Finite Group Databasebeta

← #27297462

Subgroups — #27297462

Order 13568.

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

normal subgroups only

This group has 274 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
3#20276295C211yesmaximal, characteristic, complemented
4#20272709C211yesmaximal, characteristic, complemented
5#20272706C211yesmaximal, characteristic
6#20272668C211yesmaximal, characteristic, complemented
7C53×#3263C211yesmaximal, characteristic
8#16810672C2211yescharacteristic
9#16810445—21no—
10#16810445—21no—
11#16810439—21no—
12#16810436—21no—
13#16810405C2211yescharacteristic
14#16810405C2211yescharacteristic
15#16810393—21no—
16#16810388—21no—
17#16810381C2211yescharacteristic
18C53×#542—21no—
19C53×#476—21no—
20C53×#475—21no—
21C53×#470C2211yescharacteristic
22C53×#469C2211yescharacteristic
23C53×#465—21no—
24C53×#464—21no—
25C53×#463—21no—
26C53×#424C2211yescharacteristic
27C2×#11719917—21no—
28C2×#11719917—21no—
29#13133157—21no—
30#13133157—21no—
31#13133153—21no—
32#13133153—21no—
33C2×Q16×C53D411yescharacteristic
34C2×SD16×C53—21no—
35C2×D8×C53D411yescharacteristic
36C2×M16×C53C2311yescharacteristic
37C22×C8×C53D411yescharacteristic
38C4⋊Q8×C53—21no—
39#126×C53—41no—
40#124×C53—21no—
41#124×C53—41no—
42#123×C53—41no—
43C22⋊Q8×C53—21no—
44C22⋊Q8×C53—41no—
45#121×C53—21no—
46C4×Q8×C53—21no—
47C4×D4×C53—21no—
48#117×C53D411yescharacteristic
49#106×C53—21no—
50#106×C53D411yescharacteristic
51#106×C53D411yescharacteristic
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54#103×C53—21no—
55#103×C53—21no—
56#102×C53—21no—
57#102×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64Q16×C53—41no—
65SD16×C53—81no—
66D8×C53—41no—
67M16×C53—41no—
68C2×C8×C53—21no—
69C2×C8×C53—41no—
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53C2×D411yescharacteristic
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53SD1611yescharacteristic
84C22⋊C4×C53SD1611yescharacteristic
85C42×C53—21no—
86C42×C53—41no—
87#11719917—41no—
88#11719917—41no—
89#11719917—41no—
90#11719917—41no—
91C2×#88322—41no—
92C2×#88322—41no—
93#11719908—41no—
94#11719908—41no—
95C23×C53—41no—
96C23×C53C2×SD1611yescharacteristic
97Q8×C53—41no—
98Q8×C53—41no—
99Q8×C53—161no—
100D4×C53—41no—
101D4×C53—41no—
102D4×C53—161no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—161no—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
113C8×C53—21no—
114C8×C53—21no—
115C8×C53—41no—
116#88322—81no—
117#88322—81no—
118#12306—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—41no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—81no—
124C22×C53#45011yescharacteristic
125C4×C53—21no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53—81no—
129C4×C53—81no—
130C4×C53—81no—
131C4×C53#45411yescharacteristic
132C4×C53#45811yescharacteristic
133#3263D5311yescharacteristic
134#2239—531no—
135#2238—531no—
136#2103—531no—
137#1342—531no—
138#1341—531no—
139#1328—531no—
140C2×C53—21no—
141C2×C53—41no—
142C2×C53—161no—
143C2×C53#161611yescharacteristic
144#542—21no—
145#476—21no—
146#475—21no—
147#470C2×D5311yescharacteristic
148#469C2×D5311yescharacteristic
149#465—21no—
150#464—21no—
151#463—21no—
152#430—531no—
153#424C2×D5311yescharacteristic
154#365—1061no—
155#365—1061no—
156#362—1061no—
157#361—1061no—
158#349—531no—
159#349—531no—
160#344—1061no—
161#341—1061no—
162#338—531no—
163C53#1230611yescharacteristic, complemented, Sylow
164C2×Q16C22⋊D5311yescharacteristic
165C2×SD16—21no—
166C2×D8C22⋊D5311yescharacteristic
167C2×M16—1061no—
168C2×M16—1061no—
169C2×M16C22×D5311yescharacteristic
170C22×C8C22⋊D5311yescharacteristic
171C4⋊Q8—21no—
172#126—41no—
173#124—21no—
174#124—41no—
175#123—41no—
176C22⋊Q8—21no—
177C22⋊Q8—41no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181#117C22⋊D5311yescharacteristic
182M32—1061no—
183M32—1061no—
184#108—1061no—
185#108—1061no—
186#106—21no—
187#106C22⋊D5311yescharacteristic
188#106C22⋊D5311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191#103—21no—
192#103—21no—
193#102—21no—
194#102—21no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—41no—
200C22×C4C2×C22⋊D5311yescharacteristic
201Q16—41no—
202SD16—81no—
203D8—41no—
204M16—41no—
205M16—2121no—
206M16—2121no—
207M16—2121no—
208M16—2121no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—2121no—
212C2×C8—2121no—
213C2×C8C2×C22⋊D5311yescharacteristic
214C2×C8C2×C22⋊D5311yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4#1171992311yescharacteristic
227C22⋊C4#1171992311yescharacteristic
228C42—21no—
229C42—41no—
230C16—2121no—
231C16—2121no—
232C23—41no—
233C23C2×#1171992311yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—161no—
237D4—41no—
238D4—41no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—161no—
248C2×C4#1313318611yescharacteristic
249C2×C4#1313324311yescharacteristic
250C8—21no—
251C8—21no—
252C8—41no—
253C8—4241no—
254C8—4241no—
255C22—21no—
256C22—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#1681077411yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4#1681079611yescharacteristic
268C4#1681081311yescharacteristic
269C2—21no—
270C2—41no—
271C2—161no—
272C2#2027375211yescharacteristic
subgroupG/H
centerC2#20273752
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#3263C2
socleC2×C53#1616
prime radicalC53×#469C22
hypercenter#3263D53
hypocenterC53#12306
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3263

Complements

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

NH ≅ G/Ntype
1. #20276932C2semidirect
3. #20276295C2semidirect
4. #20272709C2semidirect
6. #20272668C2semidirect
163. C53#12306semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202769272
#202762952
#202727092
#202727062
#202726682
C53×#32632
#1230653