Finite Group Databasebeta

← #27297374

Subgroups — #27297374

Order 13568.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286025C211yesmaximal, characteristic
2#20276901C212yesmaximal, complemented
3#20276208C211yesmaximal, characteristic, complemented
4#20272725C212yesmaximal, complemented
5C53×#1322C211yesmaximal, characteristic, complemented
6#16811173—21no—
7#16810904—22no—
8#16810859—21no—
9#16810859C2212yes—
10#16810819—22no—
11C2×#13133151C2211yescharacteristic
12#16810521—21no—
13#16810451—22no—
14#16810425—22no—
15#16810386—21no—
16#16810386C2211yescharacteristic
17C2×C4⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×#11719945—41no—
20C2×D424—21no—
21C2×D424D412yes—
22#13133217—21no—
23#13133214—21no—
24#13133205—41no—
25#13133191—22no—
26#13133191—41no—
27C4×D212—22no—
28C4×D212—41no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—41no—
32#13133151—21no—
33#13133151—22no—
34#13133151D412yes—
35#13133139—22no—
36#13133139—22no—
37#13133124—41no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C42×C53D411yescharacteristic
41C2×C16×C53—22no—
42C4⋊C8×C53—22no—
43C22×C4×C53D811yescharacteristic
44C22×C4×C53D811yescharacteristic
45C22×C4×C53C2×D411yescharacteristic
46C2×C22⋊D53—42no—
47C22×Dic53—41no—
48C22×Dic53—41no—
49#11719945—82no—
50#11719945—82no—
51C2×D212—22no—
52C2×D212—41no—
53C2×D212—41no—
54C2×C4×D53—42no—
55C2×Dic106—41no—
56C2×C8×C53—21no—
57C2×C8×C53—41no—
58C2×C8×C53—41no—
59C2×C8×C53C2×D412yes—
60C42×C53—21no—
61C42×C53—21no—
62#11719921—42no—
63#11719921—42no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53—41no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70#11719919—82no—
71D424—42no—
72D424—81no—
73C16×C53—42no—
74C23×C53C2×D811yescharacteristic
75C22×D53—42no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53SD3212yes—
83C2×C4×C53#12011yescharacteristic
84C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
85C22⋊D53—82no—
86C22⋊D53—82no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—81no—
90C2×Dic53—81no—
91C2×Dic53—81no—
92C2×Dic53—81no—
93C2×Dic53—81no—
94D212—42no—
95D212—42no—
96D212—82no—
97D212—82no—
98C4×D53—84no—
99Dic106—82no—
100C8×C53—22no—
101C8×C53—41no—
102#12291—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53#44711yescharacteristic
106C22×C53C2×SD3211yescharacteristic
107C22×C53#50911yescharacteristic
108C2×D53—82no—
109C2×D53—82no—
110C2×D53—82no—
111C4×C53—21no—
112C4×C53—22no—
113C4×C53—41no—
114C4×C53#45611yescharacteristic
115C4×C53#45811yescharacteristic
116Dic53—82no—
117Dic53—82no—
118Dic53—161no—
119#3245—531no—
120#2233—532no—
121#2087—531no—
122#1347—532no—
123#1322D5311yescharacteristic, complemented
124C2×C53—21no—
125C2×C53—21no—
126C2×C53#161011yescharacteristic
127C2×C53#218711yescharacteristic
128C2×C53#219011yescharacteristic
129D53—162no—
130#532—1061no—
131#474—1062no—
132#466—532no—
133#466—1061no—
134#459—1062no—
135C2×#107—531no—
136C2×C4⋊C8C2×D5311yescharacteristic
137#389—1061no—
138#367—1062no—
139#357—1062no—
140C22⋊C16C2×D5312yes—
141#340—531no—
142#340—1061no—
143C53#1229111yescharacteristic, complemented, Sylow
144C2×#41—2121no—
145C2×D8—532no—
146C2×D8—1061no—
147C22×C8D21211yescharacteristic
148C22×C8C22×D5311yescharacteristic
149C4⋊Q8—1061no—
150#127—1061no—
151#124—2121no—
152#121—1062no—
153#121—2121no—
154C4×D4—1062no—
155C4×D4—2121no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—2121no—
159C2×C42D21211yescharacteristic
160C2×C16—22no—
161#107—532no—
162#107—1061no—
163#107—1062no—
164C4⋊C8—22no—
165#102—1062no—
166#102—1062no—
167#95—2121no—
168#41—4242no—
169#41—4242no—
170C2×Q8—2121no—
171C2×D4—1062no—
172C2×D4—2121no—
173C2×D4—2121no—
174C2×D4—2122no—
175C22×C4—2122no—
176C22×C4—2121no—
177C22×C4—2121no—
178C22×C4D42411yescharacteristic
179C22×C4D42411yescharacteristic
180C22×C4C2×D21211yescharacteristic
181D8—4241no—
182D8—2122no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D5312yes—
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—4242no—
190C4⋊C4—2121no—
191C4⋊C4—2121no—
192C4⋊C4—2121no—
193C4⋊C4—2121no—
194C22⋊C4—2122no—
195C22⋊C4—2122no—
196C42—21no—
197C42—21no—
198C16—42no—
199C23—2122no—
200C23C2×D42411yescharacteristic
201Q8—4242no—
202D4—4242no—
203D4—4242no—
204D4—4242no—
205D4—4242no—
206D4—2122no—
207D4—2122no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—4244no—
215C2×C4—4241no—
216C2×C4—4241no—
217C2×C4—4241no—
218C2×C4—4241no—
219C2×C4—4241no—
220C2×C4—2122no—
221C2×C4—2122no—
222C2×C4#1313316212yes—
223C2×C4#1313318711yescharacteristic
224C2×C4#1313324511yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—4242no—
230C22—4242no—
231C22—4242no—
232C22#1681075911yescharacteristic
233C22C2×#1313316211yescharacteristic
234C22#1681106111yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—8481no—
239C4—4242no—
240C4—4242no—
241C4#1681080811yescharacteristic
242C4#1681081411yescharacteristic
243C2—21no—
244C2—21no—
245C2—8482no—
246C2#2027372011yescharacteristic
247C2#2027664611yescharacteristic
248C2#2027666311yescharacteristic
subgroupG/H
centerC22#16810759
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1322C2
socleC22×C53#447
prime radical#20286025C2
hypercenter#1322D53
hypocenterC53#12291
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #20276901C2semidirect
3. #20276208C2semidirect
4. #20272725C2semidirect
5. C53×#1322C2semidirect
123. #1322D53semidirect
143. C53#12291semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202769012
#202762082
#202727252
C53×#13222
#1229153