Finite Group Databasebeta

← #27297434

Subgroups — #27297434

Order 13568.

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

normal subgroups only

This group has 342 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#20276927C212yesmaximal, complemented
2#20276268C211yesmaximal, characteristic, complemented
3#20272700C212yesmaximal, complemented
4#20272653C211yesmaximal, characteristic, complemented
5C53×#3199C211yesmaximal, characteristic
6#16810672C2211yescharacteristic, complemented
7#16810649—21no—
8#16810640—21no—
9#16810445—22no—
10#16810436—22no—
11#16810405C2212yescomplemented
12#16810380C2211yescharacteristic, complemented
13C53×#535—21no—
14C53×#482—22no—
15C53×#469C2212yes—
16C53×#466—21no—
17C53×#447—22no—
18C53×#408C2211yescharacteristic
19C2×#11719917—21no—
20#13133176—21no—
21#13133157—22no—
22#13133153—22no—
23#13133146—42no—
24#13133144—42no—
25C2×#41×C53D411yescharacteristic
26C22×D4×C53—21no—
27C2×D8×C53—21no—
28C2×D8×C53D412yes—
29C2×M16×C53C2311yescharacteristic
30C22×C8×C53D411yescharacteristic
31#127×C53—41no—
32#124×C53—41no—
33#121×C53—22no—
34#121×C53—22no—
35#120×C53—41no—
36#120×C53—41no—
37#117×C53—21no—
38#107×C53—21no—
39#106×C53D412yes—
40#102×C53—22no—
41#102×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53—41no—
44#41×C53—41no—
45#41×C53—41no—
46C2×Q8×C53SD1611yescharacteristic
47C2×D4×C53—21no—
48C2×D4×C53—22no—
49C2×D4×C53—41no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—81no—
53C2×D4×C53—81no—
54C2×D4×C53SD1611yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53C2×D411yescharacteristic
57D8×C53—42no—
58D8×C53—81no—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53—41no—
62C2×C8×C53C2×D412yes—
63C4⋊C4×C53—22no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—41no—
66C22⋊C4×C53—42no—
67C42×C53—41no—
68C22⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4×Dic53—22no—
71#11719917—41no—
72#11719917—41no—
73C2×#88322—41no—
74#11719908—42no—
75C23×C53—21no—
76C23×C53—42no—
77C23×C53—82no—
78C23×C53—162no—
79C23×C53AGL(1,ℤ/8)11yescharacteristic
80Q8×C53—41no—
81D4×C53—41no—
82D4×C53—42no—
83D4×C53—42no—
84D4×C53—81no—
85D4×C53—82no—
86D4×C53—162no—
87D4×C53—162no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53#12011yescharacteristic
95C2×C4×C53C2×SD1611yescharacteristic
96C2×Dic53—42no—
97C8×C53—22no—
98C8×C53—41no—
99#88322—81no—
100#12301—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—41no—
103C22×C53—42no—
104C22×C53—41no—
105C22×C53—82no—
106C22×C53—82no—
107C22×C53—82no—
108C22×C53—164no—
109C22×C53#45011yescharacteristic
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—82no—
113C4×C53#45311yescharacteristic
114C4×C53#45811yescharacteristic
115Dic53—82no—
116#3199D5311yescharacteristic
117#2238—532no—
118#2098—531no—
119#1339—532no—
120#1321—531no—
121C2×C53—21no—
122C2×C53—41no—
123C2×C53—81no—
124C2×C53—162no—
125C2×C53#161311yescharacteristic
126#535—21no—
127#482—22no—
128#469C2×D5312yes—
129#466—21no—
130#447—22no—
131#430—531no—
132#424—1061no—
133#421—1061no—
134#408C2×D5311yescharacteristic
135#365—1062no—
136#361—1062no—
137#349—532no—
138H(3,ℤ/4)—531no—
139C53#1230111yescharacteristic, complemented, Sylow
140C2×#41C22⋊D5311yescharacteristic
141C22×D4—21no—
142C2×D8—21no—
143C2×D8C22⋊D5312yes—
144C2×M16—1061no—
145C2×M16C22×D5311yescharacteristic
146C22×C8C22⋊D5311yescharacteristic
147#127—41no—
148#124—41no—
149#121—22no—
150#121—22no—
151#120—41no—
152#120—41no—
153#117—21no—
154#117—1061no—
155M32—1062no—
156#108—1062no—
157#107—21no—
158#106C22⋊D5312yes—
159C4⋊C8—2122no—
160#104—2122no—
161#102—22no—
162#102—22no—
163C22⋊C8—22no—
164C24—41no—
165#41—41no—
166#41—41no—
167C2×Q8#1171992311yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4#1171992311yescharacteristic
176C22×C4—21no—
177C22×C4C2×C22⋊D5311yescharacteristic
178D8—42no—
179D8—81no—
180M16—41no—
181M16—2121no—
182M16—2121no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—2121no—
186C2×C8C2×C22⋊D5312yes—
187C4⋊C4—22no—
188C4⋊C4—2121no—
189C22⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—2121no—
193C42—41no—
194C42—1062no—
195C16—2122no—
196C23—21no—
197C23—42no—
198C23—82no—
199C23—162no—
200C23#1313324311yescharacteristic
201Q8—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—82no—
207D4—162no—
208D4—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—2122no—
216C2×C4#1313318611yescharacteristic
217C2×C4C2×#1171992311yescharacteristic
218C8—22no—
219C8—41no—
220C8—4241no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—164no—
229C22#1681077411yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—4242no—
234C4#1681079111yescharacteristic
235C4#1681081311yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—162no—
240C2#2027373511yescharacteristic
subgroupG/H
centerC2#20273735
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3199C2
socleC2×C53#1613
hypercenter#3199D53
hypocenterC53#12301
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3199

Complements

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

NH ≅ G/Ntype
1. #20276927C2semidirect
2. #20276268C2semidirect
3. #20272700C2semidirect
4. #20272653C2semidirect
6. #16810672C22semidirect
11. #16810405C22semidirect
12. #16810380C22semidirect
139. C53#12301semidirect

Maximal subgroups

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

subgroupindex
#202769272
#202762682
#202727002
#202726532
C53×#31992
#1230153