Finite Group Databasebeta

← #27276338

Subgroups — #27276338

Order 13568.

682 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 139 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#20275705C212yesmaximal, complemented
2#20273453C211yesmaximal, characteristic, complemented
3#20273310C211yesmaximal, characteristic, complemented
4#20273154C212yesmaximal
5C53×#3509C211yesmaximal, characteristic
6#16810647C2212yes—
7C4×#11719917C2212yes—
8#16810392—24no—
9#16810369—22no—
10#16810365—22no—
11Q8⋊#88322—22no—
12D4⋊#88322—22no—
13C53×#557—22no—
14C53×#556—22no—
15C53×#552—24no—
16C53×#541—22no—
17C53×#536C412yes—
18C53×#533—22no—
19C53×#533C2211yescharacteristic
20C2×C4⋊Q8×C53C2211yescharacteristic
21C53×#519C412yes—
22C4×#41×C53C2211yescharacteristic
23C2×#11719917—22no—
24C2×#11719917—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133128—22no—
28C4×#88322—22no—
29C2×#41×C53C2×C411yescharacteristic
30C22×Q8×C53C2×C412yes—
31C4⋊Q8×C53—21no—
32C4⋊Q8×C53—21no—
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53D412yes—
36#126×C53—28no—
37#125×C53—24no—
38#125×C53—28no—
39#125×C53D412yes—
40#124×C53—22no—
41#123×C53—24no—
42C22⋊Q8×C53—24no—
43C22⋊Q8×C53—24no—
44C22⋊Q8×C53—22no—
45C4×Q8×C53—24no—
46C4×Q8×C53—24no—
47C4×Q8×C53—24no—
48C4×Q8×C53—22no—
49C4×Q8×C53D412yes—
50C4×D4×C53—22no—
51C4×D4×C53D412yes—
52#117×C53—24no—
53#117×C53—21no—
54#117×C53—24no—
55#117×C53C2×C411yescharacteristic
56#117×C53C2×C412yes—
57C2×C4⋊C4×C53D414yes—
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#41×C53—42no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—22no—
64C2×Q8×C53—44no—
65C2×Q8×C53C22⋊C411yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53C22⋊C411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53C22⋊C414yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C4⋊C4×C53—28no—
74C4⋊C4×C53—28no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—24no—
80C4⋊C4×C53C22⋊C412yes—
81C4⋊C4×C53C22⋊C414yes—
82C22⋊C4×C53—28no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C42×C53—28no—
87C42×C53—21no—
88C42×C53—22no—
89C42×C53—22no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D412yes—
93#11719917—24no—
94#11719917—42no—
95#11719917—44no—
96C2×#88322—42no—
97C2×#88322—42no—
98C23×C53—21no—
99C23×C53C2×C22⋊C411yescharacteristic
100Q8×C53—22no—
101Q8×C53—44no—
102Q8×C53—44no—
103Q8×C53—42no—
104Q8×C53—42no—
105D4×C53—22no—
106D4×C53—42no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—44no—
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53C2×C22⋊C411yescharacteristic
120C2×C4×C53#12012yes—
121C2×C4×C53#12012yes—
122#88322—44no—
123#88322—82no—
124#8474—531nomaximal, Sylow
125C22×C53—22no—
126C22×C53—21no—
127C22×C53—42no—
128C22×C53—41no—
129C22×C53#37911yescharacteristic
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—44no—
134C4×C53—41no—
135C4×C53—44no—
136C4×C53#41312yes—
137C4×C53#45412yes—
138C4×C53#45612yes—
139#3509D5311yescharacteristic
140#1997—532no—
141#1561—531no—
142#1526—531no—
143#1489—532no—
144C2×C53—41no—
145C2×C53—81no—
146C2×C53#178511yescharacteristic
147C2×C53#179312yes—
148#557—22no—
149#556—22no—
150#552—24no—
151#541—22no—
152#536Dic5312yes—
153#533—22no—
154#533C2×D5311yescharacteristic
155C2×C4⋊Q8C2×D5311yescharacteristic
156#519Dic5312yes—
157C4×#41C2×D5311yescharacteristic
158#423—532no—
159C4×M16—532no—
160#343—1064no—
161#332—1062no—
162#330—1062no—
163Q8⋊C8—1062no—
164D4⋊C8—1062no—
165C53#847411yescharacteristic, complemented, Sylow
166C2×#41C2×Dic5311yescharacteristic
167C22×Q8C2×Dic5312yes—
168C2×M16—1062no—
169C2×M16—1062no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊D5312yes—
175#126—28no—
176#125—24no—
177#125—28no—
178#125C22⋊D5312yes—
179#124—22no—
180#123—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8C22⋊D5312yes—
189C4×D4—22no—
190C4×D4C22⋊D5312yes—
191#117—24no—
192#117—24no—
193#117—21no—
194#117C2×Dic5311yescharacteristic
195#117C2×Dic5312yes—
196C2×C4⋊C4C22⋊D5314yes—
197C2×C42—21no—
198C2×C42C22×D5311yescharacteristic
199C4⋊C8—1062no—
200C4⋊C8—1062no—
201#97—1062no—
202C4×C8—1062no—
203#41—42no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8C22⋊Dic5311yescharacteristic
209C2×D4—21no—
210C2×D4C22⋊Dic5311yescharacteristic
211C22×C4—21no—
212C22×C4—22no—
213C22×C4C22⋊Dic5314yes—
214C22×C4C22×Dic5311yescharacteristic
215C22×C4C2×C22⋊D5312yes—
216M16—1064no—
217M16—2124no—
218M16—2122no—
219C2×C8—2122no—
220C2×C8—2122no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic5312yes—
229C4⋊C4C22⋊Dic5314yes—
230C22⋊C4—28no—
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—28no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42C2×C22⋊D5312yes—
239C42C2×C22⋊D5311yescharacteristic
240C42C2×C22⋊D5311yescharacteristic
241C23—21no—
242C23C2×C22⋊Dic5311yescharacteristic
243Q8—22no—
244Q8—44no—
245Q8—42no—
246Q8—42no—
247Q8—44no—
248D4—22no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C2×C22⋊Dic5311yescharacteristic
262C2×C4C2×C22⋊Dic5311yescharacteristic
263C2×C4#1313318612yes—
264C2×C4#1313318612yes—
265C8—4242no—
266C8—2124no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—42no—
271C22#1681047911yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4#1681060712yes—
279C4#1681079612yes—
280C4#1681080612yes—
281C2—41no—
282C2—81no—
283C2#2027464311yescharacteristic
284C2#2027467412yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3509C2
socleC22×C53#379
prime radicalC2×#41×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#3509D53
hypocenterC53#8474
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #20275705C2semidirect
2. #20273453C2semidirect
3. #20273310C2semidirect
165. C53#8474semidirect

Maximal subgroups

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

subgroupindex
#202757052
#202734532
#202733102
#202731542
C53×#35092
#847453