Finite Group Databasebeta

← #28393815

Subgroups — #28393815

Order 14464.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16813584C212yesmaximal, complemented
2C2×#16813584C211yesmaximal, characteristic, complemented
3C2×#16813580C211yesmaximal, characteristic, complemented
4#20298524C211yesmaximal, characteristic, complemented
5#20298520C212yesmaximal, complemented
6#20298516C212yesmaximal, complemented
7#20298515C211yesmaximal, characteristic, complemented
8#20298508C211yesmaximal, characteristic
9#20298449C211yesmaximal, characteristic, complemented
10#20298445C212yesmaximal, complemented
11#398×C113C211yesmaximal, characteristic, complemented
12C22×D452C2211yescharacteristic
13C22×D452C2212yes—
14C22×C4×D113C2212yes—
15C22×C4×D113C2211yescharacteristic
16#16813584—28no—
17#16813584—24no—
18#16813580—24no—
19C2×C4⋊Dic113C2211yescharacteristic
20C2×#14268626C2212yes—
21C2×#14268624C2212yes—
22C2×#14268624C2211yescharacteristic
23C2×#14268624C2212yes—
24C2×#14268624C2212yes—
25C2×#14268624C2211yescharacteristic
26C2×#14268624C2212yes—
27C2×#14268624C2212yes—
28C2×C4×Dic113C2211yescharacteristic
29C2×C4×Dic113C2212yes—
30C2×C4×Dic113C2211yescharacteristic
31#16813529C2211yescharacteristic
32#16813529C2212yes—
33C2×C4⋊C4×C113C2211yescharacteristic
34C2×C4⋊C4×C113C2212yes—
35C2×C4⋊C4×C113C2211yescharacteristic
36#95×C113C2211yescharacteristic
37#95×C113C2212yes—
38C23×D113C2311yescharacteristic
39C23×D113C2312yes—
40C2×D452—24no—
41C2×D452—24no—
42C2×D452—24no—
43C2×D452—22no—
44C2×D452—24no—
45C22×Dic113C2312yes—
46C22×Dic113C2311yescharacteristic
47C22×Dic113C2311yescharacteristic
48C22×Dic113C2311yescharacteristic
49C2×C4×D113—22no—
50C2×C4×D113—22no—
51C2×C4×D113—24no—
52C2×C4×D113—24no—
53C2×C4×D113—24no—
54C2×C4×D113D414yescomplemented
55C4⋊Dic113—22no—
56#14268626—24no—
57#14268624—24no—
58#14268624—24no—
59#14268624—22no—
60#14268624—24no—
61#14268624—24no—
62#14268624—22no—
63#14268624—24no—
64C4×Dic113—22no—
65C4×Dic113—24no—
66C4×Dic113D414yescomplemented
67C22×C4×C113C2311yescharacteristic
68C22×C4×C113C2312yes—
69C22×C4×C113C2311yescharacteristic
70C22×C4×C113C2311yescharacteristic
71C22×C4×C113C2312yes—
72C4⋊C4×C113—22no—
73C4⋊C4×C113—24no—
74C4⋊C4×C113—22no—
75C23×C113C2411yescharacteristic
76C22×D113—21no—
77C22×D113—22no—
78C22×D113—22no—
79C22×D113—22no—
80C22×D113—21no—
81C22×D113—22no—
82C22×D113—22no—
83C22×D113—22no—
84C22×D113—22no—
85C22×D113—22no—
86C22×D113C2×D412yes—
87C22×D113#4114yes—
88C2×C4×C113—22no—
89C2×C4×C113—21no—
90C2×C4×C113—21no—
91C2×C4×C113—22no—
92C2×C4×C113—22no—
93C2×C4×C113—21no—
94C2×C4×C113—22no—
95C2×C4×C113—22no—
96C2×C4×C113—22no—
97C2×C4×C113—22no—
98C2×C4×C113C2×D412yes—
99C2×C4×C113C2×D412yes—
100C2×C4×C113#4114yes—
101C2×Dic113—22no—
102C2×Dic113—22no—
103C2×Dic113—21no—
104C2×Dic113—22no—
105C2×Dic113—22no—
106C2×Dic113—22no—
107C2×Dic113C2×D412yes—
108C2×Dic113C2×D412yes—
109C2×Dic113C2×D412yes—
110C2×Dic113#4112yes—
111D452—28no—
112D452—216no—
113C4×D113—28no—
114C4×D113—48no—
115C22×C113C22×D411yescharacteristic
116C22×C113C22×D411yescharacteristic
117C22×C113C2×#4111yescharacteristic
118C22×C113C2×#4112yes—
119C22×C113C2×#4112yes—
120C2×D113—22no—
121C2×D113—24no—
122C2×D113—24no—
123C2×D113—24no—
124C2×D113—24no—
125C2×D113—42no—
126C2×D113—41no—
127C2×D113—42no—
128C2×D113—42no—
129C2×D113—42no—
130C2×D113—41no—
131C2×D113—42no—
132C4×C113—22no—
133C4×C113—22no—
134C4×C113—24no—
135C4×C113—42no—
136C4×C113—41no—
137Dic113—22no—
138Dic113—42no—
139Dic113—41no—
140Dic113#12114yes—
141C2×C113C2×#12111yescharacteristic
142C2×C113#52511yescharacteristic
143C2×C113#52512yes—
144C2×C113#52912yes—
145C2×C113#53211yescharacteristic
146D113—42no—
147D113—44no—
148#2392—1131nomaximal, Sylow
149C113#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1131no—
151C2×C4×D4—1132no—
152C2×#117—1131no—
153#398D11311yescharacteristic, complemented
154#397—1131no—
155#396—1132no—
156#394—1131no—
157#394—1132no—
158#392—1131no—
159C4×C4⋊C4—1131no—
160C4×C22⋊C4—1132no—
161C22×D4—1131no—
162C22×D4—1132no—
163C23×C4—1131no—
164C23×C4—1132no—
165C4×D4—2264no—
166C4×D4—2268no—
167#117—2264no—
168C2×C4⋊C4—1131no—
169C2×C4⋊C4—1132no—
170C2×C4⋊C4C2×D11311yescharacteristic
171C2×C4⋊C4C2×D11312yes—
172C2×C4⋊C4C2×D11311yescharacteristic
173C2×C22⋊C4—1131no—
174C2×C22⋊C4—1131no—
175C2×C22⋊C4—1132no—
176C2×C22⋊C4—1132no—
177C2×C22⋊C4—1132no—
178C2×C22⋊C4—1132no—
179C2×C22⋊C4—1132no—
180C2×C42—1131no—
181C2×C42—1131no—
182C2×C42—1132no—
183#95—1132no—
184#95—1131no—
185#95C2×D11312yes—
186#95C2×D11311yescharacteristic
187C24—1131no—
188C24—1132no—
189C2×D4—2264no—
190C2×D4—2264no—
191C2×D4—2264no—
192C2×D4—2262no—
193C2×D4—2264no—
194C22×C4—1131no—
195C22×C4—1131no—
196C22×C4—1134no—
197C22×C4—1132no—
198C22×C4—1131no—
199C22×C4—2264no—
200C22×C4—2264no—
201C22×C4—2264no—
202C22×C4—2262no—
203C22×C4—2262no—
204C22×C4C22×D11312yes—
205C22×C4C22×D11311yescharacteristic
206C22×C4C22×D11311yescharacteristic
207C22×C4C22×D11311yescharacteristic
208C22×C4C22×D11312yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—2262no—
213C4⋊C4—2264no—
214C22⋊C4—2262no—
215C22⋊C4—2264no—
216C22⋊C4—2264no—
217C22⋊C4—2264no—
218C22⋊C4—2262no—
219C22⋊C4—2264no—
220C22⋊C4—2264no—
221C42—1134no—
222C42—2262no—
223C42—2264no—
224C23—1132no—
225C23—1134no—
226C23—2261no—
227C23—2262no—
228C23—2262no—
229C23—2262no—
230C23—2261no—
231C23—2262no—
232C23—2262no—
233C23—2262no—
234C23—2262no—
235C23—2262no—
236C23C23×D11311yescharacteristic
237D4—2268no—
238D4—22616no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—1132no—
250C2×C4—1132no—
251C2×C4—1132no—
252C2×C4—1132no—
253C2×C4—4528no—
254C2×C4—2268no—
255C2×C4—2262no—
256C2×C4—2261no—
257C2×C4—2262no—
258C2×C4—2262no—
259C2×C4—2262no—
260C2×C4—2262no—
261C2×C4D4×D11312yes—
262C2×C4D4×D11312yes—
263C2×C4#1426864814yes—
264C22—4522no—
265C22—4521no—
266C22—4522no—
267C22—4522no—
268C22—4522no—
269C22—4521no—
270C22—4522no—
271C22—2262no—
272C22—2264no—
273C22—2264no—
274C22—2264no—
275C22—2264no—
276C22C2×D4×D11311yescharacteristic
277C22C2×D4×D11311yescharacteristic
278C22#41×D11311yescharacteristic
279C22#41×D11312yes—
280C22C2×#1426864812yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—1134no—
287C4—4522no—
288C4—4521no—
289C4—2262no—
290C2—4524no—
291C2—4522no—
292C2#121×D11311yescharacteristic
293C2#2029910611yescharacteristic
294C2#2029911112yes—
295C2#2029913312yes—
296C2#2029914911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D113
commutator, socle, cosocleC23×C113C24
Fitting#398×C113C2
prime radical#20298508C2
hypercenter#398D113
hypocenterC113#2392
LCSG ▷ C23×C113 ▷ C113
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16813584C2semidirect
2. C2×#16813584C2semidirect
3. C2×#16813580C2semidirect
4. #20298524C2semidirect
5. #20298520C2semidirect
6. #20298516C2semidirect
7. #20298515C2semidirect
9. #20298449C2semidirect
10. #20298445C2semidirect
11. #398×C113C2semidirect
54. C2×C4×D113D4semidirect
66. C4×Dic113D4semidirect
149. C113#2392semidirect
153. #398D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168135842
C2×#168135802
#202985242
#202985202
#202985162
#202985152
#202985082
#202984492
#202984452
#398×C1132
#2392113