Finite Group Databasebeta

← #27299589

Subgroups — #27299589

Order 13568.

1214 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1214 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×#16810418C212yesmaximal, complemented
2C2×#16810416C212yesmaximal, complemented
3#20274630C212yesmaximal, complemented
4C2×C53×#535C211yesmaximal, characteristic
5C2×#13133132—22no—
6C2×#13133132C2214yes—
7#16810479—22no—
8#16810418—24no—
9#16810416—24no—
10C22×#41×C53C411yescharacteristic, complemented
11C23×D4×C53C2211yescharacteristic
12C53×#535—24no—
13C53×#535—24no—
14C2×#127×C53C411yescharacteristic, complemented
15C2×#127×C53C2211yescharacteristic
16C2×#124×C53C411yescharacteristic, complemented
17C2×#124×C53C2211yescharacteristic
18C2×#121×C53—22no—
19C2×#120×C53—22no—
20C2×#117×C53C411yescharacteristic, complemented
21C2×C22⋊Dic53—24no—
22C2×C22⋊Dic53—44no—
23#13133132—24no—
24#13133132—24no—
25#13133132—44no—
26C25×C53D412yes—
27C2×#41×C53—22no—
28C2×#41×C53—22no—
29C2×#41×C53—42no—
30C2×#41×C53—42no—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53—22no—
34C22×D4×C53—22no—
35C22×D4×C53—22no—
36C22×D4×C53—22no—
37C22×D4×C53—42no—
38C22×D4×C53—42no—
39C22×D4×C53C2×C411yescharacteristic
40C22×D4×C53C2×C411yescharacteristic
41C22×D4×C53D412yes—
42C22×D4×C53C2311yescharacteristic
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45#127×C53—22no—
46#127×C53—22no—
47#127×C53—22no—
48#127×C53D414yes—
49#124×C53—22no—
50#124×C53—22no—
51#124×C53—22no—
52#124×C53D414yes—
53#121×C53—216no—
54#120×C53—216no—
55#117×C53—22no—
56#117×C53—22no—
57C2×C4⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53—22no—
61C2×C42×C53C2×C411yescharacteristic
62C2×C42×C53C2×C411yescharacteristic
63C24×C53—21no—
64C24×C53—24no—
65C24×C53—24no—
66C24×C53—42no—
67C24×C53—84no—
68C24×C53C22⋊C412yes—
69C24×C53C22⋊C411yescharacteristic
70C24×C53C2×D412yes—
71#41×C53—48no—
72#41×C53—48no—
73C2×Q8×C53—21no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53C22⋊C412yescomplemented
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—28no—
80C2×D4×C53—28no—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—48no—
91C2×D4×C53—48no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C2×D4×C53—84no—
97C2×D4×C53—84no—
98C2×D4×C53C22⋊C412yescomplemented
99C2×D4×C53C2×D412yes—
100C2×D4×C53C2×D412yes—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—24no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×Dic53—44no—
114C4⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—28no—
117C22⋊C4×C53—28no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53C22⋊C412yescomplemented
121C42×C53C22⋊C412yescomplemented
122C22⋊Dic53—44no—
123C22⋊Dic53—44no—
124C22⋊Dic53—88no—
125C23×C53—21no—
126C23×C53—22no—
127C23×C53—21no—
128C23×C53—22no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—42no—
134C23×C53—48no—
135C23×C53—44no—
136C23×C53—44no—
137C23×C53—44no—
138C23×C53—48no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—84no—
142C23×C53—88no—
143C23×C53—88no—
144C23×C53#9912yes—
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53#12014yes—
147Q8×C53—42no—
148Q8×C53—42no—
149D4×C53—42no—
150D4×C53—42no—
151D4×C53—48no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—42no—
155D4×C53—42no—
156D4×C53—48no—
157D4×C53—88no—
158D4×C53—88no—
159C2×C4×C53—24no—
160C2×C4×C53—24no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—48no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53#9912yes—
174C2×C4×C53C2×C22⋊C411yescharacteristic
175C2×C4×C53C2×C22⋊C411yescharacteristic
176C2×Dic53—84no—
177C2×Dic53—88no—
178#12696—531nomaximal, Sylow
179C22×C53—22no—
180C22×C53—24no—
181C22×C53—24no—
182C22×C53—22no—
183C22×C53—44no—
184C22×C53—42no—
185C22×C53—42no—
186C22×C53—44no—
187C22×C53—44no—
188C22×C53—44no—
189C22×C53—44no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—816no—
196C22×C53—82no—
197C22×C53#37911yescharacteristic
198C22×C53C2×#9911yescharacteristic
199C22×C53C2×#9911yescharacteristic
200C4×C53—22no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—42no—
205C4×C53—42no—
206Dic53—164no—
207C2×#535D5311yescharacteristic
208C2×#354—532no—
209C2×#353—532no—
210#1781—532no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—44no—
214C2×C53—42no—
215C2×C53—82no—
216C2×C53—82no—
217C2×C53#178111yescharacteristic
218C2×C53#212712yes—
219C22×#41Dic5311yescharacteristic, complemented
220C23×D4C2×D5311yescharacteristic
221#535—24no—
222#535—24no—
223C2×#127Dic5311yescharacteristic, complemented
224C2×#127C2×D5311yescharacteristic
225C2×#124Dic5311yescharacteristic, complemented
226C2×#124C2×D5311yescharacteristic
227C2×#121—22no—
228C2×#120—22no—
229C2×#117Dic5311yescharacteristic, complemented
230C2×#99—534no—
231C2×#99—1062no—
232#379—1062no—
233#354—1064no—
234#353—1064no—
235C53#1269611yescharacteristic, complemented, Sylow
236C25C22⋊D5312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C22×Q8C2×Dic5311yescharacteristic
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Dic5311yescharacteristic
250C22×D4C2×Dic5311yescharacteristic
251C22×D4C22⋊D5312yes—
252C22×D4C22×D5311yescharacteristic
253C23×C4—21no—
254C23×C4C2×Dic5311yescharacteristic
255#127—22no—
256#127—22no—
257#127—22no—
258#127C22⋊D5314yes—
259#124—22no—
260#124—22no—
261#124—22no—
262#124C22⋊D5314yes—
263#121—216no—
264#120—216no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1064no—
272C2×C22⋊C4—2124no—
273C2×C42C2×Dic5311yescharacteristic
274C2×C42C2×Dic5311yescharacteristic
275#99—1064no—
276#99—1064no—
277#99—2124no—
278C24—21no—
279C24—24no—
280C24—24no—
281C24—42no—
282C24—84no—
283C24C22⋊Dic5312yes—
284C24C22⋊Dic5311yescharacteristic
285C24C2×C22⋊D5312yes—
286#41—48no—
287#41—48no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊Dic5312yescomplemented
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4C22⋊Dic5312yescomplemented
314C2×D4C2×C22⋊D5312yes—
315C2×D4C2×C22⋊D5312yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—2124no—
327C22×C4C22⋊Dic5311yescharacteristic
328C22×C4C22×Dic5311yescharacteristic
329C4⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—4248no—
334C22⋊C4—2124no—
335C22⋊C4—2124no—
336C42—21no—
337C42—21no—
338C42C22⋊Dic5312yescomplemented
339C42C22⋊Dic5312yescomplemented
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—48no—
350C23—44no—
351C23—44no—
352C23—48no—
353C23—44no—
354C23—41no—
355C23—41no—
356C23—84no—
357C23—88no—
358C23—88no—
359C23#1313313212yes—
360C23C2×C22⋊Dic5311yescharacteristic
361C23#1313318614yes—
362Q8—42no—
363Q8—42no—
364D4—42no—
365D4—42no—
366D4—48no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—48no—
372D4—88no—
373D4—88no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—4244no—
389C2×C4—4248no—
390C2×C4#1313313212yes—
391C2×C4C2×C22⋊Dic5311yescharacteristic
392C2×C4C2×C22⋊Dic5311yescharacteristic
393C22—22no—
394C22—24no—
395C22—22no—
396C22—24no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—816no—
410C22—82no—
411C22#1681047911yescharacteristic
412C22C2×#1313313211yescharacteristic
413C22C2×#1313313211yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—8484no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2—42no—
425C2—82no—
426C2—82no—
427C2#2027463011yescharacteristic
428C2#2027638412yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC2×C53×#535C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenterC2×#535D53
hypocenterC53#12696
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16810418C2semidirect
2. C2×#16810416C2semidirect
3. #20274630C2semidirect
10. C22×#41×C53C4semidirect
14. C2×#127×C53C4semidirect
16. C2×#124×C53C4semidirect
20. C2×#117×C53C4semidirect
76. C2×Q8×C53C22⋊C4semidirect
98. C2×D4×C53C22⋊C4semidirect
120. C42×C53C22⋊C4semidirect
121. C42×C53C22⋊C4semidirect
219. C22×#41Dic53semidirect
223. C2×#127Dic53semidirect
225. C2×#124Dic53semidirect
229. C2×#117Dic53semidirect
235. C53#12696semidirect
291. C2×Q8C22⋊Dic53semidirect
313. C2×D4C22⋊Dic53semidirect
338. C42C22⋊Dic53semidirect
339. C42C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
C2×#168104162
#202746302
C2×C53×#5352
#1269653