Finite Group Databasebeta

← #27298362

Subgroups — #27298362

Order 13568.

232 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 65 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#20276972C211yesmaximal, characteristic
2#20276617C211yesmaximal, characteristic, complemented
3#20274082C211yesmaximal, characteristic, complemented
4#20274076C211yesmaximal, characteristic
5#20272788C211yesmaximal, characteristic, complemented
6#20272762C211yesmaximal, characteristic
7C53×#1362C211yesmaximal, characteristic, complemented
8C2×#13133160—21no—
9C2×#13133160—21no—
10#16811000C2211yescharacteristic
11#16810905—21no—
12#16810824—21no—
13C8×Dic106C2211yescharacteristic
14#16810441—21no—
15C4×#11719908C2211yescharacteristic
16C16×Dic53—21no—
17#16810376C2211yescharacteristic
18#16810369—21no—
19#16810366—21no—
20C53×#494C2211yescharacteristic
21C53×C4⋊C16C2211yescharacteristic
22C53×#335C2211yescharacteristic
23C2×#11719913—41no—
24C2×D424—21no—
25C2×D424—21no—
26#13133214—21no—
27C4×Dic106D411yescharacteristic
28#13133160—42no—
29#13133160—42no—
30C2×#11719908D411yescharacteristic
31C2×#11719908D411yescharacteristic
32#13133148—41no—
33C4⋊#88322—21no—
34C4⋊#88322D411yescharacteristic
35#13133146—21no—
36#13133140—41no—
37#13133139—41no—
38C8×Dic53—21no—
39C2×Q16×C53—21no—
40C2×Q16×C53—21no—
41C4⋊Q8×C53—21no—
42C2×C16×C53—21no—
43C4⋊C8×C53—21no—
44C4×C8×C53C2311yescharacteristic
45C2×Q8×C53—41no—
46C2×D212—41no—
47C2×D212—41no—
48C2×D212—41no—
49C2×Dic106D811yescharacteristic
50Q16×C53—41no—
51Q16×C53—41no—
52Q16×C53—81no—
53C2×C8×C53—41no—
54C2×C8×C53C2×D411yescharacteristic
55C2×C8×C53#4111yescharacteristic
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C42×C53C2×D411yescharacteristic
59C4⋊Dic53D811yescharacteristic
60#11719919—21no—
61C4×Dic53—21no—
62C2×#88322—21no—
63C2×#88322—41no—
64D424—41no—
65D424—41no—
66D424—81no—
67#11719913—82no—
68C16×C53—41no—
69#11719908—21no—
70#11719908SD1612yescomplemented
71C22×D53—81no—
72Q8×C53—81no—
73Q8×C53—81no—
74C2×C4×C53—81no—
75C2×C4×C53#12111yescharacteristic
76C2×C4×C53C2×D811yescharacteristic
77C2×C4×C53#13511yescharacteristic
78C2×Dic53—21no—
79C2×Dic53—21no—
80D212—81no—
81D212—81no—
82D212—81no—
83D212—81no—
84D212—161no—
85Dic106—21no—
86Dic106SD3212yescomplemented
87C8×C53—21no—
88C8×C53—81no—
89C8×C53C2×SD1611yescharacteristic
90C8×C53AGL(1,ℤ/8)11yescharacteristic
91#88322—41no—
92#88322—81no—
93#12441—531nomaximal, Sylow
94C22×C53#46611yescharacteristic
95C2×D53—161no—
96C2×D53—161no—
97C2×D53—161no—
98C4×C53—21no—
99C4×C53—161no—
100C4×C53#46011yescharacteristic
101C4×C53#47211yescharacteristic
102C4×C53C2×SD3211yescharacteristic
103C4×C53#50811yescharacteristic
104Dic53—22no—
105Dic53—41no—
106#2248—531no—
107Q8×C16—531no—
108#1670—531no—
109#1668—531no—
110#1372—531no—
111#1362D5311yescharacteristic, complemented
112#1361—531no—
113C2×C53#166811yescharacteristic
114C2×C53#221711yescharacteristic
115C2×C53#222311yescharacteristic
116D53—321no—
117C2×SD32—1061no—
118C2×SD32—1061no—
119#494C2×D5311yescharacteristic
120#493—531no—
121#475—1061no—
122#460—1061no—
123C8×Q8—531no—
124C4⋊C16—1061no—
125C4⋊C16C2×D5311yescharacteristic
126C4×C16—531no—
127C4×C16—1061no—
128#335—531no—
129#335C2×D5311yescharacteristic
130#332—1061no—
131#331—1061no—
132C53#1244111yescharacteristic, complemented, Sylow
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2121no—
136C2×D8—1061no—
137C2×D8—1061no—
138C4⋊Q8—21no—
139#127—1061no—
140C4×Q8—531no—
141SD32—2122no—
142SD32—2122no—
143C2×C16—21no—
144C2×C16—531no—
145C2×C16—531no—
146#106—2121no—
147C4⋊C8—21no—
148C4⋊C8—531no—
149C4⋊C8—1061no—
150C4⋊C8—1061no—
151#103—2121no—
152#102—2121no—
153C4×C8—1061no—
154C4×C8C22×D5311yescharacteristic
155C2×Q8—41no—
156C2×Q8—531no—
157C2×D4—2121no—
158C2×D4—2121no—
159C2×D4—2121no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—4242no—
164D8—4241no—
165D8—2121no—
166D8—2121no—
167C2×C8—41no—
168C2×C8—1061no—
169C2×C8—2121no—
170C2×C8#1171994511yescharacteristic
171C2×C8D4×D5311yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—531no—
175C4⋊C4—1061no—
176C42—1061no—
177C42D4×D5311yescharacteristic
178C16—41no—
179C16—532no—
180C16—1061no—
181C23—4241no—
182Q8—81no—
183Q8—81no—
184Q8—532no—
185Q8—1061no—
186D4—8481no—
187D4—4241no—
188D4—4241no—
189D4—4241no—
190D4—4241no—
191C2×C4—81no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4#1313319011yescharacteristic
195C2×C4D8×D5311yescharacteristic
196C2×C4#1313323811yescharacteristic
197C8—21no—
198C8—81no—
199C8—4241no—
200C8—2121no—
201C8SD16×D5311yescharacteristic
202C8#1313324111yescharacteristic
203C22—8481no—
204C22—8481no—
205C22—8481no—
206C22#1681085811yescharacteristic
207C4—21no—
208C4—161no—
209C4—1062no—
210C4—2121no—
211C4#1681082211yescharacteristic
212C4#1681089211yescharacteristic
213C4SD32×D5311yescharacteristic
214C4#1681105611yescharacteristic
215C2—16961no—
216C2#2027407411yescharacteristic
217C2#2027681411yescharacteristic
218C2#2027684811yescharacteristic
subgroupG/H
centerC22#16810858
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1362C2
socleC22×C53#466
prime radical#16811000C22
hypercenter#1362D53
hypocenterC53#12441
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
2. #20276617C2semidirect
3. #20274082C2semidirect
5. #20272788C2semidirect
7. C53×#1362C2semidirect
70. #11719908SD16semidirect
86. Dic106SD32semidirect
111. #1362D53semidirect
132. C53#12441semidirect

Maximal subgroups

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

subgroupindex
#202769722
#202766172
#202740822
#202740762
#202727882
#202727622
C53×#13622
#1244153