Finite Group Databasebeta

← #27287315

Subgroups — #27287315

Order 13568.

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

normal subgroups only

This group has 396 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#20280911C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic
4#20272622C211yesmaximal, characteristic
5#20272600C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#1511C211yesmaximal, characteristic
8#16810587C2211yescharacteristic
9#16810559—21no—
10#16810557—21no—
11#16810555—21no—
12#16810546—21no—
13#16810520—21no—
14#16810510—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—
27C2×#11719917—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53D411yescharacteristic
38C2×C22⋊Dic53D411yescharacteristic
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C22×Q8×C53Q811yescharacteristic
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
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—
85M16×C53—81no—
86C2×C8×C53—41no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—81no—
94C22⋊Dic53—22no—
95C22⋊Dic53—42no—
96C22⋊Dic53—41no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C4⋊Dic53—81no—
100C4⋊Dic53—81no—
101C4⋊Dic53—81no—
102#11719919—42no—
103#11719919—42no—
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—
154C8×C53—81no—
155#88322—81no—
156#10385—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53#39311yescharacteristic
168C22×C53#45511yescharacteristic
169C22×C53#45611yescharacteristic
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53—81no—
178Dic53—42no—
179Dic53—81no—
180Dic53—81no—
181Dic53—81no—
182Dic53—82no—
183#2667—531no—
184#1996—531no—
185#1511D5311yescharacteristic
186#1508—531no—
187#1305—531no—
188#1297—531no—
189#1284—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—41no—
193C2×C53—41no—
194C2×C53#202811yescharacteristic
195C2×C53#203111yescharacteristic
196C2×C53#203211yescharacteristic
197C2×C22⋊Q8C2×D5311yescharacteristic
198#407—531no—
199#407C2×D5311yescharacteristic
200#399—1061no—
201#398—1061no—
202#398—1061no—
203#396—1061no—
204#389—1061no—
205#387—1061no—
206#381C2×D5311yescharacteristic
207#380—531no—
208C4×C22⋊C4—531no—
209C4×C22⋊C4—531no—
210#342—1061no—
211#342—1061no—
212#328—21no—
213#328—21no—
214#328—1061no—
215#328—1061no—
216C53#1038511yescharacteristic, complemented, Sylow
217C22×Q8Dic10611yescharacteristic
218C23×C4C22×D5311yescharacteristic
219C2×M16—21no—
220C2×M16—1061no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4Dic10611yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—531no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C42—1061no—
241C2×C42—1061no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—1061no—
245C22⋊C8—1061no—
246#95—21no—
247#95—531no—
248#95—531no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95C22⋊D5311yescharacteristic
254#95C22⋊D5311yescharacteristic
255C24D4×D5311yescharacteristic
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—81no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—81no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—2121no—
269C22×C4C2×Dic10611yescharacteristic
270C22×C4#1171994511yescharacteristic
271C22×C4D4×D5311yescharacteristic
272C22×C4#1171994711yescharacteristic
273C22×C4#1171994711yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275M16—81no—
276M16—4241no—
277C2×C8—41no—
278C2×C8—2121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—4241no—
284C4⋊C4—4241no—
285C4⋊C4—4241no—
286C4⋊C4—4241no—
287C4⋊C4—4241no—
288C4⋊C4—2122no—
289C4⋊C4—2122no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—81no—
293C22⋊C4—1062no—
294C22⋊C4—2122no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C42—4241no—
299C42—4241no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23#1313318511yescharacteristic
305C23#1313319411yescharacteristic
306C23#1313320311yescharacteristic
307Q8—81no—
308Q8—81no—
309C2×C4—41no—
310C2×C4—41no—
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—81no—
324C2×C4—81no—
325C2×C4—1062no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—2121no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2121no—
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#1313319411yescharacteristic
341C2×C4#1313319611yescharacteristic
342C2×C4#1313320311yescharacteristic
343C2×C4#1313320411yescharacteristic
344C8—81no—
345C8—4241no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22#1681053511yescharacteristic
357C22#1681079911yescharacteristic
358C22#1681080411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—4241no—
368C4—4241no—
369C4—4241no—
370C4—4242no—
371C4—2122no—
372C2—21no—
373C2—21no—
374C2—41no—
375C2—41no—
376C2#2027587411yescharacteristic
377C2#2027588511yescharacteristic
378C2#2027589211yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1511C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1511D53
hypocenterC53#10385
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
216. C53#10385semidirect

Maximal subgroups

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

subgroupindex
#202809112
#202757012
#202732352
#202726222
#202726002
#202725672
C53×#15112
#1038553