Finite Group Databasebeta

← C22×#16837441 #29452341

Subgroups — C22×#16837441 #29452341

Order 15104.

3182 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11745659C211yesmaximal, characteristic
2C22×#14273685C211yesmaximal, characteristic
3C22×C4⋊C4×D59C211yesmaximal, characteristic
4C2×#16837441C2124yesmaximal, direct product factor
5C22×#14273661C211yesmaximal, characteristic, complemented
6C22×#14273660C211yesmaximal, characteristic, complemented
7C22×#14273658C211yesmaximal, characteristic, complemented
8C22×#106×C59C211yesmaximal, characteristic, complemented
9C23×C4×D59—21no—
10C23×C4×D59C2211yescharacteristic
11C22×#11745659C4116yescomplemented
12C22×#11745659C22112yes—
13C23×C59⋊C8C2211yescharacteristic
14C2×#14273685—26no—
15C2×#14273685C22112yes—
16C22×C4⋊Dic59C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C22×Dic59⋊C4—21no—
19C2×C4⋊C4×D59—26no—
20C2×C4⋊C4×D59C22112yes—
21C22×(C2×D59)⋊C4—21no—
22C22×C4×Dic59—21no—
23#16837441C22164yesdirect product factor
24C2×#14273661C22112yescomplemented
25C2×#14273660C22112yescomplemented
26C2×#14273658C22112yescomplemented
27C23×C8×C59C2211yescharacteristic
28C22×C4⋊C4×C59C2211yescharacteristic
29C22×C4⋊C4×C59C2211yescharacteristic
30C2×#106×C59C22112yescomplemented
31C24×D59D411yescharacteristic
32C23×Dic59—21no—
33C23×Dic59—21no—
34C23×Dic59D411yescharacteristic
35C22×C4×D59—212no—
36C22×C4×D59—48no—
37C22×C4×D59C2×C418yes—
38C22×C4×D59D416yes—
39C22×C4×D59Q818yes—
40C22×C4×D59C2316yes—
41C2×#11745659C2×C4196yescomplemented
42C2×#11745659C23116yes—
43C22×C59⋊C8C2×C418yescomplemented
44C22×C59⋊C8C2316yes—
45#14273685—224no—
46#14273685C23116yes—
47C2×C4⋊Dic59—23no—
48C2×C4⋊Dic59—23no—
49C2×C4⋊Dic59C2316yes—
50C2×C4⋊Dic59C2316yes—
51C2×Dic59⋊C4—212no—
52C4⋊C4×D59—224no—
53C4⋊C4×D59C23116yes—
54C2×(C2×D59)⋊C4—212no—
55C2×C4×Dic59—212no—
56#14273661C23116yescomplemented
57#14273660C23116yescomplemented
58#14273658C23116yescomplemented
59C23×C4×C59—21no—
60C23×C4×C59—21no—
61C23×C4×C59C2311yescharacteristic
62C22×C8×C59C2×C418yescomplemented
63C22×C8×C59C2316yes—
64C2×C4⋊C4×C59—23no—
65C2×C4⋊C4×C59—23no—
66C2×C4⋊C4×C59C2316yes—
67C2×C4⋊C4×C59C2316yes—
68#106×C59C23116yescomplemented
69C24×C59C2×D411yescharacteristic
70C23×D59—24no—
71C23×D59—24no—
72C23×D59C4⋊C418yes—
73C23×D59C2×D416yes—
74C22×C4×C59—26no—
75C22×C4×C59—26no—
76C22×C4×C59—44no—
77C22×C4×C59—44no—
78C22×C4×C59C22×C414yes—
79C22×C4×C59C2×D413yes—
80C22×C4×C59C2×Q814yes—
81C22×C4×C59C2413yes—
82C22×Dic59—26no—
83C22×Dic59—26no—
84C22×Dic59—44no—
85C22×Dic59—44no—
86C22×Dic59C4⋊C418yes—
87C22×Dic59C2×D416yes—
88C2×C4×D59—216no—
89C2×C4×D59—448no—
90C2×C4×D59C4⋊C4148yes—
91C2×C4×D59C22×C4124yes—
92C2×C4×D59C2×D4112yes—
93C2×C4×D59C2×Q8124yes—
94C2×C4×D59C2414yes—
95C2×C8×C59C22×C4124yescomplemented
96C2×C8×C59C2414yes—
97C4⋊C4×C59—26no—
98C4⋊C4×C59—26no—
99C4⋊C4×C59C2414yes—
100C4⋊C4×C59C2414yes—
101(C2×D59)⋊C4—216no—
102C4⋊Dic59—26no—
103C4⋊Dic59—26no—
104C4⋊Dic59C2414yes—
105C4⋊Dic59C2414yes—
106Dic59⋊C4—216no—
107C4×Dic59—216no—
108C2×C59⋊C8C22×C4124yescomplemented
109C2×C59⋊C8C2414yes—
110#11745659C22×C4164yescomplemented
111C23×C59C2×C4⋊C414yes—
112C23×C59AGL(1,ℤ/8)14yes—
113C23×C59#13714yes—
114C23×C59C22×D413yes—
115C22×D59—212no—
116C22×D59—212no—
117C22×D59—232no—
118C22×D59C2×C4⋊C4124yes—
119C22×D59C22×D414yes—
120C2×C4×C59—24no—
121C2×C4×C59—24no—
122C2×C4×C59—412no—
123C2×C4×C59—412no—
124C2×C4×C59C2×C4⋊C4112yes—
125C2×C4×C59C23×C416yes—
126C2×C4×C59C22×D413yes—
127C2×C4×C59C22×Q816yes—
128C2×C4×C59C2511yescharacteristic
129C2×Dic59—24no—
130C2×Dic59—24no—
131C2×Dic59—412no—
132C2×Dic59—412no—
133C2×Dic59C2×C4⋊C4124yes—
134C2×Dic59C22×D414yes—
135C4×D59—432no—
136C4×D59C2×C4⋊C4148yes—
137C4×D59C23×C418yes—
138C4×D59C22×Q818yes—
139C8×C59C23×C418yescomplemented
140C59⋊C8C23×C418yescomplemented
141C22×#428—591nomaximal, Sylow
142C22×C59#428116yescomplemented
143C22×C59C22×C4⋊C416yes—
144C22×C59C2×AGL(1,ℤ/8)16yes—
145C22×C59C2×#13716yes—
146C22×C59C23×D411yescharacteristic
147C2×D59—248no—
148C2×D59—24no—
149C2×D59—24no—
150C2×D59C22×C4⋊C418yes—
151C4×C59—44no—
152C4×C59—44no—
153C4×C59C22×C4⋊C416yes—
154C4×C59C24×C411yescharacteristic
155C4×C59C23×Q811yescharacteristic
156Dic59—44no—
157Dic59—44no—
158Dic59C22×C4⋊C418yes—
159C23×M16—591no—
160C22×#117—591no—
161C23×C4⋊C4—591no—
162C2×#428—5924no—
163C22×#107—591no—
164C22×#107—591no—
165C22×#106—591no—
166C22×#106D5911yescharacteristic, complemented
167C2×C59C2×#428112yescomplemented
168C2×C59C23×C4⋊C411yescharacteristic
169C2×C59C22×AGL(1,ℤ/8)11yescharacteristic
170C2×C59C22×#13711yescharacteristic
171D59—28no—
172C24×C4—591no—
173C24×C4—1181no—
174C22×M16—5916no—
175C22×M16—5912no—
176C23×C8—591no—
177C23×C8C2×D5911yescharacteristic
178C2×#117—5912no—
179C2×#117—1186no—
180C22×C4⋊C4—591no—
181C22×C4⋊C4—591no—
182C22×C4⋊C4—5912no—
183C22×C4⋊C4—1186no—
184C22×C4⋊C4—1181no—
185C22×C4⋊C4C2×D5911yescharacteristic
186C22×C4⋊C4C2×D5911yescharacteristic
187C22×C22⋊C4—1181no—
188C22×C42—1181no—
189#428—5964no—
190C2×#107—5912no—
191C2×#107—5912no—
192C2×#106—5912no—
193C2×#106C2×D59112yescomplemented
194C59C22×#42811yescharacteristic, complemented, Sylow
195C25—591no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—591no—
199C23×C4—598no—
200C23×C4—596no—
201C23×C4—598no—
202C23×C4—596no—
203C23×C4—11812no—
204C23×C4—1181no—
205C23×C4—1181no—
206C23×C4—2368no—
207C23×C4C22×D5911yescharacteristic
208C2×M16—5996no—
209C2×M16—5916no—
210C22×C8—598no—
211C22×C8—596no—
212C22×C8C4×D5918yescomplemented
213C22×C8C22×D5916yes—
214#117—5916no—
215#117—11824no—
216C2×C4⋊C4—23no—
217C2×C4⋊C4—23no—
218C2×C4⋊C4—596no—
219C2×C4⋊C4—596no—
220C2×C4⋊C4—5916no—
221C2×C4⋊C4—11824no—
222C2×C4⋊C4—11812no—
223C2×C4⋊C4—1183no—
224C2×C4⋊C4—1183no—
225C2×C4⋊C4C22×D5916yes—
226C2×C4⋊C4C22×D5916yes—
227C2×C22⋊C4—11812no—
228C2×C42—11812no—
229#107—5916no—
230#107—5916no—
231#106—5916no—
232#106C22×D59116yescomplemented
233C24—598no—
234C24—596no—
235C24—1184no—
236C24—1184no—
237C24D4×D5911yescharacteristic
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—598no—
243C22×C4—596no—
244C22×C4—5948no—
245C22×C4—5924no—
246C22×C4—5912no—
247C22×C4—5924no—
248C22×C4—594no—
249C22×C4—11816no—
250C22×C4—1186no—
251C22×C4—1186no—
252C22×C4—23648no—
253C22×C4—2364no—
254C22×C4—2364no—
255C22×C4C2×C4×D5914yes—
256C22×C4D4×D5913yes—
257C22×C4Q8×D5914yes—
258C22×C4C23×D5913yes—
259M16—5964no—
260C2×C8—5924no—
261C2×C8—594no—
262C2×C8C2×C4×D59124yescomplemented
263C2×C8C23×D5914yes—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—594no—
267C4⋊C4—594no—
268C4⋊C4—11816no—
269C4⋊C4—1186no—
270C4⋊C4—1186no—
271C4⋊C4C23×D5914yes—
272C4⋊C4C23×D5914yes—
273C22⋊C4—11816no—
274C42—11816no—
275C23—594no—
276C23—5924no—
277C23—11812no—
278C23—11812no—
279C23—11832no—
280C23C4⋊C4×D5914yes—
281C23#1427375414yes—
282C23#1427375614yes—
283C23C2×D4×D5913yes—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—412no—
287C2×C4—412no—
288C2×C4—594no—
289C2×C4—5924no—
290C2×C4—598no—
291C2×C4—598no—
292C2×C4—5948no—
293C2×C4—1184no—
294C2×C4—1184no—
295C2×C4—23632no—
296C2×C4—23612no—
297C2×C4—23612no—
298C2×C4C4⋊C4×D59112yes—
299C2×C4C22×C4×D5916yes—
300C2×C4C2×D4×D5913yes—
301C2×C4C2×Q8×D5916yes—
302C2×C4C24×D5911yescharacteristic
303C8—598no—
304C8C22×C4×D5918yescomplemented
305C22—598no—
306C22—11848no—
307C22—1184no—
308C22—1184no—
309C22#16837441116yesdirect product factor
310C22C2×C4⋊C4×D5916yes—
311C22C2×#1427375416yes—
312C22C2×#1427375616yes—
313C22C22×D4×D5911yescharacteristic
314C4—44no—
315C4—44no—
316C4—598no—
317C4—2364no—
318C4—2364no—
319C4C2×C4⋊C4×D5916yes—
320C4C23×C4×D5911yescharacteristic
321C4C22×Q8×D5911yescharacteristic
322C2—1188no—
323C2C2×#16837441112yesdirect product factor
324C2C22×C4⋊C4×D5911yescharacteristic
325C2C22×#1427375411yescharacteristic
326C2C22×#1427375611yescharacteristic
subgroupG/H
centerC24D4×D59
commutatorC4×C59C24×C4
FrattiniC2×C4C24×D59
FittingC22×#106×C59C2
socleC24×C59C2×D4
prime radicalC24×D59D4
cosocleC2×C4×C59C25
hypercenterC22×#106D59
hypocenterC59C22×#428
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
4. C2×#16837441C2direct (N × H)
5. C22×#14273661C2semidirect
6. C22×#14273660C2semidirect
7. C22×#14273658C2semidirect
8. C22×#106×C59C2semidirect
11. C22×#11745659C4semidirect
23. #16837441C22direct (N × H)
24. C2×#14273661C22semidirect
25. C2×#14273660C22semidirect
26. C2×#14273658C22semidirect
30. C2×#106×C59C22semidirect
41. C2×#11745659C2×C4semidirect
43. C22×C59⋊C8C2×C4semidirect
56. #14273661C23semidirect
57. #14273660C23semidirect
58. #14273658C23semidirect
62. C22×C8×C59C2×C4semidirect
68. #106×C59C23semidirect
95. C2×C8×C59C22×C4semidirect
108. C2×C59⋊C8C22×C4semidirect
110. #11745659C22×C4semidirect
139. C8×C59C23×C4semidirect
140. C59⋊C8C23×C4semidirect
142. C22×C59#428semidirect
166. C22×#106D59semidirect
167. C2×C59C2×#428semidirect
193. C2×#106C2×D59semidirect
194. C59C22×#428semidirect
212. C22×C8C4×D59semidirect
232. #106C22×D59semidirect
262. C2×C8C2×C4×D59semidirect
304. C8C22×C4×D59semidirect
309. C22#16837441direct (N × H)
323. C2C2×#16837441direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117456592
C22×#142736852
C22×C4⋊C4×D592
C2×#168374412
C22×#142736612
C22×#142736602
C22×#142736582
C22×#106×C592
C22×#42859