Finite Group Databasebeta

← #27298513

Subgroups — #27298513

Order 13568.

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

normal subgroups only

This group has 234 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
1C16×D212C211yesmaximal, characteristic, complemented
2#20274111C211yesmaximal, characteristic, complemented
3#20274067C211yesmaximal, characteristic
4#20272789C211yesmaximal, characteristic, complemented
5#20272773C211yesmaximal, characteristic, complemented
6#20272761C211yesmaximal, characteristic, complemented
7C53×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D53—21no—
9#16810910—21no—
10#16810815—21no—
11C8×D212C2211yescharacteristic
12C4⋊#11719908C2211yescharacteristic
13#16810402—21no—
14#16810376C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810372—21no—
17#16810367—21no—
18C53×#500C2211yescharacteristic
19C53×#493C2211yescharacteristic
20C53×#357—21no—
21C53×#357—21no—
22C4×C16×C53C2211yescharacteristic
23C2×#11719922—41no—
24C2×C8×D53—21no—
25C4×D212D411yescharacteristic
26C2×#11719908—21no—
27C16×D53—42no—
28#13133148—41no—
29C4⋊#88322—21no—
30C4⋊#88322—21no—
31C4⋊#88322D411yescharacteristic
32#13133138—41no—
33#13133137—41no—
34#13133129—21no—
35C2×D8×C53—21no—
36C2×D8×C53—21no—
37C4⋊Q8×C53—21no—
38#127×C53—21no—
39C2×C16×C53D411yescharacteristic
40C2×C16×C53D411yescharacteristic
41#107×C53—21no—
42#107×C53—21no—
43C4×C8×C53C2311yescharacteristic
44C2×Q8×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D212D811yescharacteristic
49C2×C4×D53—21no—
50D8×C53—41no—
51D8×C53—41no—
52D8×C53—81no—
53C2×C8×C53C2×D411yescharacteristic
54C2×C8×C53#4111yescharacteristic
55C4⋊C4×C53—41no—
56C4⋊C4×C53—41no—
57C42×C53C2×D411yescharacteristic
58#11719922—82no—
59#11719921—21no—
60C4⋊Dic53D811yescharacteristic
61C2×#88322—21no—
62C2×#88322—41no—
63C2×#88322—41no—
64C8×D53—41no—
65C8×D53—41no—
66C16×C53—21no—
67C16×C53—21no—
68#11719908—41no—
69C23×C53—81no—
70C22×D53—21no—
71Q8×C53—161no—
72D4×C53—81no—
73D4×C53—81no—
74D4×C53—81no—
75D4×C53—81no—
76D4×C53—161no—
77C2×C4×C53—81no—
78C2×C4×C53#12111yescharacteristic
79C2×C4×C53C2×D811yescharacteristic
80C2×C4×C53#13511yescharacteristic
81C2×Dic53—21no—
82D212—21no—
83D212SD3212yes—
84C4×D53—41no—
85C4×D53—41no—
86C8×C53—21no—
87C8×C53#13511yescharacteristic
88C8×C53AGL(1,ℤ/8)11yescharacteristic
89#88322—41no—
90#88322—81no—
91#88322—81no—
92#12468—531nomaximal, Sylow
93C22×C53—161no—
94C22×C53—161no—
95C22×C53—161no—
96C22×C53#46611yescharacteristic
97C2×D53—22no—
98C2×D53—41no—
99C2×D53—41no—
100C4×C53—21no—
101C4×C53—161no—
102C4×C53#46311yescharacteristic
103C4×C53#47111yescharacteristic
104C4×C53C2×SD3211yescharacteristic
105C4×C53#50811yescharacteristic
106Dic53—41no—
107#2240D5311yescharacteristic, complemented
108D4×C16—531no—
109#1675—531no—
110#1667—531no—
111#1372—531no—
112#1365—531no—
113#1361—531no—
114C2×C53—321no—
115C2×C53#168111yescharacteristic
116C2×C53#221711yescharacteristic
117C2×C53#222111yescharacteristic
118D53—42no—
119C22×C16—1061no—
120#500C2×D5311yescharacteristic
121#493C2×D5311yescharacteristic
122#476—1061no—
123#459—1061no—
124C8×D4—531no—
125C4⋊C16—531no—
126#357—21no—
127#357—21no—
128C22⋊C16—1061no—
129C4×C16C2×D5311yescharacteristic
130#335—531no—
131#335—531no—
132#333—1061no—
133#331—1061no—
134C53#1246811yescharacteristic, complemented, Sylow
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2121no—
138C22×C8—1061no—
139C4⋊Q8—21no—
140#127—21no—
141C4×D4—531no—
142C2×C16—1061no—
143C2×C16—2122no—
144C2×C16C22⋊D5311yescharacteristic
145C2×C16C22⋊D5311yescharacteristic
146#107—21no—
147#107—21no—
148#106—2121no—
149C4⋊C8—531no—
150C4⋊C8—1061no—
151C4⋊C8—1061no—
152#102—2121no—
153#102—2121no—
154C22⋊C8—1061no—
155C4×C8C22×D5311yescharacteristic
156C2×Q8—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—531no—
161C22×C4—1061no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—4242no—
166C2×C8—1061no—
167C2×C8—2121no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8—2121no—
171C2×C8#1171994711yescharacteristic
172C2×C8C2×C22⋊D5311yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—531no—
176C22⋊C4—1061no—
177C42D4×D5311yescharacteristic
178C16—21no—
179C16—21no—
180C16—2121no—
181C23—81no—
182C23—1061no—
183Q8—161no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—532no—
190D4—1061no—
191C2×C4—81no—
192C2×C4—1061no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4#1313318811yescharacteristic
196C2×C4D8×D5311yescharacteristic
197C2×C4#1313323611yescharacteristic
198C8—21no—
199C8—4241no—
200C8—4241no—
201C8—2121no—
202C8#1313323711yescharacteristic
203C8#1313324211yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—1062no—
208C22—2121no—
209C22—2121no—
210C22#1681085611yescharacteristic
211C4—21no—
212C4—161no—
213C4—2121no—
214C4#1681083511yescharacteristic
215C4#1681088311yescharacteristic
216C4SD32×D5311yescharacteristic
217C4#1681105411yescharacteristic
218C2—321no—
219C2—2122no—
220C2#2027414911yescharacteristic
221C2#2027680811yescharacteristic
222C2#2027683211yescharacteristic
subgroupG/H
centerC22#16810856
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2240C2
socleC22×C53#466
prime radical#20274067C2
hypercenter#2240D53
hypocenterC53#12468
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D212C2semidirect
2. #20274111C2semidirect
4. #20272789C2semidirect
5. #20272773C2semidirect
6. #20272761C2semidirect
7. C53×#2240C2semidirect
107. #2240D53semidirect
134. C53#12468semidirect

Maximal subgroups

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

subgroupindex
C16×D2122
#202741112
#202740672
#202727892
#202727732
#202727612
C53×#22402
#1246853