Finite Group Databasebeta

← #27297428

Subgroups — #27297428

Order 13568.

328 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276932C211yesmaximal, characteristic, complemented
2#20276927C211yesmaximal, characteristic, complemented
3#20276231C211yesmaximal, characteristic, complemented
4#20272709C211yesmaximal, characteristic, complemented
5#20272706C211yesmaximal, characteristic, complemented
6#20272651C211yesmaximal, characteristic, complemented
7C53×#3197C211yesmaximal, characteristic
8#16810672C2211yescharacteristic
9#16810671—21no—
10#16810587—21no—
11#16810445—21no—
12#16810445—21no—
13#16810439—21no—
14#16810436—21no—
15#16810405C2211yescharacteristic
16#16810405C2211yescharacteristic
17#16810381C2211yescharacteristic
18C53×#525—21no—
19C53×#485—21no—
20C53×#482—21no—
21C53×#470C2211yescharacteristic
22C53×#469C2211yescharacteristic
23C53×#468—21no—
24C53×#452—21no—
25C53×#449—21no—
26C53×#407C2211yescharacteristic
27C2×#11719917—21no—
28C2×#11719917—21no—
29#13133157—21no—
30#13133157—21no—
31#13133153—21no—
32#13133153—21no—
33#13133152—42no—
34C22⋊#88322—42no—
35C2×#41×C53—21no—
36C23×C4×C53D411yescharacteristic
37C2×Q16×C53D411yescharacteristic
38C2×SD16×C53—21no—
39C2×D8×C53D411yescharacteristic
40C2×M16×C53D411yescharacteristic
41C2×M16×C53C2311yescharacteristic
42#123×C53—41no—
43C22⋊Q8×C53—21no—
44C22⋊Q8×C53—21no—
45#121×C53—21no—
46#121×C53—21no—
47#120×C53—41no—
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50#117×C53—21no—
51#107×C53—21no—
52#106×C53D411yescharacteristic
53#106×C53D411yescharacteristic
54#103×C53—21no—
55#103×C53—21no—
56#102×C53—21no—
57#102×C53—21no—
58C22⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60C24×C53SD1611yescharacteristic
61#41×C53—81no—
62#41×C53—81no—
63C2×Q8×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53SD1611yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75Q16×C53—41no—
76SD16×C53—81no—
77D8×C53—41no—
78M16×C53—41no—
79M16×C53—41no—
80C2×C8×C53—21no—
81C2×C8×C53C2×D411yescharacteristic
82C2×C8×C53C2×D411yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C42×C53—41no—
93#11719917—41no—
94#11719917—41no—
95#11719917—41no—
96#11719917—41no—
97C2×#88322—22no—
98C2×#88322—41no—
99#11719908—41no—
100#11719908—41no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—81no—
105C23×C53C2×SD1611yescharacteristic
106Q8×C53—41no—
107Q8×C53—41no—
108D4×C53—41no—
109D4×C53—41no—
110D4×C53—82no—
111D4×C53—82no—
112D4×C53—82no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—81no—
123C2×C4×C53—82no—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
126C8×C53—21no—
127C8×C53—21no—
128C8×C53—41no—
129#88322—42no—
130#88322—81no—
131#12300—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53#45011yescharacteristic
143C4×C53—21no—
144C4×C53—41no—
145C4×C53—81no—
146C4×C53—81no—
147C4×C53—81no—
148C4×C53#45311yescharacteristic
149C4×C53#45811yescharacteristic
150#3197D5311yescharacteristic
151#2239—531no—
152#2238—531no—
153#2091—531no—
154#1342—531no—
155#1341—531no—
156#1320—531no—
157C2×C53—21no—
158C2×C53—41no—
159C2×C53—81no—
160C2×C53—161no—
161C2×C53#161311yescharacteristic
162#525—21no—
163#485—21no—
164#482—21no—
165#470C2×D5311yescharacteristic
166#469C2×D5311yescharacteristic
167#468—21no—
168#452—21no—
169#449—21no—
170#430—531no—
171#430—1061no—
172#407—1061no—
173#407C2×D5311yescharacteristic
174#365—1061no—
175#365—1061no—
176#362—1061no—
177#361—1061no—
178#349—531no—
179#349—531no—
180#338—531no—
181C53#1230011yescharacteristic, complemented, Sylow
182C2×#41—21no—
183C23×C4C22⋊D5311yescharacteristic
184C2×Q16C22⋊D5311yescharacteristic
185C2×SD16—21no—
186C2×D8C22⋊D5311yescharacteristic
187C2×M16—1061no—
188C2×M16—1061no—
189C2×M16C22⋊D5311yescharacteristic
190C2×M16C22×D5311yescharacteristic
191#123—41no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—21no—
195#121—21no—
196#120—41no—
197C4×D4—41no—
198C4×D4—41no—
199#117—21no—
200M32—1061no—
201M32—1061no—
202#108—1061no—
203#108—1061no—
204#108—2122no—
205#107—21no—
206#106C22⋊D5311yescharacteristic
207#106C22⋊D5311yescharacteristic
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—2122no—
215C24#1171992311yescharacteristic
216#41—81no—
217#41—81no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4#1171992311yescharacteristic
229C22×C4C2×C22⋊D5311yescharacteristic
230Q16—41no—
231SD16—81no—
232D8—41no—
233M16—41no—
234M16—41no—
235M16—2121no—
236M16—2121no—
237M16—2121no—
238M16—2121no—
239C2×C8—21no—
240C2×C8—1062no—
241C2×C8—2121no—
242C2×C8C2×C22⋊D5311yescharacteristic
243C2×C8C2×C22⋊D5311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C42—41no—
254C16—2121no—
255C16—2121no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—81no—
260C23C2×#1171992311yescharacteristic
261Q8—41no—
262Q8—41no—
263D4—41no—
264D4—41no—
265D4—82no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4#1313318611yescharacteristic
280C2×C4#1313324311yescharacteristic
281C8—21no—
282C8—21no—
283C8—41no—
284C8—4241no—
285C8—2122no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#1681077411yescharacteristic
297C4—21no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#1681079111yescharacteristic
303C4#1681081311yescharacteristic
304C2—21no—
305C2—41no—
306C2—81no—
307C2—161no—
308C2#2027373511yescharacteristic
subgroupG/H
centerC2#20273735
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3197C2
socleC2×C53#1613
hypercenter#3197D53
hypocenterC53#12300
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3197

Complements

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

NH ≅ G/Ntype
1. #20276932C2semidirect
2. #20276927C2semidirect
3. #20276231C2semidirect
4. #20272709C2semidirect
5. #20272706C2semidirect
6. #20272651C2semidirect
181. C53#12300semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202769272
#202762312
#202727092
#202727062
#202726512
C53×#31972
#1230053