Finite Group Databasebeta

← #27270770

Subgroups — #27270770

Order 13568.

286 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 49 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#20275725C211yesmaximal, characteristic
2#20272639C211yesmaximal, characteristic
3C53×#1289C211yesmaximal, characteristic
4C22×Dic212C411yescharacteristic, complemented
5C2×#13133196—21no—
6C2×#13133142—21no—
7C22⋊#11719908—21no—
8#16810386C411yescharacteristic, complemented
9C2×C22⋊C8×C53C2211yescharacteristic
10C53×#380—21no—
11C22×Dic106—21no—
12C2×Dic212—42no—
13C2×Dic212—41no—
14#13133196—42no—
15#13133196—42no—
16C2×C4⋊Dic53—21no—
17C2×#11719919—41no—
18C2×C22⋊Dic53—41no—
19C2×#11719908—41no—
20#13133142—42no—
21C23×C4×C53D411yescharacteristic
22C22×C8×C53C2×C411yescharacteristic
23C22×C8×C53D411yescharacteristic
24C2×C4⋊C4×C53—41no—
25C2×C22⋊C4×C53—41no—
26C22⋊C8×C53—41no—
27#95×C53—22no—
28C24×C53D811yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53—41no—
32C22×C4×C53—41no—
33C22×C4×C53—42no—
34C22×C4×C53C22⋊C411yescharacteristic
35C22×C4×C53SD1611yescharacteristic
36C22×Dic53—41no—
37C22×Dic53—41no—
38C2×Dic106—22no—
39C2×Dic106—41no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106—41no—
43C2×Dic106—41no—
44C2×C8×C53—21no—
45C2×C8×C53—21no—
46C2×C8×C53—41no—
47C2×C8×C53—41no—
48C4⋊C4×C53—81no—
49C4⋊C4×C53—81no—
50C22⋊C4×C53—81no—
51C22⋊C4×C53—81no—
52C22⋊Dic53—44no—
53C4⋊Dic53—22no—
54C4⋊Dic53—41no—
55#11719919—44no—
56Dic212—42no—
57Dic212—42no—
58#11719908—81no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53#10211yescharacteristic
64C23×C53SD3211yescharacteristic
65C23×C53Q3211yescharacteristic
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—81no—
73C2×C4×C53—82no—
74C2×C4×C53—82no—
75C2×C4×C53—82no—
76C2×C4×C53#9911yescharacteristic
77C2×C4×C53#10411yescharacteristic
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—81no—
81C2×Dic53—81no—
82C2×Dic53—81no—
83C2×Dic53—81no—
84Dic106—42no—
85Dic106—42no—
86Dic106—81no—
87Dic106—81no—
88C8×C53—41no—
89C8×C53—41no—
90#7078—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53#32711yescharacteristic
101C22×C53#35811yescharacteristic
102C22×C53#36011yescharacteristic
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—162no—
106C4×C53AΣL(1,𝔽16)11yescharacteristic
107C4×C53#35311yescharacteristic
108Dic53—82no—
109Dic53—82no—
110#2001—531no—
111#1314—531no—
112#1289D5311yescharacteristic
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—41no—
116C2×C53—41no—
117C2×C53#134011yescharacteristic
118C2×C53#134311yescharacteristic
119C2×C53#134811yescharacteristic
120C22×Q16—531no—
121C2×C22⋊Q8—1061no—
122C2×#103—1061no—
123C2×C22⋊C8C2×D5311yescharacteristic
124#380—21no—
125C22⋊C16—1061no—
126#340—531no—
127C53#707811yescharacteristic, complemented, Sylow
128C22×Q8—1061no—
129C23×C4C22⋊D5311yescharacteristic
130C2×Q16—2121no—
131C2×Q16—2122no—
132C22×C8C4×D5311yescharacteristic
133C22×C8D21211yescharacteristic
134C22⋊Q8—2122no—
135C22⋊Q8—2122no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—1061no—
138C2×C4⋊C4—2121no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—2121no—
141C2×C16—2121no—
142#103—2122no—
143C22⋊C8—41no—
144#95—22no—
145C24#1171992211yescharacteristic
146C2×Q8—1062no—
147C2×Q8—2121no—
148C2×Q8—2121no—
149C2×Q8—2121no—
150C2×Q8—2121no—
151C2×Q8—2121no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—2121no—
158C22×C4—2121no—
159C22×C4#1171992111yescharacteristic
160C22×C4#1171992411yescharacteristic
161Q16—2122no—
162Q16—2122no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—1062no—
170C4⋊C4—2121no—
171C4⋊C4—2124no—
172C22⋊C4—81no—
173C22⋊C4—81no—
174C22⋊C4—2124no—
175C16—4241no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#1313313711yescharacteristic
181C23#1313316111yescharacteristic
182C23#1313316311yescharacteristic
183Q8—4241no—
184Q8—4241no—
185Q8—2122no—
186Q8—2122no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—4241no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4#1313313111yescharacteristic
204C2×C4#1313314511yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#1681035611yescharacteristic
217C22#1681042611yescharacteristic
218C22#1681043211yescharacteristic
219C4—21no—
220C4—41no—
221C4—162no—
222C4—4242no—
223C4—4242no—
224C4#1681040911yescharacteristic
225C4#1681041511yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#2027270211yescharacteristic
231C2#2027271111yescharacteristic
232C2#2027272611yescharacteristic
subgroupG/H
centerC22#16810356
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#1289C2
socleC22×C53#327
prime radicalC24×C53D8
cosocleC2×C22⋊C8×C53C22
hypercenter#1289D53
hypocenterC53#7078
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1289

Complements

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

NH ≅ G/Ntype
4. C22×Dic212C4semidirect
8. #16810386C4semidirect
127. C53#7078semidirect

Maximal subgroups

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

subgroupindex
#202757252
#202726392
C53×#12892
#707853