Finite Group Databasebeta

← #28299213

Subgroups — #28299213

Order 13568.

262 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C2×#16811533C211yesmaximal, characteristic
2#20290248C211yesmaximal, characteristic
3#20275108C211yesmaximal, characteristic
4#16811643—21no—
5C2×#13133278—21no—
6#16811533—21no—
7#16811533C412yescomplemented
8#16811483—21no—
9#16811483C412yescomplemented
10C2×#13133215C2211yescharacteristic
11#16810631—21no—
12C2×#13133136C411yescharacteristic
13#16810382—21no—
14C53×#519C411yescharacteristic
15C2×#11719940—41no—
16#13133284—22no—
17#13133278—22no—
18#13133278—41no—
19C22⋊#88324—41no—
20#13133272—41no—
21C22×Dic106D411yescharacteristic
22C2×#11719917—21no—
23#13133215—41no—
24#13133215C2×C411yescharacteristic
25#13133215C2×C411yescharacteristic
26#13133215D411yescharacteristic
27#13133215Q811yescharacteristic
28C2×#11719919—41no—
29C2×C4×Dic53D411yescharacteristic
30#13133140—42no—
31#13133136—41no—
32C22×Q8×C53C2×C411yescharacteristic
33C4×Q8×C53—42no—
34#117×C53—21no—
35#117×C53—41no—
36C2×Q8×C53—21no—
37C2×Q8×C53—41no—
38C2×Q8×C53—41no—
39C2×Q8×C53C4211yescharacteristic
40C2×Q8×C53C4⋊C411yescharacteristic
41C22×C4×C53—21no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×Dic53—21no—
44C22×Dic53—21no—
45C2×Dic106—21no—
46C2×Dic106—41no—
47C2×Dic106—41no—
48C2×Dic106C22⋊C411yescharacteristic
49C2×Dic106C4⋊C411yescharacteristic
50#11719940—42no—
51#11719940—81no—
52C2×#88324—42no—
53C2×#88324—81no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—42no—
56C22⋊C4×C53—41no—
57C42×C53—41no—
58C42×C53—42no—
59#11719919—41no—
60#11719919—41no—
61#11719919—41no—
62#11719919—41no—
63C4×Dic53—21no—
64C4×Dic53C22⋊C411yescharacteristic
65C4×Dic53C4⋊C411yescharacteristic
66#11719917—41no—
67#11719917—41no—
68C2×#88322—41no—
69C23×C53#9911yescharacteristic
70Q8×C53—42no—
71Q8×C53—41no—
72Q8×C53—41no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—42no—
80C2×C4×C53#9511yescharacteristic
81C2×C4×C53#9911yescharacteristic
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—42no—
87C2×Dic53—41no—
88C2×Dic53—41no—
89Dic106—41no—
90Dic106—41no—
91Dic106—81no—
92#88324—82no—
93#88324—82no—
94#88322—81no—
95#7483—531nomaximal, Sylow
96C22×C53—41no—
97C22×C53#34211yescharacteristic
98C22×C53AΣL(1,𝔽16)11yescharacteristic
99C22×C53#35211yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—82no—
105Dic53—22no—
106Dic53—41no—
107Dic53—41no—
108Dic53—41no—
109C2×#356—531no—
110#1886—531no—
111#1558—531no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#139011yescharacteristic
116C2×C4⋊Q8—531no—
117#519#615711yescharacteristic
118#432—1061no—
119#419—1061no—
120C2×#101—531no—
121C2×#101—1061no—
122#356—532no—
123#356—1061no—
124#339—1061no—
125#333—532no—
126#333—1061no—
127C53#748311yescharacteristic, complemented, Sylow
128C22×Q8—531no—
129C22×Q8C2×#615711yescharacteristic
130C2×M16—1061no—
131C2×M16—2121no—
132C4⋊Q8—531no—
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8—531no—
136C4⋊Q8—2121no—
137C4×Q8—42no—
138#117—21no—
139#117—41no—
140C2×C4⋊C4—2121no—
141C2×C42—531no—
142C4⋊C8—1062no—
143#103—2122no—
144#101—1062no—
145#101—2121no—
146#101—2121no—
147C22⋊C8—2121no—
148#97—2121no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—531no—
153C2×Q8—531no—
154C2×Q8—1061no—
155C2×Q8—2121no—
156C2×Q8—2121no—
157C2×Q8C4×#615711yescharacteristic
158C2×Q8C4⋊#615711yescharacteristic
159C22×C4—21no—
160C22×C4—1061no—
161C22×C4—1061no—
162C22×C4C22⋊#615711yescharacteristic
163M16—4241no—
164M16—2121no—
165M16—2122no—
166M16—2121no—
167C2×C8—4241no—
168C2×C8—2121no—
169C2×C8—2122no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C4⋊C4—2121no—
176C22⋊C4—41no—
177C42—41no—
178C42—42no—
179C42—531no—
180C42—531no—
181C42—1061no—
182C23#1313327511yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—42no—
186Q8—4241no—
187Q8—2121no—
188Q8—2121no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—2122no—
201C2×C4—2121no—
202C2×C4—2121no—
203C2×C4(C2×#6157)⋊C411yescharacteristic
204C2×C4#1313327611yescharacteristic
205C8—4241no—
206C8—4242no—
207C8—4242no—
208C22—41no—
209C22#1681150111yescharacteristic
210C22#1681152311yescharacteristic
211C22#1681152511yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—82no—
217C4—1062no—
218C4—2121no—
219C4—2121no—
220C4—2121no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2028979911yescharacteristic
subgroupG/H
centerC2#20289799
commutatorC2×Q8×C53C42
FrattiniC22×Q8C2×#6157
FittingC53×#519C4
socleC2×C53#1390
prime radicalC23×C53#99
cosocleC2×#13133215C22
hypercenter#519#6157
hypocenterC53#7483
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C22×Q8 ◁ #519

Complements

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

NH ≅ G/Ntype
7. #16811533C4semidirect
9. #16811483C4semidirect
127. C53#7483semidirect

Maximal subgroups

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

subgroupindex
C2×#168115332
#202902482
#202751082
#748353