Finite Group Databasebeta

← #28309607

Subgroups — #28309607

Order 13568.

272 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 62 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#20291090C211yesmaximal, characteristic
2#20290937C211yesmaximal, characteristic
3#20289722C212yesmaximal, complemented
4#20289721C212yesmaximal, complemented
5#20285375C211yesmaximal, characteristic
6D53⋊M32—21no—
7#16811545—21no—
8#16811520—21no—
9#16811516—21no—
10#16811515C2212yes—
11#16811515C2211yescharacteristic
12#16811515C2211yescharacteristic
13#16811236C412yes—
14#16811003C411yescharacteristic
15#16810993C2212yes—
16#16810882—21no—
17#16810865—21no—
18#16810729—21no—
19#16810717—21no—
20#16810577C2211yescharacteristic
21C53×#499C411yescharacteristic
22#13133292—22no—
23#13133291—21no—
24#13133291—21no—
25#13133291—22no—
26D53⋊C16—22no—
27C2×Dic212—21no—
28C2×Dic212—21no—
29C2×#11719912D411yescharacteristic
30C2×C8×D53D411yescharacteristic
31#13133217C2×C412yes—
32#13133212—42no—
33#13133208—42no—
34#13133197—22no—
35#13133182—22no—
36#13133174C2×C411yescharacteristic
37#13133140—22no—
38#13133140—22no—
39#13133128C2311yescharacteristic
40#13133127D411yescharacteristic
41C8×Dic53D411yescharacteristic
42#125×C53C2×C412yes—
43#107×C53—21no—
44#106×C53—21no—
45C4×C8×C53C2×C411yescharacteristic
46C2×C4×D53C22⋊C411yescharacteristic
47C2×Dic106—22no—
48C2×C8×C53C22⋊C411yescharacteristic
49C2×C8×C53C22⋊C411yescharacteristic
50C4⋊C4×C53—22no—
51C4⋊C4×C53—42no—
52C42×C53C22×C411yescharacteristic
53#11719921—21no—
54#11719921—42no—
55C4⋊Dic53—42no—
56C4⋊Dic53—42no—
57#11719919—21no—
58#11719919—42no—
59C4×Dic53—42no—
60C4×Dic53C22⋊C411yescharacteristic
61C2×#88322C2×D411yescharacteristic
62C2×#88322C2×D411yescharacteristic
63Dic212—42no—
64Dic212—41no—
65Dic212—41no—
66#11719912—41no—
67C8×D53—21no—
68C8×D53—21no—
69#11719910—22no—
70#11719910—22no—
71#11719910—42no—
72C22×D53#10011yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—42no—
75C2×C4×C53C2×C22⋊C411yescharacteristic
76C2×Dic53—21no—
77C2×Dic53—42no—
78C2×Dic53#10011yescharacteristic
79C4×D53—21no—
80C4×D53—21no—
81Dic106—42no—
82Dic106—42no—
83C8×C53—21no—
84C8×C53#10411yescharacteristic
85C8×C53#10411yescharacteristic
86#88322—21no—
87#88322—21no—
88#11914—531nomaximal, Sylow
89C22×C53C2×#10011yescharacteristic
90C2×D53—21no—
91C2×D53—41no—
92C4×C53—41no—
93C4×C53—82no—
94C4×C53C2×#10411yescharacteristic
95C4×C53#42111yescharacteristic
96Dic53—21no—
97Dic53—41no—
98Dic53—82no—
99#3177—531no—
100#2166—531no—
101#2110—531no—
102#1356—532no—
103#1356—532no—
104C2×C53—21no—
105C2×C53#153011yescharacteristic
106D53—41no—
107#541—532no—
108C2×M32—1061no—
109#499#615711yescharacteristic
110#494—531no—
111#491—532no—
112#470—1061no—
113#467—1061no—
114#441—1061no—
115C4×Q16—1061no—
116#405—531no—
117#364—1061no—
118#349—1061no—
119#347—531no—
120#347—531no—
121#347—532no—
122#347—1061no—
123C53#1191411yescharacteristic, complemented, Sylow
124C2×Q16—1061no—
125C2×Q16—1061no—
126C2×M16—531no—
127C22×C8—531no—
128C4⋊Q8—532no—
129#126—2122no—
130#125—2122no—
131#125C2×#615712yes—
132C22⋊Q8—1062no—
133C4×Q8—1062no—
134#117—531no—
135M32—1061no—
136M32—1061no—
137M32—1062no—
138M32—1062no—
139C2×C16—1062no—
140#107—21no—
141#106—21no—
142#103—1062no—
143#103—1062no—
144#97—531no—
145#97—531no—
146C4×C8—531no—
147C4×C8C2×#615711yescharacteristic
148C2×Q8—1062no—
149C22×C4—531no—
150Q16—2122no—
151Q16—2121no—
152Q16—2121no—
153M16—2121no—
154C2×C8—531no—
155C2×C8—531no—
156C2×C8—1061no—
157C2×C8—1061no—
158C2×C8C22⋊#615711yescharacteristic
159C2×C8C22⋊#615711yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—1061no—
163C4⋊C4—2122no—
164C4⋊C4—2122no—
165C4⋊C4—2122no—
166C22⋊C4—1061no—
167C22⋊C4—2122no—
168C42—531no—
169C42—2122no—
170C42C22×#615711yescharacteristic
171C16—1062no—
172C16—1062no—
173C16—2122no—
174C23—531no—
175Q8—2122no—
176Q8—2122no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—531no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—2122no—
184C2×C4C2×C22⋊#615711yescharacteristic
185C8—21no—
186C8—1061no—
187C8—1061no—
188C8#1313328211yescharacteristic
189C8#1313328211yescharacteristic
190C22—1061no—
191C22—2121no—
192C22D53⋊#10011yescharacteristic
193C4—41no—
194C4—82no—
195C4—1061no—
196C4—4242no—
197C4—2121no—
198C4C2×#1313328211yescharacteristic
199C4#1681162211yescharacteristic
200C2—21no—
201C2—2121no—
202C2#2029018611yescharacteristic
subgroupG/H
centerC2#20290186
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#499C4
socleC2×C53#1530
prime radicalC22×D53#100
cosocle#13133128C23
hypercenter#499#6157
hypocenterC53#11914
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #499

Complements

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

NH ≅ G/Ntype
3. #20289722C2semidirect
4. #20289721C2semidirect
123. C53#11914semidirect

Maximal subgroups

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

subgroupindex
#202910902
#202909372
#202897222
#202897212
#202853752
#1191453