Finite Group Databasebeta

← #27293200

Subgroups — #27293200

Order 13568.

594 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277308C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic, complemented
3#20272648C212yesmaximal, complemented
4#20272646C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic, complemented
6C22×#11719945C411yescharacteristic
7C2×#13133176C411yescharacteristic
8C2×C22⋊#88322—21no—
9#16810528C411yescharacteristic
10#16810528C2211yescharacteristic
11#16810524C411yescharacteristic
12#16810524C2211yescharacteristic
13#16810502—21no—
14#16810502—21no—
15#16810480—21no—
16#16810480—21no—
17#16810349—21no—
18#16810349C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C2×#11719945—22no—
24C2×#11719945—41no—
25C2×#11719945—41no—
26C2×#11719945—42no—
27C2×#11719945C814yescomplemented
28C22×Dic106C2×C411yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C22×C4×D53—21no—
32C22×#88322—41no—
33#13133176—42no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C22⋊#88322—22no—
43C22⋊#88322—41no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C22×C8×C53—41no—
52C2×C4⋊C4×C53—41no—
53C2×C4⋊C4×C53—41no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53C2311yescharacteristic
58C22⋊C8×C53—22no—
59C22⋊C8×C53—41no—
60C24×C53C22⋊C411yescharacteristic
61C23×D53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C2×C812yes—
75C22×C4×C53M1612yes—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C2×C22⋊D53—24no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—42no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86#11719945—44no—
87#11719945—44no—
88#11719945—44no—
89#11719945—44no—
90C2×D212—21no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212C2×C812yescomplemented
94C2×C4×D53—24no—
95C2×C4×D53—42no—
96C2×C4×D53—42no—
97C2×Dic106—21no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106C2×C812yescomplemented
101C2×C8×C53—42no—
102C2×C8×C53—82no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C22⋊Dic53—42no—
108#11719921—81no—
109#11719921—81no—
110#11719921—81no—
111#11719921—81no—
112C4⋊Dic53—42no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C2×#88322—42no—
116C2×#88322—82no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53C22⋊C812yes—
120C23×C53#9911yescharacteristic
121C23×C53#9911yescharacteristic
122C23×C53#10012yes—
123C23×C53C2×C22⋊C411yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53C22⋊C812yes—
152C2×C4×C53C22×C811yescharacteristic
153C2×C4×C53C2×M1611yescharacteristic
154C22⋊D53—44no—
155C22⋊D53—44no—
156C2×Dic53—22no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165D212—42no—
166D212—42no—
167C4×D53—48no—
168Dic106—42no—
169Dic106—42no—
170C8×C53—82no—
171#88322—82no—
172#11432—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53#32312yes—
182C22×C53#32312yes—
183C22×C53C2×C22⋊C811yescharacteristic
184C22×C53C2×#9911yescharacteristic
185C22×C53C2×#10011yescharacteristic
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—81no—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—22no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200Dic53—42no—
201Dic53—81no—
202Dic53—81no—
203#2311—531no—
204C2×#324—531no—
205C2×#324D5311yescharacteristic, complemented
206#1319—532no—
207#1318—532no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—22no—
211C2×C53—22no—
212C2×C53C2×#32311yescharacteristic
213C2×C53#212711yescharacteristic
214C2×C53#213311yescharacteristic
215D53—82no—
216C22×#41—531no—
217C2×#117—531no—
218C22×C4⋊C4C2×D5311yescharacteristic
219C2×C22⋊C8—21no—
220C2×C22⋊C8—1061no—
221#391—531no—
222#391—531no—
223#390—531no—
224#390—531no—
225#386—1061no—
226#386—1061no—
227#380—1061no—
228#380—1061no—
229#324—21no—
230#324—532no—
231#324—1061no—
232#324C2×D5312yes—
233C53#1143211yescharacteristic, complemented, Sylow
234C2×#41—534no—
235C2×#41—1062no—
236C2×#41—2121no—
237C2×#41—2121no—
238C2×#41—2122no—
239C22×Q8—531no—
240C22×D4—531no—
241C22×D4—1061no—
242C23×C4—1061no—
243C23×C4C4×D5311yescharacteristic
244C23×C4D21211yescharacteristic
245C23×C4C22⋊D5311yescharacteristic
246C22×C8—41no—
247C22×C8—2121no—
248#117—2122no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4C4×D5311yescharacteristic
253C2×C4⋊C4D21211yescharacteristic
254C2×C4⋊C4C22⋊D5311yescharacteristic
255C2×C4⋊C4C22×D5311yescharacteristic
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C42—531no—
262C2×C42—531no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—1062no—
266C22⋊C8—2121no—
267#95—1061no—
268#95—1061no—
269#95—1061no—
270#95—1061no—
271C24—1061no—
272C24#1171992111yescharacteristic
273#41—2124no—
274#41—2124no—
275#41—2124no—
276#41—2124no—
277C2×Q8—532no—
278C2×Q8—1061no—
279C2×Q8—2121no—
280C2×Q8—2121no—
281C2×D4—532no—
282C2×D4—1064no—
283C2×D4—1061no—
284C2×D4—2121no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2122no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—1061no—
299C22×C4—1064no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—2122no—
303C22×C4—2122no—
304C22×C4C8×D5312yes—
305C22×C4#1171991212yes—
306C22×C4#1171992111yescharacteristic
307C22×C4#1171992111yescharacteristic
308C22×C4#1171992111yescharacteristic
309C22×C4C2×C4×D5311yescharacteristic
310C22×C4C2×D21211yescharacteristic
311C22×C4C2×C22⋊D5311yescharacteristic
312C2×C8—42no—
313C2×C8—82no—
314C2×C8—4242no—
315C2×C8—2122no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—2122no—
321C22⋊C4—4241no—
322C22⋊C4—4241no—
323C22⋊C4—4241no—
324C22⋊C4—4241no—
325C22⋊C4—2122no—
326C42—2121no—
327C42—2121no—
328C23—41no—
329C23—41no—
330C23—1062no—
331C23—2121no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23—2121no—
337C23#1313312912yes—
338C23#1313313111yescharacteristic
339C23#1313313111yescharacteristic
340C23#1313313312yes—
341C23C2×#1171992111yescharacteristic
342Q8—2122no—
343Q8—2122no—
344D4—2122no—
345D4—2122no—
346D4—2124no—
347D4—2124no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1062no—
369C2×C4—2128no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313312912yes—
379C2×C4C2×C8×D5311yescharacteristic
380C2×C4C2×#1171991211yescharacteristic
381C8—82no—
382C8—4242no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22—4241no—
392C22—4241no—
393C22—4241no—
394C22—4241no—
395C22—2122no—
396C22—2122no—
397C22—2122no—
398C22#1681034612yes—
399C22#1681034612yes—
400C22C2×#1313312911yescharacteristic
401C22C2×#1313313111yescharacteristic
402C22C2×#1313313311yescharacteristic
403C4—22no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4241no—
411C4—4241no—
412C4—2122no—
413C2—21no—
414C2—21no—
415C2—22no—
416C2—22no—
417C2—4242no—
418C2C2×#1681034611yescharacteristic
419C2#2027638211yescharacteristic
420C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC22×#11719945C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11432
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16810349C2semidirect
3. #20272648C2semidirect
4. #20272646C2semidirect
5. C2×C53×#324C2semidirect
27. C2×#11719945C8semidirect
93. C2×D212C2×C8semidirect
100. C2×Dic106C2×C8semidirect
205. C2×#324D53semidirect
233. C53#11432semidirect

Maximal subgroups

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

subgroupindex
#202773082
C2×#168103492
#202726482
#202726462
C2×C53×#3242
#1143253