Finite Group Databasebeta

← #28302337

Subgroups — #28302337

Order 13568.

802 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 128 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#20290572C212yesmaximal
2C2×#16811482C214yesmaximal, complemented
3C2×#16811179C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×#13133274—22no—
6#16811489—22no—
7#16811482—28no—
8#16811179—28no—
9C2×#13133216C2212yes—
10C2×#13133209C411yescharacteristic
11C2×#13133195—22no—
12C2×#13133182—22no—
13C2×#13133175C412yes—
14C2×C42×D53C2211yescharacteristic
15C2×#125×C53C411yescharacteristic
16C22×#88324—24no—
17#13133284—28no—
18#13133274—44no—
19C22×Dic106—22no—
20C22×C4×D53C2×C411yescharacteristic
21C22×C4×D53D412yes—
22#13133216—24no—
23#13133216D418yes—
24#13133209—24no—
25#13133195—216no—
26#13133182—216no—
27#13133175—24no—
28#13133175—24no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×#11719921—22no—
33C2×C4×Dic53—22no—
34C2×C4×Dic53D412yes—
35C2×C4×Dic53C2311yescharacteristic
36C42×D53—24no—
37#125×C53—24no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C42×C53C2×C411yescharacteristic
41C23×D53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53C22⋊C412yes—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53—22no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C2×D412yes—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53—24no—
52C2×C4×D53—24no—
53C2×C4×D53—22no—
54C2×C4×D53—22no—
55C2×Dic106—28no—
56C2×Dic106—44no—
57C2×Dic106—42no—
58C2×Dic106—42no—
59C2×#88324—44no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—28no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53C22⋊C414yes—
65C42×C53—22no—
66#11719921—28no—
67C4⋊Dic53—22no—
68C4⋊Dic53C22⋊C414yes—
69#11719919—28no—
70#11719919—28no—
71C4×Dic53—28no—
72C4×Dic53—24no—
73C4×Dic53C2×D414yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—21no—
76C22×D53—21no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53#10112yes—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—24no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53C2×C22⋊C412yes—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53#10112yes—
95C2×Dic53#12014yes—
96C2×Dic53#13514yes—
97C2×Dic53AGL(1,ℤ/8)12yes—
98C2×Dic53#13712yes—
99C4×D53—28no—
100C4×D53—44no—
101C4×D53—44no—
102Dic106—48no—
103Dic106—48no—
104#88324—84no—
105#8345—531nomaximal, Sylow
106C22×C53#37911yescharacteristic
107C22×C53C2×#10111yescharacteristic
108C22×C53C2×#10111yescharacteristic
109C22×C53#41612yes—
110C22×C53#41712yes—
111C2×D53—24no—
112C2×D53—22no—
113C2×D53—41no—
114C2×D53—41no—
115C2×D53—42no—
116C4×C53—22no—
117C4×C53—44no—
118C4×C53—42no—
119Dic53—22no—
120Dic53—44no—
121Dic53#44914yes—
122Dic53#45214yes—
123C2×#533—531no—
124#1797—532no—
125C2×#332—534no—
126C2×C53#154411yescharacteristic
127C2×C53#155212yes—
128C2×C53#156111yescharacteristic
129C2×C53#178411yescharacteristic
130C2×C53#178812yes—
131D53—42no—
132#533—1068no—
133C2×C4⋊Q8—532no—
134C2×#125—531no—
135C2×#125#615711yescharacteristic
136C2×C22⋊Q8—1062no—
137C2×C4×Q8—1062no—
138C2×#117—532no—
139C22×C42—531no—
140C2×C4⋊C8—534no—
141C2×C22⋊C8—1062no—
142#336—1062no—
143#332—1068no—
144C53#834511yescharacteristic, complemented, Sylow
145C22×Q8—1062no—
146C23×C4—532no—
147C23×C4—531no—
148C22×C8—1064no—
149C4⋊Q8—538no—
150C4⋊Q8—1064no—
151#125—24no—
152#125—1064no—
153C22⋊Q8—10616no—
154C4×Q8—10616no—
155#117—1064no—
156#117—1064no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4C2×#615712yes—
162C2×C22⋊C4—1062no—
163C2×C42—531no—
164C2×C42—532no—
165C2×C42—1064no—
166C2×C42—1062no—
167C2×C42C2×#615711yescharacteristic
168C4⋊C8—1068no—
169C22⋊C8—2124no—
170C24—531no—
171C2×Q8—1068no—
172C2×Q8—2124no—
173C2×Q8—2122no—
174C2×Q8—2122no—
175C22×C4—22no—
176C22×C4—532no—
177C22×C4—531no—
178C22×C4—1062no—
179C22×C4—1062no—
180C22×C4—1064no—
181C22×C4—1064no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4C22⋊#615712yes—
187C22×C4C22×#615711yescharacteristic
188C2×C8—2124no—
189C2×C8—2124no—
190C2×C8—2124no—
191C4⋊C4—28no—
192C4⋊C4—22no—
193C4⋊C4—534no—
194C4⋊C4—1068no—
195C4⋊C4—1068no—
196C4⋊C4—1062no—
197C4⋊C4C22⋊#615714yes—
198C22⋊C4—1068no—
199C42—22no—
200C42—534no—
201C42—1064no—
202C42—1068no—
203C23—532no—
204C23—1061no—
205C23—1061no—
206C23—1062no—
207C23—1062no—
208C23C2×C22⋊#615711yescharacteristic
209Q8—2128no—
210Q8—2128no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—534no—
220C2×C4—534no—
221C2×C4—532no—
222C2×C4—532no—
223C2×C4—532no—
224C2×C4—1068no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1064no—
228C2×C4—2124no—
229C2×C4—2124no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4C2×C22⋊#615712yes—
233C8—4244no—
234C22—1064no—
235C22—1062no—
236C22—2121no—
237C22—2121no—
238C22—2122no—
239C22#1681156411yescharacteristic
240C22D53⋊#10111yescharacteristic
241C22D53⋊#10111yescharacteristic
242C22#1681161312yes—
243C22#1681161412yes—
244C4—22no—
245C4—42no—
246C4—44no—
247C4—534no—
248C4—534no—
249C4—1062no—
250C4—2124no—
251C2—2122no—
252C2#2029022111yescharacteristic
253C2#2029023712yes—
254C2#2029025511yescharacteristic
255C2#2029055011yescharacteristic
256C2#2029055512yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#125×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#125#6157
hypocenterC53#8345
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. C2×#16811482C2semidirect
144. C53#8345semidirect

Maximal subgroups

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

subgroupindex
#202905722
C2×#168114822
C2×#168111792
#834553