Finite Group Databasebeta

← #28320224

Subgroups — #28320224

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811633C211yesmaximal, characteristic
2C2×#16811586C211yesmaximal, characteristic
3#20291195C212yesmaximal
4#20290820C214yesmaximal, complemented
5#20290819C214yesmaximal, complemented
6#20290402C212yesmaximal
7C2×#107×D53C211yesmaximal, characteristic
8C2×D53⋊M16C2211yescharacteristic
9C2×#13133296C2212yes—
10C2×#13133296C2211yescharacteristic
11#16811633—24no—
12C2×#13133286C2212yes—
13C2×#13133285C2212yes—
14#16811586—24no—
15C2×#13133271C2212yes—
16C2×C8×#6157C2212yes—
17#16811574—24no—
18#16811574C2218yes—
19#16811565—24no—
20#16811496C2218yes—
21C22×C8×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2212yes—
23C2×#13133151C411yescharacteristic, complemented
24C2×#13133150C412yescomplemented
25#107×D53C412yescomplemented
26#107×D53C412yescomplemented
27#107×D53C2212yes—
28#107×D53C2212yes—
29C2×#107×C53C411yescharacteristic, complemented
30C2×#11719936—21no—
31D53⋊M16—22no—
32D53⋊M16Q814yes—
33C2×#11719940—21no—
34C2×D53⋊C8C2312yes—
35#13133296—22no—
36#13133296—22no—
37#13133296—24no—
38#13133296—24no—
39C2×C4⋊#6157—28no—
40C2×C4⋊#6157D412yes—
41C2×C4⋊#6157C2314yes—
42C2×C22⋊#6157—22no—
43C2×C22⋊#6157D412yes—
44C2×C4×#6157—22no—
45C2×C4×#6157C2312yes—
46#13133286—24no—
47#13133285—24no—
48#13133271—24no—
49C8×#6157—24no—
50(C2×#6157)⋊C4—28no—
51C22×C4×D53—22no—
52C22×C4×D53C2311yescharacteristic
53C22×#88322C2×C411yescharacteristic
54C2×C8×D53C2×C411yescharacteristic
55C2×C8×D53C2×C411yescharacteristic
56C2×C8×D53D414yes—
57C2×C8×D53Q814yes—
58C2×C8×D53C2311yescharacteristic
59C2×C8×D53C2311yescharacteristic
60C2×C4⋊Dic53C2×C412yes—
61C2×#11719919—22no—
62C4⋊C4×D53—22no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53C2×C414yes—
65C4⋊C4×D53C2314yes—
66#13133151C2×C412yes—
67#13133151C2×C412yes—
68#13133150C2×C414yes—
69#13133150C2×C414yes—
70C22×C8×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53C2×C412yes—
72#107×C53C2×C412yes—
73#107×C53C2×C412yes—
74C23×D53C2×D411yescharacteristic
75C22×#6157—24no—
76C22×#6157C2×D414yes—
77C22×C4×C53—22no—
78C22×C4×C53C22×C411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53#4111yescharacteristic
81C2×C4×D53—24no—
82C2×C4×D53—24no—
83C2×C4×D53—84no—
84C2×C4×D53C22×C411yescharacteristic
85C2×C4×D53C2×D412yes—
86C2×C4×D53C2×D411yescharacteristic
87C2×C4×D53C2×Q812yes—
88C2×C4×D53C2×Q812yes—
89C2×C4×D53#4112yes—
90C2×C4×D53#4111yescharacteristic
91C2×C4×D53C2411yescharacteristic
92C22⋊#6157—24no—
93C22⋊#6157—44no—
94#11719940—24no—
95C2×#88324—22no—
96C4⋊#6157—24no—
97C4⋊#6157—24no—
98C4⋊#6157—24no—
99C4⋊#6157—88no—
100C4×#6157—22no—
101C4×#6157—22no—
102C4×#6157—44no—
103#11719936—24no—
104D53⋊C8—22no—
105D53⋊C8C2×Q814yes—
106C2×C8×C53C4⋊C414yescomplemented
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C22×C411yescharacteristic
109C4⋊C4×C53—22no—
110C4⋊C4×C53C22×C414yes—
111C4⋊Dic53—22no—
112C4⋊Dic53C22×C414yes—
113#11719919—24no—
114#11719919—24no—
115C2×#88322C4⋊C414yescomplemented
116C2×#88322C22×C411yescharacteristic
117C2×#88322C22×C411yescharacteristic
118C8×D53C4⋊C414yescomplemented
119C8×D53C4⋊C414yescomplemented
120C8×D53C2×D412yes—
121C8×D53C2×D412yes—
122C8×D53C2×Q812yes—
123C8×D53C2×Q812yes—
124C23×C53C4×D411yescharacteristic
125C22×D53—22no—
126C22×D53—22no—
127C22×D53C4×D411yescharacteristic
128C22×D53#12112yes—
129C22×D53C4⋊Q812yes—
130C22×D53C22×D411yescharacteristic
131C2×#6157—24no—
132C2×#6157—24no—
133C2×#6157—24no—
134C2×#6157—44no—
135C2×#6157—44no—
136C2×#6157—44no—
137C2×C4×C53—24no—
138C2×C4×C53—82no—
139C2×C4×C53C2×C4⋊C412yes—
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C4×Q812yes—
142C2×C4×C53C23×C411yescharacteristic
143C2×Dic53—24no—
144C2×Dic53—82no—
145C2×Dic53C4×D411yescharacteristic
146C2×Dic53C22⋊Q812yes—
147C2×Dic53#12512yes—
148C2×Dic53C2×#4111yescharacteristic
149C4×D53—48no—
150C4×D53—84no—
151C4×D53C2×C4⋊C412yes—
152C4×D53C4×Q812yes—
153C4×D53#12112yes—
154C4×D53C22⋊Q812yes—
155C4×D53#12512yes—
156C4×D53C4⋊Q812yes—
157C4×D53C22×D411yescharacteristic
158C4×D53C22×Q811yescharacteristic
159C4×D53C22×Q811yescharacteristic
160C4×D53C2×#4111yescharacteristic
161#88324—24no—
162C8×C53C2×C4⋊C412yes—
163C8×C53C2×C4⋊C412yes—
164#88322C2×C4⋊C412yes—
165#88322C2×C4⋊C412yes—
166#21158—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#38912yes—
170C22×C53C2×C4×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—24no—
174C2×D53—24no—
175C2×D53#38912yes—
176C2×D53C2×#12111yescharacteristic
177C2×D53C2×C4⋊Q811yescharacteristic
178C2×D53#57612yes—
179C2×D53#57812yes—
180#6157—44no—
181#6157—84no—
182C4×C53—44no—
183C4×C53#38912yes—
184C4×C53C22×C4⋊C411yescharacteristic
185C4×C53C2×C4×Q811yescharacteristic
186Dic53—44no—
187Dic53#38912yes—
188Dic53C2×C22⋊Q811yescharacteristic
189Dic53C2×#12511yescharacteristic
190C2×#427—531no—
191C22×#107—531no—
192C2×#405—531no—
193#2316—532no—
194#1943—534no—
195#1942—534no—
196#1736—532no—
197C2×C53—22no—
198C2×C53C2×#38911yescharacteristic
199C2×C53#294811yescharacteristic
200C2×C53#295011yescharacteristic
201D53—22no—
202D53#306112yes—
203D53#316412yes—
204C22×M16—531no—
205C23×C8—531no—
206C2×#117—531no—
207C2×#117—532no—
208C22×C4⋊C4—532no—
209#427—1064no—
210C2×#107—531no—
211C2×#107—532no—
212C2×#107—532no—
213C2×#107—532no—
214C2×#107—532no—
215C2×#107—532no—
216C2×#107—532no—
217C2×#107#615711yescharacteristic, complemented
218C2×#106—532no—
219#405—1064no—
220C2×#97—532no—
221C2×C4×C8—532no—
222#389—538no—
223#389—1064no—
224#380—1064no—
225#340—538no—
226C53#2115811yescharacteristic, complemented, Sylow
227C23×C4—531no—
228C23×C4—1062no—
229C2×M16—534no—
230C2×M16—1062no—
231C2×M16—1061no—
232C2×M16—1061no—
233C22×C8—531no—
234C22×C8—534no—
235C22×C8—534no—
236C22×C8—531no—
237C22×C8—531no—
238C22×C8—531no—
239C22×C8—531no—
240C22×C8—532no—
241C22×C8C2×#615711yescharacteristic
242#117—1062no—
243#117—1062no—
244#117—1064no—
245#117—1064no—
246C2×C4⋊C4—532no—
247C2×C4⋊C4—534no—
248C2×C4⋊C4—534no—
249C2×C4⋊C4—532no—
250C2×C4⋊C4—534no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4—1062no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1068no—
255C2×C4⋊C4C2×#615712yes—
256C2×C22⋊C4—532no—
257C2×C22⋊C4—1062no—
258C2×C42—532no—
259C2×C42—1062no—
260#107—532no—
261#107—532no—
262#107—534no—
263#107—534no—
264#107—1064no—
265#107C2×#615712yes—
266#107C2×#615712yes—
267#106—1064no—
268#97—1064no—
269C4×C8—1064no—
270#95—1068no—
271C24—531no—
272C22×C4—22no—
273C22×C4—531no—
274C22×C4—532no—
275C22×C4—532no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—534no—
283C22×C4—1064no—
284C22×C4—1064no—
285C22×C4—1062no—
286C22×C4—1064no—
287C22×C4—4244no—
288C22×C4C22×#615711yescharacteristic
289M16—1064no—
290M16—1064no—
291C2×C8—534no—
292C2×C8—531no—
293C2×C8—531no—
294C2×C8—532no—
295C2×C8—532no—
296C2×C8—534no—
297C2×C8—534no—
298C2×C8—532no—
299C2×C8—532no—
300C2×C8—534no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8C4⋊#615714yescomplemented
304C2×C8C22×#615711yescharacteristic
305C2×C8C22×#615711yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—534no—
308C4⋊C4—1064no—
309C4⋊C4—1064no—
310C4⋊C4—1064no—
311C4⋊C4—1064no—
312C4⋊C4—1064no—
313C4⋊C4—1062no—
314C4⋊C4—4248no—
315C4⋊C4C22×#615714yes—
316C22⋊C4—1064no—
317C22⋊C4—2124no—
318C42—1062no—
319C42—1062no—
320C42—2124no—
321C23—531no—
322C23—532no—
323C23—532no—
324C23—531no—
325C23—1062no—
326C23—1062no—
327C23D4×#615711yescharacteristic
328C2×C4—24no—
329C2×C4—82no—
330C2×C4—532no—
331C2×C4—532no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—532no—
335C2×C4—532no—
336C2×C4—532no—
337C2×C4—532no—
338C2×C4—532no—
339C2×C4—531no—
340C2×C4—531no—
341C2×C4—532no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—1064no—
345C2×C4—1064no—
346C2×C4—1064no—
347C2×C4—1064no—
348C2×C4—4244no—
349C2×C4—4242no—
350C2×C4—2128no—
351C2×C4—2124no—
352C2×C4—2124no—
353C2×C4—2124no—
354C2×C4C2×C4⋊#615712yes—
355C2×C4D4×#615711yescharacteristic
356C2×C4Q8×#615712yes—
357C2×C4C23×#615711yescharacteristic
358C8—532no—
359C8—532no—
360C8—1064no—
361C8C2×C4⋊#615712yes—
362C8C2×C4⋊#615712yes—
363C22—21no—
364C22—21no—
365C22—531no—
366C22—531no—
367C22—532no—
368C22—532no—
369C22—532no—
370C22—1064no—
371C22—1064no—
372C22—1061no—
373C22—1061no—
374C22#1681157412yes—
375C22C2×D4×#615711yescharacteristic
376C4—44no—
377C4—531no—
378C4—531no—
379C4—532no—
380C4—4244no—
381C4—2124no—
382C4—2124no—
383C4#1681157412yes—
384C4C22×C4⋊#615711yescharacteristic
385C4C2×Q8×#615711yescharacteristic
386C2—22no—
387C2—532no—
388C2—532no—
389C2—1062no—
390C2C2×#1681157411yescharacteristic
391C2#2029147411yescharacteristic
392C2#2029147611yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#107×C53C4
socleC22×C53C2×C4×D4
prime radicalC23×D53C2×D4
cosocleC2×C4×D53C24
hypercenterC2×#107#6157
hypocenterC53#21158
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
4. #20290820C2semidirect
5. #20290819C2semidirect
23. C2×#13133151C4semidirect
24. C2×#13133150C4semidirect
25. #107×D53C4semidirect
26. #107×D53C4semidirect
29. C2×#107×C53C4semidirect
106. C2×C8×C53C4⋊C4semidirect
115. C2×#88322C4⋊C4semidirect
118. C8×D53C4⋊C4semidirect
119. C8×D53C4⋊C4semidirect
217. C2×#107#6157semidirect
226. C53#21158semidirect
303. C2×C8C4⋊#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168116332
C2×#168115862
#202911952
#202908202
#202908192
#202904022
C2×#107×D532
#2115853