Finite Group Databasebeta

← #27299922

Subgroups — #27299922

Order 13568.

768 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16810418C211yesmaximal, characteristic
2C2×#16810413C212yesmaximal
3#20275129C212yesmaximal
4#20275104C211yesmaximal, characteristic
5C2×C53×#563C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810632—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810486—22no—
11#16810418—22no—
12#16810413—24no—
13#16810389—22no—
14#16810383—21no—
15C53×#563—22no—
16C53×#563—24no—
17C53×#563C414yescomplemented
18C2×C4⋊Q8×C53—21no—
19C2×#126×C53—22no—
20C2×#125×C53—21no—
21C2×#124×C53C2211yescharacteristic
22C2×#123×C53C2212yes—
23C2×C22⋊Q8×C53—22no—
24C2×#11719917—21no—
25D4×Dic53—42no—
26#13133176—21no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29#13133141—42no—
30#13133134—22no—
31#13133132—22no—
32#13133132—22no—
33#13133124—44no—
34C22×Q8×C53—21no—
35C22×Q8×C53D411yescharacteristic
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53D412yes—
38C4⋊Q8×C53—24no—
39C4⋊Q8×C53—24no—
40#126×C53—28no—
41#126×C53—28no—
42#125×C53—24no—
43#125×C53—24no—
44#124×C53—22no—
45#124×C53C2×C412yes—
46#124×C53D412yes—
47#123×C53—22no—
48#123×C53—22no—
49#123×C53C2×C414yes—
50#123×C53D414yes—
51C22⋊Q8×C53—28no—
52C22⋊Q8×C53—28no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53D412yes—
61C2×C42×C53—21no—
62C2×C42×C53D411yescharacteristic
63C24×C53C2×D412yes—
64C2×Q8×C53—24no—
65C2×Q8×C53—21no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—81no—
69C2×Q8×C53—81no—
70C2×Q8×C53C22⋊C412yes—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53C22×C411yescharacteristic
74C2×D4×C53C2×D411yescharacteristic
75C2×D4×C53C2×D412yes—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—42no—
82C22×C4×C53—42no—
83C22×C4×C53C22⋊C414yes—
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—24no—
86C22×Dic53—42no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—28no—
89C4⋊C4×C53—28no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—24no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—24no—
95C22⋊C4×C53—24no—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53C22⋊C414yes—
99C42×C53—24no—
100C42×C53—21no—
101C42×C53C22⋊C412yes—
102C22⋊Dic53—42no—
103C22⋊Dic53—41no—
104C22⋊Dic53—42no—
105C22⋊Dic53—42no—
106C4⋊Dic53—22no—
107C4×Dic53—41no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—41no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—44no—
114C23×C53C2×C22⋊C412yes—
115C23×C53#12012yes—
116C23×C53#12011yescharacteristic
117Q8×C53—42no—
118Q8×C53—81no—
119Q8×C53—84no—
120D4×C53—41no—
121D4×C53—42no—
122D4×C53—41no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—24no—
128C2×C4×C53—22no—
129C2×C4×C53—44no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—44no—
136C2×C4×C53—82no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×Dic53—42no—
140C2×Dic53—84no—
141C2×Dic53—82no—
142C2×Dic53—84no—
143#88322—81no—
144#12764—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—44no—
151C22×C53—82no—
152C22×C53#37911yescharacteristic
153C22×C53B(4,𝔽2)11yescharacteristic
154C22×C53#45811yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—44no—
158C4×C53—42no—
159C4×C53—44no—
160C4×C53—42no—
161Dic53—82no—
162Dic53—162no—
163C2×#563D5311yescharacteristic
164C2×#354—531no—
165C2×#352—532no—
166#1890—532no—
167#1885—531no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—42no—
171C2×C53—42no—
172C2×C53#179011yescharacteristic
173C2×C53#220112yes—
174#563—22no—
175#563—24no—
176#563Dic5314yescomplemented
177C2×C4⋊Q8—21no—
178C2×#126—22no—
179C2×#125—21no—
180C2×#124C2×D5311yescharacteristic
181C2×#123C2×D5312yes—
182C2×C22⋊Q8—22no—
183#518—531no—
184#419—1061no—
185C2×#100—531no—
186C2×#99—532no—
187#381—1062no—
188#354—1062no—
189#352—1064no—
190#342—1062no—
191#339—1061no—
192C53#1276411yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D5311yescharacteristic
195C22×D4C22×D5311yescharacteristic
196C23×C4C22⋊D5312yes—
197C2×M16—1061no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200#126—28no—
201#126—28no—
202#125—24no—
203#125—24no—
204#124—22no—
205#124C2×Dic5312yes—
206#124C22⋊D5312yes—
207#123—22no—
208#123—22no—
209#123C2×Dic5314yes—
210#123C22⋊D5314yes—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C4×D4—2122no—
214#117—1061no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—1062no—
223C2×C22⋊C4—1062no—
224C2×C22⋊C4C22⋊D5312yes—
225C2×C42—21no—
226C2×C42C22⋊D5311yescharacteristic
227#103—2122no—
228#100—1062no—
229#99—1062no—
230#99—1062no—
231#95—2124no—
232C24C2×C22⋊D5312yes—
233C2×Q8—24no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×Q8C22⋊Dic5312yes—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4C22×Dic5311yescharacteristic
243C2×D4C2×C22⋊D5311yescharacteristic
244C2×D4C2×C22⋊D5312yes—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—1064no—
253C22×C4—2122no—
254C22×C4C22⋊Dic5314yes—
255C22×C4C2×C22⋊D5311yescharacteristic
256M16—2121no—
257M16—2121no—
258C2×C8—2121no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—1062no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C22⋊C4—2121no—
275C22⋊C4C22⋊Dic5314yes—
276C42—24no—
277C42—21no—
278C42—2121no—
279C42C22⋊Dic5312yes—
280C23—22no—
281C23—22no—
282C23—44no—
283C23C2×C22⋊Dic5312yes—
284C23#1313318612yes—
285C23#1313318611yescharacteristic
286Q8—42no—
287Q8—81no—
288Q8—84no—
289D4—41no—
290D4—42no—
291D4—41no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—82no—
306C2×C4—4242no—
307C2×C4—4244no—
308C2×C4—4244no—
309C2×C4—2122no—
310C2×C4C2×C22⋊Dic5311yescharacteristic
311C2×C4#1313318611yescharacteristic
312C8—4241no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—82no—
320C22#1681047911yescharacteristic
321C22#1681081011yescharacteristic
322C22#1681081311yescharacteristic
323C4—21no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—8482no—
330C4—4242no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—42no—
335C2#2027466311yescharacteristic
336C2#2027672812yes—
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#563C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#563D53
hypocenterC53#12764
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
17. C53×#563C4semidirect
176. #563Dic53semidirect
192. C53#12764semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
C2×#168104132
#202751292
#202751042
C2×C53×#5632
#1276453