Finite Group Databasebeta

← #28402810

Subgroups — #28402810

Order 14464.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D452C211yesmaximal, characteristic, complemented
2#20298946C211yesmaximal, characteristic, complemented
3#20298922C212yesmaximal, complemented
4#20298885C211yesmaximal, characteristic, complemented
5#20298849C212yesmaximal, complemented
6#20298793C211yesmaximal, characteristic, complemented
7#20298731C212yesmaximal, complemented
8C4×#14268632C211yesmaximal, characteristic, complemented
9#20298676C211yesmaximal, characteristic, complemented
10#20298602C212yesmaximal, complemented
11#555×C113C211yesmaximal, characteristic, complemented
12C22×D452C2212yes—
13C2×D4×D113—21no—
14C2×#14268632C2211yescharacteristic
15C2×#14268632C2212yescomplemented
16C2×#14268631C2212yescomplemented
17#16813619C2211yescharacteristic
18#16813597—21no—
19#16813596C2212yes—
20#16813592—22no—
21C4×D452C2211yescharacteristic
22#16813555C2211yescharacteristic, complemented
23#16813553C2212yescomplemented
24C4⋊#11744751C2211yescharacteristic, complemented
25#16813543C2211yescharacteristic, complemented
26#16813542C2211yescharacteristic
27#16813542C2212yes—
28#16813542C2212yescomplemented
29#16813542C2212yescomplemented
30#16813542C2212yescomplemented
31C22⋊#11744751C2212yescomplemented
32C4×#11744751C2211yescharacteristic, complemented
33C4⋊Q8×C113C2211yescharacteristic, complemented
34#124×C113—22no—
35#121×C113—21no—
36#121×C113C2212yescomplemented
37C4×D4×C113C2211yescharacteristic
38C4×D4×C113C2211yescharacteristic, complemented
39#117×C113—21no—
40#117×C113C2212yescomplemented
41C23×D113—22no—
42C2×D452—24no—
43C2×D452—22no—
44C2×D452—21no—
45C2×D452—21no—
46C2×D452—42no—
47C2×D452C2312yes—
48C2×D452C2311yescharacteristic
49C2×C22⋊D113—22no—
50D4×D113—44no—
51C2×C4×D113—21no—
52#14268632—22no—
53#14268632—42no—
54#14268631—42no—
55C2×#11744751C2311yescharacteristic
56C2×#11744751C2312yes—
57C2×#11744751C2311yescharacteristic
58C4⋊Dic113C2311yescharacteristic
59#14268624—21no—
60#14268624—22no—
61C2×Q8×C113—21no—
62C2×D4×C113—22no—
63C2×D4×C113D411yescharacteristic
64C2×D4×C113C2311yescharacteristic
65C22×C4×C113—21no—
66C22×C4×C113C2312yes—
67C4⋊C4×C113—21no—
68C4⋊C4×C113D411yescharacteristic
69C4⋊C4×C113C2311yescharacteristic
70C4⋊C4×C113C2312yes—
71C22⋊C4×C113—22no—
72C22⋊C4×C113—22no—
73C22⋊C4×C113—21no—
74C22⋊C4×C113D412yes—
75C42×C113—22no—
76C42×C113C2311yescharacteristic
77C23×C113—21no—
78C23×C113C2×D412yes—
79C22×D113—21no—
80C22×D113—22no—
81C22×D113—44no—
82C22×D113—44no—
83C22×D113—82no—
84Q8×C113—81no—
85D4×C113—22no—
86D4×C113—21no—
87D4×C113—21no—
88D4×C113—82no—
89C2×C4×C113—22no—
90C2×C4×C113—22no—
91C2×C4×C113—21no—
92C2×C4×C113—22no—
93C2×C4×C113—42no—
94C2×C4×C113C2×D412yes—
95C2×C4×C113C2×D411yescharacteristic
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113C2411yescharacteristic
98C22⋊D113—44no—
99C2×Dic113—21no—
100D452—22no—
101D452—21no—
102D452—22no—
103D452—44no—
104D452—42no—
105D452—81no—
106D452#4112yes—
107C4×D113—42no—
108#11744751—21no—
109#11744751—21no—
110#11744751—22no—
111C22×C113—22no—
112C22×C113—22no—
113C22×C113—22no—
114C22×C113—41no—
115C22×C113—41no—
116C22×C113—41no—
117C22×C113C22×D411yescharacteristic
118C2×D113—22no—
119C2×D113—41no—
120C2×D113—41no—
121C2×D113—42no—
122C2×D113—42no—
123C2×D113—42no—
124C2×D113—88no—
125C4×C113—22no—
126C4×C113—21no—
127C4×C113—41no—
128C4×C113—42no—
129C4×C113AGL(1,ℤ/8)12yes—
130C4×C113C2×#4111yescharacteristic
131C4×C113H(5,𝔽2)11yescharacteristic
132Dic113—41no—
133C2×C113—42no—
134C2×C113—81no—
135C2×C113#54611yescharacteristic
136C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C113#57611yescharacteristic
138D113—42no—
139D113—82no—
140#3325—1131nomaximal, Sylow
141C113#332511yescharacteristic, complemented, Sylow
142#555D11311yescharacteristic, complemented
143D42—1131no—
144#486—1131no—
145#482—1132no—
146#476—1131no—
147#469—1132no—
148#459—1131no—
149#447—1132no—
150C4×D8—1131no—
151#436—1131no—
152#418—1132no—
153C22×D4—1132no—
154C22×D4—2261no—
155C2×D8—1132no—
156C2×D8—1131no—
157C2×M16—1132no—
158C4⋊Q8C2×D11311yescharacteristic, complemented
159#127—1131no—
160#124—22no—
161#121—21no—
162#121—1132no—
163#121—2261no—
164#121C2×D11312yescomplemented
165#120—2262no—
166C4×D4—1131no—
167C4×D4C2×D11311yescharacteristic
168C4×D4C2×D11311yescharacteristic, complemented
169#117—21no—
170#117C2×D11312yescomplemented
171#107—1131no—
172#106—1132no—
173C4⋊C8—1131no—
174#102—1131no—
175#102—1131no—
176#102—1132no—
177#102—1132no—
178#102—1132no—
179#102—1132no—
180C22⋊C8—1132no—
181C4×C8—1131no—
182C24—2262no—
183C2×Q8—21no—
184C2×D4—22no—
185C2×D4—1131no—
186C2×D4—1132no—
187C2×D4—4522no—
188C2×D4—4524no—
189C2×D4—2264no—
190C2×D4—2262no—
191C2×D4—2261no—
192C2×D4—2262no—
193C2×D4—2261no—
194C2×D4C22⋊D11311yescharacteristic
195C2×D4C22×D11311yescharacteristic
196C22×C4—21no—
197C22×C4—2261no—
198C22×C4C22×D11312yes—
199D8—4522no—
200D8—2262no—
201M16—4522no—
202C2×C8—1131no—
203C2×C8—1131no—
204C2×C8—1132no—
205C4⋊C4—21no—
206C4⋊C4—1131no—
207C4⋊C4C22⋊D11311yescharacteristic
208C4⋊C4C22×D11312yes—
209C4⋊C4C22×D11311yescharacteristic
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—2261no—
214C22⋊C4—2262no—
215C22⋊C4C22⋊D11312yes—
216C42—22no—
217C42C22×D11311yescharacteristic
218C23—21no—
219C23—9042no—
220C23—4524no—
221C23—4524no—
222C23—2262no—
223C23—2261no—
224C23C2×C22⋊D11312yes—
225Q8—81no—
226D4—22no—
227D4—21no—
228D4—21no—
229D4—82no—
230D4—1132no—
231D4—9041no—
232D4—4524no—
233D4—4524no—
234D4—4522no—
235D4—2262no—
236D4—2262no—
237D4—2261no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—4522no—
244C2×C4—2261no—
245C2×C4C2×C22⋊D11312yes—
246C2×C4C2×C22⋊D11311yescharacteristic
247C2×C4C2×C22⋊D11311yescharacteristic
248C2×C4C23×D11311yescharacteristic
249C8—2261no—
250C8—2261no—
251C8—2262no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—9048no—
259C22—4522no—
260C22—4522no—
261C22—4522no—
262C22—4521no—
263C22—4521no—
264C22—2262no—
265C22C22×C22⋊D11311yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—4521no—
271C4#1681364712yes—
272C4#41×D11311yescharacteristic
273C4#1681366811yescharacteristic
274C2—42no—
275C2—81no—
276C2—9042no—
277C2—4522no—
278C2#2029923611yescharacteristic
279C2C2×#1681364711yescharacteristic
280C2#2029940111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#555×C113C2
socleC22×C113C22×D4
hypercenter#555D113
hypocenterC113#3325
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. D4×D452C2semidirect
2. #20298946C2semidirect
3. #20298922C2semidirect
4. #20298885C2semidirect
5. #20298849C2semidirect
6. #20298793C2semidirect
7. #20298731C2semidirect
8. C4×#14268632C2semidirect
9. #20298676C2semidirect
10. #20298602C2semidirect
11. #555×C113C2semidirect
15. C2×#14268632C22semidirect
16. C2×#14268631C22semidirect
22. #16813555C22semidirect
23. #16813553C22semidirect
24. C4⋊#11744751C22semidirect
25. #16813543C22semidirect
28. #16813542C22semidirect
29. #16813542C22semidirect
30. #16813542C22semidirect
31. C22⋊#11744751C22semidirect
32. C4×#11744751C22semidirect
33. C4⋊Q8×C113C22semidirect
36. #121×C113C22semidirect
38. C4×D4×C113C22semidirect
40. #117×C113C22semidirect
141. C113#3325semidirect
142. #555D113semidirect
158. C4⋊Q8C2×D113semidirect
164. #121C2×D113semidirect
168. C4×D4C2×D113semidirect
170. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
D4×D4522
#202989462
#202989222
#202988852
#202988492
#202987932
#202987312
C4×#142686322
#202986762
#202986022
#555×C1132
#3325113