Finite Group Databasebeta

← #27297417

Subgroups — #27297417

Order 13568.

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

normal subgroups only

This group has 376 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#20276231C211yesmaximal, characteristic, complemented
3#20272706C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic, complemented
5C53×#3192C211yesmaximal, characteristic
6#16810672C2211yescharacteristic, complemented
7#16810671—21no—
8#16810587—21no—
9#16810445—22no—
10#16810436—22no—
11#16810405C2212yescomplemented
12#16810381C2211yescharacteristic, complemented
13C2×#121×C53—21no—
14C53×#481—22no—
15C53×#469—21no—
16C53×#469C2212yes—
17C53×#447—22no—
18C53×#407C2211yescharacteristic
19C2×#11719917—21no—
20C2×#11719917—21no—
21#13133157—22no—
22#13133153—22no—
23#13133152—42no—
24C22⋊#88322—42no—
25C22×D4×C53—21no—
26C22×D4×C53—41no—
27C23×C4×C53D411yescharacteristic
28C2×D8×C53—21no—
29C2×D8×C53D412yes—
30C2×M16×C53D411yescharacteristic
31C2×M16×C53C2311yescharacteristic
32#121×C53—22no—
33#121×C53—22no—
34#121×C53—41no—
35#121×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C22⋊C4×C53—41no—
38#106×C53—21no—
39#106×C53D412yes—
40#102×C53—22no—
41#102×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53—41no—
44C24×C53SD1611yescharacteristic
45C2×D4×C53—22no—
46C2×D4×C53—41no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—81no—
50C2×D4×C53—81no—
51C2×D4×C53—84no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53SD1611yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59D8×C53—42no—
60D8×C53—81no—
61M16×C53—41no—
62M16×C53—41no—
63C2×C8×C53—21no—
64C2×C8×C53C2×D412yes—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69#11719917—41no—
70#11719917—41no—
71#11719917—41no—
72#11719917—41no—
73C2×#88322—22no—
74C2×#88322—41no—
75#11719908—42no—
76C23×C53—21no—
77C23×C53—41no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53—82no—
81C23×C53—82no—
82C23×C53—82no—
83C23×C53C2×SD1611yescharacteristic
84D4×C53—42no—
85D4×C53—42no—
86D4×C53—82no—
87D4×C53—84no—
88D4×C53—84no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—81no—
97C2×C4×C53—82no—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C8×C53—22no—
101C8×C53—41no—
102#88322—42no—
103#88322—81no—
104#12298—531nomaximal, Sylow
105C22×C53—22no—
106C22×C53—21no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53—82no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53—82no—
115C22×C53—164no—
116C22×C53#45011yescharacteristic
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—82no—
120C4×C53#45311yescharacteristic
121C4×C53#45811yescharacteristic
122#3192D5311yescharacteristic
123#2238—532no—
124#2091—531no—
125#1341—532no—
126#1320—531no—
127C2×C53—21no—
128C2×C53—41no—
129C2×C53—42no—
130C2×C53—162no—
131C2×C53#161311yescharacteristic
132C2×#121—21no—
133#481—22no—
134#469—21no—
135#469C2×D5312yes—
136#447—22no—
137#430—531no—
138#430—1061no—
139#407—1061no—
140#407C2×D5311yescharacteristic
141#365—1062no—
142#361—1062no—
143#349—532no—
144#338—531no—
145C53#1229811yescharacteristic, complemented, Sylow
146C22×D4—21no—
147C22×D4—41no—
148C23×C4C22⋊D5311yescharacteristic
149C2×D8—21no—
150C2×D8C22⋊D5312yes—
151C2×M16—1061no—
152C2×M16—1061no—
153C2×M16C22⋊D5311yescharacteristic
154C2×M16C22×D5311yescharacteristic
155#121—22no—
156#121—22no—
157#121—41no—
158#121—41no—
159C2×C4⋊C4—21no—
160C2×C22⋊C4—41no—
161M32—1062no—
162#108—1062no—
163#108—2122no—
164#106—21no—
165#106C22⋊D5312yes—
166#102—22no—
167#102—22no—
168C22⋊C8—22no—
169C22⋊C8—2122no—
170C24—41no—
171C24#1171992311yescharacteristic
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—84no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4#1171992311yescharacteristic
185C22×C4C2×C22⋊D5311yescharacteristic
186D8—42no—
187D8—81no—
188M16—41no—
189M16—41no—
190M16—2121no—
191M16—2121no—
192M16—2121no—
193M16—2121no—
194C2×C8—21no—
195C2×C8—1062no—
196C2×C8—2121no—
197C2×C8C2×C22⋊D5312yes—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C16—2122no—
203C23—21no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—82no—
208C23—82no—
209C23—82no—
210C23C2×#1171992311yescharacteristic
211D4—42no—
212D4—42no—
213D4—82no—
214D4—84no—
215D4—84no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4#1313318611yescharacteristic
226C2×C4#1313324311yescharacteristic
227C8—22no—
228C8—41no—
229C8—4241no—
230C8—2122no—
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—82no—
238C22—82no—
239C22—81no—
240C22—82no—
241C22—164no—
242C22#1681077411yescharacteristic
243C4—21no—
244C4—41no—
245C4—82no—
246C4#1681079111yescharacteristic
247C4#1681081311yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—162no—
252C2#2027373511yescharacteristic
subgroupG/H
centerC2#20273735
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3192C2
socleC2×C53#1613
hypercenter#3192D53
hypocenterC53#12298
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. #20276927C2semidirect
2. #20276231C2semidirect
3. #20272706C2semidirect
4. #20272651C2semidirect
6. #16810672C22semidirect
11. #16810405C22semidirect
12. #16810381C22semidirect
145. C53#12298semidirect

Maximal subgroups

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

subgroupindex
#202769272
#202762312
#202727062
#202726512
C53×#31922
#1229853