Finite Group Databasebeta

← #27279076

Subgroups — #27279076

Order 13568.

558 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277974C211yesmaximal, characteristic
2#20274812C212yesmaximal, complemented
3C4×#13133132C211yesmaximal, characteristic
4#20272600C212yesmaximal, complemented
5C53×#2975C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7C2×#13133132—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C411yescharacteristic, complemented
10#16810553C411yescharacteristic, complemented
11#16810551—21no—
12#16810548C411yescharacteristic, complemented
13#16810537—21no—
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C2212yes—
17C4×C22⋊Dic53C2211yescharacteristic
18#16810390—21no—
19#16810384—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#436—24no—
22C53×#431—21no—
23C53×#407—21no—
24C53×#407C2212yes—
25#13133176—22no—
26#13133176—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—22no—
39#13133132—24no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—22no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C2×M16×C53—22no—
51C2×M16×C53D414yes—
52C4×D4×C53—22no—
53C4×D4×C53—22no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C4⋊C8×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C4×C8×C53—22no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—22no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4112yes—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—42no—
84M16×C53—44no—
85M16×C53—82no—
86C2×C8×C53—24no—
87C4⋊C4×C53—22no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—24no—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C4⋊Dic53—22no—
99C4⋊Dic53—22no—
100#11719919—42no—
101#11719919—42no—
102C4×Dic53—42no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C4×Dic53—44no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#12112yes—
117D4×C53—44no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53#10412yes—
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—44no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149C8×C53—44no—
150#9012—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53#38611yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53#45812yes—
172Dic53—42no—
173Dic53—81no—
174Dic53—84no—
175Dic53—81no—
176Dic53—81no—
177#2975D5311yescharacteristic
178#2395—531no—
179#1826—532no—
180C4×#99—531no—
181#1297—532no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—81no—
186C2×C53#187111yescharacteristic
187C2×C53#188411yescharacteristic
188C2×C53#189011yescharacteristic
189C2×C4×D4C2×D5311yescharacteristic
190C2×#117—531no—
191#436—24no—
192#431—21no—
193C2×#99—1061no—
194#407—21no—
195#407C2×D5312yes—
196#400—531no—
197#399—531no—
198#398—531no—
199#397—531no—
200#397—1061no—
201#394—1061no—
202C4×C4⋊C4—1061no—
203C4×C22⋊C4—531no—
204C4×C22⋊C4—532no—
205C4×C22⋊C4—1061no—
206#342—1061no—
207#339—1064no—
208C53#901211yescharacteristic, complemented, Sylow
209C22×D4D21211yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C23×C4C22×D5311yescharacteristic
212C2×M16—22no—
213C2×M16C22⋊D5314yes—
214C4×D4—22no—
215C4×D4—22no—
216#117—1062no—
217#117—1062no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4C4×D5311yescharacteristic
223C2×C22⋊C4—1062no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4C4×D5311yescharacteristic
229C2×C22⋊C4D21211yescharacteristic
230C2×C42—1061no—
231C2×C42—1061no—
232C2×C42—1062no—
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—22no—
235#99—1064no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C4×C8—22no—
240#95—531no—
241#95—531no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246#95—1062no—
247C24#1171992111yescharacteristic
248C24C2×D21211yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—2122no—
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4C2×C4×D5311yescharacteristic
265C22×C4#1171994512yes—
266C22×C4C2×C22⋊D5312yes—
267C22×C4C2×C22⋊D5311yescharacteristic
268M16—44no—
269M16—82no—
270C2×C8—24no—
271C4⋊C4—22no—
272C4⋊C4—1062no—
273C4⋊C4—1062no—
274C4⋊C4—2122no—
275C4⋊C4—2122no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—1064no—
279C22⋊C4—2122no—
280C22⋊C4—2122no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C22⋊C4—2122no—
284C42—21no—
285C42—21no—
286C42—2124no—
287C42—2122no—
288C42—2121no—
289C42—2121no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×#1171992111yescharacteristic
300C23#1313319112yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—1062no—
315C2×C4—4242no—
316C2×C4—4242no—
317C2×C4—2121no—
318C2×C4—2124no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4#1313314312yes—
330C2×C4C4×C22⋊D5312yes—
331C2×C4#1313318611yescharacteristic
332C2×C4#1313320011yescharacteristic
333C8—44no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1681050211yescharacteristic
347C22C2×#1313314311yescharacteristic
348C22#1681063911yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—4241no—
355C4—4244no—
356C4—4241no—
357C4—4241no—
358C4—2122no—
359C4#1681081312yes—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2027501711yescharacteristic
365C2#2027509811yescharacteristic
366C2#2027512811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2975C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2975D53
hypocenterC53#9012
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #20274812C2semidirect
4. #20272600C2semidirect
8. #16810562C4semidirect
9. #16810559C4semidirect
10. #16810553C4semidirect
12. #16810548C4semidirect
208. C53#9012semidirect

Maximal subgroups

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

subgroupindex
#202779742
#202748122
C4×#131331322
#202726002
C53×#29752
#901253