Finite Group Databasebeta

← C2×#21423213 #28564300

Subgroups — C2×#21423213 #28564300

Order 15104.

1722 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 219 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×#16838037C211yesmaximal, characteristic
2C2×#16837774C211yesmaximal, characteristic, complemented
3C2×#16837693C212yesmaximal, complemented
4#21423213C218yesmaximal, direct product factor
5C2×#16837129C212yesmaximal, complemented
6C2×C59×#331C211yesmaximal, characteristic, complemented
7C22×D4×D59—22no—
8C22×#11745670—22no—
9#16838037—24no—
10#16838037—22no—
11#16838037C2214yes—
12C2×#14273727C2211yescharacteristic, complemented
13C2×#14273698—22no—
14C2×#14273695—22no—
15C2×C4×D236C2212yes—
16#16837774—22no—
17#16837774C2214yescomplemented
18#16837693—24no—
19#16837693C2218yescomplemented
20C2×#14273659—22no—
21C2×#14273649—22no—
22C2×C4×C59⋊C8C2211yescharacteristic, complemented
23#16837129C2218yescomplemented
24C2×#127×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2212yescomplemented
26C59×#331C2214yescomplemented
27C24×D59—22no—
28C22×Dic118—21no—
29C22×D236D412yes—
30C22×C22⋊D59—24no—
31C2×D4×D59—28no—
32C2×D4×D59—28no—
33C2×D4×D59—48no—
34C2×D4×D59—48no—
35C22×C4×D59—22no—
36C2×#11745670—28no—
37C2×#11745670—48no—
38C22×C59⋊C8—21no—
39#14273727—22no—
40#14273727—21no—
41#14273727C2312yescomplemented
42#14273698—28no—
43#14273698—28no—
44#14273695—28no—
45#14273695—28no—
46C4×D236—22no—
47C4×D236—22no—
48C4×D236D414yes—
49C4×D236C2314yes—
50C2×C4⋊Dic59—22no—
51C2×C4⋊Dic59D412yes—
52C2×C22⋊Dic59—22no—
53C2×(C2×D59)⋊C4—22no—
54#14273659—28no—
55#14273649—28no—
56C4×C59⋊C8—21no—
57C4×C59⋊C8C2312yescomplemented
58C22×D4×C59—22no—
59C22×D4×C59D412yes—
60C22×C8×C59—22no—
61#127×C59—22no—
62#127×C59D412yes—
63#127×C59C2312yes—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—22no—
66C4⋊C8×C59C2314yescomplemented
67C24×C59—22no—
68C23×D59—22no—
69C23×D59—28no—
70C23×D59—416no—
71C23×D59—84no—
72C2×D4×C59—24no—
73C2×D4×C59—22no—
74C2×D4×C59—24no—
75C2×D4×C59—44no—
76C2×D4×C59—44no—
77C2×D4×C59—44no—
78C2×D4×C59C2×D414yes—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D412yes—
81C2×C22⋊D59—216no—
82C2×C22⋊D59—416no—
83C22×Dic59—22no—
84C22×Dic59—41no—
85D4×D59—416no—
86D4×D59—432no—
87D4×D59—416no—
88C2×D236—22no—
89C2×D236—24no—
90C2×D236SD1618yescomplemented
91C2×D236C2×D414yes—
92C2×C4×D59—24no—
93C2×C4×D59—24no—
94C2×C4×D59—44no—
95C2×C4×D59—44no—
96C2×Dic118—22no—
97C2×Dic118—44no—
98C2×Dic118—41no—
99C2×C8×C59—24no—
100C2×C8×C59—44no—
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D412yes—
103C42×C59C2411yescharacteristic
104C22⋊Dic59—28no—
105#11745670—416no—
106(C2×D59)⋊C4—24no—
107(C2×D59)⋊C4—24no—
108C4⋊Dic59—22no—
109C4⋊Dic59—24no—
110C4⋊Dic59—42no—
111C4⋊Dic59C2×D414yes—
112C2×C59⋊C8—24no—
113C2×C59⋊C8—22no—
114C23×C59—24no—
115C23×C59—48no—
116C23×C59—44no—
117C23×C59#12011yescharacteristic
118C22×D59—24no—
119C22×D59—28no—
120C22×D59—432no—
121C22×D59—44no—
122C22×D59—44no—
123C22×D59—88no—
124C22×D59—832no—
125D4×C59—44no—
126D4×C59—44no—
127D4×C59—44no—
128D4×C59—48no—
129D4×C59—82no—
130C2×C4×C59—22no—
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12012yes—
133C2×C4×C59C2×SD1614yes—
134C2×C4×C59AGL(1,ℤ/8)14yes—
135C2×C4×C59C22×D412yes—
136C2×C4×C59C22×D411yescharacteristic
137C22⋊D59—432no—
138C2×Dic59—24no—
139C2×Dic59—44no—
140C2×Dic59—42no—
141C2×Dic59—82no—
142D236—24no—
143D236—28no—
144D236C2×SD1618yescomplemented
145C4×D59—44no—
146C4×D59—44no—
147C4×D59—48no—
148Dic118—44no—
149Dic118—82no—
150C8×C59—44no—
151C59⋊C8—24no—
152C2×#1629—591nomaximal, Sylow
153C22×C59—48no—
154C22×C59—44no—
155C22×C59—88no—
156C22×C59#45014yes—
157C22×C59#45312yes—
158C22×C59C2×#12011yescharacteristic
159C2×D59—28no—
160C2×D59—416no—
161C2×D59—44no—
162C2×D59—44no—
163C2×D59—832no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59#45014yes—
167C4×C59C22×SD1612yes—
168C4×C59C2×AGL(1,ℤ/8)12yes—
169Dic59—44no—
170Dic59—82no—
171C2×D42—591no—
172C2×#492—591no—
173C2×#476—592no—
174#1629—598no—
175C2×#331D5911yescharacteristic, complemented
176C2×D4⋊C8—592no—
177C2×C59—84no—
178C2×C59#162914yescomplemented
179C2×C59C2×#45012yes—
180C2×C59C2×#45311yescharacteristic
181D59—48no—
182C23×D4—1182no—
183C22×SD16—1182no—
184D42—594no—
185D42—1184no—
186D42—1182no—
187C2×C4⋊Q8—591no—
188C2×#127C2×D5911yescharacteristic
189C2×#121—1182no—
190C2×#120—1182no—
191C2×C4×D4—592no—
192#492—594no—
193#492—1182no—
194#476—598no—
195#476—1184no—
196C2×#106—1182no—
197C2×C4⋊C8C2×D5912yescomplemented
198C2×#102—1182no—
199C2×C4×C8—591no—
200#331C2×D5914yescomplemented
201D4⋊C8—598no—
202C59C2×#162911yescharacteristic, complemented, Sylow
203C25—1182no—
204C22×Q8—1181no—
205C22×D4—22no—
206C22×D4—592no—
207C22×D4—1188no—
208C22×D4—1188no—
209C22×D4—1184no—
210C22×D4—2368no—
211C22×D4—2368no—
212C22×D4D23612yes—
213C23×C4—1182no—
214C2×SD16—1188no—
215C2×SD16—2368no—
216C22×C8—22no—
217C22×C8—1181no—
218C4⋊Q8—592no—
219C4⋊Q8—1181no—
220C4⋊Q8—1182no—
221#127—22no—
222#127D23612yes—
223#127C22×D5912yes—
224#121—1188no—
225#121—1188no—
226#120—1188no—
227#120—1188no—
228C4×D4—594no—
229C4×D4—594no—
230C4×D4—1182no—
231C4×D4—1182no—
232C2×C4⋊C4—592no—
233C2×C4⋊C4—1182no—
234C2×C22⋊C4—1182no—
235C2×C22⋊C4—1182no—
236C2×C42C22×D5911yescharacteristic
237#106—1188no—
238C4⋊C8—22no—
239C4⋊C8C22×D5914yescomplemented
240#102—1188no—
241C4×C8—592no—
242C4×C8—1181no—
243C24—22no—
244C24—1188no—
245C24—1182no—
246C24—4724no—
247C24—23616no—
248C2×Q8—1182no—
249C2×Q8—2364no—
250C2×Q8—2361no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—598no—
258C2×D4—594no—
259C2×D4—11816no—
260C2×D4—1182no—
261C2×D4—1184no—
262C2×D4—23616no—
263C2×D4—23616no—
264C2×D4—23632no—
265C2×D4—23616no—
266C2×D4C2×D23614yes—
267C22×C4—1182no—
268C22×C4—1184no—
269C22×C4—1184no—
270C22×C4—2361no—
271C22×C4—2364no—
272C22×C4—2364no—
273C22×C4C2×D23611yescharacteristic
274C22×C4D4×D5912yes—
275SD16—23616no—
276C2×C8—24no—
277C2×C8—44no—
278C2×C8—1184no—
279C2×C8—1182no—
280C4⋊C4—594no—
281C4⋊C4—1182no—
282C4⋊C4—1184no—
283C4⋊C4—2362no—
284C22⋊C4—1188no—
285C22⋊C4—1184no—
286C22⋊C4—1184no—
287C42C2×D23611yescharacteristic
288C42D4×D5912yes—
289C42C23×D5911yescharacteristic
290C23—24no—
291C23—48no—
292C23—44no—
293C23—1184no—
294C23—1188no—
295C23—4728no—
296C23—47232no—
297C23—23632no—
298C23—2364no—
299C23—2364no—
300C23#1427369711yescharacteristic
301Q8—4722no—
302Q8—2364no—
303D4—44no—
304D4—44no—
305D4—44no—
306D4—48no—
307D4—82no—
308D4—598no—
309D4—1184no—
310D4—1188no—
311D4—23632no—
312C2×C4—22no—
313C2×C4—1184no—
314C2×C4—4722no—
315C2×C4—2362no—
316C2×C4—2364no—
317C2×C4—2364no—
318C2×C4—2364no—
319C2×C4—2368no—
320C2×C4#1427369711yescharacteristic
321C2×C4#1427369712yes—
322C2×C4SD16×D5914yes—
323C2×C4#1427375114yes—
324C2×C4C2×D4×D5912yes—
325C2×C4C22×D23611yescharacteristic
326C8—44no—
327C8—1184no—
328C22—44no—
329C22—48no—
330C22—88no—
331C22—1188no—
332C22—47232no—
333C22—2364no—
334C22—23616no—
335C22—2364no—
336C22#1683755414yes—
337C22#1683757212yes—
338C22C2×#1427369711yescharacteristic
339C4—21no—
340C4—21no—
341C4—4722no—
342C4—2364no—
343C4#1683755414yes—
344C4C2×SD16×D5912yes—
345C4C2×#1427375112yes—
346C2—84no—
347C2—2368no—
348C2#2142321314yesdirect product factor
349C2C2×#1683755412yes—
350C2C2×#1683757211yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#331C2
socleC23×C59#120
prime radicalC2×#16838037C2
hypercenterC2×#331D59
hypocenterC59C2×#1629
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. C2×#16837774C2semidirect
3. C2×#16837693C2semidirect
4. #21423213C2direct (N × H)
5. C2×#16837129C2semidirect
6. C2×C59×#331C2semidirect
12. C2×#14273727C22semidirect
17. #16837774C22semidirect
19. #16837693C22semidirect
22. C2×C4×C59⋊C8C22semidirect
23. #16837129C22semidirect
25. C2×C4⋊C8×C59C22semidirect
26. C59×#331C22semidirect
41. #14273727C23semidirect
57. C4×C59⋊C8C23semidirect
66. C4⋊C8×C59C23semidirect
90. C2×D236SD16semidirect
144. D236C2×SD16semidirect
175. C2×#331D59semidirect
178. C2×C59#1629semidirect
197. C2×C4⋊C8C2×D59semidirect
200. #331C2×D59semidirect
202. C59C2×#1629semidirect
239. C4⋊C8C22×D59semidirect
348. C2#21423213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168380372
C2×#168377742
C2×#168376932
#214232132
C2×#168371292
C2×C59×#3312
C2×#162959