Finite Group Databasebeta

← #28299092

Subgroups — #28299092

Order 13568.

236 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289810C211yesmaximal, characteristic
2#20289532C211yesmaximal, characteristic
3#20275868C211yesmaximal, characteristic
4C2×#13133276—21no—
5#16811489—21no—
6#16810627—21no—
7#16810524C2211yescharacteristic
8#16810392C411yescharacteristic, complemented
9#16810382—21no—
10C53×#544C411yescharacteristic, complemented
11C22×#88324—41no—
12C2×C22⋊#6157—41no—
13#13133276—22no—
14#13133276—41no—
15C22×D212D411yescharacteristic
16C2×#11719917—21no—
17C2×#11719921—41no—
18C2×C4×Dic53D411yescharacteristic
19#13133137—41no—
20#13133137—41no—
21C4⋊Q8×C53—41no—
22#125×C53—41no—
23C22⋊Q8×C53—41no—
24#117×C53—21no—
25C2×C4⋊C4×C53C2×C411yescharacteristic
26C23×D53—21no—
27C22×#6157—81no—
28C2×Q8×C53—41no—
29C22×C4×C53—21no—
30C22×C4×C53C22⋊C411yescharacteristic
31C22×Dic53—21no—
32C2×D212—21no—
33C2×D212—41no—
34C2×D212—41no—
35C2×D212SD1611yescharacteristic
36C2×D212Q1611yescharacteristic
37C22⋊#6157—42no—
38C22⋊#6157—81no—
39C2×#88324—42no—
40C2×#88324—81no—
41C2×#88324—81no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—41no—
45C4⋊C4×C53—42no—
46C4⋊C4×C53—41no—
47C22⋊C4×C53—41no—
48C42×C53—41no—
49#11719921—81no—
50#11719921—81no—
51C4×Dic53—41no—
52#11719917—41no—
53#11719917—41no—
54C2×#88322—41no—
55C23×C53#9911yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—41no—
59C22×D53—41no—
60C22×D53—81no—
61C22×D53—81no—
62C2×#6157—82no—
63C2×#6157—161no—
64C2×#6157—161no—
65Q8×C53—161no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—81no—
72C2×C4×C53#10311yescharacteristic
73C2×C4×C53#10411yescharacteristic
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78D212—41no—
79D212—41no—
80D212—81no—
81#88324—82no—
82#88322—81no—
83#7430—531nomaximal, Sylow
84C22×C53—21no—
85C22×C53—21no—
86C22×C53#32811yescharacteristic
87C22×C53AΣL(1,𝔽16)11yescharacteristic
88C22×C53#35411yescharacteristic
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—81no—
92C2×D53—81no—
93C2×D53—81no—
94C2×D53—81no—
95C2×D53—161no—
96#6157—162no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—81no—
100C4×C53—81no—
101C4×C53—81no—
102Dic53—41no—
103Dic53—41no—
104#2027—531no—
105#1393—531no—
106#1270—531no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#135111yescharacteristic
110C2×C53#140211yescharacteristic
111C2×C53#140411yescharacteristic
112D53—81no—
113D53—81no—
114#544#615711yescharacteristic, complemented
115#418—1061no—
116C2×#99—1061no—
117#390—531no—
118#343—531no—
119#339—1061no—
120#336—1061no—
121C53#743011yescharacteristic, complemented, Sylow
122C22×D4—531no—
123C2×M16—1061no—
124C22×C8—2121no—
125C4⋊Q8—41no—
126#125—41no—
127C22⋊Q8—41no—
128#117—21no—
129C2×C4⋊C4C2×#615711yescharacteristic
130C2×C22⋊C4—2121no—
131C2×C22⋊C4—2121no—
132C2×C42—531no—
133#102—2121no—
134#102—2121no—
135#99—1062no—
136#99—2121no—
137C24—1061no—
138C2×Q8—41no—
139C2×D4—531no—
140C2×D4—531no—
141C2×D4—1061no—
142C2×D4—2121no—
143C2×D4—2121no—
144C22×C4—21no—
145C22×C4—1061no—
146C22×C4—4241no—
147C22×C4C22⋊#615711yescharacteristic
148M16—2121no—
149M16—2121no—
150C2×C8—4241no—
151C2×C8—4241no—
152C2×C8—2121no—
153C2×C8—2122no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C22⋊C4—41no—
160C22⋊C4—4241no—
161C22⋊C4—4241no—
162C22⋊C4—4241no—
163C22⋊C4—2122no—
164C42—41no—
165C42—2121no—
166C23—1061no—
167C23—1061no—
168C23—4241no—
169C23—4241no—
170C23—2121no—
171C23—2121no—
172C23#1313327511yescharacteristic
173Q8—161no—
174D4—4241no—
175D4—2121no—
176D4—2121no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—8481no—
184C2×C4—8481no—
185C2×C4—1061no—
186C2×C4—1061no—
187C2×C4—4242no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4#1313328011yescharacteristic
191C2×C4#1313328211yescharacteristic
192C8—4241no—
193C8—4242no—
194C22—21no—
195C22—21no—
196C22—8481no—
197C22—4241no—
198C22—4241no—
199C22—4241no—
200C22—4241no—
201C22—2121no—
202C22—2121no—
203C22#1681147711yescharacteristic
204C22#1681152311yescharacteristic
205C22#1681152911yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—81no—
210C4—81no—
211C4—8482no—
212C4—2121no—
213C4—2121no—
214C2—21no—
215C2—21no—
216C2—4241no—
217C2—4241no—
218C2#2028971011yescharacteristic
219C2#2028983411yescharacteristic
220C2#2028983811yescharacteristic
subgroupG/H
centerC22#16811523
commutatorC2×C4⋊C4×C53C2×C4
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#544C4
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D212D4
cosocle#16810524C22
hypercenter#544#6157
hypocenterC53#7430
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #544

Complements

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

NH ≅ G/Ntype
8. #16810392C4semidirect
10. C53×#544C4semidirect
114. #544#6157semidirect
121. C53#7430semidirect

Maximal subgroups

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

subgroupindex
#202898102
#202895322
#202758682
#743053