Finite Group Databasebeta

← #27355607

Subgroups — #27355607

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11719917C212yesmaximal, complemented
2#20273629C212yesmaximal, complemented
3#20273580C214yesmaximal, complemented
4#20273539C212yesmaximal, complemented
5#20273471C211yesmaximal, characteristic, complemented
6#20273140C212yesmaximal, complemented
7#20273004C211yesmaximal, characteristic, complemented
8C2×C4×D8×C53C211yesmaximal, characteristic
9C22×#11719917—22no—
10#16810701C2214yescomplemented
11#16810695—24no—
12D4×#88322C2214yescomplemented
13#16810676C2211yescharacteristic
14#16810647C2212yes—
15#16810587—22no—
16C4×#11719917C2211yescharacteristic
17#16810374C2212yescomplemented
18#16810374C2212yescomplemented
19D4⋊#88322C2214yescomplemented
20D4⋊#88322C2214yescomplemented
21#16810345C2212yescomplemented
22#16810344C2212yescomplemented
23C22×D8×C53C411yescharacteristic
24C2×C4×D4×C53C2212yes—
25C4×D8×C53C412yes—
26C4×D8×C53C411yescharacteristic
27C4×D8×C53C411yescharacteristic
28C4×D8×C53C2211yescharacteristic
29C4×D8×C53C2212yes—
30C4×D8×C53C2211yescharacteristic
31C2×#107×C53C411yescharacteristic
32C2×#102×C53C412yes—
33C2×C4×C8×C53C2211yescharacteristic
34C2×#11719917—24no—
35C2×#11719917—22no—
36C2×#11719917—48no—
37C2×#11719917D412yes—
38C22×#88322—24no—
39C4⋊#88322—22no—
40C4⋊#88322—21no—
41C4⋊#88322C2314yes—
42C22⋊#88322—24no—
43C22⋊#88322—24no—
44C22⋊#88322—21no—
45#13133128D412yes—
46C4×#88322C2312yes—
47C4×#88322C2312yes—
48C22×D4×C53C2×C412yes—
49C23×C4×C53—22no—
50C2×D8×C53—24no—
51C2×D8×C53C2×C411yescharacteristic
52C2×D8×C53C2×C412yes—
53C2×D8×C53C2×C411yescharacteristic
54C22×C8×C53C2×C411yescharacteristic
55C22×C8×C53C2×C411yescharacteristic
56C4×D4×C53—22no—
57C4×D4×C53—22no—
58C4×D4×C53C2×C412yes—
59C4×D4×C53C2×C412yes—
60C4×D4×C53C2312yes—
61C4×D4×C53C2312yes—
62C2×C4⋊C4×C53C2×C412yes—
63C2×C22⋊C4×C53—22no—
64C2×C42×C53C2311yescharacteristic
65#107×C53C2×C412yes—
66#107×C53C2×C411yescharacteristic
67#107×C53C2×C411yescharacteristic
68#102×C53C2×C412yes—
69#102×C53C2×C412yes—
70#102×C53C2×C412yes—
71#102×C53C2×C412yes—
72C4×C8×C53C2×C411yescharacteristic
73C4×C8×C53C2×C411yescharacteristic
74C4×C8×C53C2311yescharacteristic
75C4×C8×C53C2311yescharacteristic
76C24×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—22no—
79C2×D4×C53—42no—
80C2×D4×C53C22×C412yes—
81C2×D4×C53C22×C412yes—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—44no—
88C22×C4×C53—44no—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92D8×C53—24no—
93D8×C53—22no—
94D8×C53M1614yescomplemented
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53C22×C411yescharacteristic
98C2×C8×C53C22×C411yescharacteristic
99C2×C8×C53C22×C411yescharacteristic
100C2×C8×C53C22×C411yescharacteristic
101C4⋊C4×C53—22no—
102C4⋊C4×C53C22×C412yes—
103C4⋊C4×C53C22×C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53C2411yescharacteristic
112#11719917—24no—
113#11719917—48no—
114#11719917—44no—
115C2×#88322—24no—
116C2×#88322—22no—
117C2×#88322—48no—
118C2×#88322C2×D414yes—
119C23×C53—22no—
120C23×C53—22no—
121C23×C53—44no—
122C23×C53—44no—
123C23×C53—82no—
124C23×C53C4×D411yescharacteristic
125D4×C53—22no—
126D4×C53—22no—
127D4×C53—22no—
128D4×C53—44no—
129D4×C53C2×M1614yescomplemented
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—44no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—44no—
141C2×C4×C53—44no—
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)14yes—
146C2×C4×C53C23×C411yescharacteristic
147C2×C4×C53C22×D411yescharacteristic
148C2×C4×C53C2×#4111yescharacteristic
149C8×C53—21no—
150C8×C53—21no—
151C8×C53—21no—
152C8×C53C2×M1612yescomplemented
153#88322—24no—
154#88322—42no—
155#88322—44no—
156#18044—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—24no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—88no—
166C22×C53#44212yes—
167C22×C53C2×C4×D411yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53#44212yes—
174C4×C53C22×M1611yescharacteristic
175C4×C53#56811yescharacteristic
176C4×C53C2×AGL(1,ℤ/8)12yes—
177C2×C4×D8D5311yescharacteristic
178D4×M16—532no—
179#1594—532no—
180#1586—534no—
181#1580—532no—
182#1566—531no—
183#1486—532no—
184#1451—531no—
185C2×C53—22no—
186C2×C53—44no—
187C2×C53—82no—
188C2×C53D4×M1611yescharacteristic
189C2×C53C2×#44211yescharacteristic
190C2×C53#295711yescharacteristic
191C22×D8Dic5311yescharacteristic
192C22×M16—1062no—
193C2×C4×D4C2×D5312yes—
194C4×D8Dic5312yes—
195C4×D8Dic5311yescharacteristic
196C4×D8Dic5311yescharacteristic
197C4×D8C2×D5311yescharacteristic
198C4×D8C2×D5312yes—
199C4×D8C2×D5311yescharacteristic
200#436—534no—
201#435—1064no—
202C8×D4—534no—
203#431—531no—
204C2×#107Dic5311yescharacteristic
205#423—532no—
206C2×#102Dic5312yes—
207#407—1062no—
208C4×M16—531no—
209C2×C4×C8C2×D5311yescharacteristic
210#334—532no—
211#334—532no—
212D4⋊C8—534no—
213D4⋊C8—534no—
214#322—532no—
215#322—532no—
216C53#1804411yescharacteristic, complemented, Sylow
217C22×D4C2×Dic5312yes—
218C23×C4—22no—
219C2×D8—24no—
220C2×D8C2×Dic5312yes—
221C2×D8C2×Dic5311yescharacteristic
222C2×D8C2×Dic5311yescharacteristic
223C2×M16—532no—
224C2×M16—1064no—
225C2×M16—1062no—
226C2×M16—2128no—
227C22×C8—1064no—
228C22×C8C2×Dic5311yescharacteristic
229C22×C8C2×Dic5311yescharacteristic
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×Dic5312yes—
233C4×D4C2×Dic5312yes—
234C4×D4C22×D5312yes—
235C4×D4C22×D5312yes—
236C2×C4⋊C4C2×Dic5312yes—
237C2×C22⋊C4—22no—
238C2×C42C22×D5311yescharacteristic
239#107C2×Dic5312yes—
240#107C2×Dic5311yescharacteristic
241#107C2×Dic5311yescharacteristic
242C4⋊C8—534no—
243C4⋊C8—1062no—
244C4⋊C8—1061no—
245#102C2×Dic5312yes—
246#102C2×Dic5312yes—
247#102C2×Dic5312yes—
248#102C2×Dic5312yes—
249C22⋊C8—1064no—
250C22⋊C8—1064no—
251C22⋊C8—1061no—
252#97—532no—
253C4×C8—532no—
254C4×C8—532no—
255C4×C8C2×Dic5311yescharacteristic
256C4×C8C2×Dic5311yescharacteristic
257C4×C8C22×D5311yescharacteristic
258C4×C8C22×D5311yescharacteristic
259C24—22no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4C22×Dic5312yes—
264C2×D4C22×Dic5312yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4D4×D5311yescharacteristic
273C22×C4#1171994711yescharacteristic
274C22×C4C22×Dic5311yescharacteristic
275D8—24no—
276D8—22no—
277D8#1171991714yescomplemented
278M16—1064no—
279M16—2128no—
280M16—2124no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—534no—
284C2×C8—1064no—
285C2×C8—1062no—
286C2×C8—2128no—
287C2×C8C22×Dic5311yescharacteristic
288C2×C8C22×Dic5311yescharacteristic
289C2×C8C22×Dic5311yescharacteristic
290C2×C8C22×Dic5311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4C22×Dic5312yes—
293C4⋊C4C22×Dic5312yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C42D4×D5311yescharacteristic
299C42#1171994711yescharacteristic
300C42C22×Dic5311yescharacteristic
301C42C23×D5311yescharacteristic
302C23—22no—
303C23—22no—
304C23—44no—
305C23—44no—
306C23—82no—
307C23D4×Dic5311yescharacteristic
308D4—22no—
309D4—22no—
310D4—22no—
311D4—44no—
312D4C2×#1171991714yescomplemented
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4D4×Dic5311yescharacteristic
326C2×C4D4×Dic5311yescharacteristic
327C2×C4D4×Dic5311yescharacteristic
328C2×C4#1313324114yes—
329C2×C4C23×Dic5311yescharacteristic
330C2×C4C2×D4×D5311yescharacteristic
331C2×C4C2×#1171994711yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—1064no—
336C8—2122no—
337C8—2124no—
338C8C2×#1171991712yescomplemented
339C22—21no—
340C22—21no—
341C22—24no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—88no—
348C22#1681073512yes—
349C22C2×D4×Dic5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4#1681073512yes—
356C4C22×#1171991711yescharacteristic
357C4#1681139011yescharacteristic
358C4C2×#1313324112yes—
359C2—22no—
360C2—44no—
361C2—82no—
362C2D4×#1171991711yescharacteristic
363C2C2×#1681073511yescharacteristic
364C2#2028363211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C4×D8×C53C2
socleC22×C53C2×C4×D4
prime radicalC22×D8×C53C4
cosocleC42×C53C24
hypercenterC2×C4×D8D53
hypocenterC53#18044
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. D4×#11719917C2semidirect
2. #20273629C2semidirect
3. #20273580C2semidirect
4. #20273539C2semidirect
5. #20273471C2semidirect
6. #20273140C2semidirect
7. #20273004C2semidirect
10. #16810701C22semidirect
12. D4×#88322C22semidirect
17. #16810374C22semidirect
18. #16810374C22semidirect
19. D4⋊#88322C22semidirect
20. D4⋊#88322C22semidirect
21. #16810345C22semidirect
22. #16810344C22semidirect
94. D8×C53M16semidirect
129. D4×C53C2×M16semidirect
152. C8×C53C2×M16semidirect
216. C53#18044semidirect
277. D8#11719917semidirect
312. D4C2×#11719917semidirect
338. C8C2×#11719917semidirect

Maximal subgroups

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

subgroupindex
D4×#117199172
#202736292
#202735802
#202735392
#202734712
#202731402
#202730042
C2×C4×D8×C532
#1804453