Finite Group Databasebeta

← #27297479

Subgroups — #27297479

Order 13568.

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

normal subgroups only

This group has 280 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#20276290C211yesmaximal, characteristic, complemented
4#20272703C211yesmaximal, characteristic, complemented
5#20272700C211yesmaximal, characteristic
6#20272666C211yesmaximal, characteristic, complemented
7C53×#3256C211yesmaximal, characteristic
8#16810672C2211yescharacteristic
9#16810445—21no—
10#16810445—21no—
11#16810439—21no—
12#16810436—21no—
13#16810405C2211yescharacteristic
14#16810405C2211yescharacteristic
15#16810395—21no—
16#16810383—21no—
17#16810380C2211yescharacteristic
18C53×#529—21no—
19C53×#479—21no—
20C53×#478—21no—
21C53×#470C2211yescharacteristic
22C53×#469C2211yescharacteristic
23C53×#468—21no—
24C53×#464—21no—
25C53×#463—21no—
26C53×#423C2211yescharacteristic
27C2×#11719917—21no—
28#13133176—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×C53D411yescharacteristic
37C2×M16×C53C2311yescharacteristic
38#126×C53—41no—
39#125×C53—21no—
40#124×C53—21no—
41#123×C53—41no—
42C22⋊Q8×C53—21no—
43#121×C53—21no—
44C4×Q8×C53—21no—
45C4×D4×C53—21no—
46C4×D4×C53—41no—
47#117×C53—41no—
48C2×C42×C53D411yescharacteristic
49#107×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×D4×C53—21no—
60C2×D4×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64Q16×C53—41no—
65SD16×C53—81no—
66D8×C53—41no—
67M16×C53—41no—
68M16×C53—41no—
69C2×C8×C53—21no—
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53C2×D411yescharacteristic
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
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—
83C42×C53—21no—
84C42×C53—41no—
85C42×C53SD1611yescharacteristic
86C42×C53SD1611yescharacteristic
87C22⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4×Dic53—41no—
90#11719917—41no—
91#11719917—41no—
92C2×#88322—41no—
93#11719908—41no—
94#11719908—41no—
95C23×C53—41no—
96C23×C53AGL(1,ℤ/8)11yescharacteristic
97Q8×C53—41no—
98Q8×C53—41no—
99D4×C53—41no—
100D4×C53—41no—
101D4×C53—82no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—81no—
110C2×C4×C53—82no—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53C2×SD1611yescharacteristic
113C2×Dic53—81no—
114C8×C53—21no—
115C8×C53—21no—
116C8×C53—41no—
117#88322—81no—
118#12309—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
133Dic53—161no—
134#3256D5311yescharacteristic
135#2239—531no—
136#2238—531no—
137#2102—531no—
138#1340—531no—
139#1339—531no—
140#1327—531no—
141C2×C53—21no—
142C2×C53—41no—
143C2×C53—161no—
144C2×C53#161611yescharacteristic
145#529—21no—
146#479—21no—
147#478—21no—
148#470C2×D5311yescharacteristic
149#469C2×D5311yescharacteristic
150#468—21no—
151#464—21no—
152#463—21no—
153#430—531no—
154#423C2×D5311yescharacteristic
155#365—1061no—
156#365—1061no—
157#362—1061no—
158#361—1061no—
159#349—531no—
160#349—531no—
161#344—1061no—
162#339—1061no—
163H(3,ℤ/4)—531no—
164C53#1230911yescharacteristic, complemented, Sylow
165C2×Q16C22⋊D5311yescharacteristic
166C2×SD16—21no—
167C2×D8C22⋊D5311yescharacteristic
168C2×M16—1061no—
169C2×M16C22⋊D5311yescharacteristic
170C2×M16C22×D5311yescharacteristic
171#126—41no—
172#125—21no—
173#124—21no—
174#123—41no—
175C22⋊Q8—21no—
176#121—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—41no—
180#117—41no—
181#117—1061no—
182C2×C42C22⋊D5311yescharacteristic
183M32—1061no—
184M32—1061no—
185#108—1061no—
186#108—1061no—
187#107—21no—
188#106C22⋊D5311yescharacteristic
189#106C22⋊D5311yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—21no—
192#103—21no—
193#103—21no—
194#102—21no—
195#102—21no—
196C2×Q8—21no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4C2×C22⋊D5311yescharacteristic
202Q16—41no—
203SD16—81no—
204D8—41no—
205M16—41no—
206M16—41no—
207M16—2121no—
208M16—2121no—
209C2×C8—21no—
210C2×C8—2121no—
211C2×C8C2×C22⋊D5311yescharacteristic
212C2×C8C2×C22⋊D5311yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—2121no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—2121no—
226C42—21no—
227C42—41no—
228C42—2121no—
229C42#1171992311yescharacteristic
230C42#1171992311yescharacteristic
231C16—2121no—
232C16—2121no—
233C23—41no—
234C23#1313324311yescharacteristic
235Q8—41no—
236Q8—41no—
237D4—41no—
238D4—41no—
239D4—82no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—4241no—
250C2×C4#1313318611yescharacteristic
251C2×C4C2×#1171992311yescharacteristic
252C8—21no—
253C8—21no—
254C8—41no—
255C8—4241no—
256C22—21no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#1681077411yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—8481no—
269C4#1681079611yescharacteristic
270C4#1681081311yescharacteristic
271C2—21no—
272C2—41no—
273C2—161no—
274C2#2027375211yescharacteristic
subgroupG/H
centerC2#20273752
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#3256C2
socleC2×C53#1616
prime radicalC53×#469C22
hypercenter#3256D53
hypocenterC53#12309
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3256

Complements

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

NH ≅ G/Ntype
1. #20276932C2semidirect
3. #20276290C2semidirect
4. #20272703C2semidirect
6. #20272666C2semidirect
164. C53#12309semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202769272
#202762902
#202727032
#202727002
#202726662
C53×#32562
#1230953