Finite Group Databasebeta

← #28308229

Subgroups — #28308229

Order 13568.

284 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 64 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#20291084C212yesmaximal, complemented
2#20289662C212yesmaximal, complemented
3#20289648C212yesmaximal
4#20283351C211yesmaximal, characteristic
5#16811546—22no—
6#16811519C2214yes—
7#16811515—22no—
8C2×#13133174C411yescharacteristic
9#16810685—22no—
10#16810682—22no—
11#16810646C411yescharacteristic
12#16810646C2211yescharacteristic
13#16810586C411yescharacteristic
14#16810586C2211yescharacteristic
15C4×#11719917C2211yescharacteristic
16C4×M16×C53C411yescharacteristic
17#13133291—24no—
18C22×C4×D53C2×C411yescharacteristic
19C2×#11719917C2×C411yescharacteristic
20C2×#11719917C2311yescharacteristic
21#13133174—21no—
22#13133174—21no—
23#13133174—22no—
24C2×#11719919—21no—
25C2×#11719921—21no—
26C2×C4×Dic53C2×C411yescharacteristic
27#13133146—22no—
28#13133146—22no—
29#13133129—22no—
30#13133129—22no—
31#13133128D412yes—
32C4×#88322D412yes—
33C2×M16×C53C2×C411yescharacteristic
34C2×M16×C53C2×C411yescharacteristic
35C2×C42×C53C2×C411yescharacteristic
36#97×C53—21no—
37C4×C8×C53—21no—
38C23×D53M1611yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—21no—
42C22×Dic53M1611yescharacteristic
43C2×C4×D53—21no—
44C2×C4×D53—21no—
45C2×C4×D53—41no—
46C2×C4×D53—41no—
47M16×C53—41no—
48M16×C53—41no—
49C2×C8×C53—21no—
50C2×C8×C53—21no—
51C42×C53—21no—
52C42×C53C22⋊C411yescharacteristic
53C42×C53C22⋊C411yescharacteristic
54#11719921—22no—
55#11719921—22no—
56#11719919—22no—
57#11719919—22no—
58C4×Dic53—21no—
59C4×Dic53M1612yes—
60#11719917—41no—
61#11719917—41no—
62C2×#88322C22⋊C412yes—
63C2×#88322C2×D412yes—
64#11719910—44no—
65C23×C53C2×M1611yescharacteristic
66C22×D53—21no—
67C22×D53—41no—
68C22×D53—81no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—81no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53C2×M1611yescharacteristic
76C2×Dic53—21no—
77C2×Dic53—22no—
78C2×Dic53—41no—
79C2×Dic53—81no—
80C4×D53—42no—
81C4×D53—41no—
82C4×D53—41no—
83C8×C53—41no—
84C8×C53—41no—
85#88322—22no—
86#88322—22no—
87#11522—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—41no—
90C22×C53#40711yescharacteristic
91C2×D53—42no—
92C2×D53—41no—
93C2×D53—82no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53#41011yescharacteristic
98C4×C53#41311yescharacteristic
99Dic53—42no—
100Dic53—41no—
101#2921—531no—
102#2165—532no—
103#1327—532no—
104#1320—532no—
105C2×C53—21no—
106C2×C53—41no—
107C2×C53#146411yescharacteristic
108D53—81no—
109C2×#117—531no—
110#433—1062no—
111#433—1062no—
112#423—531no—
113#423—531no—
114#407—531no—
115#407—531no—
116C4×M16—531no—
117C4×M16#615711yescharacteristic
118#364—1062no—
119#349—534no—
120#347—1062no—
121C53#1152211yescharacteristic, complemented, Sylow
122C23×C4—531no—
123C2×M16—531no—
124C2×M16—531no—
125C2×M16C2×#615711yescharacteristic
126C2×M16C2×#615711yescharacteristic
127#117—1062no—
128#117—1061no—
129#117—1061no—
130C2×C4⋊C4—1061no—
131C2×C22⋊C4—1061no—
132C2×C42—531no—
133C2×C42C2×#615711yescharacteristic
134M32—1064no—
135C4⋊C8—1062no—
136C4⋊C8—1062no—
137C22⋊C8—1062no—
138C22⋊C8—1062no—
139#97—21no—
140#97—532no—
141C4×C8—21no—
142C4×C8—532no—
143C24—531no—
144C22×C4—21no—
145C22×C4—531no—
146C22×C4—1061no—
147C22×C4—1061no—
148C22×C4—1061no—
149C22×C4—2121no—
150C22×C4—2121no—
151C22×C4C22×#615711yescharacteristic
152M16—41no—
153M16—41no—
154M16—2121no—
155M16—2121no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—532no—
159C2×C8—532no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C22⋊C4—1062no—
163C22⋊C4—1062no—
164C42—21no—
165C42—532no—
166C42—1061no—
167C42C22⋊#615711yescharacteristic
168C42C22⋊#615711yescharacteristic
169C16—2124no—
170C23—1061no—
171C23—4241no—
172C23—2121no—
173C23D53⋊M1611yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—81no—
179C2×C4—1062no—
180C2×C4—1061no—
181C2×C4—4241no—
182C2×C4—2122no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4C2×C22⋊#615711yescharacteristic
187C2×C4D53⋊M1611yescharacteristic
188C8—41no—
189C8—41no—
190C8—1062no—
191C8—1062no—
192C22—21no—
193C22—41no—
194C22—4242no—
195C22—2122no—
196C22—2121no—
197C22#1681159511yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—2122no—
202C4—2121no—
203C4#1681160211yescharacteristic
204C4#1681160911yescharacteristic
205C2—21no—
206C2—41no—
207C2—4241no—
208C2#2028996811yescharacteristic
subgroupG/H
centerC4#16811602
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×M16×C53C4
socleC2×C53#1464
prime radicalC23×D53M16
cosocleC2×#11719917C23
hypercenterC4×M16#6157
hypocenterC53#11522
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289662C2semidirect
121. C53#11522semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202896622
#202896482
#202833512
#1152253