Finite Group Databasebeta

← #28308235

Subgroups — #28308235

Order 13568.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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#20289663C212yesmaximal, complemented
3#20289649C212yesmaximal
4#20283359C211yesmaximal, characteristic
5#16811546—22no—
6#16811519C2214yes—
7#16811515—22no—
8#16810684—22no—
9#16810676—22no—
10#16810650C411yescharacteristic
11#16810650C2211yescharacteristic
12#16810590C411yescharacteristic
13#16810590C2211yescharacteristic
14C4×#11719917C2211yescharacteristic
15C2×C4×#88322C411yescharacteristic
16C4×#41×C53C411yescharacteristic
17#13133291—24no—
18C2×#11719917C2×C411yescharacteristic
19C2×#11719917C2311yescharacteristic
20C22×#88322C2×C411yescharacteristic
21C22×#88322C2×C411yescharacteristic
22C4⋊#88322—22no—
23C4⋊#88322—22no—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133128D412yes—
27C4×#88322—21no—
28C4×#88322—21no—
29C4×#88322D412yes—
30C2×#41×C53C2×C411yescharacteristic
31C4×Q8×C53—21no—
32C4×D4×C53—22no—
33C4×D4×C53—21no—
34#117×C53—21no—
35#117×C53C2×C411yescharacteristic
36C2×C42×C53—21no—
37C2×C42×C53C2×C411yescharacteristic
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53M1611yescharacteristic
41C2×D4×C53—21no—
42C2×D4×C53M1611yescharacteristic
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22×C411yescharacteristic
47C4⋊C4×C53—22no—
48C4⋊C4×C53—21no—
49C22⋊C4×C53—22no—
50C22⋊C4×C53M1612yes—
51C42×C53—22no—
52C42×C53—21no—
53C42×C53—21no—
54C42×C53C22⋊C411yescharacteristic
55C42×C53C22⋊C411yescharacteristic
56#11719917—41no—
57#11719917—41no—
58C2×#88322—21no—
59C2×#88322—21no—
60C2×#88322—41no—
61C2×#88322—41no—
62C2×#88322C22⋊C412yes—
63C2×#88322C2×D412yes—
64#11719910—44no—
65C23×C53—21no—
66C23×C53C2×M1611yescharacteristic
67Q8×C53—41no—
68D4×C53—41no—
69D4×C53—42no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—44no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C2×M1611yescharacteristic
82#88322—22no—
83#88322—22no—
84#88322—41no—
85#88322—41no—
86#11524—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—41no—
89C22×C53—42no—
90C22×C53—41no—
91C22×C53#40711yescharacteristic
92C4×C53—22no—
93C4×C53—21no—
94C4×C53—42no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53#41011yescharacteristic
98C4×C53#41311yescharacteristic
99#2922—531no—
100#2165—532no—
101#1328—532no—
102#1321—532no—
103C2×C53—21no—
104C2×C53—41no—
105C2×C53—81no—
106C2×C53#146411yescharacteristic
107C4×#41#615711yescharacteristic
108#433—1062no—
109#431—1062no—
110#424—531no—
111#424—531no—
112#408—531no—
113#408—531no—
114C4×M16—531no—
115C2×C4×C8—531no—
116#364—1062no—
117#349—534no—
118#347—1062no—
119C53#1152411yescharacteristic, complemented, Sylow
120C2×#41C2×#615711yescharacteristic
121C2×M16—531no—
122C2×M16—531no—
123C22×C8—531no—
124C22×C8—531no—
125C4×Q8—21no—
126C4×D4—21no—
127C4×D4—22no—
128#117—21no—
129#117C2×#615711yescharacteristic
130C2×C42—21no—
131C2×C42C2×#615711yescharacteristic
132M32—1064no—
133C4⋊C8—1062no—
134C4⋊C8—1062no—
135C22⋊C8—1062no—
136C22⋊C8—1062no—
137#97—532no—
138C4×C8—532no—
139C4×C8—1061no—
140C4×C8—1061no—
141#41—41no—
142#41—41no—
143C2×Q8#1171994011yescharacteristic
144C2×D4—21no—
145C2×D4#1171994011yescharacteristic
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4C22×#615711yescharacteristic
150M16—2121no—
151M16—2121no—
152C2×C8—532no—
153C2×C8—532no—
154C2×C8—1061no—
155C2×C8—1061no—
156C2×C8—2121no—
157C2×C8—2121no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C22⋊C4—22no—
161C22⋊C4#1171994012yes—
162C42—22no—
163C42—21no—
164C42—21no—
165C42C22⋊#615711yescharacteristic
166C42C22⋊#615711yescharacteristic
167C16—2124no—
168C23—21no—
169C23C2×#1171994011yescharacteristic
170Q8—41no—
171D4—41no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4C2×C22⋊#615711yescharacteristic
184C2×C4C2×#1171994011yescharacteristic
185C8—1062no—
186C8—1062no—
187C8—2121no—
188C8—2121no—
189C22—21no—
190C22—41no—
191C22—42no—
192C22—41no—
193C22#1681159311yescharacteristic
194C4—22no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4#1681160211yescharacteristic
200C4#1681160911yescharacteristic
201C2—21no—
202C2—41no—
203C2—81no—
204C2#2028996611yescharacteristic
subgroupG/H
centerC4#16811602
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×#41×C53C4
socleC2×C53#1464
prime radicalC2×#41×C53C2×C4
cosocleC2×#11719917C23
hypercenterC4×#41#6157
hypocenterC53#11524
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289663C2semidirect
119. C53#11524semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202896632
#202896492
#202833592
#1152453