Finite Group Databasebeta

← #27298373

Subgroups — #27298373

Order 13568.

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

normal subgroups only

This group has 228 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#20276623C211yesmaximal, characteristic, complemented
2#20274082C211yesmaximal, characteristic, complemented
3#20274072C211yesmaximal, characteristic
4#20272787C211yesmaximal, characteristic, complemented
5#20272764C211yesmaximal, characteristic, complemented
6#20272761C211yesmaximal, characteristic
7C53×#2248C211yesmaximal, characteristic
8#16810905—21no—
9#16810820—21no—
10C8×Dic106C2211yescharacteristic
11C4⋊#11719908C2211yescharacteristic
12#16810441—21no—
13#16810398—21no—
14#16810376C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810369—21no—
17#16810367—21no—
18C2×SD32×C53—21no—
19C2×SD32×C53—21no—
20C53×#494C2211yescharacteristic
21C53×#493C2211yescharacteristic
22C4×C16×C53C2211yescharacteristic
23C2×#11719923—41no—
24C4×Dic106D411yescharacteristic
25C2×#11719908—21no—
26#13133148—41no—
27C4⋊#88322—21no—
28C4⋊#88322—21no—
29C4⋊#88322D411yescharacteristic
30#13133146—21no—
31#13133140—41no—
32#13133138—41no—
33C8×Dic53—21no—
34C2×Q16×C53—21no—
35C2×Q16×C53—21no—
36C2×D8×C53—21no—
37C2×D8×C53—21no—
38C4⋊Q8×C53—21no—
39#127×C53—21no—
40SD32×C53—42no—
41SD32×C53—42no—
42C2×C16×C53D411yescharacteristic
43C2×C16×C53D411yescharacteristic
44C4×C8×C53C2311yescharacteristic
45C2×Q8×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×Dic106D811yescharacteristic
50Q16×C53—41no—
51Q16×C53—41no—
52Q16×C53—81no—
53D8×C53—41no—
54D8×C53—41no—
55D8×C53—81no—
56C2×C8×C53C2×D411yescharacteristic
57C2×C8×C53#4111yescharacteristic
58C4⋊C4×C53—41no—
59C4⋊C4×C53—41no—
60C42×C53C2×D411yescharacteristic
61#11719923—82no—
62C4⋊Dic53D811yescharacteristic
63#11719919—21no—
64C4×Dic53—21no—
65C2×#88322—21no—
66C2×#88322—41no—
67C2×#88322—41no—
68C16×C53—21no—
69C16×C53SD1612yescomplemented
70#11719908—41no—
71C23×C53—81no—
72Q8×C53—81no—
73Q8×C53—81no—
74D4×C53—81no—
75D4×C53—81no—
76D4×C53—81no—
77D4×C53—81no—
78D4×C53—161no—
79C2×C4×C53—81no—
80C2×C4×C53#12111yescharacteristic
81C2×C4×C53C2×D811yescharacteristic
82C2×C4×C53#13511yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85Dic106—21no—
86Dic106—21no—
87C8×C53—21no—
88C8×C53C2×SD1611yescharacteristic
89C8×C53AGL(1,ℤ/8)11yescharacteristic
90#88322—41no—
91#88322—81no—
92#88322—81no—
93#12444—531nomaximal, Sylow
94C22×C53—161no—
95C22×C53—161no—
96C22×C53—161no—
97C22×C53#46611yescharacteristic
98C4×C53—21no—
99C4×C53—161no—
100C4×C53#46011yescharacteristic
101C4×C53#47211yescharacteristic
102C4×C53#50911yescharacteristic
103C4×C53#51011yescharacteristic
104Dic53—41no—
105Dic53—41no—
106#2248D5311yescharacteristic
107#2183—531no—
108#1670—531no—
109#1668—531no—
110#1371—531no—
111#1362—531no—
112#1361—531no—
113C2×C53—321no—
114C2×C53#166811yescharacteristic
115C2×C53#222011yescharacteristic
116C2×C53#222311yescharacteristic
117C2×SD32—21no—
118C2×SD32—21no—
119#494C2×D5311yescharacteristic
120#493C2×D5311yescharacteristic
121#475—1061no—
122#460—1061no—
123C8×Q8—531no—
124C4⋊C16—531no—
125C4⋊C16—1061no—
126#346—1061no—
127C4×C16C2×D5311yescharacteristic
128#335—531no—
129#335—531no—
130#332—1061no—
131#331—1061no—
132C53#1244411yescharacteristic, complemented, Sylow
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2121no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8—531no—
141SD32—42no—
142SD32—42no—
143C2×C16—1061no—
144C2×C16C22⋊D5311yescharacteristic
145C2×C16C22⋊D5311yescharacteristic
146#106—2121no—
147C4⋊C8—531no—
148C4⋊C8—1061no—
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—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—4242no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—1061no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8#1171994711yescharacteristic
171C2×C8C2×C22⋊D5311yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—531no—
175C4⋊C4—1061no—
176C42—1061no—
177C42D4×D5311yescharacteristic
178C16—21no—
179C16—2121no—
180C16#1171992312yescomplemented
181C23—81no—
182Q8—81no—
183Q8—81no—
184Q8—1061no—
185Q8—1061no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—81no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4#1313318811yescharacteristic
195C2×C4D8×D5311yescharacteristic
196C2×C4#1313323611yescharacteristic
197C8—21no—
198C8—4241no—
199C8—4241no—
200C8—2121no—
201C8C2×#1171992311yescharacteristic
202C8#1313324311yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1681085611yescharacteristic
207C4—21no—
208C4—161no—
209C4—2121no—
210C4—2121no—
211C4#1681082011yescharacteristic
212C4#1681089011yescharacteristic
213C4#1681106011yescharacteristic
214C4#1681106211yescharacteristic
215C2—321no—
216C2#2027407211yescharacteristic
217C2#2027682511yescharacteristic
218C2#2027684211yescharacteristic
subgroupG/H
centerC22#16810856
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2248C2
socleC22×C53#466
prime radicalC53×#493C22
hypercenter#2248D53
hypocenterC53#12444
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #20276623C2semidirect
2. #20274082C2semidirect
4. #20272787C2semidirect
5. #20272764C2semidirect
69. C16×C53SD16semidirect
132. C53#12444semidirect
180. C16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202766232
#202740822
#202740722
#202727872
#202727642
#202727612
C53×#22482
#1244453