Finite Group Databasebeta

← #28308081

Subgroups — #28308081

Order 13568.

428 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 90 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#20289957C212yesmaximal
2#20289657C212yesmaximal
3#20289646C212yesmaximal
4C4×#13133217C211yesmaximal, characteristic
5#16811640—22no—
6#16811469—22no—
7#16811468C2214yes—
8C2×#13133217C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810529C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810521—22no—
14#16810493C2211yescharacteristic
15C4×C4⋊Dic53—22no—
16C43×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20C22×Dic106C2×C411yescharacteristic
21C22×Dic106C2×C411yescharacteristic
22#13133217—22no—
23#13133217C814yescomplemented
24C4×Dic106—41no—
25C4×Dic106—41no—
26C4×Dic106—41no—
27C4×Dic106—41no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2311yescharacteristic
32C2×#11719919—21no—
33C2×#11719919D412yes—
34C2×C4×Dic53—21no—
35C2×C4×Dic53D412yes—
36#13133124—22no—
37#13133124—22no—
38C2×C42×C53—21no—
39C2×C42×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—21no—
49C22×Dic53—21no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×Dic106—21no—
53C2×Dic106—41no—
54C2×Dic106—41no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106C2×C812yes—
58C2×Dic106C2×C812yes—
59C2×Dic106M1612yes—
60C2×#88324—44no—
61C42×C53—24no—
62C42×C53—21no—
63C42×C53—21no—
64C42×C53—21no—
65C42×C53—42no—
66C42×C53—41no—
67C42×C53C2×C812yes—
68C4⋊Dic53—28no—
69C4⋊Dic53—41no—
70C4⋊Dic53—44no—
71C4⋊Dic53—41no—
72#11719919—42no—
73#11719919—42no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—41no—
77C4×Dic53—41no—
78C23×C53C2×C22⋊C411yescharacteristic
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53#9911yescharacteristic
91C2×C4×C53#9911yescharacteristic
92C2×C4×C53C22×C811yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—41no—
103C2×Dic53C22⋊C814yes—
104Dic106—42no—
105Dic106—42no—
106Dic106—42no—
107Dic106—42no—
108#88324—84no—
109#11478—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53C2×C22⋊C811yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53#41311yescharacteristic
115C4×C53—24no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—41no—
121C4×C53#35612yes—
122Dic53—24no—
123Dic53—42no—
124Dic53—42no—
125Dic53—81no—
126C4×C4⋊Q8—531no—
127#1461—532no—
128#1325—532no—
129#1319—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#145911yescharacteristic
133C2×C53#212611yescharacteristic
134C2×C53C2×#35611yescharacteristic
135C2×C4⋊Q8—531no—
136C2×C4×Q8—531no—
137C2×C4×Q8—531no—
138#431—1062no—
139#391—531no—
140#391—531no—
141#389—1062no—
142#384—531no—
143C4×C4⋊C4—1062no—
144C43#615711yescharacteristic
145#324—534no—
146#324—1062no—
147C53#1147811yescharacteristic, complemented, Sylow
148C22×Q8—531no—
149C22×Q8—531no—
150C4⋊Q8—534no—
151C4⋊Q8—1062no—
152C4×Q8—2121no—
153C4×Q8—2121no—
154C4×Q8—2121no—
155C4×Q8—2121no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1061no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—532no—
165C2×C42—1061no—
166C2×C42C2×#615711yescharacteristic
167C2×C42C2×#615711yescharacteristic
168C2×C42C2×#615711yescharacteristic
169C4⋊C8—2122no—
170C22⋊C8—1064no—
171C4×C8—2122no—
172#95—1062no—
173#95—1062no—
174C2×Q8—532no—
175C2×Q8—532no—
176C2×Q8—532no—
177C2×Q8—1061no—
178C2×Q8—2121no—
179C2×Q8—2121no—
180C2×Q8—2121no—
181C2×Q8—2121no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—532no—
185C22×C4—532no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4C22⋊#615711yescharacteristic
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C4⋊C4—1068no—
193C4⋊C4—2122no—
194C4⋊C4—2121no—
195C4⋊C4—2121no—
196C4⋊C4—2122no—
197C4⋊C4—2124no—
198C42—24no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—41no—
203C42—42no—
204C42—1062no—
205C42—1062no—
206C42—2121no—
207C42—2121no—
208C42D53⋊C812yes—
209C23C2×C22⋊#615711yescharacteristic
210Q8—2122no—
211Q8—2122no—
212Q8—2122no—
213Q8—2122no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—534no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—2122no—
231C2×C4—2121no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2122no—
235C2×C4#1313327611yescharacteristic
236C2×C4#1313327611yescharacteristic
237C2×C4C2×D53⋊C811yescharacteristic
238C2×C4D53⋊M1611yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22D53⋊(C22⋊C8)11yescharacteristic
243C22C2×#1313327611yescharacteristic
244C22#1681160911yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—41no—
251C4—1064no—
252C4—4241no—
253C4—2122no—
254C4—2122no—
255C4#1681153412yes—
256C2—21no—
257C2—21no—
258C2#2028995211yescharacteristic
259C2#2029098811yescharacteristic
260C2C2×#1681153411yescharacteristic
subgroupG/H
centerC2×C4#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC43×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC43#6157
hypocenterC53#11478
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
23. #13133217C8semidirect
147. C53#11478semidirect

Maximal subgroups

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

subgroupindex
#202899572
#202896572
#202896462
C4×#131332172
#1147853