Finite Group Databasebeta

← #27273172

Subgroups — #27273172

Order 13568.

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

normal subgroups only

This group has 394 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#20281746C211yesmaximal, characteristic
2#20276274C211yesmaximal, characteristic
3#20276186C211yesmaximal, characteristic
4#20276125C211yesmaximal, characteristic
5#20272540C211yesmaximal, characteristic
6#20272539C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133216—21no—
9C2×#13133150—21no—
10C2×#13133142—21no—
11#16810556—21no—
12#16810544—21no—
13#16810543—21no—
14#16810529C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810509—21no—
17#16810506—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C22×Dic106D411yescharacteristic
29C22×#88322—21no—
30#13133216—42no—
31#13133216—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×C4×Dic53—21no—
43#13133150—42no—
44#13133142—42no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C2×Dic106—21no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106—42no—
72C2×Dic106—81no—
73C2×Dic106—81no—
74C2×Dic106D812yes—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53SD1612yes—
86C42×C53—41no—
87C4⋊Dic53—21no—
88C4⋊Dic53—42no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—42no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53Q1612yes—
97#11719919—42no—
98#11719919—41no—
99#11719919—41no—
100#11719919—81no—
101C4×Dic53—42no—
102C2×#88322—22no—
103C2×#88322—41no—
104C2×#88322—41no—
105C23×C53#12011yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53C2×D811yescharacteristic
119C2×C4×C53C2×SD1611yescharacteristic
120C2×C4×C53C2×Q1611yescharacteristic
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53#13711yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139Dic106—42no—
140Dic106—81no—
141Dic106—84no—
142C8×C53—81no—
143C8×C53—81no—
144#88322—42no—
145#7794—531nomaximal, Sylow
146C22×C53#44711yescharacteristic
147C22×C53#44811yescharacteristic
148C22×C53#45111yescharacteristic
149C22×C53#45411yescharacteristic
150C22×C53#45511yescharacteristic
151C22×C53#45611yescharacteristic
152C22×C53#45811yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—42no—
160C4×C53—81no—
161Dic53—42no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165Dic53—161no—
166#2734—531no—
167#2099—531no—
168#2083—531no—
169#2072—531no—
170#1272—531no—
171#1271—531no—
172#1271D5311yescharacteristic
173C2×C53#160711yescharacteristic
174C2×C53#161111yescharacteristic
175C2×C53#161711yescharacteristic
176C2×C53#162711yescharacteristic
177C2×C53#164511yescharacteristic
178C2×C53#164911yescharacteristic
179C2×C53#166411yescharacteristic
180C2×C4⋊Q8—1061no—
181C2×#107—1061no—
182C2×#103—1061no—
183#398—1061no—
184#395—1061no—
185#395—1061no—
186#391—531no—
187#389—531no—
188#389C2×D5311yescharacteristic
189#387—1061no—
190#387—1061no—
191#384—531no—
192#340—1061no—
193#340—1061no—
194#340—1061no—
195#340—1061no—
196#336—531no—
197#336C2×D5311yescharacteristic
198#336C2×D5311yescharacteristic
199C53#779411yescharacteristic, complemented, Sylow
200C22×Q8—531no—
201C22×Q8—1061no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—1061no—
205C4⋊Q8—2122no—
206C4⋊Q8—2122no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4D21211yescharacteristic
219C2×C4⋊C4D21211yescharacteristic
220C2×C42—1061no—
221C2×C42C22×D5311yescharacteristic
222#107—2122no—
223#103—2122no—
224#95—21no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C2×Q8—532no—
232C2×Q8—1061no—
233C2×Q8—4241no—
234C2×Q8—4241no—
235C2×Q8—2121no—
236C2×Q8—2121no—
237C2×Q8—2122no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—2121no—
245C22×C4C2×D21211yescharacteristic
246C22×C4D4×D5311yescharacteristic
247C22×C4D4×D5311yescharacteristic
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—1062no—
255C2×C8—2121no—
256C2×C8—2121no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—532no—
262C4⋊C4—1061no—
263C4⋊C4—4241no—
264C4⋊C4—2122no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4#1171991312yes—
276C42—41no—
277C42—2122no—
278C23#1313318711yescharacteristic
279Q8—4241no—
280Q8—4244no—
281Q8—2122no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—4241no—
297C2×C4—4241no—
298C2×C4—4241no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4D8×D5311yescharacteristic
310C2×C4C2×#1171991311yescharacteristic
311C2×C4Q16×D5311yescharacteristic
312C2×C4#1313324411yescharacteristic
313C2×C4#1313324611yescharacteristic
314C2×C4#1313325011yescharacteristic
315C8—81no—
316C8—81no—
317C8—2122no—
318C22#1681075811yescharacteristic
319C22#1681076411yescharacteristic
320C22#1681078011yescharacteristic
321C22#1681079711yescharacteristic
322C22#1681080211yescharacteristic
323C22#1681080811yescharacteristic
324C22#1681081411yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—81no—
333C4—8481no—
334C4—4241no—
335C4—4241no—
336C4—2122no—
337C4—2122no—
338C2#2027370211yescharacteristic
339C2#2027372511yescharacteristic
340C2#2027375811yescharacteristic
341C2#2027382111yescharacteristic
342C2#2027393811yescharacteristic
343C2#2027396511yescharacteristic
344C2#2027405111yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53#120
hypercenter#1271D53
hypocenterC53#7794
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
199. C53#7794semidirect

Maximal subgroups

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

subgroupindex
#202817462
#202762742
#202761862
#202761252
#202725402
#202725392
C53×#12712
#779453