Finite Group Databasebeta

← #27271683

Subgroups — #27271683

Order 13568.

236 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 43 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#20272840C211yesmaximal, characteristic
2#20272536C211yesmaximal, characteristic
3C53×#2027C211yesmaximal, characteristic
4C2×#13133132—21no—
5#16810378—21no—
6C53×#544C411yescharacteristic, complemented
7C53×#418—21no—
8C53×#390C2211yescharacteristic
9C53×#343C411yescharacteristic, complemented
10C53×#339—21no—
11C22×#88322—41no—
12C2×C22⋊Dic53—41no—
13#13133132—22no—
14#13133132—41no—
15C22×D4×C53D411yescharacteristic
16C2×M16×C53—21no—
17C4⋊Q8×C53—41no—
18#125×C53—41no—
19C22⋊Q8×C53—41no—
20#117×C53—21no—
21C2×C4⋊C4×C53C2×C411yescharacteristic
22C2×C22⋊C4×C53—41no—
23C2×C42×C53D411yescharacteristic
24#102×C53—41no—
25#102×C53—41no—
26C24×C53—21no—
27C2×Q8×C53—41no—
28C2×D4×C53—21no—
29C2×D4×C53—41no—
30C2×D4×C53—41no—
31C2×D4×C53SD1611yescharacteristic
32C2×D4×C53Q1611yescharacteristic
33C22×C4×C53—21no—
34C22×C4×C53—21no—
35C22×C4×C53C22⋊C411yescharacteristic
36C22×Dic53—81no—
37M16×C53—41no—
38M16×C53—41no—
39C2×C8×C53—41no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—41no—
43C4⋊C4×C53—42no—
44C4⋊C4×C53—41no—
45C22⋊C4×C53—41no—
46C22⋊C4×C53—81no—
47C22⋊C4×C53—81no—
48C42×C53—41no—
49C42×C53—41no—
50C22⋊Dic53—42no—
51C22⋊Dic53—81no—
52C2×#88322—42no—
53C2×#88322—81no—
54C2×#88322—81no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—41no—
58C23×C53—41no—
59C23×C53—81no—
60C23×C53—81no—
61C23×C53#9911yescharacteristic
62Q8×C53—161no—
63D4×C53—41no—
64D4×C53—41no—
65D4×C53—81no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—81no—
76C2×C4×C53#10311yescharacteristic
77C2×C4×C53#10411yescharacteristic
78C2×Dic53—82no—
79C2×Dic53—161no—
80C2×Dic53—161no—
81C8×C53—81no—
82#88322—82no—
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—
101C4×C53—81no—
102C4×C53—81no—
103Dic53—162no—
104#2027D5311yescharacteristic
105#1393—531no—
106#1270—531no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—81no—
110C2×C53—81no—
111C2×C53#135111yescharacteristic
112C2×C53#140211yescharacteristic
113C2×C53#140411yescharacteristic
114#544Dic5311yescharacteristic, complemented
115#418—21no—
116C2×#99—1061no—
117#390C2×D5311yescharacteristic
118#343Dic5311yescharacteristic, complemented
119#339—21no—
120#336—1061no—
121C53#743011yescharacteristic, complemented, Sylow
122C22×D4C22⋊D5311yescharacteristic
123C2×M16—21no—
124C22×C8—2121no—
125C4⋊Q8—41no—
126#125—41no—
127C22⋊Q8—41no—
128#117—21no—
129C2×C4⋊C4C2×Dic5311yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—2121no—
132C2×C42C22⋊D5311yescharacteristic
133#102—41no—
134#102—41no—
135#99—1062no—
136#99—2121no—
137C24—21no—
138C2×Q8—41no—
139C2×D4—21no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4#1171992411yescharacteristic
143C2×D4#1171992511yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—4241no—
147C22×C4C22⋊Dic5311yescharacteristic
148M16—41no—
149M16—41no—
150C2×C8—41no—
151C2×C8—4241no—
152C2×C8—4241no—
153C2×C8—2122no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C22⋊C4—41no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C22⋊C4—4241no—
163C22⋊C4—2122no—
164C42—41no—
165C42—41no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23—81no—
171C23—81no—
172C23#1313313211yescharacteristic
173Q8—161no—
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—41no—
185C2×C4—41no—
186C2×C4—81no—
187C2×C4—8481no—
188C2×C4—8481no—
189C2×C4—4242no—
190C2×C4#1313314111yescharacteristic
191C2×C4#1313314411yescharacteristic
192C8—81no—
193C8—4242no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#1681036011yescharacteristic
204C22#1681041011yescharacteristic
205C22#1681041811yescharacteristic
206C4—21no—
207C4—21no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—8482no—
214C2—21no—
215C2—21no—
216C2—81no—
217C2—81no—
218C2#2027273611yescharacteristic
219C2#2027286011yescharacteristic
220C2#2027286611yescharacteristic
subgroupG/H
centerC22#16810410
commutatorC2×C4⋊C4×C53C2×C4
Frattini#390C2×D53
FittingC53×#2027C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53D4
cosocleC53×#390C22
hypercenter#2027D53
hypocenterC53#7430
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2027

Complements

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

NH ≅ G/Ntype
6. C53×#544C4semidirect
9. C53×#343C4semidirect
114. #544Dic53semidirect
118. #343Dic53semidirect
121. C53#7430semidirect

Maximal subgroups

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

subgroupindex
#202728402
#202725362
C53×#20272
#743053