Finite Group Databasebeta

← #27299574

Subgroups — #27299574

Order 13568.

976 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16810413C214yesmaximal, complemented
2#20274643C211yesmaximal, characteristic
3#20274633C211yesmaximal, characteristic
4C2×C53×#536C211yesmaximal, characteristic
5C2×#13133136—21no—
6C2×#13133134C2212yes—
7C2×#13133132—21no—
8C2×#13133132C2212yes—
9#16810587—21no—
10#16810486—21no—
11#16810413—28no—
12C22×#41×C53C2211yescharacteristic
13C23×Q8×C53C411yescharacteristic, complemented
14C53×#536—28no—
15C2×C4⋊Q8×C53—21no—
16C2×#124×C53—21no—
17C2×#123×C53C412yescomplemented
18C2×#123×C53C2212yes—
19C2×C22⋊Q8×C53—21no—
20C2×C22⋊Q8×C53—21no—
21C2×#117×C53C411yescharacteristic, complemented
22C2×#11719917—22no—
23C2×C22⋊Dic53—22no—
24#13133136—42no—
25#13133134—24no—
26#13133132—24no—
27#13133132—42no—
28C22⋊#88322—42no—
29#13133124—42no—
30C2×#41×C53—24no—
31C2×#41×C53—42no—
32C2×#41×C53—42no—
33C22×Q8×C53—21no—
34C22×Q8×C53—21no—
35C22×Q8×C53—24no—
36C22×Q8×C53—42no—
37C22×Q8×C53—42no—
38C22×Q8×C53D411yescharacteristic
39C22×D4×C53C2×C411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C22×D4×C53C2311yescharacteristic
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2×C412yes—
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C4⋊Q8×C53—28no—
47#124×C53—28no—
48#123×C53—24no—
49#123×C53—24no—
50#123×C53—24no—
51#123×C53D418yes—
52C22⋊Q8×C53—28no—
53C22⋊Q8×C53—28no—
54#117×C53—24no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53C2×C412yes—
61C2×C42×C53—21no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65#41×C53—416no—
66C2×Q8×C53—24no—
67C2×Q8×C53—24no—
68C2×Q8×C53—22no—
69C2×Q8×C53—22no—
70C2×Q8×C53—41no—
71C2×Q8×C53—416no—
72C2×Q8×C53—42no—
73C2×Q8×C53—42no—
74C2×Q8×C53—42no—
75C2×Q8×C53—42no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53C2×D414yes—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53C22⋊C411yescharacteristic
100C22×C4×C53C22⋊C411yescharacteristic
101C22×C4×C53C22⋊C411yescharacteristic
102C22×C4×C53C22⋊C414yes—
103C22×C4×C53C2×D411yescharacteristic
104C22×Dic53—42no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—28no—
107C4⋊C4×C53—28no—
108C22⋊C4×C53—28no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—28no—
111C22⋊C4×C53C22⋊C414yes—
112C42×C53—24no—
113C22⋊Dic53—42no—
114C22⋊Dic53—42no—
115#11719917—42no—
116#11719917—42no—
117C2×#88322—42no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C2×C22⋊C412yes—
130C23×C53#12012yes—
131Q8×C53—44no—
132Q8×C53—48no—
133Q8×C53—48no—
134D4×C53—44no—
135D4×C53—44no—
136D4×C53—44no—
137C2×C4×C53—28no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—24no—
146C2×C4×C53—24no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—44no—
150C2×C4×C53—44no—
151C2×C4×C53—44no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53#9912yes—
157C2×C4×C53#9912yes—
158C2×C4×C53#12012yes—
159C2×Dic53—82no—
160C2×Dic53—84no—
161#88322—82no—
162#12692—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—41no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53#37911yescharacteristic
177C22×C53C2×#9911yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—22no—
182C4×C53—44no—
183C4×C53—41no—
184C4×C53—44no—
185Dic53—162no—
186C2×#536D5311yescharacteristic
187C2×#352—534no—
188#1785—531no—
189#1782—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—42no—
193C2×C53—42no—
194C2×C53—81no—
195C2×C53#178111yescharacteristic
196C2×C53#212312yes—
197C22×#41C2×D5311yescharacteristic
198C23×Q8Dic5311yescharacteristic, complemented
199#536—28no—
200C2×C4⋊Q8—21no—
201C2×#124—21no—
202C2×#123Dic5312yescomplemented
203C2×#123C2×D5312yes—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8—21no—
206C2×#117Dic5311yescharacteristic, complemented
207C2×#101—1061no—
208C2×#100—532no—
209C2×#99—532no—
210C2×#99—1061no—
211#407—1061no—
212#381—1061no—
213#352—1068no—
214C53#1269211yescharacteristic, complemented, Sylow
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8C22⋊D5311yescharacteristic
224C22×D4C2×Dic5311yescharacteristic
225C22×D4C22⋊D5311yescharacteristic
226C22×D4C22×D5311yescharacteristic
227C23×C4C2×Dic5311yescharacteristic
228C23×C4C2×Dic5312yes—
229C23×C4C22⋊D5311yescharacteristic
230C23×C4C22⋊D5311yescharacteristic
231C2×M16—1062no—
232C4⋊Q8—28no—
233#124—28no—
234#123—24no—
235#123—24no—
236#123—24no—
237#123C22⋊D5318yes—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—1062no—
247C2×C22⋊C4C2×Dic5312yes—
248C2×C42—21no—
249#101—2122no—
250#100—1064no—
251#99—1064no—
252#99—2122no—
253C22⋊C8—2122no—
254#95—2122no—
255C24C22⋊Dic5311yescharacteristic
256C24C22×Dic5311yescharacteristic
257C24C2×C22⋊D5311yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C2×C22⋊D5314yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—2122no—
293C22×C4C22⋊Dic5311yescharacteristic
294C22×C4C22⋊Dic5311yescharacteristic
295C22×C4C22⋊Dic5314yes—
296C22×C4C22⋊Dic5311yescharacteristic
297C22×C4C2×C22⋊D5311yescharacteristic
298M16—2122no—
299M16—2122no—
300C2×C8—2122no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—28no—
307C22⋊C4—2122no—
308C22⋊C4—2122no—
309C22⋊C4C22⋊Dic5314yes—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊Dic5311yescharacteristic
322C23C2×C22⋊Dic5312yes—
323C23#1313318612yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—4242no—
350C2×C4—4244no—
351C2×C4#1313313212yes—
352C2×C4#1313313212yes—
353C2×C4#1313318612yes—
354C8—4242no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22#1681047911yescharacteristic
369C22C2×#1313313211yescharacteristic
370C22C2×#1313313211yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—8482no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#2027463011yescharacteristic
384C2#2027636912yes—
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#536C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#536D53
hypocenterC53#12692
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810413C2semidirect
13. C23×Q8×C53C4semidirect
17. C2×#123×C53C4semidirect
21. C2×#117×C53C4semidirect
198. C23×Q8Dic53semidirect
202. C2×#123Dic53semidirect
206. C2×#117Dic53semidirect
214. C53#12692semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202746432
#202746332
C2×C53×#5362
#1269253