Finite Group Databasebeta

← #27281438

Subgroups — #27281438

Order 13568.

496 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283353C211yesmaximal, characteristic, complemented
2#20273313C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic, complemented
4#20272608C212yesmaximal, complemented
5C53×#2610C211yesmaximal, characteristic
6#16810684—24no—
7#16810680C2212yes—
8#16810647—21no—
9#16810587—21no—
10C4×#11719917C2212yes—
11#16810363—24no—
12#16810349—21no—
13#16810347—21no—
14C2×#124×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#395C411yescharacteristic
17C53×#394C411yescharacteristic
18C53×#393C411yescharacteristic
19C53×#392C411yescharacteristic
20C53×#390—21no—
21C53×#388—21no—
22C53×#388—21no—
23C53×#386—21no—
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C4⋊#88322—24no—
27C4⋊#88322—22no—
28C22⋊#88322—24no—
29C22⋊#88322—22no—
30#13133128—22no—
31#13133128D414yescomplemented
32C4×#88322—22no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37#124×C53—22no—
38#124×C53—22no—
39#117×C53—22no—
40#117×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53—21no—
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic
59#95×C53C2×C411yescharacteristic
60C24×C53—21no—
61C24×C53C22⋊C411yescharacteristic
62C2×Q8×C53—22no—
63C2×Q8×C53—81no—
64C2×D4×C53—22no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—81no—
68C2×D4×C53—81no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—81no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—81no—
93C22⋊C4×C53—81no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—42no—
97C42×C53C2×D412yes—
98C42×C53#4112yes—
99#11719917—42no—
100#11719917—42no—
101C2×#88322—24no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116Q8×C53—82no—
117D4×C53—82no—
118D4×C53—84no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12112yes—
144C2×C4×C53#12311yescharacteristic
145#88322—44no—
146#88322—82no—
147#9389—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53#38611yescharacteristic
163C22×C53#42111yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53—81no—
175#2921—531no—
176#2610D5311yescharacteristic
177#1527—532no—
178#1464—531no—
179#1300—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—81no—
183C2×C53—161no—
184C2×C53#188411yescharacteristic
185C2×C53#189711yescharacteristic
186C2×C53#191411yescharacteristic
187C2×#124C2×D5311yescharacteristic
188C2×#117C2×D5311yescharacteristic
189#433—1064no—
190#432—532no—
191#423—1061no—
192#407—1061no—
193C4×M16—532no—
194#395Dic5311yescharacteristic
195#394Dic5311yescharacteristic
196#393Dic5311yescharacteristic
197#392Dic5311yescharacteristic
198#390—21no—
199#388—21no—
200#388—21no—
201#386—21no—
202#376C2×D5311yescharacteristic
203#329—1064no—
204#324—1061no—
205#323—1061no—
206C53#938911yescharacteristic, complemented, Sylow
207C22×Q8C2×Dic5311yescharacteristic
208C22×D4—21no—
209C22×D4C2×Dic5311yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C2×M16—1062no—
212#124—22no—
213#124—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic5311yescharacteristic
218C2×C4⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic5311yescharacteristic
226C2×C22⋊C4C22⋊D5311yescharacteristic
227C2×C42—21no—
228C2×C42C22⋊D5311yescharacteristic
229C2×C42C22×D5311yescharacteristic
230C4⋊C8—1064no—
231C4⋊C8—1062no—
232C22⋊C8—1064no—
233C22⋊C8—1062no—
234#97—534no—
235#97—1062no—
236C4×C8—1062no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×Dic5311yescharacteristic
241#95C2×Dic5311yescharacteristic
242C24—21no—
243C24C22⋊Dic5311yescharacteristic
244C2×Q8—22no—
245C2×Q8—81no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C22×Dic5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264M16—2122no—
265M16—2122no—
266C2×C8—1064no—
267C2×C8—2122no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—81no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42D4×D5312yes—
284C42#1171994712yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic5311yescharacteristic
298Q8—82no—
299D4—82no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4D4×Dic5312yes—
324C2×C4#1313318511yescharacteristic
325C2×C4#1313318912yes—
326C2×C4#1313319811yescharacteristic
327C8—4242no—
328C8—2124no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#1681050311yescharacteristic
344C22#1681064011yescharacteristic
345C22#1681064011yescharacteristic
346C4—22no—
347C4—22no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—161no—
360C2#2027509911yescharacteristic
361C2#2027516411yescharacteristic
362C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2610C2
socleC22×C53#386
prime radicalC53×#392C4
cosocleC2×C42×C53C23
hypercenter#2610D53
hypocenterC53#9389
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #20283353C2semidirect
2. #20273313C2semidirect
3. #20273049C2semidirect
4. #20272608C2semidirect
31. #13133128D4semidirect
206. C53#9389semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202733132
#202730492
#202726082
C53×#26102
#938953