Finite Group Databasebeta

← #28298479

Subgroups — #28298479

Order 13568.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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#20290885C211yesmaximal, characteristic
2#20290468C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4C2×C4⋊#88324C411yescharacteristic
5C2×#13133278C412yes—
6C4×#11719940—21no—
7C2×C4×#88324C411yescharacteristic
8#16811503—21no—
9#16811489—21no—
10#16811489—21no—
11C2×#13133182C2211yescharacteristic
12#16810673—21no—
13#16810349C411yescharacteristic
14#16810348—21no—
15C53×#324C411yescharacteristic
16C2×#11719940—22no—
17C22×#88324—21no—
18C22×#88324—21no—
19C4⋊#88324—41no—
20#13133278—22no—
21#13133278C814yescomplemented
22#13133272—41no—
23C4×#88324—41no—
24C4×#88324—41no—
25C22×Dic106C2×C411yescharacteristic
26#13133182—41no—
27#13133182—41no—
28C2×#11719919D411yescharacteristic
29C2×#11719919Q811yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33#13133146—41no—
34C22⋊#88322—21no—
35C8×Dic53—41no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C22⋊C8×C53—21no—
38C22×C4×C53C4211yescharacteristic
39C22×C4×C53C22⋊C411yescharacteristic
40C22×C4×C53C4⋊C411yescharacteristic
41C22×Dic53C22⋊C411yescharacteristic
42C22×Dic53C22⋊C411yescharacteristic
43C22×Dic53C4⋊C411yescharacteristic
44C22×Dic53C4⋊C411yescharacteristic
45C2×Dic106—41no—
46C2×Dic106—41no—
47C2×Dic106C2×C812yes—
48C2×Dic106M1612yes—
49#11719940—24no—
50#11719940—42no—
51C2×#88324—22no—
52C2×#88324—42no—
53C2×#88324—41no—
54C2×#88324—41no—
55C2×#88324—42no—
56C2×#88324—41no—
57C2×C8×C53—41no—
58C4⋊C4×C53—41no—
59#11719919—41no—
60#11719919—41no—
61C4×Dic53—21no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C4×Dic53—21no—
67C2×#88322—41no—
68C23×C53#9511yescharacteristic
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53C4×C811yescharacteristic
76C2×C4×C53#9711yescharacteristic
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—41no—
85C2×Dic53C22⋊C812yes—
86C2×Dic53#9911yescharacteristic
87C2×Dic53#9911yescharacteristic
88C2×Dic53C4⋊C812yes—
89Dic106—42no—
90Dic106—42no—
91#88324—42no—
92#88324—44no—
93#88324—81no—
94C8×C53—81no—
95#88322—81no—
96#7206—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—21no—
99C22×C53#33611yescharacteristic
100C22×C53#34211yescharacteristic
101C22×C53#34411yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—22no—
107Dic53—22no—
108Dic53—21no—
109Dic53—41no—
110Dic53#35511yescharacteristic
111Dic53#35611yescharacteristic
112#1984—531no—
113C4×#101—531no—
114#1461—531no—
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#128211yescharacteristic
118C2×C53#139411yescharacteristic
119C2×C53#139511yescharacteristic
120C2×C4×Q8—531no—
121#431—1061no—
122C2×C4⋊C8—531no—
123C2×#101—532no—
124C4×M16—1061no—
125C2×C4×C8—531no—
126#343—1061no—
127#336—1061no—
128#336—1061no—
129#324—531no—
130#324—1061no—
131#324#615711yescharacteristic
132C53#720611yescharacteristic, complemented, Sylow
133C22×Q8—531no—
134C2×M16—1062no—
135C22×C8—1061no—
136C22×C8—1061no—
137C4×Q8—2121no—
138C4×Q8—2121no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4C2×#615711yescharacteristic
142C2×C42—531no—
143C2×C42—531no—
144C2×C42—531no—
145C4⋊C8—2121no—
146C4⋊C8—2121no—
147#101—534no—
148#101—1062no—
149C22⋊C8—21no—
150C22⋊C8—1061no—
151#97—2121no—
152C4×C8—2121no—
153C4×C8—2121no—
154C4×C8—2121no—
155C2×Q8—532no—
156C2×Q8—532no—
157C2×Q8—2121no—
158C2×Q8—2121no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4C4×#615711yescharacteristic
164C22×C4C4⋊#615711yescharacteristic
165C22×C4C22⋊#615711yescharacteristic
166M16—1064no—
167M16—2122no—
168C2×C8—41no—
169C2×C8—1062no—
170C2×C8—2121no—
171C2×C8—2121no—
172C2×C8—2122no—
173C2×C8—2121no—
174C2×C8—2122no—
175C2×C8—2121no—
176C4⋊C4—41no—
177C4⋊C4—2121no—
178C4⋊C4—2121no—
179C42—1061no—
180C42—1061no—
181C42—1061no—
182C42—1061no—
183C42—1061no—
184C42—1061no—
185C23(C2×#6157)⋊C411yescharacteristic
186Q8—2122no—
187Q8—2122no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—531no—
195C2×C4—532no—
196C2×C4—532no—
197C2×C4—531no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2121no—
206C2×C4C8×#615711yescharacteristic
207C2×C4#1313327111yescharacteristic
208C8—81no—
209C8—4241no—
210C8—4241no—
211C8—2122no—
212C8—2124no—
213C22—21no—
214C22—21no—
215C22(C2×#6157)⋊C811yescharacteristic
216C22#1681150011yescharacteristic
217C22#1681150611yescharacteristic
218C4—21no—
219C4—21no—
220C4—41no—
221C4—41no—
222C4—531no—
223C4—531no—
224C4—1061no—
225C4—1062no—
226C4—1062no—
227C4—2121no—
228C2—21no—
229C2—21no—
230C2#2028955611yescharacteristic
231C2#2028981111yescharacteristic
232C2#2028981411yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC2×C4×C53C4×C8
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×C53#95
cosocleC2×#13133182C22
hypercenter#324#6157
hypocenterC53#7206
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
21. #13133278C8semidirect
132. C53#7206semidirect

Maximal subgroups

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

subgroupindex
#202908852
#202904682
#202730372
#720653