Finite Group Databasebeta

← #28309433

Subgroups — #28309433

Order 13568.

344 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 74 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#20291078C211yesmaximal, characteristic, complemented
2#20290640C211yesmaximal, characteristic, complemented
3#20289706C214yesmaximal, complemented
4#20285295C211yesmaximal, characteristic
5#16811628—21no—
6#16811544—21no—
7#16811518C2212yes—
8#16811511—21no—
9#16811499—21no—
10#16811499C2212yes—
11#16811174—21no—
12#16811000C411yescharacteristic
13#16810971—21no—
14#16810860—21no—
15#16810860C2212yes—
16#16810705—21no—
17#16810705C412yes—
18C4×C8×D53C2211yescharacteristic
19C53×#500C411yescharacteristic
20C2×#11719936—22no—
21C2×#11719910—22no—
22#13133284—41no—
23#13133284—41no—
24C2×#11719949—41no—
25C2×D424—21no—
26C2×D424C2×C412yes—
27C2×C8×D53D411yescharacteristic
28C2×C8×D53C2311yescharacteristic
29#13133214—21no—
30#13133206—41no—
31#13133192—22no—
32#13133192—41no—
33#13133179—22no—
34#13133179—41no—
35C42×D53D411yescharacteristic
36#13133137—22no—
37#13133137—22no—
38C4×#88322D411yescharacteristic
39C8×Dic53C2×C411yescharacteristic
40C8×Dic53D411yescharacteristic
41C4⋊Q8×C53—21no—
42#107×C53—21no—
43#107×C53C2×C412yes—
44C4×C8×C53C2×C411yescharacteristic
45C2×Q8×C53—41no—
46#11719949—84no—
47C2×D212—22no—
48C2×D212—42no—
49C2×D212—41no—
50C2×D212—41no—
51C2×C4×D53—42no—
52C2×C4×D53SD1611yescharacteristic
53C2×C4×D53SD1611yescharacteristic
54C2×C4×D53C2×D411yescharacteristic
55C2×#88324—42no—
56#11719936—42no—
57#11719936—42no—
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C22×C411yescharacteristic
60C4⋊C4×C53—22no—
61C4⋊C4×C53—41no—
62C42×C53C22⋊C411yescharacteristic
63#11719921—42no—
64#11719921—42no—
65C4×Dic53C22⋊C411yescharacteristic
66C4×Dic53D811yescharacteristic
67C4×Dic53D811yescharacteristic
68C2×#88322C22⋊C411yescharacteristic
69C2×#88322C2×D411yescharacteristic
70D424—42no—
71D424—42no—
72C8×D53—41no—
73C8×D53—41no—
74#11719910—42no—
75C22×D53—42no—
76C22×D53C2×SD1611yescharacteristic
77Q8×C53—82no—
78C2×C4×C53—42no—
79C2×C4×C53#10211yescharacteristic
80C2×C4×C53#10211yescharacteristic
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C2×Dic53#10211yescharacteristic
83C2×Dic53#10211yescharacteristic
84C2×Dic53C2×D811yescharacteristic
85D212—42no—
86D212—42no—
87D212—82no—
88D212—82no—
89D212—84no—
90C4×D53—21no—
91C4×D53—21no—
92C4×D53—22no—
93C4×D53—41no—
94C4×D53—84no—
95#88324—82no—
96C8×C53—21no—
97C8×C53—21no—
98#88322—21no—
99#88322—21no—
100#11864—531nomaximal, Sylow
101C22×C53C2×#10211yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—21no—
105C2×D53—82no—
106C2×D53—82no—
107C2×D53—82no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—82no—
111C4×C53#41011yescharacteristic
112C4×C53#42111yescharacteristic
113Dic53—21no—
114Dic53—21no—
115Dic53#36112yes—
116#3169—531no—
117#2163—531no—
118#1837—531no—
119#1349—534no—
120C2×C53#151711yescharacteristic
121C2×C53#214311yescharacteristic
122C2×C53C2×#36111yescharacteristic
123D53—41no—
124D53—162no—
125#532—1061no—
126#500#615711yescharacteristic
127#493—531no—
128#486—1061no—
129#466—532no—
130#466—1061no—
131C4×D8—532no—
132C4×D8—1061no—
133#423—1061no—
134C2×C4×C8—531no—
135C4⋊C16—1061no—
136C22⋊C16—532no—
137#346—1061no—
138#341—532no—
139#341—1061no—
140C53#1186411yescharacteristic, complemented, Sylow
141C2×#41—2121no—
142C2×D8—532no—
143C2×D8—1061no—
144C2×M16—1062no—
145C22×C8—531no—
146C22×C8—531no—
147C4⋊Q8—21no—
148#127—1061no—
149#124—2121no—
150#121—1062no—
151#121—2121no—
152C4×D4—1062no—
153C4×D4—2121no—
154C2×C42—531no—
155C2×C16—1062no—
156#107—21no—
157#107C2×#615712yes—
158C4⋊C8—2121no—
159C4⋊C8—2121no—
160#102—1062no—
161#102—1062no—
162C4×C8—531no—
163C4×C8—531no—
164C4×C8—531no—
165C4×C8C2×#615711yescharacteristic
166#41—4244no—
167C2×Q8—41no—
168C2×D4—1062no—
169C2×D4—2122no—
170C2×D4—2121no—
171C2×D4—2121no—
172C22×C4—531no—
173C22×C4—531no—
174C22×C4—531no—
175C22×C4—2122no—
176D8—2122no—
177D8—2122no—
178M16—2122no—
179M16—2122no—
180C2×C8—531no—
181C2×C8—531no—
182C2×C8—2121no—
183C2×C8—2121no—
184C2×C8—2122no—
185C2×C8C22⋊#615711yescharacteristic
186C2×C8C22×#615711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—41no—
189C22⋊C4—2122no—
190C22⋊C4—2122no—
191C42—531no—
192C42—531no—
193C42—531no—
194C42C22⋊#615711yescharacteristic
195C16—2122no—
196C23—531no—
197C23—2122no—
198Q8—82no—
199D4—4242no—
200D4—4242no—
201D4—4244no—
202D4—2122no—
203D4—2122no—
204C2×C4—42no—
205C2×C4—531no—
206C2×C4—531no—
207C2×C4—531no—
208C2×C4—1062no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—4244no—
212C2×C4—2121no—
213C2×C4#1313327911yescharacteristic
214C2×C4#1313327911yescharacteristic
215C2×C4C2×C22⋊#615711yescharacteristic
216C8—21no—
217C8—21no—
218C8—1061no—
219C8—1061no—
220C8—4242no—
221C22—1061no—
222C22—1061no—
223C22—1061no—
224C22—4242no—
225C22—4242no—
226C22—4242no—
227C22C2×#1313327911yescharacteristic
228C4—21no—
229C4—21no—
230C4—82no—
231C4—532no—
232C4—1061no—
233C4—1061no—
234C4#1681160411yescharacteristic
235C4#1681162211yescharacteristic
236C2—8482no—
237C2—2121no—
238C2#2029014611yescharacteristic
239C2#2029103111yescharacteristic
240C2D53⋊#36111yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#500C4
socleC22×C53C2×#102
prime radical#20285295C2
cosocleC2×C8×D53C23
hypercenter#500#6157
hypocenterC53#11864
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #20291078C2semidirect
2. #20290640C2semidirect
3. #20289706C2semidirect
140. C53#11864semidirect

Maximal subgroups

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

subgroupindex
#202910782
#202906402
#202897062
#202852952
#1186453