Finite Group Databasebeta

← #28309744

Subgroups — #28309744

Order 13568.

296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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#20291074C211yesmaximal, characteristic
2#20290271C211yesmaximal, characteristic
3#20289731C214yesmaximal, complemented
4#20285226C211yesmaximal, characteristic
5#16811627—21no—
6#16811544C2212yes—
7#16811518—21no—
8C4×#11719910—21no—
9#16811485—21no—
10#16811485C2212yes—
11#16811178—21no—
12#16811028C411yescharacteristic
13#16811024—21no—
14#16811024C2212yes—
15#16811019—21no—
16#16810660—21no—
17#16810660C412yes—
18C4×C8×D53C2211yescharacteristic
19C53×#500C411yescharacteristic
20C2×#11719936—41no—
21C2×#11719910—22no—
22C4⋊#88324—22no—
23C4⋊#88324—41no—
24C2×C8×D53C2×C411yescharacteristic
25C2×C8×D53D411yescharacteristic
26#13133217—21no—
27#13133208—22no—
28#13133197—41no—
29#13133193—41no—
30Q8×Dic53—41no—
31#13133182—41no—
32#13133175—22no—
33C42×D53D411yescharacteristic
34#13133151—21no—
35#13133151C2×C412yes—
36#13133148—22no—
37#13133148—22no—
38C4×#88322D411yescharacteristic
39C8×Dic53D411yescharacteristic
40C8×Dic53C2311yescharacteristic
41C4⋊Q8×C53—21no—
42#107×C53—21no—
43#107×C53C2×C412yes—
44C4×C8×C53C2×C411yescharacteristic
45C2×Q8×C53—41no—
46C2×C4×D53C22⋊C411yescharacteristic
47C2×C4×D53D811yescharacteristic
48C2×C4×D53SD1611yescharacteristic
49C2×Dic106—41no—
50C2×#88324—42no—
51#11719936—82no—
52C2×C8×C53C22⋊C411yescharacteristic
53C2×C8×C53C22×C411yescharacteristic
54C4⋊C4×C53—22no—
55C4⋊C4×C53—41no—
56C42×C53C22⋊C411yescharacteristic
57#11719921—42no—
58C4⋊Dic53—22no—
59C4⋊Dic53—41no—
60C4⋊Dic53—42no—
61#11719919—42no—
62C4×Dic53—42no—
63C4×Dic53D811yescharacteristic
64C4×Dic53SD1611yescharacteristic
65C4×Dic53C2×D411yescharacteristic
66C2×#88322C22⋊C411yescharacteristic
67C2×#88322C2×D411yescharacteristic
68C8×D53—22no—
69C8×D53—22no—
70#11719910—42no—
71C22×D53#10211yescharacteristic
72Q8×C53—82no—
73C2×C4×C53—42no—
74C2×C4×C53#10211yescharacteristic
75C2×C4×C53#10211yescharacteristic
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C2×Dic53—42no—
78C2×Dic53#10211yescharacteristic
79C2×Dic53C2×D811yescharacteristic
80C2×Dic53C2×SD1611yescharacteristic
81C4×D53—22no—
82C4×D53—22no—
83C4×D53—21no—
84C4×D53—21no—
85Dic106—82no—
86#88324—82no—
87C8×C53—21no—
88C8×C53—21no—
89#88322#10112yes—
90#88322#13712yes—
91#11968—531nomaximal, Sylow
92C22×C53C2×#10211yescharacteristic
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C4×C53—82no—
97C4×C53#35912yes—
98C4×C53#36012yes—
99C4×C53C2×#10111yescharacteristic
100C4×C53#41911yescharacteristic
101Dic53—21no—
102Dic53—21no—
103Dic53—21no—
104Dic53—82no—
105#3160—531no—
106#2162—531no—
107#1566—531no—
108#1363—534no—
109C2×C53#154611yescharacteristic
110C2×C53C2×#35911yescharacteristic
111C2×C53C2×#36011yescharacteristic
112D53—41no—
113#533—1061no—
114#500—531no—
115#500#615711yescharacteristic
116#499—532no—
117#499—1061no—
118#498—1061no—
119#427—532no—
120#427—1061no—
121#423—1061no—
122C2×C4×C8—531no—
123C4⋊C16—532no—
124C22⋊C16—1061no—
125C4×C16—1061no—
126#334—532no—
127#334—1061no—
128C53#1196811yescharacteristic, complemented, Sylow
129C2×M16—2121no—
130C22×C8—531no—
131C22×C8—531no—
132C4⋊Q8—21no—
133C4⋊Q8—1061no—
134#125—1062no—
135C22⋊Q8—2121no—
136C22⋊Q8—2121no—
137C4×Q8—2121no—
138C4×Q8—2121no—
139#117—1062no—
140C2×C42—531no—
141C2×C16—1062no—
142#107—21no—
143#107—532no—
144#107—1061no—
145#107C2×#615712yes—
146#106—1062no—
147#106—1062no—
148C4⋊C8—1062no—
149C4⋊C8—2121no—
150C4×C8—531no—
151C4×C8—531no—
152C4×C8—531no—
153C4×C8C2×#615711yescharacteristic
154C2×Q8—41no—
155C2×Q8—2121no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—531no—
159M16—4242no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8—1062no—
163C2×C8—1062no—
164C2×C8—2122no—
165C2×C8C22⋊#615711yescharacteristic
166C2×C8C22×#615711yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—1062no—
170C4⋊C4—2122no—
171C4⋊C4—2122no—
172C4⋊C4—2121no—
173C22⋊C4—2122no—
174C42—531no—
175C42—531no—
176C42—531no—
177C42—2122no—
178C42C22⋊#615711yescharacteristic
179C16—2122no—
180C23—531no—
181Q8—82no—
182Q8—4242no—
183C2×C4—42no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—531no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—2122no—
192C2×C4#1313327911yescharacteristic
193C2×C4#1313327911yescharacteristic
194C2×C4C2×C22⋊#615711yescharacteristic
195C8—21no—
196C8—21no—
197C8—532no—
198C8—532no—
199C8—4242no—
200C22—1061no—
201C22—1061no—
202C22—1061no—
203C22C2×#1313327911yescharacteristic
204C4—82no—
205C4—1061no—
206C4—1061no—
207C4—1061no—
208C4—4242no—
209C4#1681153712yes—
210C4#1681154012yes—
211C4D53⋊#10111yescharacteristic
212C4#1681161611yescharacteristic
213C2—2121no—
214C2#2029022511yescharacteristic
215C2C2×#1681153711yescharacteristic
216C2C2×#1681154011yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#500C4
socleC22×C53C2×#102
prime radicalC22×D53#102
cosocleC8×Dic53C23
hypercenter#500#6157
hypocenterC53#11968
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #20289731C2semidirect
128. C53#11968semidirect

Maximal subgroups

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

subgroupindex
#202910742
#202902712
#202897312
#202852262
#1196853