Finite Group Databasebeta

← #27271454

Subgroups — #27271454

Order 13568.

342 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 47 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#20272649C211yesmaximal, characteristic, complemented
2#20272541C211yesmaximal, characteristic, complemented
3C53×#1793C211yesmaximal, characteristic
4#16810378—21no—
5#16810349—21no—
6C2×2−1+4×C53C411yescharacteristic
7C2×#104×C53—21no—
8C53×#417—21no—
9C2×#101×C53C411yescharacteristic
10C53×#391C2211yescharacteristic
11C22×#88322—41no—
12C22⋊#88322—41no—
132−1+4×C53—21no—
142−1+4×C53—41no—
152−1+4×C53—42no—
162−1+4×C53C812yescomplemented
17C2×#41×C53—21no—
18C2×#41×C53—41no—
19C2×#41×C53—41no—
20C22×Q8×C53—41no—
21C22×Q8×C53C2×C411yescharacteristic
22C2×M16×C53—21no—
23C2×C4⋊C4×C53D411yescharacteristic
24C2×C42×C53D411yescharacteristic
25#104×C53—42no—
26#103×C53—41no—
27#102×C53—41no—
28#101×C53—41no—
29#95×C53—41no—
30#41×C53—22no—
31#41×C53—22no—
32#41×C53—41no—
33#41×C53—42no—
34#41×C53—42no—
35#41×C53—42no—
36#41×C53—42no—
37#41×C53—44no—
38#41×C53—42no—
39#41×C53—42no—
40#41×C53—41no—
41C2×Q8×C53—21no—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53—42no—
46C2×Q8×C53—42no—
47C2×Q8×C53—44no—
48C2×Q8×C53—41no—
49C2×Q8×C53C2×C811yescharacteristic
50C2×Q8×C53M1611yescharacteristic
51C2×D4×C53—21no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60M16×C53—41no—
61M16×C53—41no—
62C2×C8×C53—41no—
63C4⋊C4×C53—41no—
64C42×C53—41no—
65C2×#88322—42no—
66C2×#88322—81no—
67C2×#88322—81no—
68C2×#88322—81no—
69C23×C53—41no—
70C23×C53#9911yescharacteristic
71Q8×C53—22no—
72Q8×C53—42no—
73Q8×C53—41no—
74Q8×C53—41no—
75Q8×C53—41no—
76Q8×C53—42no—
77Q8×C53—42no—
78D4×C53—22no—
79D4×C53—41no—
80D4×C53—42no—
81D4×C53—42no—
82D4×C53—42no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—44no—
94C2×C4×C53—44no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—81no—
101C2×C4×C53C22⋊C811yescharacteristic
102C2×C4×C53#10011yescharacteristic
103C8×C53—81no—
104#88322—82no—
105#88322—161no—
106#7345—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—42no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53#32311yescharacteristic
113C22×C53AΣL(1,𝔽16)11yescharacteristic
114C22×C53AΣL(1,𝔽16)11yescharacteristic
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53—41no—
121C4×C53—42no—
122C4×C53—81no—
123#1793D5311yescharacteristic
124#1319—531no—
125#1272—531no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—82no—
129C2×C53#131611yescharacteristic
130C2×C53#140411yescharacteristic
131C2×C53#140511yescharacteristic
132C2×2−1+4Dic5311yescharacteristic
133C2×#104—21no—
134#417—21no—
135C2×#101Dic5311yescharacteristic
136#391C2×D5311yescharacteristic
137#336—1061no—
138#324—1061no—
139C53#734511yescharacteristic, complemented, Sylow
1402−1+4—21no—
1412−1+4—41no—
1422−1+4—42no—
1432−1+4#8832212yescomplemented
144C2×#41—21no—
145C2×#41—41no—
146C2×#41—41no—
147C22×Q8—41no—
148C22×Q8C2×Dic5311yescharacteristic
149C2×M16—21no—
150C22×C8—2121no—
151C2×C4⋊C4C22⋊D5311yescharacteristic
152C2×C42C22⋊D5311yescharacteristic
153#104—42no—
154#103—41no—
155#102—41no—
156#101—41no—
157C22⋊C8—2121no—
158#95—41no—
159#41—22no—
160#41—22no—
161#41—41no—
162#41—42no—
163#41—42no—
164#41—42no—
165#41—42no—
166#41—44no—
167#41—42no—
168#41—42no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—41no—
178C2×Q8C2×#8832211yescharacteristic
179C2×Q8#1171991711yescharacteristic
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4C22⋊Dic5311yescharacteristic
189M16—41no—
190M16—41no—
191C2×C8—41no—
192C2×C8—4241no—
193C2×C8—4241no—
194C2×C8—4241no—
195C2×C8—2122no—
196C4⋊C4—41no—
197C42—41no—
198C23—41no—
199C23#1313313211yescharacteristic
200Q8—22no—
201Q8—41no—
202Q8—41no—
203Q8—42no—
204Q8—41no—
205Q8—42no—
206Q8—42no—
207D4—22no—
208D4—41no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—42no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4C22⋊#8832211yescharacteristic
231C2×C4#1313313411yescharacteristic
232C8—81no—
233C8—8481no—
234C8—4242no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—81no—
239C22—81no—
240C22#1681034711yescharacteristic
241C22#1681041011yescharacteristic
242C22#1681041011yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—42no—
250C4—81no—
251C2—21no—
252C2—21no—
253C2—82no—
254C2#2027264311yescharacteristic
255C2#2027286611yescharacteristic
256C2#2027286911yescharacteristic
subgroupG/H
centerC22#16810410
commutatorC2×Q8×C53C2×C8
Frattini#391C2×D53
FittingC53×#1793C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC2×2−1+4×C53C4
cosocleC53×#391C22
hypercenter#1793D53
hypocenterC53#7345
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #20272649C2semidirect
2. #20272541C2semidirect
16. 2−1+4×C53C8semidirect
139. C53#7345semidirect
143. 2−1+4#88322semidirect

Maximal subgroups

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

subgroupindex
#202726492
#202725412
C53×#17932
#734553