Finite Group Databasebeta

← #27281943

Subgroups — #27281943

Order 13568.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#20283795C211yesmaximal, characteristic, complemented
2#20279624C211yesmaximal, characteristic
3#20274708C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic
5#20272575C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#1486C211yesmaximal, characteristic
8C2×#13133196—21no—
9C2×C4×Dic106C2211yescharacteristic
10C8×Dic106—22no—
11#16810747—22no—
12#16810676C2211yescharacteristic
13#16810673—22no—
14#16810647C2211yescharacteristic
15C2×#13133135—21no—
16#16810554C412yes—
17#16810537C412yes—
18#16810511—22no—
19#16810489C2211yescharacteristic
20#16810483—21no—
21#16810391—21no—
22D4⋊#88322—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C53×D4⋊C8—22no—
27C22×Dic106D411yescharacteristic
28C2×#11719917—21no—
29#13133196—42no—
30#13133196—42no—
31C4×Dic106—21no—
32C4×Dic106—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—22no—
38C2×#11719919C2×C412yes—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—22no—
41C2×C4×Dic53D412yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133146—24no—
46#13133135—42no—
47C22⋊#88322—21no—
48C4×#88322D412yes—
49C8×Dic53—24no—
50#13133124—21no—
51#13133124—22no—
52#13133124C2×C412yes—
53C22×D4×C53C2×C411yescharacteristic
54C23×C4×C53—21no—
55C2×M16×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67C4×C8×C53D412yes—
68C24×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—42no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—22no—
83C22×Dic53C22⋊C412yes—
84C22×Dic53C22⋊C412yes—
85C2×Dic106—21no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—21no—
89C2×Dic106D812yes—
90C2×Dic106SD1612yes—
91M16×C53—41no—
92M16×C53—41no—
93C2×C8×C53—22no—
94C2×C8×C53—41no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102C42×C53#4111yescharacteristic
103C22⋊Dic53—44no—
104C22⋊Dic53—44no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109#11719919—22no—
110#11719919—22no—
111#11719919—44no—
112C4×Dic53—22no—
113C4×Dic53—22no—
114#11719917—41no—
115#11719917—41no—
116C2×#88322—22no—
117C2×#88322—41no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53C2×C22⋊C411yescharacteristic
126D4×C53—22no—
127D4×C53—22no—
128D4×C53—41no—
129D4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53#10214yes—
157Dic106—22no—
158Dic106—22no—
159Dic106—22no—
160Dic106—22no—
161C8×C53—42no—
162C8×C53—81no—
163#88322—42no—
164#88322—81no—
165#9486—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#38611yescharacteristic
176C22×C53C2×#10211yescharacteristic
177C22×C53#41611yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—81no—
183C4×C53#46511yescharacteristic
184C4×C53#46511yescharacteristic
185C4×C53#46611yescharacteristic
186C4×C53#46611yescharacteristic
187Dic53—24no—
188Dic53—42no—
189Dic53—82no—
190#2982—531no—
191#2536—531no—
192#1801—531no—
193#1486—531no—
194#1486D5311yescharacteristic
195#1286—531no—
196#1286—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—82no—
200C2×C53#189111yescharacteristic
201C2×C53#189311yescharacteristic
202C2×C53#191211yescharacteristic
203C2×C22⋊Q8—1061no—
204C2×C4×Q8—531no—
205C2×C4×D4C2×D5311yescharacteristic
206C8×Q8—1062no—
207C8×Q8—1062no—
208#431—531no—
209#431—1062no—
210#431C2×D5311yescharacteristic
211#423—531no—
212#423C2×D5311yescharacteristic
213C2×#101—1061no—
214#398—532no—
215#394—532no—
216#388—1062no—
217#382—531no—
218#380—1061no—
219#343—1061no—
220D4⋊C8—22no—
221D4⋊C8—1062no—
222C53#948611yescharacteristic, complemented, Sylow
223C22×Q8—531no—
224C22×D4C4×D5311yescharacteristic
225C23×C4—21no—
226C2×M16—21no—
227C2×M16—1061no—
228C22⋊Q8—2122no—
229C22⋊Q8—2122no—
230C4×Q8—1061no—
231C4×Q8—1061no—
232C4×Q8—1061no—
233C4×Q8—1061no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—532no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1062no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4C4×D5311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C42—532no—
247C2×C42C22×D5311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1064no—
252C4⋊C8—1061no—
253C4⋊C8—1061no—
254C4⋊C8—1061no—
255#101—2122no—
256C22⋊C8—21no—
257C22⋊C8—1061no—
258C4×C8—532no—
259C4×C8—1064no—
260C4×C8C22⋊D5312yes—
261#95—532no—
262#95—1062no—
263#95—1061no—
264C24—21no—
265C2×Q8—532no—
266C2×Q8—532no—
267C2×Q8—1061no—
268C2×Q8—1061no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—1062no—
284C22×C4C2×C4×D5311yescharacteristic
285C22×C4D4×D5311yescharacteristic
286C22×C4D4×D5311yescharacteristic
287M16—41no—
288M16—41no—
289M16—2121no—
290M16—2121no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—1062no—
294C2×C8—2121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—1062no—
298C4⋊C4—1062no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—2124no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—2124no—
307C22⋊C4—2124no—
308C42—1062no—
309C42—1062no—
310C42#1171994511yescharacteristic
311C42D4×D5311yescharacteristic
312C42#1171994711yescharacteristic
313C42C2×C22⋊D5311yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22⋊C4×D5311yescharacteristic
322Q8—1062no—
323Q8—1062no—
324Q8—1062no—
325Q8—1062no—
326D4—22no—
327D4—22no—
328D4—41no—
329D4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—534no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4#1313317711yescharacteristic
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318511yescharacteristic
350C2×C4#1313318811yescharacteristic
351C2×C4#1313319011yescharacteristic
352C2×C4#1313319911yescharacteristic
353C2×C4D8×D5311yescharacteristic
354C2×C4SD16×D5311yescharacteristic
355C2×C4#1313323611yescharacteristic
356C2×C4#1313323811yescharacteristic
357C8—42no—
358C8—81no—
359C8—4241no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1681049911yescharacteristic
371C22#102×D5311yescharacteristic
372C22#1681061611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—1064no—
379C4—4242no—
380C4—2122no—
381C4#1681084911yescharacteristic
382C4#1681085111yescharacteristic
383C4#1681085611yescharacteristic
384C4#1681085811yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#2027513011yescharacteristic
389C2#2027514011yescharacteristic
390C2#2027524911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1486C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1486D53
hypocenterC53#9486
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #20283795C2semidirect
3. #20274708C2semidirect
5. #20272575C2semidirect
6. #20272574C2semidirect
222. C53#9486semidirect

Maximal subgroups

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

subgroupindex
#202837952
#202796242
#202747082
#202731402
#202725752
#202725742
C53×#14862
#948653