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53C4×#9911yescharacteristic
129C2×C53#213311yescharacteristic
130C2×C53#213511yescharacteristic
131C2×C4⋊Q8—531no—
132C2×C4×Q8—531no—
133C2×C4×Q8—531no—
134C2×#101—534no—
135C4×M16—1062no—
136#391—531no—
137#391#615711yescharacteristic
138#387—1062no—
139#387—1062no—
140#382—531no—
141#376—531no—
142#356—538no—
143#343—1062no—
144C53#1263111yescharacteristic, complemented, Sylow
145C22×Q8—531no—
146C22×Q8C2×#615711yescharacteristic
147C2×M16—1064no—
148C4⋊Q8—534no—
149C4⋊Q8—1062no—
150C4×Q8—2122no—
151C4×Q8—2122no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4C2×#615711yescharacteristic
157C2×C42—531no—
158C2×C42—532no—
159C2×C42—531no—
160C2×C42—1061no—
161C2×C42C2×#615711yescharacteristic
162#101—538no—
163#101—1064no—
164#97—2122no—
165C4×C8—2122no—
166#95—22no—
167#95—1061no—
168#95—1061no—
169#95—1062no—
170#95—1062no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—532no—
175C2×Q8—532no—
176C2×Q8—2121no—
177C2×Q8—2121no—
178C2×Q8C4×#615712yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—532no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4C22⋊#615712yes—
187C22×C4C22×#615711yescharacteristic
188M16—1068no—
189M16—2124no—
190C2×C8—2124no—
191C4⋊C4—41no—
192C4⋊C4—1068no—
193C4⋊C4—2122no—
194C4⋊C4—2121no—
195C4⋊C4—2122no—
196C42—41no—
197C42—532no—
198C42—1062no—
199C42—1062no—
200C42—1062no—
201C42—1061no—
202C42—2122no—
203C23C2×C22⋊#615711yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—2122no—
207Q8—2122no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—534no—
219C2×C4—532no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4—2121no—
225C2×C4—2122no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4—2121no—
230C2×C4C2×C4×#615711yescharacteristic
231C2×C4#1313329611yescharacteristic
232C8—2128no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊#615711yescharacteristic
236C22D53⋊#9911yescharacteristic
237C22#1681160311yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—1064no—
244C4—1062no—
245C4—4241no—
246C4—2122no—
247C4—2121no—
248C2—21no—
249C2—21no—
250C2#2029046211yescharacteristic
251C2#2029101311yescharacteristic
252C2#2029101811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenter#391#6157
hypocenterC53#12631
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
7. #16811533C4semidirect
144. C53#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#168115332
#202904682
#202898122
#202774332
#1263153