Finite Group Databasebeta

← #27416059

Subgroups — #27416059

Order 13568.

1158 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 195 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×#16811235C211yesmaximal, characteristic
2C2×#16810779C212yesmaximal
3#20275866C212yesmaximal, complemented
4#20275814C212yesmaximal, complemented
5#20275059C214yesmaximal, complemented
6#20275027C212yesmaximal
7#20274786C211yesmaximal, characteristic
8C53×#2310C211yesmaximal, characteristic
9C22×#11719925C2214yes—
10#16811235—28no—
11C2×#13133217C2211yescharacteristic
12C2×#13133210—22no—
13C2×#13133209C2211yescharacteristic
14C2×#13133207—22no—
15C2×#13133196C2212yes—
16C2×C4×Dic106C2212yes—
17#16810779—28no—
18C2×#13133141C2214yes—
19C2×#13133140C2214yes—
20C2×C22⋊#88322C2212yes—
21#16810385C2214yes—
22#16810378C2212yes—
23#16810378C2212yes—
24C23×Q8×C53C2211yescharacteristic
25C2×#117×C53C2211yescharacteristic
26C2×#117×C53C2211yescharacteristic
27C53×#391—22no—
28C53×#391C2212yes—
29C53×#391C2212yes—
30C53×#381—22no—
31C22×Dic106C2312yes—
32C2×#11719925—28no—
33C2×#11719925—28no—
34C2×#11719925—44no—
35C22×#88322C2314yes—
36#13133217—24no—
37#13133210—216no—
38#13133209—24no—
39#13133207—216no—
40#13133196—24no—
41#13133196D418yes—
42C4×Dic106—24no—
43C4×Dic106—24no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53C2312yes—
46C2×#11719919—22no—
47C2×#11719919—22no—
48C2×#11719919—22no—
49C2×C22⋊Dic53—22no—
50C2×C4×Dic53—22no—
51#13133141—28no—
52#13133140—44no—
53C22⋊#88322—24no—
54C22×Q8×C53—24no—
55C22×Q8×C53—21no—
56C22×Q8×C53—24no—
57C22×Q8×C53—42no—
58C22×Q8×C53C2312yes—
59C23×C4×C53—21no—
60C23×C4×C53C2311yescharacteristic
61#117×C53—24no—
62#117×C53—42no—
63C2×C4⋊C4×C53D412yes—
64C2×C4⋊C4×C53C2312yes—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53D412yes—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C2×C42×C53C2311yescharacteristic
70#95×C53—24no—
71#95×C53—24no—
72C24×C53C2×D411yescharacteristic
73C2×Q8×C53—24no—
74C2×Q8×C53—22no—
75C2×Q8×C53—24no—
76C2×Q8×C53—24no—
77C2×Q8×C53—416no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×Q8×C53#4114yes—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—24no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D414yes—
92C22×C4×C53C2411yescharacteristic
93C22×Dic53—22no—
94C22×Dic53—22no—
95C2×Dic106—24no—
96C2×Dic106—24no—
97C2×Dic106—22no—
98C2×Dic106C2×D414yes—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—42no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—42no—
106C22⋊Dic53—28no—
107#11719925—416no—
108C4⋊Dic53—28no—
109C4⋊Dic53—22no—
110C4⋊Dic53C2×D414yes—
111#11719919—28no—
112#11719919—28no—
113#11719919—28no—
114C4×Dic53—28no—
115C2×#88322—24no—
116C2×#88322—24no—
117C2×#88322—24no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—24no—
121C23×C53#12012yes—
122C23×C53C22×D411yescharacteristic
123Q8×C53—28no—
124Q8×C53—48no—
125Q8×C53—44no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—28no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—24no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—48no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—44no—
139C2×C4×C53#12012yes—
140C2×C4×C53C22×D412yes—
141C2×C4×C53C2×#4112yes—
142C2×C4×C53H(5,𝔽2)12yes—
143C2×Dic53—24no—
144C2×Dic53—24no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—44no—
148Dic106—28no—
149Dic106—44no—
150#88322—44no—
151#23943—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—24no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53C2×#12011yescharacteristic
157C22×C53#52511yescharacteristic
158C22×C53#53511yescharacteristic
159C22×C53#57814yes—
160C4×C53—22no—
161C4×C53—44no—
162C4×C53—42no—
163C4×C53—42no—
164C4×C53—82no—
165C4×C53#54614yes—
166Dic53—44no—
167Dic53—44no—
168C2×#541—531no—
169C2×#451—532no—
170#2310D5311yescharacteristic
171#2026—532no—
172#2016—532no—
173#1878—534no—
174#1873—532no—
175#1818—531no—
176C2×C53—42no—
177C2×C53#240411yescharacteristic
178C2×C53#301112yes—
179C2×C53#312712yes—
180C2×C53#322312yes—
181C23×Q8C2×D5311yescharacteristic
182C22×Q16—534no—
183#541—1068no—
184C2×C4⋊Q8—531no—
185C2×#126—1062no—
186C2×#125—531no—
187C2×#125—1062no—
188C2×C22⋊Q8—532no—
189C2×C4×Q8—532no—
190C2×#117C2×D5311yescharacteristic
191C2×#117C2×D5311yescharacteristic
192#451—1068no—
193C2×#103—534no—
194C2×#103—534no—
195C2×C22⋊C8—532no—
196#391—22no—
197#391C2×D5312yes—
198#391C2×D5312yes—
199#381—22no—
200#340—534no—
201#336—532no—
202#336—532no—
203C53#2394311yescharacteristic, complemented, Sylow
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8—532no—
209C22×Q8C22×D5312yes—
210C23×C4—21no—
211C23×C4C22×D5311yescharacteristic
212C2×Q16—1068no—
213C2×Q16—1068no—
214C2×Q16—2124no—
215C22×C8—534no—
216C4⋊Q8—1064no—
217#126—10616no—
218#125—1064no—
219#125—10616no—
220C22⋊Q8—538no—
221C22⋊Q8—1064no—
222C4×Q8—1064no—
223C4×Q8—1064no—
224#117—24no—
225#117—42no—
226C2×C4⋊C4—532no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4C22⋊D5312yes—
232C2×C4⋊C4C22×D5312yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1062no—
235C2×C22⋊C4C22⋊D5312yes—
236C2×C42—21no—
237C2×C42—1062no—
238C2×C42C22×D5311yescharacteristic
239C2×C42C22×D5311yescharacteristic
240#103—1068no—
241#103—2124no—
242C22⋊C8—1064no—
243#95—24no—
244#95—24no—
245C24C2×C22⋊D5311yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—416no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—534no—
254C2×Q8—1062no—
255C2×Q8—1064no—
256C2×Q8—1064no—
257C2×Q8#1171994514yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4D4×D5314yes—
268C22×C4C2×C22⋊D5312yes—
269C22×C4C2×C22⋊D5312yes—
270C22×C4C2×C22⋊D5311yescharacteristic
271C22×C4C23×D5311yescharacteristic
272Q16—21216no—
273C2×C8—1064no—
274C2×C8—1064no—
275C2×C8—1064no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—534no—
279C4⋊C4—1068no—
280C4⋊C4—1068no—
281C4⋊C4—1068no—
282C4⋊C4—1068no—
283C4⋊C4—1062no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—1068no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—1068no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1313318512yes—
295C23C22×C22⋊D5311yescharacteristic
296Q8—28no—
297Q8—48no—
298Q8—44no—
299Q8—1068no—
300Q8—2124no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—1064no—
315C2×C4—1064no—
316C2×C4—2122no—
317C2×C4—2124no—
318C2×C4—2122no—
319C2×C4#1313318512yes—
320C2×C4C2×D4×D5312yes—
321C2×C4C2×#1171994512yes—
322C2×C4#1313326212yes—
323C8—2124no—
324C22—22no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22C2×#1313318511yescharacteristic
329C22#1681113411yescharacteristic
330C22#1681119011yescharacteristic
331C22#1681143814yes—
332C4—22no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—82no—
337C4—2124no—
338C4—2124no—
339C4#1681126314yes—
340C2—42no—
341C2#2027803411yescharacteristic
342C2#2028400812yes—
343C2#2028501512yes—
344C2#2028579812yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2310C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2310D53
hypocenterC53#23943
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #20275866C2semidirect
4. #20275814C2semidirect
5. #20275059C2semidirect
203. C53#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#168112352
C2×#168107792
#202758662
#202758142
#202750592
#202750272
#202747862
C53×#23102
#2394353