Finite Group Databasebeta

← #27272974

Subgroups — #27272974

Order 13568.

436 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 87 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#20281747C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic
3#20275958C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272539C211yesmaximal, characteristic
7C53×#2024C211yesmaximal, characteristic
8C2×#13133215—21no—
9C2×#13133148—21no—
10C2×#13133141—21no—
11C2×#13133141—21no—
12#16810553—21no—
13#16810544—21no—
14#16810544—21no—
15#16810521C2211yescharacteristic
16#16810521C2211yescharacteristic
17#16810508—21no—
18#16810508—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#103×C53—21no—
24C53×#391C2211yescharacteristic
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30#13133215—42no—
31#13133215—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C4×Dic53—21no—
42#13133148—42no—
43#13133141—42no—
44#13133141—42no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×Q8×C53—21no—
51C22×Q8×C53D411yescharacteristic
52C22×C8×C53—21no—
53C4⋊Q8×C53—42no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#103×C53—42no—
59#95×C53—21no—
60#95×C53—21no—
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53SD1612yes—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C2×Dic106—42no—
82C2×Dic106—81no—
83C2×Dic106—81no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C42×C53—41no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—81no—
98C4⋊Dic53SD1612yes—
99C4⋊Dic53Q1612yes—
100#11719919—42no—
101#11719919—42no—
102#11719919—41no—
103#11719919—41no—
104#11719919—41no—
105#11719919—41no—
106C4×Dic53—42no—
107C2×#88322—22no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C23×C53#12011yescharacteristic
114Q8×C53—42no—
115Q8×C53—42no—
116Q8×C53—42no—
117Q8×C53—81no—
118Q8×C53—81no—
119Q8×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53C2×SD1611yescharacteristic
137C2×C4×C53C2×Q1611yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153Dic106—84no—
154C8×C53—81no—
155#88322—42no—
156#88322—81no—
157#7766—531nomaximal, Sylow
158C22×C53#44811yescharacteristic
159C22×C53#45011yescharacteristic
160C22×C53#45111yescharacteristic
161C22×C53#45411yescharacteristic
162C22×C53#45511yescharacteristic
163C22×C53#45611yescharacteristic
164C22×C53B(4,𝔽2)11yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—82no—
173C4×C53—81no—
174Dic53—42no—
175Dic53—42no—
176Dic53—81no—
177Dic53—81no—
178#2734—531no—
179#2058—531no—
180#2043—531no—
181#2024D5311yescharacteristic
182#1272—531no—
183#1271—531no—
184#1271—531no—
185C2×C53#159911yescharacteristic
186C2×C53#160111yescharacteristic
187C2×C53#160511yescharacteristic
188C2×C53#163711yescharacteristic
189C2×C53#164511yescharacteristic
190C2×C53#165111yescharacteristic
191C2×C53#166411yescharacteristic
192C2×C4⋊Q8—1061no—
193C2×C4⋊Q8C2×D5311yescharacteristic
194C2×#106—1061no—
195C2×#103—21no—
196C2×#103—1061no—
197C2×#103—1061no—
198#398—1061no—
199#395—1061no—
200#395—1061no—
201#391C2×D5311yescharacteristic
202#389—531no—
203#389—531no—
204#387—1061no—
205#387—1061no—
206#340—21no—
207#340—1061no—
208#336—531no—
209#336—531no—
210#336C2×D5311yescharacteristic
211C53#776611yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C22×Q8—1061no—
214C22×Q8C22⋊D5311yescharacteristic
215C22×C8—21no—
216C22×C8—1061no—
217C22×C8—1061no—
218C4⋊Q8—42no—
219C4⋊Q8—2122no—
220C4⋊Q8—2122no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C42—1061no—
234C2×C42C22×D5311yescharacteristic
235#106—2122no—
236#103—42no—
237#103—2122no—
238#103—2122no—
239#95—21no—
240#95—21no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—4241no—
256C2×Q8—4241no—
257C2×Q8—2122no—
258C2×Q8#1171992412yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4D4×D5311yescharacteristic
267C22×C4D4×D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—1062no—
273C2×C8—2121no—
274C2×C8—2121no—
275C2×C8—2121no—
276C2×C8—2121no—
277C2×C8—2121no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—532no—
284C4⋊C4—532no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—4241no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C42—41no—
297C42—2122no—
298C23#1313318511yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304Q8—81no—
305Q8—4244no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4SD16×D5311yescharacteristic
334C2×C4C2×#1171992411yescharacteristic
335C2×C4Q16×D5311yescharacteristic
336C2×C4#1313324211yescharacteristic
337C2×C4#1313324611yescharacteristic
338C2×C4#1313324911yescharacteristic
339C8—81no—
340C8—4241no—
341C8—2122no—
342C22#1681076011yescharacteristic
343C22#1681077311yescharacteristic
344C22#1681077711yescharacteristic
345C22#1681079411yescharacteristic
346C22#1681080011yescharacteristic
347C22#1681080411yescharacteristic
348C22#1681080911yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—82no—
358C4—4241no—
359C4—4241no—
360C4—2122no—
361C4—2122no—
362C2#2027365311yescharacteristic
363C2#2027366611yescharacteristic
364C2#2027368711yescharacteristic
365C2#2027387811yescharacteristic
366C2#2027393311yescharacteristic
367C2#2027397711yescharacteristic
368C2#2027404811yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2024C2
socle, prime radicalC23×C53#120
hypercenter#2024D53
hypocenterC53#7766
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
211. C53#7766semidirect

Maximal subgroups

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

subgroupindex
#202817472
#202760452
#202759582
#202725412
#202725392
C53×#20242
#776653