Finite Group Databasebeta

← #28298899

Subgroups — #28298899

Order 13568.

318 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 38 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#20289649C211yesmaximal, characteristic, complemented
2#20289637C211yesmaximal, characteristic, complemented
3#20274679C211yesmaximal, characteristic
4#16811519—21no—
5C22⋊#11719910—21no—
6#16810640—21no—
7#16810619—21no—
8#16810597C411yescharacteristic
9#16810590C2211yescharacteristic
10C53×#585C411yescharacteristic
11#13133291—41no—
12C2×#11719910—41no—
13C2×#11719917D411yescharacteristic
14C22×#88322D411yescharacteristic
15#13133176—21no—
16#13133144—42no—
17#13133141—41no—
18#13133138—41no—
19#13133132—41no—
20C22⋊#88322—41no—
212−1+4×C53—21no—
222−1+4×C53—41no—
23H(5,𝔽2)×C53—41no—
24H(5,𝔽2)×C53—41no—
25H(5,𝔽2)×C53C812yes—
26C2×#41×C53—21no—
27C2×#41×C53—41no—
28C2×#41×C53—42no—
29C2×#41×C53C2×C411yescharacteristic
30#41×C53—22no—
31#41×C53—22no—
32#41×C53—41no—
33#41×C53—42no—
34#41×C53—41no—
35#41×C53—41no—
36#41×C53—42no—
37#41×C53—42no—
38#41×C53—44no—
39#41×C53—42no—
40#41×C53—41no—
41#41×C53—42no—
42C2×Q8×C53—21no—
43C2×Q8×C53—42no—
44C2×Q8×C53—41no—
45C2×Q8×C53M1611yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—42no—
49C2×D4×C53—44no—
50C2×D4×C53—41no—
51C2×D4×C53—42no—
52C2×D4×C53—41no—
53C2×D4×C53C2×C811yescharacteristic
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53—42no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22⋊Dic53—41no—
59C4⋊Dic53—41no—
60C4×Dic53—41no—
61#11719917—41no—
62C2×#88322—21no—
63C2×#88322—21no—
64C2×#88322—41no—
65#11719910—81no—
66#11719910—81no—
67C23×C53—21no—
68C23×C53—41no—
69C23×C53—42no—
70C23×C53#10011yescharacteristic
71Q8×C53—22no—
72Q8×C53—41no—
73Q8×C53—42no—
74Q8×C53—41no—
75D4×C53—22no—
76D4×C53—41no—
77D4×C53—42no—
78D4×C53—42no—
79D4×C53—42no—
80D4×C53—44no—
81D4×C53—42no—
82D4×C53—41no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—44no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53C22⋊C811yescharacteristic
94C2×C4×C53#9911yescharacteristic
95C2×Dic53—81no—
96#88322—41no—
97#88322—41no—
98#7346—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—42no—
102C22×C53—41no—
103C22×C53—81no—
104C22×C53—161no—
105C22×C53—162no—
106C22×C53#32311yescharacteristic
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—42no—
110C4×C53AΣL(1,𝔽16)11yescharacteristic
111C4×C53AΣL(1,𝔽16)11yescharacteristic
112Dic53—161no—
113#1794—531no—
114#1321—531no—
115#1315—531no—
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—81no—
119C2×C53—82no—
120C2×C53#131611yescharacteristic
121#585#615711yescharacteristic
122#421—1061no—
123#416—1061no—
124#410—531no—
125#408—531no—
126#349—1061no—
127C22⋊C16—1061no—
128C53#734611yescharacteristic, complemented, Sylow
1292−1+4—21no—
1302−1+4—41no—
131H(5,𝔽2)—41no—
132H(5,𝔽2)—41no—
133H(5,𝔽2)#8832412yes—
134C2×#41—21no—
135C2×#41—41no—
136C2×#41—42no—
137C2×#41C2×#615711yescharacteristic
138C2×M16—531no—
139C22×C8—531no—
140#117—1061no—
141M32—2121no—
142C2×C16—2121no—
143#104—2122no—
144#103—2121no—
145#102—2121no—
146#99—2121no—
147C22⋊C8—2121no—
148#41—22no—
149#41—22no—
150#41—41no—
151#41—42no—
152#41—41no—
153#41—41no—
154#41—42no—
155#41—42no—
156#41—44no—
157#41—42no—
158#41—41no—
159#41—42no—
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8#1171994011yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4C2×#8832411yescharacteristic
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4C22⋊#615711yescharacteristic
176M16—2121no—
177C2×C8—1061no—
178C2×C8—1061no—
179C2×C8—2121no—
180C4⋊C4—2121no—
181C22⋊C4—2121no—
182C42—2121no—
183C16—4241no—
184C16—4241no—
185C23—21no—
186C23—41no—
187C23—42no—
188C23#1313327711yescharacteristic
189Q8—22no—
190Q8—41no—
191Q8—42no—
192Q8—41no—
193D4—22no—
194D4—41no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—44no—
199D4—42no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—4241no—
212C2×C4C22⋊#8832411yescharacteristic
213C2×C4#1313327511yescharacteristic
214C8—2121no—
215C8—2121no—
216C22—21no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22—161no—
222C22—162no—
223C22#1681146611yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—8481no—
228C4#1681152311yescharacteristic
229C4#1681152311yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—82no—
234C2#2028963911yescharacteristic
subgroupG/H
centerC4#16811523
commutatorC2×D4×C53C2×C8
FrattiniC2×#41C2×#6157
Fitting, prime radicalC53×#585C4
socleC2×C53#1316
cosocle#16810590C22
hypercenter#585#6157
hypocenterC53#7346
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20289649C2semidirect
2. #20289637C2semidirect
128. C53#7346semidirect

Maximal subgroups

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

subgroupindex
#202896492
#202896372
#202746792
#734653