Finite Group Databasebeta

← #28391600

Subgroups — #28391600

Order 14464.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#20299162C211yesmaximal, characteristic
2C2×#16813611C211yesmaximal, characteristic, complemented
3C2×#16813549C211yesmaximal, characteristic, complemented
4#20298607C211yesmaximal, characteristic, complemented
5C2×#16813538C211yesmaximal, characteristic, complemented
6#20298359C211yesmaximal, characteristic, complemented
7C2×#137×C113C211yesmaximal, characteristic, complemented
8C22×D452C2211yescharacteristic
9C2×#14268631C2211yescharacteristic
10#16813618—21no—
11#16813611—22no—
12#16813611—21no—
13#16813611—22no—
14#16813596—22no—
15#16813590—22no—
16#16813581C2211yescharacteristic
17C2×#14268624—22no—
18C2×C4×Dic113C2211yescharacteristic
19#16813549—22no—
20#16813546—22no—
21#16813538—22no—
22C2×#41×C113C2211yescharacteristic
23C22×Q8×C113C2211yescharacteristic
24#137×C113—22no—
25#137×C113—22no—
26C2×Q16×C113—21no—
27C2×SD16×C113—21no—
28C2×M16×C113C2211yescharacteristic
29C23×D113D412yes—
30C2×D452—21no—
31C2×D452—21no—
32C2×D452—21no—
33C2×D452—21no—
34C2×D452—21no—
35C2×D452—21no—
36C2×C22⋊D113—24no—
37C22×Dic113—21no—
38#14268631—21no—
39#14268631—21no—
40C2×#11744751—21no—
41C4⋊Dic113D412yes—
42C22⋊Dic113—21no—
43#14268624—28no—
44#14268624—24no—
45C4×Dic113—21no—
46C4×Dic113—21no—
47C4×Dic113D412yescomplemented
48#41×C113—21no—
49#41×C113—21no—
50#41×C113—21no—
51#41×C113—21no—
52C2×Q8×C113—21no—
53C2×Q8×C113—21no—
54C2×Q8×C113—21no—
55C2×Q8×C113—21no—
56C2×Q8×C113—21no—
57C2×Q8×C113D411yescharacteristic
58C2×Q8×C113D412yescomplemented
59C2×D4×C113—21no—
60C2×D4×C113D411yescharacteristic
61C22×C4×C113—21no—
62C22×C4×C113—21no—
63C22×C4×C113C2311yescharacteristic
64Q16×C113—24no—
65SD16×C113—24no—
66M16×C113—21no—
67M16×C113D412yescomplemented
68C2×C8×C113—21no—
69C23×C113—21no—
70C23×C113C2×D411yescharacteristic
71C22×D113—22no—
72C22×D113—22no—
73C22×D113—22no—
74C22×D113—44no—
75Q8×C113—22no—
76Q8×C113—21no—
77Q8×C113—22no—
78Q8×C113—21no—
79Q8×C113—41no—
80Q8×C113—41no—
81D4×C113—21no—
82D4×C113—21no—
83D4×C113—41no—
84D4×C113—41no—
85C2×C4×C113—21no—
86C2×C4×C113—22no—
87C2×C4×C113—41no—
88C2×C4×C113—41no—
89C2×C4×C113—41no—
90C2×C4×C113—41no—
91C2×C4×C113C2×D411yescharacteristic
92C2×C4×C113C2×D411yescharacteristic
93C2×C4×C113C2×D411yescharacteristic
94C2×C4×C113C2×D411yescharacteristic
95C2×C4×C113C2×D411yescharacteristic
96C2×C4×C113#4111yescharacteristic
97C22⋊D113—84no—
98C2×Dic113—22no—
99C2×Dic113—22no—
100C2×Dic113—42no—
101D452—41no—
102D452—41no—
103D452—41no—
104D452—41no—
105C8×C113—22no—
106#11744751—41no—
107C22×C113—22no—
108C22×C113—41no—
109C22×C113—42no—
110C22×C113#12011yescharacteristic
111C22×C113#12011yescharacteristic
112C22×C113#12111yescharacteristic
113C2×D113—44no—
114C2×D113—42no—
115C2×D113—44no—
116C2×D113—82no—
117C4×C113—41no—
118C4×C113—42no—
119C4×C113#12011yescharacteristic
120C4×C113#12111yescharacteristic
121C4×C113#12111yescharacteristic
122C4×C113#12711yescharacteristic
123Dic113—42no—
124Dic113—42no—
125C2×C113—21no—
126C2×C113—21no—
127C2×C113—81no—
128C2×C113#39211yescharacteristic
129C2×C113#45512yes—
130D113—82no—
131#2004—1131nomaximal, Sylow
132C113#200411yescharacteristic, complemented, Sylow
133C2×#137D11311yescharacteristic, complemented
134#535—1131no—
135C2×#124—1131no—
136C2×#104—1131no—
137#419—1131no—
138C2×#100—1131no—
139#339—1131no—
140C2×#41C2×D11311yescharacteristic
141C22×Q8C2×D11311yescharacteristic
142C22×D4—1131no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1131no—
148C2×M16C2×D11311yescharacteristic
149#127—2261no—
150#124—2262no—
151#124—2261no—
152#124—2262no—
153#121—2262no—
154#120—2262no—
155#117—1131no—
156C2×C22⋊C4—2262no—
157C2×C42—1131no—
158#104—2262no—
159#103—2262no—
160#100—2262no—
161C24—1132no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D11311yescharacteristic
172C2×Q8C22⋊D11312yescomplemented
173C2×D4—21no—
174C2×D4—2264no—
175C2×D4—2261no—
176C2×D4—2261no—
177C2×D4—2261no—
178C2×D4—2261no—
179C2×D4—2261no—
180C2×D4—2261no—
181C2×D4C22⋊D11311yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2261no—
185C22×C4C22×D11311yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—2261no—
190M16—2261no—
191M16C22⋊D11312yescomplemented
192C2×C8—21no—
193C2×C8—2261no—
194C4⋊C4—1132no—
195C22⋊C4—2268no—
196C22⋊C4—2264no—
197C22⋊C4—2261no—
198C42—1132no—
199C42—2261no—
200C42—2261no—
201C23—21no—
202C23—4524no—
203C23—2262no—
204C23—2262no—
205C23—2262no—
206C23D4×D11311yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—9044no—
218D4—4521no—
219D4—4521no—
220D4—4521no—
221D4—4521no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4522no—
229C2×C4—2262no—
230C2×C4—2262no—
231C2×C4D4×D11311yescharacteristic
232C2×C4D4×D11311yescharacteristic
233C2×C4C2×C22⋊D11311yescharacteristic
234C2×C4C2×C22⋊D11311yescharacteristic
235C2×C4C2×C22⋊D11311yescharacteristic
236C2×C4#1426864611yescharacteristic
237C8—22no—
238C8—4521no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—9042no—
243C22—4524no—
244C22—4522no—
245C22—4524no—
246C22#1681359011yescharacteristic
247C22#1681359011yescharacteristic
248C22#1681359311yescharacteristic
249C4—41no—
250C4—42no—
251C4—4522no—
252C4—4522no—
253C4#1681359111yescharacteristic
254C4#1681359411yescharacteristic
255C4#1681359411yescharacteristic
256C4#1681361811yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—9042no—
261C2#2029850611yescharacteristic
262C2#2029877412yes—
subgroupG/H
centerC22#16813590
commutator, cosocleC22×C4×C113C23
FrattiniC22×C4C22×D113
FittingC2×#137×C113C2
socleC22×C113#120
prime radical#20299162C2
hypercenterC2×#137D113
hypocenterC113#2004
LCSG ▷ C22×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16813611C2semidirect
3. C2×#16813549C2semidirect
4. #20298607C2semidirect
5. C2×#16813538C2semidirect
6. #20298359C2semidirect
7. C2×#137×C113C2semidirect
47. C4×Dic113D4semidirect
58. C2×Q8×C113D4semidirect
67. M16×C113D4semidirect
132. C113#2004semidirect
133. C2×#137D113semidirect
172. C2×Q8C22⋊D113semidirect
191. M16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
#202991622
C2×#168136112
C2×#168135492
#202986072
C2×#168135382
#202983592
C2×#137×C1132
#2004113