Finite Group Databasebeta

← #28302336

Subgroups — #28302336

Order 13568.

802 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290571C212yesmaximal
2C2×#16811482C214yesmaximal, complemented
3C2×#16811176C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×C22⋊#88324—22no—
6#16811489—22no—
7#16811482—28no—
8#16811176—28no—
9C2×#13133216C2212yes—
10C2×#13133208C412yes—
11C2×#13133194—22no—
12C2×C4×Dic106—22no—
13C2×#13133176C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15C2×#117×C53C411yescharacteristic
16C22×#88324—24no—
17#13133284—28no—
18C22⋊#88324—44no—
19C22×Dic106—22no—
20C23×Dic53D412yes—
21#13133216—24no—
22#13133216D418yes—
23#13133208—24no—
24#13133208—24no—
25#13133194—216no—
26C4×Dic106—216no—
27#13133176—24no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—24no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37C2×C4×Dic53C2311yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39#117×C53—24no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—21no—
43C24×C53C22⋊C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—24no—
52C22×Dic53—24no—
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C2×D412yes—
57C2×Dic106—28no—
58C2×Dic106—44no—
59C2×Dic106—42no—
60C2×Dic106—42no—
61C2×#88324—44no—
62C2×#88324—44no—
63C2×#88324—44no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53C22⋊C414yes—
66C22⋊C4×C53—24no—
67C42×C53—24no—
68C22⋊Dic53—24no—
69C4⋊Dic53—28no—
70C4⋊Dic53—22no—
71C4⋊Dic53C22⋊C414yes—
72#11719919—28no—
73#11719919—28no—
74C4×Dic53—24no—
75C4×Dic53—22no—
76C4×Dic53—24no—
77C4×Dic53C2×D414yes—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53#10112yes—
83C23×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53—28no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—24no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53#10112yes—
93C2×C4×C53C2×C22⋊C412yes—
94C2×Dic53—24no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—44no—
101C2×Dic53—44no—
102C2×Dic53#12014yes—
103C2×Dic53#13514yes—
104C2×Dic53AGL(1,ℤ/8)12yes—
105C2×Dic53#13712yes—
106Dic106—48no—
107Dic106—48no—
108#88324—84no—
109#8345—531nomaximal, Sylow
110C22×C53—24no—
111C22×C53—22no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53#37911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C22×C53C2×#10111yescharacteristic
118C22×C53#41612yes—
119C22×C53#41712yes—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—44no—
123Dic53—44no—
124Dic53—42no—
125Dic53#44914yes—
126Dic53#45214yes—
127C2×#533—531no—
128#1797—532no—
129C2×#332—534no—
130C2×C53—42no—
131C2×C53#154411yescharacteristic
132C2×C53#155212yes—
133C2×C53#156111yescharacteristic
134C2×C53#178411yescharacteristic
135C2×C53#178812yes—
136#533—1068no—
137C2×C4⋊Q8—532no—
138C2×#125—532no—
139C2×C22⋊Q8—1062no—
140C2×C4×Q8—1062no—
141C2×#117—531no—
142C2×#117#615711yescharacteristic
143C22×C42—531no—
144C2×C4⋊C8—534no—
145C2×C22⋊C8—1062no—
146#336—1062no—
147#332—1068no—
148C53#834511yescharacteristic, complemented, Sylow
149C22×Q8—1062no—
150C23×C4—532no—
151C23×C4C2×#615711yescharacteristic
152C22×C8—1064no—
153C4⋊Q8—538no—
154C4⋊Q8—1064no—
155#125—1064no—
156#125—1064no—
157C22⋊Q8—10616no—
158C4×Q8—10616no—
159#117—24no—
160#117—1064no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4C2×#615712yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—1061no—
168C2×C42—21no—
169C2×C42—531no—
170C2×C42—532no—
171C2×C42—531no—
172C2×C42—1064no—
173C2×C42—1061no—
174C4⋊C8—1068no—
175C22⋊C8—2124no—
176C24C22⋊#615711yescharacteristic
177C2×Q8—1068no—
178C2×Q8—2124no—
179C2×Q8—2122no—
180C2×Q8—2122no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4—1062no—
188C22×C4—1064no—
189C22×C4—1064no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4C22⋊#615711yescharacteristic
193C22×C4C22×#615711yescharacteristic
194C2×C8—2124no—
195C2×C8—2124no—
196C2×C8—2124no—
197C4⋊C4—22no—
198C4⋊C4—534no—
199C4⋊C4—1068no—
200C4⋊C4—1068no—
201C4⋊C4—1068no—
202C4⋊C4—1062no—
203C4⋊C4C22⋊#615714yes—
204C22⋊C4—24no—
205C22⋊C4—1064no—
206C42—24no—
207C42—534no—
208C42—1064no—
209C42—1064no—
210C42—1062no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23#1313327812yes—
216C23C2×C22⋊#615711yescharacteristic
217Q8—2128no—
218Q8—2128no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—534no—
228C2×C4—534no—
229C2×C4—532no—
230C2×C4—532no—
231C2×C4—1062no—
232C2×C4—1064no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—2124no—
236C2×C4—2124no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4#1313327812yes—
240C2×C4C2×C22⋊#615712yes—
241C8—4244no—
242C22—24no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22#1681156411yescharacteristic
248C22C2×#1313327811yescharacteristic
249C22C2×#1313327811yescharacteristic
250C22#1681161312yes—
251C22#1681161412yes—
252C4—22no—
253C4—22no—
254C4—44no—
255C4—534no—
256C4—534no—
257C4—2122no—
258C4—2124no—
259C2—42no—
260C2#2029022011yescharacteristic
261C2#2029023612yes—
262C2#2029025411yescharacteristic
263C2#2029054911yescharacteristic
264C2#2029055512yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8345
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

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

Maximal subgroups

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

subgroupindex
#202905712
C2×#168114822
C2×#168111762
#834553