Finite Group Databasebeta

← #28308227

Subgroups — #28308227

Order 13568.

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

normal subgroups only

This group has 284 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#20289661C212yesmaximal, complemented
3#20289647C212yesmaximal
4#20283353C211yesmaximal, characteristic
5#16811546—22no—
6#16811519C2214yes—
7#16811515—22no—
8#16810684—22no—
9#16810684—22no—
10#16810647C411yescharacteristic
11#16810647C2211yescharacteristic
12#16810587C411yescharacteristic
13#16810587C2211yescharacteristic
14C4×#11719917C411yescharacteristic
15C4×#11719917C2211yescharacteristic
16C2×#117×C53C411yescharacteristic
17#13133291—24no—
18C2×#11719917C2×C411yescharacteristic
19C2×#11719917C2×C411yescharacteristic
20C2×#11719917C2×C411yescharacteristic
21C2×#11719917C2311yescharacteristic
22C4⋊#88322—22no—
23C4⋊#88322—22no—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133128—21no—
27#13133128D412yes—
28C4×#88322—21no—
29C4×#88322D412yes—
30C23×C4×C53C2×C411yescharacteristic
31#117×C53—21no—
32#117×C53—21no—
33#117×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C22⋊C4×C53—21no—
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C24×C53M1611yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53M1611yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47C4⋊C4×C53—22no—
48C4⋊C4×C53—22no—
49C22⋊C4×C53—22no—
50C22⋊C4×C53—22no—
51C42×C53—21no—
52C42×C53—21no—
53C42×C53C22⋊C411yescharacteristic
54C42×C53C22⋊C411yescharacteristic
55C42×C53M1612yes—
56#11719917—41no—
57#11719917—41no—
58#11719917—41no—
59#11719917—41no—
60C2×#88322—21no—
61C2×#88322—21no—
62C2×#88322C22⋊C412yes—
63C2×#88322C2×D412yes—
64#11719910—44no—
65C23×C53—21no—
66C23×C53—41no—
67C23×C53—81no—
68C23×C53C2×M1611yescharacteristic
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—42no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—81no—
80C2×C4×C53—81no—
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53C2×M1611yescharacteristic
83#88322—22no—
84#88322—22no—
85#88322—41no—
86#88322—41no—
87#11522—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—42no—
91C22×C53—41no—
92C22×C53—82no—
93C22×C53#40711yescharacteristic
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53—42no—
98C4×C53—41no—
99C4×C53#41011yescharacteristic
100C4×C53#41311yescharacteristic
101#2921—531no—
102#2165—532no—
103#1327—532no—
104#1320—532no—
105C2×C53—21no—
106C2×C53—41no—
107C2×C53—81no—
108C2×C53#146411yescharacteristic
109C2×#117#615711yescharacteristic
110#433—1062no—
111#433—1062no—
112#423—531no—
113#423—531no—
114#407—531no—
115#407—531no—
116C4×M16—531no—
117C4×M16—531no—
118#364—1062no—
119#349—534no—
120#347—1062no—
121C53#1152211yescharacteristic, complemented, Sylow
122C23×C4C2×#615711yescharacteristic
123C2×M16—531no—
124C2×M16—531no—
125C2×M16—531no—
126C2×M16—531no—
127#117—22no—
128#117—21no—
129#117—21no—
130C2×C4⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42C2×#615711yescharacteristic
133C2×C42C2×#615711yescharacteristic
134M32—1064no—
135C4⋊C8—1062no—
136C4⋊C8—1062no—
137C22⋊C8—1062no—
138C22⋊C8—1062no—
139#97—532no—
140#97—1061no—
141C4×C8—532no—
142C4×C8—1061no—
143C24#1171994011yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#1171994011yescharacteristic
151C22×C4C22×#615711yescharacteristic
152M16—2121no—
153M16—2121no—
154M16—2121no—
155M16—2121no—
156C2×C8—532no—
157C2×C8—532no—
158C2×C8—1061no—
159C2×C8—1061no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—21no—
165C42—21no—
166C42#1171994012yes—
167C42C22⋊#615711yescharacteristic
168C42C22⋊#615711yescharacteristic
169C16—2124no—
170C23—21no—
171C23—41no—
172C23—81no—
173C23C2×#1171994011yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4C2×C22⋊#615711yescharacteristic
187C2×C4C2×#1171994011yescharacteristic
188C8—1062no—
189C8—1062no—
190C8—2121no—
191C8—2121no—
192C22—21no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—82no—
197C22#1681159311yescharacteristic
198C4—21no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4#1681160211yescharacteristic
204C4#1681160911yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2#2028996611yescharacteristic
subgroupG/H
centerC4#16811602
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC2×C53#1464
prime radicalC24×C53M16
cosocleC2×#11719917C23
hypercenterC2×#117#6157
hypocenterC53#11522
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289661C2semidirect
121. C53#11522semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202896612
#202896472
#202833532
#1152253