Finite Group Databasebeta

← #27273092

Subgroups — #27273092

Order 13568.

424 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 81 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#20282670C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic
3#20276040C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2016C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133148—21no—
10#16810565—21no—
11#16810563—21no—
12#16810542—21no—
13#16810521C2211yescharacteristic
14#16810521C2211yescharacteristic
15#16810520—21no—
16#16810506—21no—
17#16810504—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C22×Q16×C53—21no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C53×#391C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53D411yescharacteristic
30C2×C4⋊Dic53D411yescharacteristic
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C4×Dic53—21no—
38#13133150—42no—
39#13133148—42no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×Q8×C53—21no—
48C22×Q8×C53D411yescharacteristic
49C2×Q16×C53—42no—
50C2×Q16×C53—42no—
51C2×Q16×C53—42no—
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—22no—
62C2×Q8×C53—21no—
63C2×Q8×C53—21no—
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×C53—41no—
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—
81Q16×C53—48no—
82C2×C8×C53—22no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C42×C53—41no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53Q1612yescomplemented
97#11719919—41no—
98#11719919—41no—
99#11719919—41no—
100#11719919—41no—
101#11719919—81no—
102#11719919—81no—
103C4×Dic53—81no—
104C2×#88322—22no—
105C2×#88322—22no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C23×C53#12011yescharacteristic
111Q8×C53—42no—
112Q8×C53—42no—
113Q8×C53—42no—
114Q8×C53—81no—
115Q8×C53—81no—
116Q8×C53—81no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
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—42no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C2×Q1611yescharacteristic
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C2×Dic53—22no—
139C2×Dic53—41no—
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—41no—
149C2×Dic53—41no—
150C8×C53—42no—
151#88322—42no—
152#88322—42no—
153#7783—531nomaximal, Sylow
154C22×C53#44911yescharacteristic
155C22×C53#45111yescharacteristic
156C22×C53#45211yescharacteristic
157C22×C53#45311yescharacteristic
158C22×C53#45611yescharacteristic
159C22×C53#45611yescharacteristic
160C22×C53B(4,𝔽2)11yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—81no—
168C4×C53—82no—
169C4×C53—81no—
170Dic53—42no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174#2805—531no—
175#2058—531no—
176#2057—531no—
177#2016D5311yescharacteristic
178#1272—531no—
179#1271—531no—
180#1271—531no—
181C2×C53#160011yescharacteristic
182C2×C53#160311yescharacteristic
183C2×C53#160511yescharacteristic
184C2×C53#163211yescharacteristic
185C2×C53#164811yescharacteristic
186C2×C53#165211yescharacteristic
187C2×C53#166011yescharacteristic
188C22×Q16—21no—
189C2×C4⋊Q8C2×D5311yescharacteristic
190C2×#107—1061no—
191C2×#106—1061no—
192C2×#103—21no—
193#400—1061no—
194#400—1061no—
195#395—1061no—
196#391C2×D5311yescharacteristic
197#389—531no—
198#389—531no—
199#389—1061no—
200#387—1061no—
201#387—1061no—
202#340—1061no—
203#340—1061no—
204#336—531no—
205#336—531no—
206#336C2×D5311yescharacteristic
207C53#778311yescharacteristic, complemented, Sylow
208C22×Q8—21no—
209C22×Q8C22⋊D5311yescharacteristic
210C2×Q16—42no—
211C2×Q16—42no—
212C2×Q16—42no—
213C22×C8—21no—
214C22×C8—1061no—
215C22×C8—1061no—
216C4⋊Q8—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C42—1061no—
229C2×C42C22×D5311yescharacteristic
230#107—2122no—
231#106—2122no—
232#103—42no—
233#95—21no—
234#95—21no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C2×Q8—22no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4D4×D5311yescharacteristic
260C22×C4D4×D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262Q16—48no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—1062no—
267C2×C8—1062no—
268C2×C8—2121no—
269C2×C8—2121no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—532no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—4241no—
282C4⋊C4—4241no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C42—41no—
290C42—4241no—
291C23#1313318511yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—42no—
295Q8—81no—
296Q8—81no—
297Q8—81no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—1062no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4Q16×D5311yescharacteristic
326C2×C4#1313323711yescharacteristic
327C2×C4#1313324011yescharacteristic
328C2×C4#1313324311yescharacteristic
329C2×C4#1313324911yescharacteristic
330C2×C4#1313324911yescharacteristic
331C8—42no—
332C8—2122no—
333C8—2122no—
334C22#1681076611yescharacteristic
335C22#1681077711yescharacteristic
336C22#1681078411yescharacteristic
337C22#1681079011yescharacteristic
338C22#1681080411yescharacteristic
339C22#1681080411yescharacteristic
340C22#1681080911yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—82no—
350C4—4241no—
351C4—4241no—
352C4—4241no—
353C4—2122no—
354C2#2027365911yescharacteristic
355C2#2027367711yescharacteristic
356C2#2027368711yescharacteristic
357C2#2027384611yescharacteristic
358C2#2027395511yescharacteristic
359C2#2027398311yescharacteristic
360C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2016C2
socle, prime radicalC23×C53#120
hypercenter#2016D53
hypocenterC53#7783
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
96. C4⋊Dic53Q16semidirect
207. C53#7783semidirect

Maximal subgroups

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

subgroupindex
#202826702
#202760452
#202760402
#202725412
#202725392
#202725372
C53×#20162
#778353