Finite Group Databasebeta

← #27287301

Subgroups — #27287301

Order 13568.

404 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 75 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#20280887C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic
4#20272622C211yesmaximal, characteristic
5#20272600C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#1511C211yesmaximal, characteristic
8#16810587C2211yescharacteristic
9#16810565—21no—
10#16810546—21no—
11#16810544—21no—
12#16810535—21no—
13#16810527—21no—
14#16810506—21no—
15#16810483C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810390—21no—
19#16810390—21no—
20#16810360—21no—
21#16810360—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#407C2211yescharacteristic
24C53×#381C2211yescharacteristic
25C53×#328—21no—
26C53×#328—21no—
27C22×Dic106—21no—
28C2×#11719917—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53D411yescharacteristic
36C2×C22⋊Dic53D411yescharacteristic
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C22×Q8×C53Q811yescharacteristic, complemented
50C23×C4×C53C2311yescharacteristic
51C2×M16×C53—21no—
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53Q811yescharacteristic, complemented
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62#95×C53—21no—
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C2×Q8×C53—21no—
67C2×Q8×C53—21no—
68C2×Q8×C53—81no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—81no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×Q811yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—41no—
85C2×Dic106—42no—
86C2×Dic106—81no—
87C2×Dic106—81no—
88M16×C53—81no—
89C2×C8×C53—41no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—81no—
97C22⋊Dic53—22no—
98C22⋊Dic53—42no—
99C22⋊Dic53—41no—
100C22⋊Dic53—41no—
101C22⋊Dic53—41no—
102C4⋊Dic53—42no—
103C4⋊Dic53—81no—
104#11719919—81no—
105#11719919—81no—
106C4×Dic53—81no—
107C4×Dic53—81no—
108#11719917—81no—
109C2×#88322—41no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C23×C53C22⋊Q811yescharacteristic
116C23×C53#12411yescharacteristic
117Q8×C53—81no—
118Q8×C53—81no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C22⋊Q811yescharacteristic
136C2×C4×C53C22⋊Q811yescharacteristic
137C2×C4×C53#12411yescharacteristic
138C2×C4×C53#12411yescharacteristic
139C2×Dic53—22no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154Dic106—84no—
155C8×C53—81no—
156#88322—81no—
157#10383—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#39311yescharacteristic
169C22×C53#45511yescharacteristic
170C22×C53#45611yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179Dic53—42no—
180Dic53—81no—
181Dic53—81no—
182Dic53—81no—
183Dic53—82no—
184#2665—531no—
185#1996—531no—
186#1511D5311yescharacteristic
187#1508—531no—
188#1305—531no—
189#1297—531no—
190#1284—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#202811yescharacteristic
196C2×C53#203111yescharacteristic
197C2×C53#203211yescharacteristic
198C2×C22⋊Q8C2×D5311yescharacteristic
199#407—531no—
200#407C2×D5311yescharacteristic
201#400—1061no—
202#396—1061no—
203#395—1061no—
204#393—1061no—
205#391—1061no—
206#387—1061no—
207#381C2×D5311yescharacteristic
208#380—531no—
209C4×C22⋊C4—531no—
210C4×C22⋊C4—531no—
211#342—1061no—
212#342—1061no—
213#328—21no—
214#328—21no—
215#328—1061no—
216#328—1061no—
217C53#1038311yescharacteristic, complemented, Sylow
218C22×Q8—1061no—
219C22×Q8Dic10611yescharacteristic, complemented
220C23×C4C22×D5311yescharacteristic
221C2×M16—21no—
222C2×M16—1061no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4Dic10611yescharacteristic, complemented
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—531no—
236C2×C22⋊C4—531no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4—1061no—
239C2×C42—1061no—
240C2×C42—1061no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—1061no—
244C22⋊C8—1061no—
245#95—21no—
246#95—531no—
247#95—531no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95C22⋊D5311yescharacteristic
255#95C22⋊D5311yescharacteristic
256C24D4×D5311yescharacteristic
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—81no—
260C2×Q8—4241no—
261C2×Q8—4241no—
262C2×Q8—2122no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4C2×Dic10611yescharacteristic
274C22×C4#1171994511yescharacteristic
275C22×C4D4×D5311yescharacteristic
276C22×C4#1171994711yescharacteristic
277C22×C4#1171994711yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279M16—81no—
280M16—4241no—
281C2×C8—41no—
282C2×C8—2121no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—4241no—
288C4⋊C4—4241no—
289C4⋊C4—4241no—
290C4⋊C4—2122no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—1062no—
295C22⋊C4—2122no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C42—4241no—
300C42—4241no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23#1313318511yescharacteristic
306C23#1313319411yescharacteristic
307C23#1313320311yescharacteristic
308Q8—81no—
309Q8—81no—
310Q8—4244no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—1062no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—2121no—
331C2×C4—2122no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4#1313319411yescharacteristic
343C2×C4#1313319611yescharacteristic
344C2×C4#1313320311yescharacteristic
345C2×C4#1313320411yescharacteristic
346C8—81no—
347C8—4241no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#1681053511yescharacteristic
359C22#1681079911yescharacteristic
360C22#1681080411yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—4241no—
370C4—4241no—
371C4—4241no—
372C4—4242no—
373C4—2122no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2027587411yescharacteristic
379C2#2027588511yescharacteristic
380C2#2027589211yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1511C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1511D53
hypocenterC53#10383
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
49. C22×Q8×C53Q8semidirect
57. C2×C4⋊C4×C53Q8semidirect
217. C53#10383semidirect
219. C22×Q8Dic106semidirect
232. C2×C4⋊C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808872
#202757012
#202732352
#202726222
#202726002
#202725672
C53×#15112
#1038353