Finite Group Databasebeta

← #28393932

Subgroups — #28393932

Order 14464.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D113C211yesmaximal, characteristic, complemented
2C2×#16813599C212yesmaximal, complemented
3C2×#16813582C211yesmaximal, characteristic
4#20298519C211yesmaximal, characteristic, complemented
5#20298517C211yesmaximal, characteristic, complemented
6#20298510C211yesmaximal, characteristic, complemented
7#20298502C212yesmaximal, complemented
8#20298474C212yesmaximal
9#20298472C212yesmaximal, complemented
10#20298457C211yesmaximal, characteristic, complemented
11#393×C113C211yesmaximal, characteristic, complemented
12C22×Dic226C2211yescharacteristic
13C22×Dic226C2212yes—
14C2×Q8×D113—28no—
15C2×Q8×D113—24no—
16C22×C22⋊D113C2211yescharacteristic
17C23×Dic113C2211yescharacteristic
18C22×C4×D113C2212yes—
19C22×C4×D113C2211yescharacteristic
20#16813599—28no—
21#16813582—24no—
22C2×C4⋊Dic113C2212yes—
23C2×#14268626C2212yes—
24C2×#14268626C2211yescharacteristic
25C2×#14268626C2212yes—
26C2×C22⋊Dic113C2212yes—
27C2×#14268624C2211yescharacteristic
28C2×#14268624C2212yes—
29C2×C4×Dic113C2211yescharacteristic
30#16813529C2212yes—
31#16813529C2212yes—
32#16813529C2212yes—
33#16813529C2211yescharacteristic
34C22×Q8×C113C2211yescharacteristic
35C2×C22⋊C4×C113C2211yescharacteristic
36C2×C22⋊C4×C113C2212yes—
37#95×C113C2212yes—
38#95×C113C2211yescharacteristic
39C23×D113C2311yescharacteristic
40C2×Dic226—22no—
41C2×Dic226—24no—
42C2×Dic226—24no—
43C2×Dic226—24no—
44C2×Dic226D418yescomplemented
45Q8×D113—232no—
46C2×C22⋊D113—22no—
47C2×C22⋊D113—22no—
48C2×C22⋊D113Q814yescomplemented
49C22×Dic113—22no—
50C22×Dic113—24no—
51C22×Dic113C2312yes—
52C22×Dic113C2311yescharacteristic
53C22×Dic113C2312yes—
54C22×Dic113C2311yescharacteristic
55C22×Dic113C2311yescharacteristic
56C2×C4×D113—24no—
57C2×C4×D113—24no—
58C2×C4×D113—24no—
59C2×C4×D113—24no—
60C2×C4×D113D414yes—
61C4⋊Dic113—24no—
62#14268626—22no—
63#14268626—24no—
64#14268626—24no—
65C22⋊Dic113—24no—
66#14268624—22no—
67#14268624—24no—
68C4×Dic113—22no—
69C24×C113C2311yescharacteristic
70C2×Q8×C113—24no—
71C2×Q8×C113—22no—
72C22×C4×C113C2311yescharacteristic
73C22×C4×C113C2312yes—
74C22×C4×C113C2312yes—
75C22×C4×C113C2311yescharacteristic
76C22⋊C4×C113—22no—
77C22⋊C4×C113—24no—
78C23×C113—22no—
79C23×C113—21no—
80C23×C113—21no—
81C23×C113—22no—
82C23×C113C2×Q812yescomplemented
83C23×C113C2411yescharacteristic
84C22×D113—22no—
85C22×D113—22no—
86C22×D113C2×D412yes—
87C22×D113C2×Q812yes—
88C22×D113#4112yes—
89Q8×C113—28no—
90C2×C4×C113—22no—
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×D414yes—
99C2×C4×C113C2×D412yes—
100C22⋊D113—28no—
101C2×Dic113—21no—
102C2×Dic113—22no—
103C2×Dic113—22no—
104C2×Dic113—21no—
105C2×Dic113—21no—
106C2×Dic113—22no—
107C2×Dic113—22no—
108C2×Dic113—22no—
109C2×Dic113—22no—
110C2×Dic113—44no—
111C2×Dic113C2×D414yes—
112C2×Dic113C2×D412yes—
113C2×Dic113C2×D414yes—
114C2×Dic113C2×Q812yescomplemented
115C4×D113—216no—
116C4×D113—28no—
117Dic226—28no—
118Dic226—216no—
119C22×C113—22no—
120C22×C113—24no—
121C22×C113—41no—
122C22×C113—41no—
123C22×C113—42no—
124C22×C113C22×D411yescharacteristic
125C22×C113C22×D412yes—
126C22×C113C22×Q811yescharacteristic
127C22×C113C2×#4111yescharacteristic
128C22×C1132−1+412yes—
129C2×D113—22no—
130C2×D113—22no—
131C2×D113—24no—
132C2×D113—24no—
133C2×D113C22⋊Q814yescomplemented
134C4×C113—22no—
135C4×C113—24no—
136C4×C113—41no—
137C4×C113—42no—
138Dic113—24no—
139Dic113—22no—
140Dic113—41no—
141Dic113—42no—
142Dic113#12014yes—
143C2×C113—42no—
144C2×C113C2×#12011yescharacteristic
145C2×C113C2×C22⋊Q811yescharacteristic
146C2×C113#53611yescharacteristic
147C2×C113#54812yes—
148C2×C113D4×Q812yes—
149D113—24no—
150#2409—1131nomaximal, Sylow
151C113#240911yescharacteristic, complemented, Sylow
152C23×Q8—1131no—
153C2×C22⋊Q8—1132no—
154C2×C4×D4—1131no—
155#395—1131no—
156#395—1131no—
157#393—1131no—
158#393D11311yescharacteristic, complemented
159#391—1132no—
160#386—1132no—
161#385—1132no—
162#380—1131no—
163C22×Q8—1132no—
164C22×Q8—1131no—
165C22×Q8—2264no—
166C22×Q8—2268no—
167C22×Q8C2×D11311yescharacteristic
168C22×D4—1131no—
169C23×C4—1131no—
170C23×C4—1131no—
171C23×C4—1132no—
172C22⋊Q8—2268no—
173C4×D4—2264no—
174C2×C4⋊C4—1131no—
175C2×C4⋊C4—1132no—
176C2×C4⋊C4—1132no—
177C2×C4⋊C4—1132no—
178C2×C22⋊C4—1132no—
179C2×C22⋊C4—1131no—
180C2×C22⋊C4—1132no—
181C2×C22⋊C4C2×D11311yescharacteristic
182C2×C22⋊C4C2×D11312yes—
183C2×C42—1131no—
184#95—1131no—
185#95—1132no—
186#95—1132no—
187#95—1132no—
188#95C2×D11311yescharacteristic
189#95C2×D11312yes—
190C24—1131no—
191C24C22×D11311yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—1138no—
195C2×Q8—2264no—
196C2×Q8—2264no—
197C2×Q8—2264no—
198C2×Q8—22632no—
199C2×Q8—2262no—
200C2×D4—1134no—
201C2×D4—2262no—
202C2×D4—2262no—
203C22×C4—1131no—
204C22×C4—1131no—
205C22×C4—1132no—
206C22×C4—1134no—
207C22×C4—1132no—
208C22×C4—1131no—
209C22×C4—2264no—
210C22×C4—2264no—
211C22×C4—2264no—
212C22×C4—2262no—
213C22×C4—2264no—
214C22×C4—2264no—
215C22×C4C22×D11311yescharacteristic
216C22×C4C22×D11312yes—
217C22×C4C22×D11311yescharacteristic
218C22×C4C22×D11312yes—
219C4⋊C4—2264no—
220C4⋊C4—2264no—
221C4⋊C4—2262no—
222C4⋊C4—2264no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—2262no—
226C22⋊C4—2264no—
227C22⋊C4—2264no—
228C42—2262no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—1132no—
234C23—1132no—
235C23—1132no—
236C23—2262no—
237C23—2262no—
238C23Q8×D11312yescomplemented
239C23C23×D11311yescharacteristic
240Q8—28no—
241Q8—22616no—
242Q8—2268no—
243D4—2268no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—1132no—
253C2×C4—1134no—
254C2×C4—1134no—
255C2×C4—1132no—
256C2×C4—4524no—
257C2×C4—2268no—
258C2×C4—22616no—
259C2×C4—2262no—
260C2×C4—2261no—
261C2×C4—2262no—
262C2×C4—2262no—
263C2×C4—2262no—
264C2×C4—2261no—
265C2×C4—2262no—
266C2×C4—2262no—
267C2×C4—2261no—
268C2×C4D4×D11312yes—
269C2×C4D4×D11314yes—
270C22—22no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—1134no—
276C22—2262no—
277C22—2262no—
278C22—2264no—
279C22—2264no—
280C22C2×D4×D11311yescharacteristic
281C22C2×D4×D11312yes—
282C22C2×Q8×D11311yescharacteristic
283C22#41×D11311yescharacteristic
284C22#1681366912yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—1134no—
290C4—4521no—
291C4—4522no—
292C4—2264no—
293C4—2262no—
294C2—42no—
295C2—2264no—
296C2#120×D11311yescharacteristic
297C2C22⋊Q8×D11311yescharacteristic
298C2#2029916811yescharacteristic
299C2#2029925612yes—
300C2#2029926112yes—
subgroupG/H
center, FrattiniC23C23×D113
commutator, socle, cosocleC23×C113C24
Fitting#393×C113C2
prime radicalC22×C22⋊D113C22
hypercenter#393D113
hypocenterC113#2409
LCSG ▷ C23×C113 ▷ C113
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D113C2semidirect
2. C2×#16813599C2semidirect
4. #20298519C2semidirect
5. #20298517C2semidirect
6. #20298510C2semidirect
7. #20298502C2semidirect
9. #20298472C2semidirect
10. #20298457C2semidirect
11. #393×C113C2semidirect
44. C2×Dic226D4semidirect
48. C2×C22⋊D113Q8semidirect
82. C23×C113C2×Q8semidirect
114. C2×Dic113C2×Q8semidirect
133. C2×D113C22⋊Q8semidirect
151. C113#2409semidirect
158. #393D113semidirect
238. C23Q8×D113semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1132
C2×#168135992
C2×#168135822
#202985192
#202985172
#202985102
#202985022
#202984742
#202984722
#202984572
#393×C1132
#2409113