Finite Group Databasebeta

← #27358007

Subgroups — #27358007

Order 13568.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 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×#16811254C211yesmaximal, characteristic, complemented
2#20285472C212yesmaximal
3#20273722C214yesmaximal, complemented
4#20273654C214yesmaximal, complemented
5#20273286C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C53×#2618C211yesmaximal, characteristic
8#16811254—24no—
9#16811254—22no—
10#16811254C2214yes—
11C2×#13133216—22no—
12C2×#13133208—22no—
13C2×#13133196—21no—
14C2×#13133196C2212yes—
15C2×#13133176—21no—
16#16810953C2214yes—
17#16810939—22no—
18#16810850—24no—
19#16810774C2214yes—
20#16810761—22no—
21C2×C22⋊#88322C2212yes—
22#16810359C2218yes—
23#16810349C2214yes—
24C2×#123×C53—21no—
25C2×#121×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic
27C53×#398C2212yes—
28C53×#396—22no—
29C53×#394C2212yes—
30C53×#386—21no—
31C53×#380C2211yescharacteristic
32C22×Dic106—21no—
33C22×Dic106D412yes—
34C2×#11719923—24no—
35C22×#88322—22no—
36#13133216—28no—
37#13133216—28no—
38#13133208—28no—
39#13133208—28no—
40#13133196—24no—
41#13133196—24no—
42#13133196—22no—
43#13133196—22no—
44#13133196D414yes—
45#13133196C2314yes—
46#13133176—24no—
47#13133176—24no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53D412yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×C22⋊Dic53—22no—
55C2×C4×Dic53—22no—
56#13133148—24no—
57#13133140—24no—
58#13133138—24no—
59C22⋊#88322—22no—
60C22⋊#88322C2314yes—
61C22×D4×C53—21no—
62C22×D4×C53C2311yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53D411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66#123×C53—44no—
67#121×C53—22no—
68#121×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D412yes—
74C2×C4⋊C4×C53C2311yescharacteristic
75C2×C4⋊C4×C53C2312yes—
76C2×C22⋊C4×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—22no—
79C2×C22⋊C4×C53—22no—
80#95×C53—22no—
81#95×C53C2312yes—
82C24×C53—21no—
83C24×C53C2×D411yescharacteristic
84C2×D4×C53—22no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—81no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—81no—
98C22×C4×C53—81no—
99C22×C4×C53SD1614yescomplemented
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2411yescharacteristic
106C22×Dic53—22no—
107C22×Dic53—22no—
108C2×Dic106—24no—
109C2×Dic106—22no—
110C2×Dic106—82no—
111C2×Dic106—82no—
112C2×Dic106C2×D414yes—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—44no—
117C4⋊C4×C53—81no—
118C22⋊C4×C53—42no—
119C22⋊C4×C53—42no—
120C22⋊C4×C53—44no—
121C22⋊C4×C53—82no—
122C22⋊Dic53—24no—
123C22⋊Dic53—24no—
124#11719923—84no—
125C4⋊Dic53—24no—
126C4⋊Dic53—24no—
127C4⋊Dic53—24no—
128C4⋊Dic53—22no—
129C4⋊Dic53C2×D414yes—
130#11719919—28no—
131#11719919—24no—
132#11719919—24no—
133C4×Dic53—28no—
134C2×#88322—24no—
135C2×#88322—82no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53#12011yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53C2×SD1612yes—
146C23×C53#13712yes—
147C23×C53C22×D411yescharacteristic
148D4×C53—44no—
149D4×C53—82no—
150D4×C53—82no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—81no—
169C2×C4×C53#12012yes—
170C2×C4×C53C2×SD1614yes—
171C2×C4×C53C22×D412yes—
172C2×Dic53—24no—
173C2×Dic53—24no—
174C2×Dic53—44no—
175C2×Dic53—44no—
176Dic106—44no—
177Dic106—82no—
178Dic106—84no—
179#88322—44no—
180#18294—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53#44812yes—
195C22×C53#44812yes—
196C22×C53C2×#12011yescharacteristic
197C22×C53C22×SD1611yescharacteristic
198C22×C53C2×#13711yescharacteristic
199C4×C53—24no—
200C4×C53—22no—
201C4×C53—22no—
202C4×C53—42no—
203C4×C53—82no—
204C4×C53—81no—
205Dic53—44no—
206Dic53—44no—
207C2×#544—531no—
208#3187—532no—
209#2618D5311yescharacteristic
210#1611—534no—
211#1599—534no—
212#1520—532no—
213C2×#324—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—22no—
217C2×C53—22no—
218C2×C53—161no—
219C2×C53C2×#44811yescharacteristic
220C2×C53#302011yescharacteristic
221C2×C53#302611yescharacteristic
222#544—534no—
223#544—1064no—
224#544—1062no—
225C2×C4⋊Q8—1062no—
226C2×#125—1062no—
227C2×#123—21no—
228C2×C22⋊Q8—532no—
229C2×C22⋊Q8—1061no—
230C2×#121C2×D5311yescharacteristic
231C2×#117—1061no—
232C22×C4⋊C4C2×D5311yescharacteristic
233#483—534no—
234#481—1062no—
235#465—1064no—
236#450—534no—
237#448—1062no—
238C2×C22⋊C8—532no—
239#398C2×D5312yes—
240#396—22no—
241#394C2×D5312yes—
242#386—21no—
243#380C2×D5311yescharacteristic
244#328—538no—
245#324—534no—
246C53#1829411yescharacteristic, complemented, Sylow
247C22×Q8—532no—
248C22×Q8—1061no—
249C22×D4—21no—
250C22×D4C22×D5311yescharacteristic
251C23×C4C22⋊D5311yescharacteristic
252C23×C4C22⋊D5311yescharacteristic
253C23×C4C22×D5311yescharacteristic
254C2×SD16—1064no—
255C22×C8—1062no—
256C4⋊Q8—1068no—
257C4⋊Q8—1068no—
258#125—1068no—
259#125—1068no—
260#123—44no—
261C22⋊Q8—534no—
262C22⋊Q8—534no—
263C22⋊Q8—1064no—
264C22⋊Q8—1062no—
265C22⋊Q8—1062no—
266C22⋊Q8—1064no—
267#121—22no—
268#121—22no—
269#117—1064no—
270#117—1064no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—532no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1062no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4C22⋊D5312yes—
282C2×C4⋊C4C22×D5311yescharacteristic
283C2×C4⋊C4C22×D5312yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—1062no—
289C2×C42—1062no—
290#106—1064no—
291#103—1064no—
292#102—1064no—
293C22⋊C8—534no—
294C22⋊C8—1062no—
295#95—22no—
296#95C22×D5312yes—
297C24—21no—
298C24C2×C22⋊D5311yescharacteristic
299C2×Q8—534no—
300C2×Q8—1064no—
301C2×Q8—1062no—
302C2×Q8—4242no—
303C2×Q8—4242no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4#1171992314yescomplemented
322C22×C4D4×D5312yes—
323C22×C4D4×D5312yes—
324C22×C4C2×C22⋊D5312yes—
325C22×C4C2×C22⋊D5312yes—
326C22×C4C2×C22⋊D5311yescharacteristic
327C22×C4C23×D5311yescharacteristic
328SD16—4244no—
329C2×C8—1064no—
330C2×C8—4242no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—81no—
336C4⋊C4—534no—
337C4⋊C4—1064no—
338C4⋊C4—1064no—
339C4⋊C4—1064no—
340C4⋊C4—1068no—
341C4⋊C4—1062no—
342C4⋊C4—1064no—
343C4⋊C4—1064no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—1064no—
349C22⋊C4—1064no—
350C42—1068no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#1313318511yescharacteristic
359C23#1313318511yescharacteristic
360C23C2×#1171992312yes—
361C23#1313324712yes—
362C23C22×C22⋊D5311yescharacteristic
363Q8—4242no—
364Q8—4244no—
365Q8—2124no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—1064no—
388C2×C4—1064no—
389C2×C4—2124no—
390C2×C4—2124no—
391C2×C4#1313318512yes—
392C2×C4C2×#1171992314yes—
393C2×C4C2×D4×D5312yes—
394C8—2124no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#1681076112yes—
409C22#1681076112yes—
410C22C2×#1313318511yescharacteristic
411C22C22×#1171992311yescharacteristic
412C22C2×#1313324711yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—2124no—
420C4—2124no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#1681076111yescharacteristic
427C2#2028407811yescharacteristic
428C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2618C2
socleC22×C53C2×#120
prime radicalC2×#121×C53C22
hypercenter#2618D53
hypocenterC53#18294
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#16811254C2semidirect
3. #20273722C2semidirect
4. #20273654C2semidirect
5. #20273286C2semidirect
6. C2×#16810349C2semidirect
99. C22×C4×C53SD16semidirect
246. C53#18294semidirect
321. C22×C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202854722
#202737222
#202736542
#202732862
C2×#168103492
C53×#26182
#1829453