Finite Group Databasebeta

← C22×#16837622 #29455495

Subgroups — C22×#16837622 #29455495

Order 15104.

2918 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11745672C211yesmaximal, characteristic, complemented
2C22×#14273714C211yesmaximal, characteristic
3C22×Q8×Dic59C211yesmaximal, characteristic, complemented
4C2×#16837622C2124yesmaximal, direct product factor
5C22×Dic59⋊C8C211yesmaximal, characteristic, complemented
6C22×#14273652C211yesmaximal, characteristic, complemented
7C22×#14273648C211yesmaximal, characteristic
8C23×SD16×C59C211yesmaximal, characteristic
9C23×Dic118C2211yescharacteristic
10C22×#11745672—28no—
11C22×#11745672C22112yescomplemented
12C23×C59⋊C8C2211yescharacteristic
13C2×#14273714—26no—
14C2×#14273714C22112yes—
15C2×Q8×Dic59—26no—
16C2×Q8×Dic59C22112yescomplemented
17C22×C4⋊Dic59—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19C22×C22⋊Dic59—21no—
20C22×C22⋊Dic59—21no—
21C22×C4×Dic59—21no—
22C22×C4×Dic59C2211yescharacteristic
23#16837622C22164yesdirect product factor
24C2×Dic59⋊C8C22112yescomplemented
25C2×#14273652C22112yescomplemented
26C2×#14273648C22112yes—
27C23×Q8×C59C2211yescharacteristic
28C23×D4×C59C2211yescharacteristic
29C22×SD16×C59—28no—
30C22×SD16×C59C22112yes—
31C23×C8×C59C2211yescharacteristic
32C22×Dic118—24no—
33C22×Dic118—24no—
34C22×Dic118—23no—
35C22×Dic118C2316yes—
36C23×Dic59—21no—
37C23×Dic59—21no—
38C23×Dic59D411yescharacteristic
39C23×Dic59D411yescharacteristic
40C2×#11745672—248no—
41C2×#11745672C23116yescomplemented
42C22×C59⋊C8—24no—
43C22×C59⋊C8C2316yes—
44#14273714—224no—
45#14273714C23116yes—
46Q8×Dic59—224no—
47Q8×Dic59C23116yescomplemented
48C2×C4⋊Dic59—23no—
49C2×C4⋊Dic59—212no—
50C2×C4⋊Dic59C2316yes—
51C2×C22⋊Dic59—212no—
52C2×C22⋊Dic59—212no—
53C2×C4×Dic59—212no—
54C2×C4×Dic59D416yes—
55C2×C4×Dic59C2316yes—
56Dic59⋊C8C23116yescomplemented
57#14273652C23116yescomplemented
58#14273648C23116yes—
59C25×C59—21no—
60C22×Q8×C59—23no—
61C22×Q8×C59—24no—
62C22×Q8×C59D418yes—
63C22×Q8×C59C2316yes—
64C22×D4×C59—24no—
65C22×D4×C59—23no—
66C22×D4×C59—24no—
67C22×D4×C59C2316yes—
68C23×C4×C59—21no—
69C23×C4×C59C2311yescharacteristic
70C2×SD16×C59—248no—
71C2×SD16×C59C23116yes—
72C22×C8×C59—24no—
73C22×C8×C59C2316yes—
74C24×C59—26no—
75C24×C59—44no—
76C24×C59—44no—
77C24×C59—44no—
78C24×C59C2×D411yescharacteristic
79C2×Q8×C59—224no—
80C2×Q8×C59—26no—
81C2×Q8×C59—212no—
82C2×Q8×C59C2×D4124yes—
83C2×Q8×C59C2414yes—
84C2×D4×C59—212no—
85C2×D4×C59—26no—
86C2×D4×C59—212no—
87C2×D4×C59—412no—
88C2×D4×C59C2414yes—
89C22×C4×C59—28no—
90C22×C4×C59—26no—
91C22×C4×C59C2×D413yes—
92C22×C4×C59C2×D414yes—
93C22×C4×C59#4114yes—
94C22×C4×C59C2413yes—
95C22×Dic59—24no—
96C22×Dic59—24no—
97C22×Dic59—26no—
98C22×Dic59—26no—
99C22×Dic59—44no—
100C22×Dic59—44no—
101C22×Dic59C2×D416yes—
102C22×Dic59C2×D416yes—
103C2×Dic118—212no—
104C2×Dic118—212no—
105C2×Dic118—26no—
106C2×Dic118—412no—
107C2×Dic118C2414yes—
108SD16×C59—232no—
109C2×C8×C59—212no—
110C2×C8×C59C2414yes—
111C22⋊Dic59—216no—
112C22⋊Dic59—216no—
113#11745672—232no—
114C4⋊Dic59—26no—
115C4⋊Dic59—216no—
116C4⋊Dic59C2414yes—
117C4×Dic59—216no—
118C4×Dic59C2×D4112yes—
119C4×Dic59C2414yes—
120C2×C59⋊C8—212no—
121C2×C59⋊C8C2414yes—
122C23×C59—24no—
123C23×C59—412no—
124C23×C59—412no—
125C23×C59—412no—
126C23×C59—816no—
127C23×C59#12114yes—
128C23×C59#13514yes—
129C23×C59#13714yes—
130C23×C59C22×D413yes—
131Q8×C59—224no—
132Q8×C59—24no—
133Q8×C59C22×D418yes—
134D4×C59—24no—
135D4×C59—24no—
136D4×C59—412no—
137C2×C4×C59—224no—
138C2×C4×C59—24no—
139C2×C4×C59#121112yes—
140C2×C4×C59C22×D413yes—
141C2×C4×C59C22×D416yes—
142C2×C4×C59C2×#4116yes—
143C2×C4×C59C2511yescharacteristic
144C2×Dic59—212no—
145C2×Dic59—212no—
146C2×Dic59—24no—
147C2×Dic59—24no—
148C2×Dic59—412no—
149C2×Dic59—412no—
150C2×Dic59C22×D414yes—
151C2×Dic59C22×D414yes—
152Dic118—24no—
153Dic118—24no—
154Dic118—412no—
155C8×C59—24no—
156C59⋊C8—24no—
157C22×#464—591nomaximal, Sylow
158C22×C59—44no—
159C22×C59—44no—
160C22×C59—44no—
161C22×C59—824no—
162C22×C59#464116yescomplemented
163C22×C59C2×#12116yes—
164C22×C59C2×#13516yes—
165C22×C59C2×#13716yes—
166C22×C59C23×D411yescharacteristic
167C4×C59—28no—
168C4×C59C2×#12116yes—
169C4×C59C23×D411yescharacteristic
170C4×C59C22×#4111yescharacteristic
171Dic59—24no—
172Dic59—24no—
173Dic59—44no—
174Dic59—44no—
175C23×Q16—591no—
176C23×SD16D5911yescharacteristic
177C22×#124—591no—
178C22×C4×Q8—591no—
179C2×#464—5924no—
180C22×C4⋊C8—591no—
181C22×#103—591no—
182C22×#102—591no—
183C2×C59—84no—
184C2×C59C2×#464112yescomplemented
185C2×C59C22×#12111yescharacteristic
186C2×C59C22×#13511yescharacteristic
187C2×C59C22×#13711yescharacteristic
188C23×Q8—591no—
189C23×Q8C2×D5911yescharacteristic
190C23×D4C2×D5911yescharacteristic
191C22×Q16—5912no—
192C22×Q16—1188no—
193C22×SD16—28no—
194C22×SD16C2×D59112yes—
195C23×C8—591no—
196C23×C8C2×D5911yescharacteristic
197C2×#124—5912no—
198C2×#124—1186no—
199C2×C4×Q8—5912no—
200C2×C4×Q8—1186no—
201C22×C4⋊C4—591no—
202C22×C4⋊C4—1181no—
203C22×C22⋊C4—1181no—
204C22×C22⋊C4—1181no—
205C22×C42—591no—
206C22×C42—1181no—
207#464—5964no—
208C2×C4⋊C8—5912no—
209C2×#103—5912no—
210C2×#102—5912no—
211C59C22×#46411yescharacteristic, complemented, Sylow
212C25—21no—
213C22×Q8—23no—
214C22×Q8—24no—
215C22×Q8—596no—
216C22×Q8—1183no—
217C22×Q8—1184no—
218C22×Q8—1184no—
219C22×Q8C22⋊D5918yes—
220C22×Q8C22×D5916yes—
221C22×D4—23no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4C22×D5916yes—
225C23×C4—21no—
226C23×C4—591no—
227C23×C4—591no—
228C23×C4—1181no—
229C23×C4—1181no—
230C23×C4C22×D5911yescharacteristic
231C2×Q16—5916no—
232C2×Q16—11848no—
233C2×SD16—248no—
234C2×SD16C22×D59116yes—
235C22×C8—24no—
236C22×C8—596no—
237C22×C8—1184no—
238C22×C8C22×D5916yes—
239#124—5916no—
240#124—11824no—
241C4×Q8—5916no—
242C4×Q8—11824no—
243C2×C4⋊C4—596no—
244C2×C4⋊C4—11812no—
245C2×C4⋊C4—1183no—
246C2×C22⋊C4—11812no—
247C2×C22⋊C4—11812no—
248C2×C42—596no—
249C2×C42—596no—
250C2×C42—11812no—
251C4⋊C8—5916no—
252#103—5916no—
253#102—5916no—
254C24—26no—
255C24—44no—
256C24—44no—
257C24—44no—
258C24D4×D5911yescharacteristic
259C2×Q8—224no—
260C2×Q8—26no—
261C2×Q8—212no—
262C2×Q8—594no—
263C2×Q8—1186no—
264C2×Q8—11812no—
265C2×Q8—11812no—
266C2×Q8—23612no—
267C2×Q8C2×C22⋊D59124yes—
268C2×Q8C23×D5914yes—
269C2×D4—26no—
270C2×D4—212no—
271C2×D4—212no—
272C2×D4—412no—
273C2×D4C23×D5914yes—
274C22×C4—28no—
275C22×C4—26no—
276C22×C4—596no—
277C22×C4—596no—
278C22×C4—1184no—
279C22×C4—1184no—
280C22×C4—1186no—
281C22×C4—1186no—
282C22×C4—2364no—
283C22×C4—2364no—
284C22×C4D4×D5913yes—
285C22×C4#1174568714yes—
286C22×C4C2×C22⋊D5914yes—
287C22×C4C23×D5913yes—
288Q16—11832no—
289SD16—232no—
290C2×C8—212no—
291C2×C8—594no—
292C2×C8—11812no—
293C2×C8C23×D5914yes—
294C4⋊C4—594no—
295C4⋊C4—11816no—
296C4⋊C4—1186no—
297C22⋊C4—11816no—
298C22⋊C4—11816no—
299C42—5912no—
300C42—594no—
301C42—11816no—
302C23—24no—
303C23—412no—
304C23—412no—
305C23—412no—
306C23—816no—
307C23#1427369914yes—
308C23#1427374814yes—
309C23#1427375614yes—
310C23C2×D4×D5913yes—
311Q8—224no—
312Q8—24no—
313Q8—1184no—
314Q8—1184no—
315Q8—23612no—
316Q8C22×C22⋊D5918yes—
317D4—24no—
318D4—24no—
319D4—412no—
320C2×C4—224no—
321C2×C4—24no—
322C2×C4—594no—
323C2×C4—594no—
324C2×C4—11812no—
325C2×C4—11812no—
326C2×C4—1184no—
327C2×C4—1184no—
328C2×C4—23612no—
329C2×C4—23612no—
330C2×C4#14273699112yes—
331C2×C4C2×D4×D5913yes—
332C2×C4C22×C22⋊D5916yes—
333C2×C4C2×#1174568716yes—
334C2×C4C24×D5911yescharacteristic
335C8—24no—
336C8—1184no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—824no—
341C22#16837622116yesdirect product factor
342C22C2×#1427369916yes—
343C22C2×#1427374816yes—
344C22C2×#1427375616yes—
345C22C22×D4×D5911yescharacteristic
346C4—28no—
347C4—1184no—
348C4—1184no—
349C4—2364no—
350C4—2364no—
351C4C2×#1427369916yes—
352C4C23×C22⋊D5911yescharacteristic
353C4C22×#1174568711yescharacteristic
354C2—84no—
355C2C2×#16837622112yesdirect product factor
356C2C22×#1427369911yescharacteristic
357C2C22×#1427374811yescharacteristic
358C2C22×#1427375611yescharacteristic
subgroupG/H
centerC24D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC23×SD16×C59C2
socleC24×C59C2×D4
prime radicalC23×D4×C59C22
hypercenterC23×SD16D59
hypocenterC59C22×#464
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×#11745672C2semidirect
3. C22×Q8×Dic59C2semidirect
4. C2×#16837622C2direct (N × H)
5. C22×Dic59⋊C8C2semidirect
6. C22×#14273652C2semidirect
11. C22×#11745672C22semidirect
16. C2×Q8×Dic59C22semidirect
23. #16837622C22direct (N × H)
24. C2×Dic59⋊C8C22semidirect
25. C2×#14273652C22semidirect
41. C2×#11745672C23semidirect
47. Q8×Dic59C23semidirect
56. Dic59⋊C8C23semidirect
57. #14273652C23semidirect
162. C22×C59#464semidirect
184. C2×C59C2×#464semidirect
211. C59C22×#464semidirect
341. C22#16837622direct (N × H)
355. C2C2×#16837622direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117456722
C22×#142737142
C22×Q8×Dic592
C2×#168376222
C22×Dic59⋊C82
C22×#142736522
C22×#142736482
C23×SD16×C592
C22×#46459