Finite Group Databasebeta

← #27281972

Subgroups — #27281972

Order 13568.

544 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283134C211yesmaximal, characteristic
2#20273162C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20272579C212yesmaximal, complemented
5C53×#2599C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810680C2212yes—
8#16810579—24no—
9C4×#11719917—21no—
10C4×#11719917C2212yes—
11Q8⋊#88322—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#398C411yescharacteristic
17C53×#397C411yescharacteristic
18C53×#395C411yescharacteristic
19C53×#393C411yescharacteristic
20C53×#388—21no—
21C53×#387—21no—
22C53×#387—21no—
23C53×#385—21no—
24C53×#383C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—24no—
27C2×#11719917—24no—
28C2×#11719917—22no—
29C22×#88322—22no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133128—22no—
34#13133128—22no—
35C4×#88322—22no—
36C4×#88322D414yes—
37C22×Q8×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41#117×C53—22no—
42#117×C53—22no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63#95×C53C2×C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C2×Q8×C53—22no—
66C2×Q8×C53—42no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—42no—
95C42×C53C2×D412yes—
96C42×C53#4112yes—
97#11719917—28no—
98#11719917—24no—
99#11719917—24no—
100C2×#88322—28no—
101C2×#88322—24no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109Q8×C53—24no—
110Q8×C53—42no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53#10412yes—
137C2×C4×C53C4×D412yes—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12112yes—
140C2×C4×C53#12311yescharacteristic
141#88322—28no—
142#88322—82no—
143#9491—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53#38611yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—42no—
161C4×C53—41no—
162C4×C53—81no—
163C4×C53—81no—
164C4×C53—81no—
165#2873—531no—
166#2599D5311yescharacteristic
167#1491—532no—
168#1464—531no—
169#1288—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53#189111yescharacteristic
174C2×C53#189611yescharacteristic
175C2×C53#190911yescharacteristic
176C22×M16—1061no—
177C2×C4×Q8C2×D5311yescharacteristic
178C2×#117C2×D5311yescharacteristic
179#432—532no—
180#405—1064no—
181C4×M16—532no—
182C4×M16—1061no—
183#398Dic5311yescharacteristic
184#397Dic5311yescharacteristic
185#395Dic5311yescharacteristic
186#393Dic5311yescharacteristic
187#388—21no—
188#387—21no—
189#387—21no—
190#385—21no—
191#383C2×D5311yescharacteristic
192Q8⋊C8—1064no—
193#324—1061no—
194#323—1061no—
195C53#949111yescharacteristic, complemented, Sylow
196C22×Q8C2×Dic5311yescharacteristic
197C23×C4C22⋊D5311yescharacteristic
198C2×M16—1062no—
199C2×M16—1064no—
200C2×M16—1064no—
201C2×M16—1062no—
202C22×C8—1062no—
203C4×Q8—22no—
204C4×Q8—22no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C4⋊C4C2×Dic5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4C2×Dic5311yescharacteristic
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42—21no—
218C2×C42C22⋊D5311yescharacteristic
219C2×C42C22×D5311yescharacteristic
220C4⋊C8—1062no—
221C22⋊C8—1062no—
222#97—1062no—
223#97—1062no—
224#97—1062no—
225C4×C8—534no—
226C4×C8—1062no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic5311yescharacteristic
234#95C2×Dic5311yescharacteristic
235C24C22⋊Dic5311yescharacteristic
236C2×Q8—22no—
237C2×Q8—42no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22×Dic5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252M16—1068no—
253M16—1064no—
254M16—1064no—
255C2×C8—1068no—
256C2×C8—1064no—
257C2×C8—2122no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—21no—
270C42—21no—
271C42—42no—
272C42D4×D5312yes—
273C42#1171994712yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1313314412yes—
279C23C2×C22⋊Dic5311yescharacteristic
280Q8—24no—
281Q8—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4#1313314412yes—
308C2×C4D4×Dic5312yes—
309C2×C4#1313318511yescharacteristic
310C2×C4#1313318912yes—
311C2×C4#1313319811yescharacteristic
312C8—1068no—
313C8—4242no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1681050311yescharacteristic
322C22C2×#1313314411yescharacteristic
323C22C2×#1313314411yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2027513411yescharacteristic
339C2#2027515911yescharacteristic
340C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2599C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2599D53
hypocenterC53#9491
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
2. #20273162C2semidirect
4. #20272579C2semidirect
195. C53#9491semidirect

Maximal subgroups

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

subgroupindex
#202831342
#202731622
#202730492
#202725792
C53×#25992
#949153