Finite Group Databasebeta

← #27472500

Subgroups — #27472500

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D424C211yesmaximal, characteristic
2#20279633C212yesmaximal, complemented
3#20276203C212yesmaximal, complemented
4#20276026C212yesmaximal
5#20275398C214yesmaximal, complemented
6#20274843C211yesmaximal, characteristic, complemented
7#20274716C212yesmaximal, complemented
8C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
9C22×D424C2211yescharacteristic
10C2×#13133193—22no—
11C2×C4×D212C2212yes—
12C4×D424—24no—
13C2×#13133151C2211yescharacteristic
14C2×#13133139C2214yes—
15C2×#13133139C2212yes—
16#16810545—24no—
17#16810525C2214yes—
18#16810521C2214yes—
19#16810493C2212yes—
20#16810480—22no—
21#16810386C2214yes—
22#16810386C2212yes—
23#16810386C2212yes—
24C2×C4×Q8×C53C2211yescharacteristic
25C8×Q8×C53—24no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2212yes—
28C2×C4×C8×C53C2212yes—
29C2×C4×C8×C53C2211yescharacteristic
30C22×D212C2312yes—
31C22×C4×D53—22no—
32C2×D424—22no—
33C2×D424—24no—
34#13133193—48no—
35C4×D212—24no—
36C4×D212—24no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53C2312yes—
40C2×C4⋊Dic53C2314yes—
41C2×#11719919—24no—
42C2×#11719921—24no—
43C2×#11719921—22no—
44C2×#11719921—24no—
45#13133151—22no—
46#13133139—24no—
47#13133139—28no—
48#13133124—24no—
49C22×Q8×C53D411yescharacteristic
50C22×C8×C53C2312yes—
51C22×C8×C53C2311yescharacteristic
52C22×C8×C53C2311yescharacteristic
53C4×Q8×C53—22no—
54C4×Q8×C53—22no—
55C2×C4⋊C4×C53D412yes—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53C2312yes—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—24no—
60C4⋊C8×C53—22no—
61C4×C8×C53—24no—
62C4×C8×C53D414yes—
63C23×D53—22no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53D814yescomplemented
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D412yes—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2411yescharacteristic
72C22×Dic53—22no—
73C22×Dic53—24no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—22no—
79C2×D212#4114yes—
80C2×C4×D53—44no—
81C2×C4×D53—44no—
82C2×C4×D53—44no—
83C2×C8×C53—22no—
84C2×C8×C53—22no—
85C2×C8×C53—22no—
86C2×C8×C53—24no—
87C2×C8×C53C2×D412yes—
88C2×C8×C53C2×D412yes—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—22no—
91C42×C53—22no—
92C42×C53—22no—
93C42×C53C2×D412yes—
94C42×C53#4112yes—
95#11719921—44no—
96#11719921—48no—
97#11719921—84no—
98C4⋊Dic53—22no—
99C4⋊Dic53—24no—
100C4⋊Dic53—22no—
101C4⋊Dic53—24no—
102C4⋊Dic53—44no—
103C4⋊Dic53—82no—
104#11719919—48no—
105D424—28no—
106C23×C53C22×D411yescharacteristic
107C22×D53—24no—
108C22×D53—42no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—42no—
114Q8×C53—28no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—24no—
120C2×C4×C53#12112yes—
121C2×C4×C53#12112yes—
122C2×C4×C53C2×D814yes—
123C2×C4×C53C2×D812yes—
124C2×C4×C53#13512yes—
125C2×C4×C53C22×D411yescharacteristic
126C2×C4×C53C2×#4112yes—
127C2×C4×C53C2×#4111yescharacteristic
128C2×C4×C532−1+412yes—
129C2×Dic53—42no—
130C2×Dic53—44no—
131C2×Dic53—42no—
132C2×Dic53—44no—
133C2×Dic53—42no—
134C2×Dic53—44no—
135D212—24no—
136D212—24no—
137D212—42no—
138D212—42no—
139C4×D53—88no—
140C8×C53—22no—
141C8×C53—22no—
142C8×C53—42no—
143#28185—531nomaximal, Sylow
144C22×C53C2×#12111yescharacteristic
145C22×C53#54812yes—
146C22×C53C22×D811yescharacteristic
147C22×C53C2×#13511yescharacteristic
148C22×C53#57712yes—
149C2×D53—44no—
150C2×D53—44no—
151C2×D53—44no—
152C2×D53—82no—
153C2×D53—82no—
154C2×D53—82no—
155C2×D53—82no—
156C4×C53—24no—
157C4×C53—22no—
158C4×C53—42no—
159C4×C53#46614yes—
160C4×C53#52511yescharacteristic
161C4×C53#53611yescharacteristic
162C4×C53#55212yes—
163Dic53—84no—
164Dic53—82no—
165C2×C8×Q8D5311yescharacteristic, complemented
166C2×C4×D8—531no—
167#2538—532no—
168#2086—532no—
169#2054—532no—
170#1934—534no—
171#1834—531no—
172#1802—532no—
173C2×C53#253611yescharacteristic
174C2×C53C2×#46611yescharacteristic
175C2×C53#306011yescharacteristic
176C2×C53#335812yes—
177C2×C53#336012yes—
178D53—84no—
179C22×D8—531no—
180C2×C22⋊Q8—1062no—
181C2×C4×Q8C2×D5311yescharacteristic
182C2×C4×D4—532no—
183C8×Q8—24no—
184C4×D8—1064no—
185C2×#107—531no—
186C2×C4⋊C8C2×D5312yes—
187C2×C4⋊C8C2×D5311yescharacteristic
188C2×#102—534no—
189C2×#102—532no—
190C2×C4×C8C2×D5312yes—
191C2×C4×C8C2×D5311yescharacteristic
192#396—1064no—
193#390—534no—
194#389—534no—
195#384—532no—
196#380—1062no—
197#340—532no—
198#340—534no—
199#340—532no—
200C53#2818511yescharacteristic, complemented, Sylow
201C22×Q8D21211yescharacteristic
202C22×D4—532no—
203C23×C4—1062no—
204C2×D8—1064no—
205C2×D8—1062no—
206C22×C8C22×D5312yes—
207C22×C8C22×D5311yescharacteristic
208C22×C8C22×D5311yescharacteristic
209C22⋊Q8—2128no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×D4—1064no—
213C4×D4—1064no—
214C2×C4⋊C4—532no—
215C2×C4⋊C4—534no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4—1064no—
218C2×C4⋊C4—1062no—
219C2×C4⋊C4D21211yescharacteristic
220C2×C4⋊C4D21212yes—
221C2×C22⋊C4—1064no—
222C2×C22⋊C4—1062no—
223C2×C22⋊C4—1064no—
224C2×C42C22×D5312yes—
225C2×C42C22×D5311yescharacteristic
226#107—1062no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229#102—1068no—
230#102—1064no—
231C4×C8—24no—
232C4×C8C22⋊D5314yes—
233#95—1064no—
234C24—1062no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8D42414yescomplemented
238C2×D4—534no—
239C2×D4—1062no—
240C2×D4—1062no—
241C2×D4—1062no—
242C2×D4—1062no—
243C2×D4—1062no—
244C22×C4—1062no—
245C22×C4—1064no—
246C22×C4—2124no—
247C22×C4—2124no—
248C22×C4—2124no—
249C22×C4C2×D21212yes—
250C22×C4C2×D21211yescharacteristic
251C22×C4C2×D21212yes—
252C22×C4C2×D21211yescharacteristic
253C22×C4C23×D5311yescharacteristic
254D8—1068no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8C2×C22⋊D5312yes—
260C2×C8C2×C22⋊D5312yes—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—1062no—
264C4⋊C4—1064no—
265C4⋊C4—1062no—
266C4⋊C4—1064no—
267C4⋊C4—4242no—
268C4⋊C4—2128no—
269C4⋊C4—2124no—
270C22⋊C4—4244no—
271C22⋊C4—2124no—
272C22⋊C4—2128no—
273C42—22no—
274C42—22no—
275C42#1171994512yes—
276C42C2×C22⋊D5312yes—
277C23—1064no—
278C23—2122no—
279C23—2122no—
280C23—2122no—
281C23—2122no—
282C23—2122no—
283C23—2122no—
284C23C22×D21211yescharacteristic
285Q8—28no—
286D4—1064no—
287D4—1064no—
288D4—2122no—
289D4—2122no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—4248no—
296C2×C4—2124no—
297C2×C4—2122no—
298C2×C4—2124no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2124no—
302C2×C4#1313319112yes—
303C2×C4#1313319112yes—
304C2×C4C2×D42414yes—
305C2×C4C2×D42412yes—
306C2×C4#1313323912yes—
307C2×C4C22×C22⋊D5311yescharacteristic
308C2×C4#41×D5312yes—
309C2×C4C2×#1171994511yescharacteristic
310C2×C4#1313326512yes—
311C8—22no—
312C8—22no—
313C8—42no—
314C22—4242no—
315C22—4242no—
316C22—4242no—
317C22—4242no—
318C22—2124no—
319C22—2124no—
320C22—2124no—
321C22C2×#1313319111yescharacteristic
322C22#1681128512yes—
323C22C22×D42411yescharacteristic
324C22C2×#1313323911yescharacteristic
325C22#1681143512yes—
326C4—24no—
327C4—22no—
328C4—42no—
329C4—4244no—
330C4—4242no—
331C4#1681085914yes—
332C4#1681113011yescharacteristic
333C4#1681119711yescharacteristic
334C4#1681131412yes—
335C2—4244no—
336C2#2027962011yescharacteristic
337C2C2×#1681085911yescharacteristic
338C2#2028437711yescharacteristic
339C2#2028735112yes—
340C2#2028736612yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C8×Q8×C53C2
socleC23×C53C22×D4
prime radicalC22×D424C22
hypercenterC2×C8×Q8D53
hypocenterC53#28185
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20279633C2semidirect
3. #20276203C2semidirect
5. #20275398C2semidirect
6. #20274843C2semidirect
7. #20274716C2semidirect
8. C2×C8×Q8×C53C2semidirect
66. C2×Q8×C53D8semidirect
165. C2×C8×Q8D53semidirect
200. C53#28185semidirect
237. C2×Q8D424semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4242
#202796332
#202762032
#202760262
#202753982
#202748432
#202747162
C2×C8×Q8×C532
#2818553