Finite Group Databasebeta

← #27301646

Subgroups — #27301646

Order 13568.

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

normal subgroups only

This group has 694 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#20278339C211yesmaximal, characteristic
2C2×#16810421C212yesmaximal, complemented
3#20274643C211yesmaximal, characteristic
4#20272841C212yesmaximal, complemented
5C2×C53×#413C211yesmaximal, characteristic
6C2×#13133215C2211yescharacteristic
7C2×#13133204C411yescharacteristic
8C2×#13133196C411yescharacteristic
9C2×#13133136C2212yes—
10C2×#13133134—21no—
11#16810587—21no—
12#16810537—21no—
13#16810537—21no—
14#16810530C2211yescharacteristic
15#16810526C411yescharacteristic
16#16810496—21no—
17#16810496—21no—
18C4×C22⋊Dic53C411yescharacteristic
19#16810421—24no—
20C22×#41×C53C2211yescharacteristic
21C22×M16×C53—21no—
22C53×#413—24no—
23C2×#101×C53C2212yes—
24C2×#100×C53—21no—
25C22×Dic106C2×C411yescharacteristic
26C2×#11719917—22no—
27#13133215—22no—
28#13133215—22no—
29#13133204—42no—
30#13133196—41no—
31#13133196—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44#13133136—22no—
45#13133136—22no—
46#13133134—42no—
47C22⋊#88322—41no—
48C22⋊#88322—41no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C2×#41×C53—22no—
53C2×#41×C53—22no—
54C2×#41×C53—41no—
55C2×#41×C53—41no—
56C2×#41×C53—42no—
57C22×Q8×C53C2311yescharacteristic
58C22×D4×C53C2×C411yescharacteristic
59C22×D4×C53D411yescharacteristic
60C22×D4×C53D411yescharacteristic
61C23×C4×C53C2×C411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C2×M16×C53—24no—
65C2×M16×C53—22no—
66C2×M16×C53—42no—
67C2×M16×C53—41no—
68C22×C8×C53—41no—
69#101×C53—22no—
70#101×C53D414yes—
71#100×C53—24no—
72C24×C53C22⋊C411yescharacteristic
73C24×C53C22⋊C411yescharacteristic
74C24×C53C22⋊C411yescharacteristic
75#41×C53—44no—
76#41×C53—44no—
77#41×C53—44no—
78#41×C53—44no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53C2×D412yes—
82C2×Q8×C53#4112yes—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—22no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C2×Dic106—22no—
115C2×Dic106—41no—
116C2×Dic106—41no—
117M16×C53—24no—
118M16×C53—44no—
119M16×C53—42no—
120C2×C8×C53—44no—
121C2×C8×C53—42no—
122C22⋊Dic53—42no—
123C22⋊Dic53—42no—
124C22⋊Dic53—42no—
125C22⋊Dic53—42no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128C4⋊Dic53—41no—
129#11719919—41no—
130#11719919—41no—
131#11719919—41no—
132#11719919—42no—
133#11719919—41no—
134C4×Dic53—22no—
135C4×Dic53—41no—
136#11719917—42no—
137#11719917—42no—
138C2×#88322—42no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53#9912yes—
154C23×C53C2×C22⋊C411yescharacteristic
155Q8×C53—42no—
156Q8×C53—42no—
157D4×C53—44no—
158D4×C53—44no—
159D4×C53—42no—
160D4×C53—42no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53#9911yescharacteristic
178C2×C4×C53#9911yescharacteristic
179C2×C4×C53C4×D412yes—
180C2×C4×C53#12011yescharacteristic
181C2×C4×C53#12112yes—
182C2×C4×C53#12311yescharacteristic
183C2×Dic53—22no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195Dic106—82no—
196C8×C53—44no—
197#88322—82no—
198#13095—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—22no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53#38611yescharacteristic
217C22×C53C2×#9911yescharacteristic
218C22×C53C2×#9911yescharacteristic
219C4×C53—22no—
220C4×C53—21no—
221C4×C53—22no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225Dic53—42no—
226Dic53—81no—
227Dic53—81no—
228Dic53—81no—
229C2×#413D5311yescharacteristic
230#2432—531no—
231C2×#356—532no—
232#1785—531no—
233#1394—532no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—42no—
237C2×C53—81no—
238C2×C53—81no—
239C2×C53#189811yescharacteristic
240C2×C53#213211yescharacteristic
241C2×C53#213511yescharacteristic
242C22×#41C2×D5311yescharacteristic
243C22×M16—21no—
244C2×C4⋊Q8—531no—
245C2×#124—531no—
246C2×C22⋊Q8—531no—
247#413—24no—
248C2×#101—532no—
249C2×#101C2×D5312yes—
250C2×#100—21no—
251C2×#100—1061no—
252#407—1061no—
253#394—1061no—
254#394—1061no—
255#391—531no—
256#390—531no—
257#385—1061no—
258#385—1061no—
259C4×C22⋊C4—531no—
260#356—1064no—
261C53#1309511yescharacteristic, complemented, Sylow
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—41no—
265C2×#41—41no—
266C2×#41—42no—
267C22×Q8—531no—
268C22×Q8C22×D5311yescharacteristic
269C22×D4C4×D5311yescharacteristic
270C22×D4D21211yescharacteristic
271C22×D4C22⋊D5311yescharacteristic
272C23×C4C4×D5311yescharacteristic
273C23×C4D21211yescharacteristic
274C23×C4C22⋊D5311yescharacteristic
275C2×M16—24no—
276C2×M16—22no—
277C2×M16—42no—
278C2×M16—41no—
279C2×M16—1062no—
280C22×C8—41no—
281C4⋊Q8—1062no—
282C4⋊Q8—1062no—
283#124—2122no—
284C22⋊Q8—2121no—
285C22⋊Q8—2121no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295C2×C22⋊C4—1061no—
296C2×C42—531no—
297C2×C42—531no—
298#101—22no—
299#101—1062no—
300#101—1062no—
301#101C22⋊D5314yes—
302#100—24no—
303#100—2122no—
304C22⋊C8—2121no—
305C22⋊C8—2121no—
306#95—1061no—
307#95—1061no—
308#95—1061no—
309C24#1171992111yescharacteristic
310C24#1171992111yescharacteristic
311C24#1171992111yescharacteristic
312#41—44no—
313#41—44no—
314#41—44no—
315#41—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—1062no—
319C2×Q8—2121no—
320C2×Q8—2121no—
321C2×Q8#1171994512yes—
322C2×Q8C2×C22⋊D5312yes—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—1061no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4#1171992111yescharacteristic
351C22×C4C2×C4×D5311yescharacteristic
352C22×C4C2×D21211yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354M16—24no—
355M16—44no—
356M16—42no—
357M16—2122no—
358M16—2122no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—2122no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C4⋊C4—2121no—
365C4⋊C4—2121no—
366C4⋊C4—2122no—
367C4⋊C4—2121no—
368C22⋊C4—2122no—
369C22⋊C4—2122no—
370C22⋊C4—2122no—
371C22⋊C4—2122no—
372C22⋊C4—2122no—
373C22⋊C4—2122no—
374C42—1062no—
375C42—2121no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23#1313313112yes—
391C23C2×#1171992111yescharacteristic
392Q8—42no—
393Q8—42no—
394Q8—4242no—
395D4—44no—
396D4—44no—
397D4—42no—
398D4—42no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—1062no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4#1313313111yescharacteristic
428C2×C4#1313313111yescharacteristic
429C2×C4C4×C22⋊D5312yes—
430C2×C4#1313318611yescharacteristic
431C2×C4#1313319112yes—
432C2×C4#1313320011yescharacteristic
433C8—44no—
434C8—4242no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#1681050211yescharacteristic
453C22C2×#1313313111yescharacteristic
454C22C2×#1313313111yescharacteristic
455C4—21no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—4241no—
462C4—4241no—
463C4—4241no—
464C4—2122no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—81no—
469C2—81no—
470C2#2027516811yescharacteristic
471C2#2027639811yescharacteristic
472C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#413C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×Q8×C53C23
hypercenterC2×#413D53
hypocenterC53#13095
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16810421C2semidirect
4. #20272841C2semidirect
261. C53#13095semidirect

Maximal subgroups

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

subgroupindex
#202783392
C2×#168104212
#202746432
#202728412
C2×C53×#4132
#1309553