Finite Group Databasebeta

← #28400196

Subgroups — #28400196

Order 14464.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16813647C211yesmaximal, characteristic, complemented
2C2×M16×D113C211yesmaximal, characteristic, complemented
3#20299105C211yesmaximal, characteristic, complemented
4C2×#16813597C211yesmaximal, characteristic, complemented
5#20298850C212yesmaximal, complemented
6#20298847C212yesmaximal, complemented
7#20298833C212yesmaximal, complemented
8#20298830C212yesmaximal, complemented
9#20298638C211yesmaximal, characteristic, complemented
10#20298604C211yesmaximal, characteristic, complemented
11#417×C113C211yesmaximal, characteristic, complemented
12C22×D452—21no—
13C22×D452C2211yescharacteristic
14C22×C4×D113C2211yescharacteristic
15#16813647—24no—
16C2×#14268634C2212yescomplemented
17C2×#14268632C2212yescomplemented
18C2×#14268631C2211yescharacteristic
19C2×#14268630C2212yescomplemented
20M16×D113—24no—
21C2×C8×D113C2212yescomplemented
22#16813603—21no—
23#16813598C2211yescharacteristic, complemented
24#16813598C2211yescharacteristic, complemented
25#16813597—21no—
26#16813597—21no—
27#16813597C2212yes—
28#16813597C2212yes—
29#16813593C2211yescharacteristic, complemented
30#16813593C2211yescharacteristic, complemented
31#16813590—21no—
32C4×C22⋊D113—21no—
33C4×C22⋊D113—21no—
34#16813581C2211yescharacteristic
35C2×#14268624—21no—
36#16813555C2212yescomplemented
37#16813553C2212yescomplemented
38#16813544C2212yescomplemented
39#16813544C2212yescomplemented
40C2×#41×C113C2211yescharacteristic
41C2×M16×C113C2211yescharacteristic
42C2×C4⋊C4×C113C2211yescharacteristic
43#103×C113C2212yescomplemented
44#102×C113C2212yescomplemented
45C23×D113—21no—
46C23×D113D411yescharacteristic
47C2×D452—22no—
48C2×D452—22no—
49C2×D452—22no—
50C2×D452—22no—
51C2×D452—21no—
52C2×D452—44no—
53C2×D452C2312yes—
54C2×C22⋊D113—21no—
55C2×C22⋊D113—21no—
56C22×Dic113D411yescharacteristic
57C2×C4×D113—22no—
58C2×C4×D113—22no—
59C2×C4×D113D411yescharacteristic
60C2×C4×D113D411yescharacteristic
61C2×C4×D113C2311yescharacteristic
62C2×C4×D113C2311yescharacteristic
63#14268634—24no—
64#14268632—24no—
65#14268631D414yescomplemented
66#14268630—24no—
67C2×#11744751C2312yes—
68C8×D113—24no—
69C4⋊Dic113C2311yescharacteristic
70C4⋊Dic113C2311yescharacteristic
71#14268626—21no—
72#14268626—21no—
73C22⋊Dic113—21no—
74C22⋊Dic113—21no—
75C22⋊Dic113—21no—
76#14268624—21no—
77#14268624—21no—
78#14268624—22no—
79#14268624—22no—
80C4×Dic113—21no—
81#41×C113—22no—
82#41×C113—22no—
83C2×Q8×C113C2311yescharacteristic
84C2×D4×C113—21no—
85C2×D4×C113C2311yescharacteristic
86C22×C4×C113—21no—
87C22×C4×C113—21no—
88C22×C4×C113C2311yescharacteristic
89M16×C113—22no—
90C2×C8×C113C2312yes—
91C4⋊C4×C113—21no—
92C4⋊C4×C113C2312yes—
93C23×C113—21no—
94C23×C113C2×D411yescharacteristic
95C22×D113—22no—
96C22×D113—22no—
97C22×D113—22no—
98C22×D113—41no—
99C22×D113—42no—
100C22×D113—42no—
101C22×D113—42no—
102C22×D113C2×D411yescharacteristic
103C22×D113C2×D411yescharacteristic
104Q8×C113—21no—
105Q8×C113—21no—
106D4×C113—21no—
107D4×C113—21no—
108D4×C113—42no—
109C2×C4×C113—22no—
110C2×C4×C113—21no—
111C2×C4×C113—42no—
112C2×C4×C113—42no—
113C2×C4×C113C2×D411yescharacteristic
114C2×C4×C113C2×D412yes—
115C2×C4×C113#4112yes—
116C2×C4×C113C2411yescharacteristic
117C22⋊D113—42no—
118C22⋊D113—42no—
119C2×Dic113—22no—
120C2×Dic113—21no—
121C2×Dic113—21no—
122C2×Dic113C2×D411yescharacteristic
123C2×Dic113C2×D411yescharacteristic
124D452—22no—
125D452—22no—
126D452—44no—
127D452—44no—
128D452—42no—
129C4×D113—22no—
130C4×D113—22no—
131C4×D113—21no—
132C4×D113—21no—
133C4×D113—21no—
134C4×D113—21no—
135C8×C113—22no—
136#11744751C2×D414yescomplemented
137C22×C113—21no—
138C22×C113—21no—
139C22×C113—41no—
140C22×C113—41no—
141C22×C113—41no—
142C22×C113#12112yes—
143C22×C113C22×D411yescharacteristic
144C2×D113—21no—
145C2×D113—21no—
146C2×D113—21no—
147C2×D113—21no—
148C2×D113—21no—
149C2×D113—44no—
150C2×D113—42no—
151C2×D113—42no—
152C2×D113—42no—
153C2×D113—84no—
154C2×D113AGL(1,ℤ/8)12yescomplemented
155C4×C113—41no—
156C4×C113—42no—
157C4×C113#12112yes—
158C4×C113C22×D411yescharacteristic
159C4×C113C2×#4111yescharacteristic
160Dic113—21no—
161Dic113—21no—
162Dic113—41no—
163Dic113—41no—
164C2×C113—22no—
165C2×C113—81no—
166C2×C113C2×#12111yescharacteristic
167C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C113#57511yescharacteristic
169D113—22no—
170D113—41no—
171D113—82no—
172#3055—1131nomaximal, Sylow
173C113#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1131no—
175C22×M16—1131no—
176#525—1131no—
177C2×#121—1131no—
178#469—1132no—
179#468—1132no—
180#466—1132no—
181#465—1132no—
182#428—1131no—
183#418—1131no—
184#417D11311yescharacteristic, complemented
185C2×#41C2×D11311yescharacteristic
186C22×D4—1131no—
187C22×D4—2261no—
188C23×C4—1131no—
189AGL(1,ℤ/8)—2264no—
190C2×SD16—1132no—
191C2×D8—1132no—
192C2×M16—1131no—
193C2×M16—1132no—
194C2×M16—2264no—
195C2×M16C2×D11311yescharacteristic
196C22×C8—1132no—
197#123—2261no—
198C22⋊Q8—1131no—
199C22⋊Q8—1131no—
200#121—1131no—
201#121—1131no—
202#121—1132no—
203#121—1132no—
204#121—2261no—
205#121—2261no—
206#120—2261no—
207C4×D4—2261no—
208C4×D4—2261no—
209#117—1131no—
210C2×C4⋊C4C2×D11311yescharacteristic
211C2×C22⋊C4—2261no—
212#107—1132no—
213#106—1132no—
214#103C2×D11312yescomplemented
215#102—1132no—
216#102—1132no—
217#102C2×D11312yescomplemented
218C24—1131no—
219C24—2261no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D11311yescharacteristic
223C2×D4—21no—
224C2×D4—1132no—
225C2×D4—4524no—
226C2×D4—2262no—
227C2×D4—2262no—
228C2×D4—2261no—
229C2×D4—2261no—
230C2×D4—2261no—
231C2×D4—2262no—
232C2×D4—2262no—
233C2×D4C22×D11311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1131no—
237C22×C4—1131no—
238C22×C4—1131no—
239C22×C4—1131no—
240C22×C4—1131no—
241C22×C4—2262no—
242C22×C4—2262no—
243C22×C4C22×D11311yescharacteristic
244SD16—2264no—
245D8—2264no—
246M16—22no—
247M16—1134no—
248M16—2264no—
249C2×C8—1132no—
250C2×C8—2264no—
251C2×C8C22×D11312yes—
252C4⋊C4—21no—
253C4⋊C4—1131no—
254C4⋊C4—1131no—
255C4⋊C4—2261no—
256C4⋊C4—2261no—
257C4⋊C4C22×D11312yes—
258C22⋊C4—2262no—
259C22⋊C4—2262no—
260C22⋊C4—2261no—
261C22⋊C4—2261no—
262C22⋊C4—2261no—
263C22⋊C4—2261no—
264C22⋊C4—2261no—
265C42—2261no—
266C23—21no—
267C23—1131no—
268C23—1131no—
269C23—4522no—
270C23—4522no—
271C23—4521no—
272C23—4522no—
273C23—2262no—
274C23—2262no—
275C23—2262no—
276C23D4×D11311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—4524no—
283D4—4524no—
284D4—4522no—
285D4—4522no—
286D4—4522no—
287D4—2262no—
288D4—2262no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1131no—
294C2×C4—1131no—
295C2×C4—2261no—
296C2×C4—2261no—
297C2×C4—2262no—
298C2×C4—2262no—
299C2×C4—2262no—
300C2×C4—2261no—
301C2×C4—2261no—
302C2×C4—2261no—
303C2×C4—2261no—
304C2×C4D4×D11311yescharacteristic
305C2×C4D4×D11312yes—
306C2×C4#1426864712yes—
307C2×C4C23×D11311yescharacteristic
308C8—22no—
309C8—1134no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—1132no—
316C22—9044no—
317C22—4524no—
318C22—4522no—
319C22—4522no—
320C22—4522no—
321C22—2261no—
322C22—2261no—
323C22—2261no—
324C22—2261no—
325C22—2261no—
326C22#1681359512yes—
327C22C2×D4×D11311yescharacteristic
328C4—41no—
329C4—42no—
330C4—4521no—
331C4—4521no—
332C4—2261no—
333C4—2261no—
334C4#1681359512yes—
335C4C2×D4×D11311yescharacteristic
336C4C2×#1426864711yescharacteristic
337C2—22no—
338C2—81no—
339C2—9042no—
340C2—4521no—
341C2—2262no—
342C2C2×#1681359511yescharacteristic
343C2AGL(1,ℤ/8)×D11311yescharacteristic
344C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#417×C113C2
socleC22×C113C22×D4
hypercenter#417D113
hypocenterC113#3055
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16813647C2semidirect
2. C2×M16×D113C2semidirect
3. #20299105C2semidirect
4. C2×#16813597C2semidirect
5. #20298850C2semidirect
6. #20298847C2semidirect
7. #20298833C2semidirect
8. #20298830C2semidirect
9. #20298638C2semidirect
10. #20298604C2semidirect
11. #417×C113C2semidirect
16. C2×#14268634C22semidirect
17. C2×#14268632C22semidirect
19. C2×#14268630C22semidirect
21. C2×C8×D113C22semidirect
23. #16813598C22semidirect
24. #16813598C22semidirect
29. #16813593C22semidirect
30. #16813593C22semidirect
36. #16813555C22semidirect
37. #16813553C22semidirect
38. #16813544C22semidirect
39. #16813544C22semidirect
43. #103×C113C22semidirect
44. #102×C113C22semidirect
65. #14268631D4semidirect
136. #11744751C2×D4semidirect
154. C2×D113AGL(1,ℤ/8)semidirect
173. C113#3055semidirect
184. #417D113semidirect
214. #103C2×D113semidirect
217. #102C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136472
C2×M16×D1132
#202991052
C2×#168135972
#202988502
#202988472
#202988332
#202988302
#202986382
#202986042
#417×C1132
#3055113