Finite Group Databasebeta

← #27297422

Subgroups — #27297422

Order 13568.

324 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276932C212yesmaximal, complemented
2#20276231C211yesmaximal, characteristic
3#20272709C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic
5C53×#3195C211yesmaximal, characteristic
6#16810672C2211yescharacteristic
7#16810671—21no—
8#16810587—21no—
9#16810445—22no—
10#16810439—22no—
11#16810405C2212yes—
12#16810381C2211yescharacteristic
13C2×C22⋊Q8×C53—21no—
14C53×#483—22no—
15C53×#470—21no—
16C53×#470C2212yes—
17C53×#451—22no—
18C53×#407C2211yescharacteristic
19C2×#11719917—21no—
20C2×#11719917—21no—
21#13133157—22no—
22#13133153—22no—
23#13133152—42no—
24C22⋊#88322—42no—
25C22×Q8×C53—21no—
26C23×C4×C53D411yescharacteristic
27C2×Q16×C53—21no—
28C2×Q16×C53D412yes—
29C2×M16×C53D411yescharacteristic
30C2×M16×C53C2311yescharacteristic
31C22⋊Q8×C53—22no—
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—41no—
34C22⋊Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—41no—
37C2×C22⋊C4×C53—41no—
38#106×C53—21no—
39#106×C53D412yes—
40#103×C53—22no—
41#103×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53SD1611yescharacteristic
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—81no—
47C2×Q8×C53—81no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53SD1611yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56Q16×C53—42no—
57Q16×C53—81no—
58M16×C53—41no—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53C2×D412yes—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68#11719917—41no—
69#11719917—41no—
70#11719917—41no—
71#11719917—41no—
72C2×#88322—22no—
73C2×#88322—41no—
74#11719908—42no—
75C23×C53—21no—
76C23×C53—41no—
77C23×C53—42no—
78C23×C53C2×SD1611yescharacteristic
79Q8×C53—42no—
80Q8×C53—42no—
81Q8×C53—82no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—81no—
91C2×C4×C53—82no—
92C2×C4×C53—82no—
93C2×C4×C53#12011yescharacteristic
94C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
95C8×C53—22no—
96C8×C53—41no—
97#88322—42no—
98#88322—81no—
99#12299—531nomaximal, Sylow
100C22×C53—22no—
101C22×C53—21no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53—81no—
107C22×C53#45011yescharacteristic
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—82no—
111C4×C53—82no—
112C4×C53#45311yescharacteristic
113C4×C53#45811yescharacteristic
114#3195D5311yescharacteristic
115#2239—532no—
116#2091—531no—
117#1342—532no—
118#1320—531no—
119C2×C53—21no—
120C2×C53—42no—
121C2×C53—41no—
122C2×C53#161311yescharacteristic
123C2×C22⋊Q8—21no—
124#483—22no—
125#470—21no—
126#470C2×D5312yes—
127#451—22no—
128#430—531no—
129#430—1061no—
130#407—1061no—
131#407C2×D5311yescharacteristic
132#365—1062no—
133#362—1062no—
134#349—532no—
135#338—531no—
136C53#1229911yescharacteristic, complemented, Sylow
137C22×Q8—21no—
138C23×C4C22⋊D5311yescharacteristic
139C2×Q16—21no—
140C2×Q16C22⋊D5312yes—
141C2×M16—1061no—
142C2×M16—1061no—
143C2×M16C22⋊D5311yescharacteristic
144C2×M16C22×D5311yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152M32—1062no—
153#108—1062no—
154#108—2122no—
155#106—21no—
156#106C22⋊D5312yes—
157#103—22no—
158#103—22no—
159C22⋊C8—22no—
160C22⋊C8—2122no—
161C24#1171992311yescharacteristic
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4#1171992311yescharacteristic
173C22×C4C2×C22⋊D5311yescharacteristic
174Q16—42no—
175Q16—81no—
176M16—41no—
177M16—41no—
178M16—2121no—
179M16—2121no—
180M16—2121no—
181M16—2121no—
182C2×C8—21no—
183C2×C8—1062no—
184C2×C8—2121no—
185C2×C8C2×C22⋊D5312yes—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C16—2122no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1171992311yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#1313318611yescharacteristic
212C2×C4#1313324311yescharacteristic
213C8—22no—
214C8—41no—
215C8—4241no—
216C8—2122no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1681077411yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4#1681079111yescharacteristic
230C4#1681081311yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2027373511yescharacteristic
subgroupG/H
centerC2#20273735
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#3195C2
socleC2×C53#1613
prime radicalC24×C53SD16
hypercenter#3195D53
hypocenterC53#12299
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. #20276932C2semidirect
3. #20272709C2semidirect
136. C53#12299semidirect

Maximal subgroups

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

subgroupindex
#202769322
#202762312
#202727092
#202726512
C53×#31952
#1229953