Finite Group Databasebeta

← #27271123

Subgroups — #27271123

Order 13568.

422 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 75 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#20274795C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3C53×#1436C211yesmaximal, characteristic
4#16810587—21no—
5#16810587—21no—
6#16810392—21no—
7#16810392C412yescomplemented
8#16810349—21no—
9#16810349C412yescomplemented
10C22×C4⋊C4×C53C2211yescharacteristic
11C22×C42×C53C411yescharacteristic
12C2×#95×C53—21no—
13C2×#95×C53—21no—
14C2×#95×C53C411yescharacteristic
15C2×#11719917—22no—
16C2×#11719917—41no—
17C22⋊#88322—22no—
18C22⋊#88322—41no—
19C22⋊#88322—41no—
20C22⋊#88322—41no—
21C23×C4×C53—21no—
22C23×C4×C53—21no—
23C23×C4×C53C2×C411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C2×C4⋊C4×C53—41no—
27C2×C4⋊C4×C53—41no—
28C2×C4⋊C4×C53C2×C411yescharacteristic
29C2×C4⋊C4×C53C2×C411yescharacteristic
30C2×C4⋊C4×C53D411yescharacteristic
31C2×C4⋊C4×C53Q811yescharacteristic
32C2×C42×C53—42no—
33C2×C42×C53—41no—
34#95×C53—24no—
35#95×C53—42no—
36#95×C53—41no—
37C24×C53C22⋊C411yescharacteristic
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C4211yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C4⋊C411yescharacteristic
68C22×C4×C53C4⋊C411yescharacteristic
69C22×C4×C53C4⋊C411yescharacteristic
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—42no—
75C42×C53—42no—
76#11719917—42no—
77#11719917—42no—
78#11719917—81no—
79#11719917—81no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53#9511yescharacteristic
89C23×C53#9911yescharacteristic
90C23×C53#9911yescharacteristic
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—24no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53#10412yes—
127#88322—82no—
128#88322—82no—
129#7215—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53H(3,ℤ/4)11yescharacteristic
145C22×C53#33911yescharacteristic
146C22×C53#34211yescharacteristic
147C22×C53AΣL(1,𝔽16)11yescharacteristic
148C22×C53#35211yescharacteristic
149C22×C53#35511yescharacteristic
150C22×C53#35611yescharacteristic
151C4×C53—22no—
152C4×C53—44no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—82no—
159#1821—531no—
160#1463—531no—
161#1436D5311yescharacteristic
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#128411yescharacteristic
169C2×C53#139011yescharacteristic
170C2×C53#139411yescharacteristic
171C22×C4⋊C4C2×D5311yescharacteristic
172C22×C42Dic5311yescharacteristic
173#407—1061no—
174#407—1061no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95Dic5311yescharacteristic
178#343—532no—
179#343—1061no—
180#324—532no—
181#324—1061no—
182C53#721511yescharacteristic, complemented, Sylow
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C2×Dic5311yescharacteristic
186C23×C4C22⋊D5311yescharacteristic
187C23×C4C22⋊D5311yescharacteristic
188C2×M16—1062no—
189C2×M16—2121no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4Dic10611yescharacteristic
193C2×C4⋊C4C4×D5311yescharacteristic
194C2×C4⋊C4C4×D5311yescharacteristic
195C2×C4⋊C4D21211yescharacteristic
196C2×C42—41no—
197C2×C42—42no—
198C22⋊C8—1062no—
199C22⋊C8—2121no—
200C22⋊C8—2121no—
201C22⋊C8—2121no—
202#95—24no—
203#95—41no—
204#95—42no—
205C24C22⋊Dic5311yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4C4×Dic5311yescharacteristic
233C22×C4#1171991911yescharacteristic
234C22×C4#1171991911yescharacteristic
235C22×C4C4⋊Dic5311yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238M16—4241no—
239M16—4241no—
240M16—2122no—
241M16—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—42no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#1313312411yescharacteristic
257C23#1313313211yescharacteristic
258C23#1313313211yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4#1313314512yes—
295C8—4242no—
296C8—4242no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1681037911yescharacteristic
312C22#1681038311yescharacteristic
313C22#1681039011yescharacteristic
314C22#1681041011yescharacteristic
315C22#1681041311yescharacteristic
316C22#1681042011yescharacteristic
317C22#1681042211yescharacteristic
318C4—22no—
319C4—44no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2027256811yescharacteristic
333C2#2027283311yescharacteristic
334C2#2027284211yescharacteristic
subgroupG/H
centerC22#16810390
commutatorC22×C4×C53C42
FrattiniC22×C4⋊C4C2×D53
FittingC53×#1436C2
socleC22×C53#342
prime radicalC24×C53C22⋊C4
cosocleC22×C4⋊C4×C53C22
hypercenter#1436D53
hypocenterC53#7215
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1436

Complements

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

NH ≅ G/Ntype
7. #16810392C4semidirect
9. #16810349C4semidirect
182. C53#7215semidirect

Maximal subgroups

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

subgroupindex
#202747952
#202730452
C53×#14362
#721553