Finite Group Databasebeta

← #27297089

Subgroups — #27297089

Order 13568.

394 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 59 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#20280088C211yesmaximal, characteristic
2#20273723C212yesmaximal
3#20272720C212yesmaximal
4#20272661C211yesmaximal, characteristic
5C53×#2060C211yesmaximal, characteristic
6C2×C4×Dic106—21no—
7#16810951—22no—
8#16810553—21no—
9#16810553C2212yes—
10#16810528—22no—
11#16810522—22no—
12#16810360—22no—
13#16810349C2212yes—
14C53×#544C2211yescharacteristic
15C53×#428—21no—
16C53×#384C2211yescharacteristic
17C53×#343C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106—21no—
20C4×Dic106—42no—
21C4×Dic106—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53D412yes—
25C2×#11719919—22no—
26C2×C4×Dic53—22no—
27#13133148—42no—
28#13133141—42no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—21no—
32#13133124—22no—
33#13133124D412yes—
34C2×M16×C53—21no—
35C4⋊Q8×C53—41no—
36#125×C53—41no—
37C22⋊Q8×C53—22no—
38#117×C53—21no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53D411yescharacteristic
43#107×C53—41no—
44#106×C53—41no—
45C2×Q8×C53—22no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C2×Dic106—24no—
53C2×Dic106—42no—
54C2×Dic106—42no—
55M16×C53—22no—
56M16×C53—41no—
57C2×C8×C53—41no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—21no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—44no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—41no—
65C22⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—41no—
68C42×C53SD1612yes—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71C4⋊Dic53—81no—
72#11719919—42no—
73#11719919—42no—
74C4×Dic53—44no—
75C2×#88322—42no—
76C23×C53#12011yescharacteristic
77Q8×C53—82no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53C2×SD1611yescharacteristic
90C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97Dic106—48no—
98C8×C53—42no—
99#88322—82no—
100#12247—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—21no—
103C22×C53#45011yescharacteristic
104C22×C53#45611yescharacteristic
105C22×C53B(4,𝔽2)11yescharacteristic
106C4×C53—24no—
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—42no—
110C4×C53—81no—
111C4×C53—81no—
112C4×C53—81no—
113Dic53—44no—
114Dic53—82no—
115#2582—531no—
116#2060D5311yescharacteristic
117#1611—532no—
118#1346—532no—
119#1325—531no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#160411yescharacteristic
123C2×C53#219811yescharacteristic
124C2×C53#220511yescharacteristic
125#544C2×D5311yescharacteristic
126C2×C4×Q8—1061no—
127#483—1062no—
128#428—21no—
129#398—532no—
130#398—1061no—
131#391—1062no—
132#389—1062no—
133#384C2×D5311yescharacteristic
134#343C2×D5311yescharacteristic
135#339—21no—
136#328—1062no—
137#324—532no—
138C53#1224711yescharacteristic, complemented, Sylow
139C22×Q8—1061no—
140C2×M16—21no—
141C4⋊Q8—41no—
142#125—41no—
143C22⋊Q8—22no—
144C4×Q8—2122no—
145C4×Q8—2122no—
146#117—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—1061no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4C22⋊D5311yescharacteristic
153C2×C4⋊C4C22×D5311yescharacteristic
154C2×C42—1062no—
155C2×C42C22⋊D5311yescharacteristic
156#107—41no—
157#106—41no—
158#106—2122no—
159#103—2122no—
160C22⋊C8—1062no—
161#95—532no—
162#95—1062no—
163#95—1061no—
164#95—1062no—
165C2×Q8—22no—
166C2×Q8—1064no—
167C2×Q8—2122no—
168C2×Q8—2122no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4D4×D5312yes—
174C22×C4C2×C22⋊D5311yescharacteristic
175M16—22no—
176M16—41no—
177C2×C8—41no—
178C2×C8—2122no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—4241no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C4⋊C4—2122no—
191C22⋊C4—41no—
192C42—21no—
193C42—41no—
194C42—2124no—
195C42#1171992412yes—
196C23#1313318511yescharacteristic
197Q8—82no—
198Q8—2128no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1064no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4C2×#1171992411yescharacteristic
217C2×C4#1313324211yescharacteristic
218C8—42no—
219C8—4242no—
220C22—21no—
221C22—21no—
222C22#1681077311yescharacteristic
223C22#1681080511yescharacteristic
224C22#1681080911yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—4242no—
233C4—2124no—
234C2—21no—
235C2—21no—
236C2#2027368311yescharacteristic
237C2#2027671011yescharacteristic
238C2#2027674411yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2060C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2060D53
hypocenterC53#12247
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
138. C53#12247semidirect

Maximal subgroups

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

subgroupindex
#202800882
#202737232
#202727202
#202726612
C53×#20602
#1224753