Finite Group Databasebeta

← #27271684

Subgroups — #27271684

Order 13568.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#20275871C211yesmaximal, characteristic
2#20272840C211yesmaximal, characteristic
3C53×#1270C211yesmaximal, characteristic
4#16811254C411yescharacteristic, complemented
5#16810628—21no—
6C2×#13133132—21no—
7#16810392C411yescharacteristic, complemented
8#16810383—21no—
9C53×#390C2211yescharacteristic
10C53×#336—21no—
11C2×#11719917—21no—
12#13133216—41no—
13#13133208—41no—
14#13133196—41no—
15#13133176—21no—
16C2×C22⋊Dic53—41no—
17#13133138—41no—
18#13133138—41no—
19#13133132—22no—
20#13133132—41no—
21C22×D4×C53D411yescharacteristic
22C22×C8×C53—41no—
23C2×C4⋊C4×C53C2×C411yescharacteristic
24C2×C22⋊C4×C53—41no—
25C2×C42×C53D411yescharacteristic
26C24×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—41no—
29C2×D4×C53—41no—
30C2×D4×C53SD1611yescharacteristic
31C2×D4×C53Q1611yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53C22⋊C411yescharacteristic
35C22×Dic53—81no—
36C2×Dic106—41no—
37C2×C8×C53—42no—
38C2×C8×C53—81no—
39C2×C8×C53—81no—
40C4⋊C4×C53—41no—
41C22⋊C4×C53—81no—
42C22⋊C4×C53—81no—
43C42×C53—41no—
44C22⋊Dic53—41no—
45C22⋊Dic53—42no—
46C22⋊Dic53—81no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—41no—
50#11719919—42no—
51C4×Dic53—41no—
52#11719917—41no—
53#11719917—41no—
54C2×#88322—41no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—41no—
58C23×C53—41no—
59C23×C53—81no—
60C23×C53—81no—
61C23×C53#9911yescharacteristic
62D4×C53—41no—
63D4×C53—41no—
64D4×C53—81no—
65C2×C4×C53—21no—
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#10311yescharacteristic
74C2×C4×C53#10411yescharacteristic
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—82no—
78C2×Dic53—161no—
79C2×Dic53—161no—
80Dic106—161no—
81C8×C53—82no—
82#88322—81no—
83#7430—531nomaximal, Sylow
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—81no—
89C22×C53—81no—
90C22×C53—81no—
91C22×C53—81no—
92C22×C53—161no—
93C22×C53#32811yescharacteristic
94C22×C53AΣL(1,𝔽16)11yescharacteristic
95C22×C53#35411yescharacteristic
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—41no—
99C4×C53—41no—
100C4×C53—81no—
101Dic53—81no—
102Dic53—81no—
103Dic53—162no—
104#2027—531no—
105#1393—531no—
106#1270D5311yescharacteristic
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—81no—
110C2×C53—81no—
111C2×C53#135111yescharacteristic
112C2×C53#140211yescharacteristic
113C2×C53#140411yescharacteristic
114#544—531no—
115#418—1061no—
116C2×#99—1061no—
117#390C2×D5311yescharacteristic
118#343—531no—
119#339—1061no—
120#336—21no—
121C53#743011yescharacteristic, complemented, Sylow
122C22×D4D21211yescharacteristic
123C2×M16—1061no—
124C22×C8—41no—
125C4⋊Q8—2121no—
126#125—2121no—
127C22⋊Q8—2121no—
128#117—1061no—
129C2×C4⋊C4C4×D5311yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—2121no—
132C2×C42C22⋊D5311yescharacteristic
133#102—2121no—
134#102—2121no—
135#99—1062no—
136#99—2121no—
137C24—21no—
138C2×Q8—2121no—
139C2×D4—21no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4#1171991311yescharacteristic
143C2×D4Dic21211yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—4241no—
147C22×C4#1171992111yescharacteristic
148M16—2121no—
149M16—2121no—
150C2×C8—42no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—2121no—
154C4⋊C4—41no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—2121no—
158C4⋊C4—2122no—
159C22⋊C4—81no—
160C22⋊C4—81no—
161C22⋊C4—4241no—
162C22⋊C4—2122no—
163C22⋊C4—2121no—
164C42—41no—
165C42—2121no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23—81no—
171C23—81no—
172C23#1313313111yescharacteristic
173Q8—8481no—
174D4—41no—
175D4—41no—
176D4—81no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—8481no—
186C2×C4—8481no—
187C2×C4—4242no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4#1313314211yescharacteristic
191C2×C4#1313314311yescharacteristic
192C8—82no—
193C8—4241no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#1681036111yescharacteristic
204C22#1681041111yescharacteristic
205C22#1681041711yescharacteristic
206C4—21no—
207C4—21no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—8482no—
212C4—4241no—
213C4—4241no—
214C2—21no—
215C2—21no—
216C2—81no—
217C2—81no—
218C2#2027273711yescharacteristic
219C2#2027286111yescharacteristic
220C2#2027286711yescharacteristic
subgroupG/H
centerC22#16810411
commutatorC2×C4⋊C4×C53C2×C4
Frattini#390C2×D53
FittingC53×#1270C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53D4
cosocleC53×#390C22
hypercenter#1270D53
hypocenterC53#7430
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1270

Complements

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

NH ≅ G/Ntype
4. #16811254C4semidirect
7. #16810392C4semidirect
121. C53#7430semidirect

Maximal subgroups

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

subgroupindex
#202758712
#202728402
C53×#12702
#743053