Finite Group Databasebeta

← #28309097

Subgroups — #28309097

Order 13568.

316 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 62 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#20291090C211yesmaximal, characteristic, complemented
2#20290436C211yesmaximal, characteristic, complemented
3#20289689C212yesmaximal, complemented
4#20289685C212yesmaximal, complemented
5#20285362C211yesmaximal, characteristic
6#16811586—21no—
7#16811545—21no—
8#16811520C2212yes—
9#16811515—21no—
10#16811492C2211yescharacteristic
11#16811491C2211yescharacteristic
12#16811490—21no—
13#16811227—21no—
14#16811025C411yescharacteristic, complemented
15#16810987—21no—
16#16810873C2212yes—
17#16810856—21no—
18#16810735C412yescomplemented
19D8×Dic53—21no—
20#16810577C2211yescharacteristic
21C53×#493C411yescharacteristic, complemented
22D53⋊M16—21no—
23#13133296—21no—
24#13133291—22no—
25#13133271—22no—
26C8×#6157—22no—
27C2×#11719912C2311yescharacteristic
28C2×C8×D53D411yescharacteristic
29#13133209—21no—
30#13133198—42no—
31#13133189—42no—
32#13133188—22no—
33D4×Dic53—22no—
34#13133174D411yescharacteristic
35#13133151—21no—
36#13133149C2×C412yes—
37#13133138—22no—
38#13133138—22no—
39#13133128D411yescharacteristic
40#13133127C2×C411yescharacteristic
41C8×Dic53D411yescharacteristic
42C2×D8×C53—21no—
43C2×D8×C53C2×C412yes—
44#127×C53—21no—
45C4×C8×C53C2×C411yescharacteristic
46C2×D4×C53—22no—
47C2×D4×C53—42no—
48C2×C22⋊D53—42no—
49C22×Dic53—42no—
50C2×C4×D53C2×D411yescharacteristic
51C22⋊#6157—41no—
52#11719940—41no—
53C4⋊#6157—41no—
54C4×#6157—22no—
55#11719936—41no—
56D53⋊C8—22no—
57D8×C53—42no—
58D8×C53—41no—
59D8×C53—41no—
60C2×C8×C53C22⋊C411yescharacteristic
61C2×C8×C53C22×C411yescharacteristic
62C42×C53C22⋊C411yescharacteristic
63C22⋊Dic53—42no—
64C22⋊Dic53—42no—
65#11719921—21no—
66C4⋊Dic53—22no—
67#11719919—21no—
68#11719919—42no—
69C4×Dic53C22⋊C411yescharacteristic
70C2×#88322C22⋊C411yescharacteristic
71C2×#88322C2×D411yescharacteristic
72#11719912—41no—
73C8×D53—21no—
74C8×D53—21no—
75#11719910—42no—
76C23×C53—42no—
77C22×D53AGL(1,ℤ/8)11yescharacteristic
78C2×#6157—42no—
79D4×C53—42no—
80D4×C53—42no—
81D4×C53—162no—
82C2×C4×C53—21no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C22⋊D53—162no—
85C2×Dic53—21no—
86C2×Dic53—42no—
87C2×Dic53—162no—
88C2×Dic53AGL(1,ℤ/8)11yescharacteristic
89C4×D53—21no—
90C4×D53—21no—
91#88324—42no—
92C8×C53—21no—
93C8×C53#10411yescharacteristic
94C8×C53#10411yescharacteristic
95#88322—21no—
96#88322—21no—
97#11770—531nomaximal, Sylow
98C22×C53—82no—
99C22×C53—82no—
100C22×C53—82no—
101C22×C53#41811yescharacteristic
102C2×D53—21no—
103C2×D53—41no—
104#6157—82no—
105C4×C53—41no—
106C4×C53#41011yescharacteristic
107C4×C53C2×#10411yescharacteristic
108Dic53—21no—
109Dic53—41no—
110Dic53—82no—
111#3176—531no—
112#2166—531no—
113#1747—531no—
114#1341—532no—
115#1339—532no—
116C2×C53—21no—
117C2×C53—162no—
118C2×C53#150511yescharacteristic
119D53—41no—
120#540—1061no—
121#499—531no—
122#493#615711yescharacteristic, complemented
123#490—1061no—
124#469—532no—
125#466—1061no—
126#442—532no—
127C4×D8—1061no—
128#405—531no—
129#405—1061no—
130#364—1061no—
131#349—532no—
132#347—1061no—
133#338—531no—
134H(3,ℤ/4)—531no—
135H(3,ℤ/4)—1061no—
136C53#1177011yescharacteristic, complemented, Sylow
137C2×D8—21no—
138C2×D8C2×#615712yes—
139C2×M16—531no—
140C2×M16—1061no—
141C22×C8—531no—
142#127—21no—
143#125—1061no—
144#123—2122no—
145#121—1062no—
146#121—2122no—
147C4×D4—1062no—
148#117—531no—
149#117—1061no—
150M32—1062no—
151#107—1061no—
152#106—532no—
153#102—1062no—
154#102—1062no—
155#97—531no—
156#97—531no—
157#97—1062no—
158C4×C8—531no—
159C4×C8—1062no—
160C4×C8C2×#615711yescharacteristic
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—2122no—
164C22×C4—531no—
165C22×C4—2122no—
166D8—42no—
167D8—41no—
168D8—41no—
169M16—2121no—
170M16—2121no—
171M16—2121no—
172C2×C8—531no—
173C2×C8—531no—
174C2×C8—1061no—
175C2×C8—1061no—
176C2×C8—1062no—
177C2×C8C22⋊#615711yescharacteristic
178C2×C8C22×#615711yescharacteristic
179C4⋊C4—1061no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C4⋊C4—2121no—
183C22⋊C4—1061no—
184C22⋊C4—2122no—
185C22⋊C4—2122no—
186C22⋊C4—2121no—
187C42—531no—
188C42—1062no—
189C42C22⋊#615711yescharacteristic
190C16—2122no—
191C23—42no—
192C23—531no—
193D4—42no—
194D4—42no—
195D4—162no—
196D4—8482no—
197C2×C4—21no—
198C2×C4—531no—
199C2×C4—8482no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4C2×C22⋊#615711yescharacteristic
206C8—21no—
207C8—1061no—
208C8—1061no—
209C8—2122no—
210C8#1313328111yescharacteristic
211C8#1313328111yescharacteristic
212C22—82no—
213C22—82no—
214C22—82no—
215C22—1061no—
216C22—2121no—
217C22#1681161511yescharacteristic
218C4—41no—
219C4—1061no—
220C4—4242no—
221C4—4242no—
222C4—2121no—
223C4#1681160311yescharacteristic
224C4C2×#1313328111yescharacteristic
225C2—21no—
226C2—162no—
227C2—2121no—
228C2#2029010411yescharacteristic
subgroupG/H
centerC2#20290104
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#493C4
socleC2×C53#1505
prime radical#20285362C2
cosocleC2×#11719912C23
hypercenter#493#6157
hypocenterC53#11770
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #20291090C2semidirect
2. #20290436C2semidirect
3. #20289689C2semidirect
4. #20289685C2semidirect
14. #16811025C4semidirect
18. #16810735C4semidirect
21. C53×#493C4semidirect
122. #493#6157semidirect
136. C53#11770semidirect

Maximal subgroups

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

subgroupindex
#202910902
#202904362
#202896892
#202896852
#202853622
#1177053