Finite Group Databasebeta

← C2×#21423115 #28560809

Subgroups — C2×#21423115 #28560809

Order 15104.

1524 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1524 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16837964C211yesmaximal, characteristic
2C2×#16837720C212yesmaximal, complemented
3#21423115C218yesmaximal, direct product factor
4C2×#16837135C212yesmaximal, complemented
5C2×#16837126C211yesmaximal, characteristic, complemented
6C2×C59×#399C211yesmaximal, characteristic, complemented
7C23×C22⋊D59—21no—
8#16837964—24no—
9#16837964—22no—
10#16837964C2214yes—
11C2×#14273711—22no—
12C2×#14273701—21no—
13C2×#14273701C2212yes—
14C2×#14273697—22no—
15#16837720—24no—
16#16837720C2218yescomplemented
17C2×#14273658—22no—
18C2×#14273647—22no—
19C2×C22⋊(C59⋊C8)C2212yescomplemented
20#16837135C2218yescomplemented
21#16837126C2214yescomplemented
22C22×C4⋊C4×C59—21no—
23C22×C4⋊C4×C59C2211yescharacteristic, complemented
24C22×C22⋊C4×C59—21no—
25C22×C22⋊C4×C59C2211yescharacteristic
26C59×#399—22no—
27C59×#399C2214yescomplemented
28C2×#95×C59C2211yescharacteristic, complemented
29C24×D59—21no—
30C22×D236—21no—
31C22×D236D412yes—
32C22×C22⋊D59—22no—
33C22×C22⋊D59—24no—
34C22×C22⋊D59—22no—
35C22×C22⋊D59—24no—
36C22×C22⋊D59—44no—
37C22×C22⋊D59—44no—
38C23×Dic59—21no—
39C22×C59⋊C8—22no—
40#14273711—28no—
41#14273711—28no—
42#14273701—24no—
43#14273701—24no—
44#14273701—22no—
45#14273701—22no—
46#14273701D414yes—
47#14273701C2314yes—
48#14273697—28no—
49#14273697—28no—
50C2×C4⋊Dic59—21no—
51C2×C4⋊Dic59D412yes—
52C2×(C2×D59)⋊C4—22no—
53C2×(C2×D59)⋊C4—22no—
54#14273658—28no—
55#14273647—28no—
56C22⋊(C59⋊C8)—22no—
57C22⋊(C59⋊C8)C2314yescomplemented
58C25×C59D411yescharacteristic
59C23×C4×C59—21no—
60C23×C4×C59—21no—
61C23×C4×C59D411yescharacteristic
62C23×C4×C59C2311yescharacteristic
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59—22no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—24no—
67C2×C4⋊C4×C59—44no—
68C2×C4⋊C4×C59C2312yescomplemented
69C2×C22⋊C4×C59—22no—
70C2×C22⋊C4×C59—22no—
71C2×C22⋊C4×C59—24no—
72C2×C22⋊C4×C59—44no—
73C2×C22⋊C4×C59D412yes—
74C2×C22⋊C4×C59C2312yes—
75#95×C59—21no—
76#95×C59C2312yescomplemented
77C24×C59—21no—
78C24×C59—21no—
79C24×C59—22no—
80C24×C59—42no—
81C24×C59—42no—
82C24×C59SD1614yes—
83C24×C59C2×D411yescharacteristic
84C24×C59C2×D412yes—
85C23×D59—22no—
86C23×D59—24no—
87C23×D59—44no—
88C23×D59—82no—
89C23×D59—82no—
90C22×C4×C59—24no—
91C22×C4×C59—21no—
92C22×C4×C59—22no—
93C22×C4×C59—21no—
94C22×C4×C59—22no—
95C22×C4×C59—42no—
96C22×C4×C59—42no—
97C22×C4×C59—42no—
98C22×C4×C59—42no—
99C22×C4×C59—42no—
100C22×C4×C59—42no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59C2×D412yes—
104C22×C4×C59C2×D412yes—
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D412yes—
107C22×C4×C59C2411yescharacteristic
108C2×C22⋊D59—28no—
109C2×C22⋊D59—24no—
110C2×C22⋊D59—24no—
111C2×C22⋊D59—48no—
112C2×C22⋊D59—416no—
113C2×C22⋊D59—48no—
114C2×C22⋊D59—48no—
115C2×C22⋊D59—44no—
116C2×C22⋊D59—44no—
117C22×Dic59—24no—
118C22×Dic59—22no—
119C22×Dic59—44no—
120C2×D236—24no—
121C2×D236—22no—
122C2×D236—44no—
123C2×D236—82no—
124C2×D236C2×D414yes—
125C4⋊C4×C59—24no—
126C4⋊C4×C59—48no—
127C4⋊C4×C59—42no—
128C22⋊C4×C59—24no—
129C22⋊C4×C59—24no—
130C22⋊C4×C59—48no—
131(C2×D59)⋊C4—28no—
132(C2×D59)⋊C4—24no—
133(C2×D59)⋊C4—24no—
134C4⋊Dic59—24no—
135C4⋊Dic59—22no—
136C4⋊Dic59C2×D414yes—
137C2×C59⋊C8—24no—
138C2×C59⋊C8—44no—
139C23×C59—22no—
140C23×C59—22no—
141C23×C59—22no—
142C23×C59—24no—
143C23×C59—22no—
144C23×C59—24no—
145C23×C59—42no—
146C23×C59—48no—
147C23×C59—42no—
148C23×C59—42no—
149C23×C59—42no—
150C23×C59—84no—
151C23×C59—81no—
152C23×C59—81no—
153C23×C59#12011yescharacteristic
154C23×C59#12011yescharacteristic
155C23×C59C2×SD1612yes—
156C23×C59C2×SD1614yes—
157C23×C59AGL(1,ℤ/8)12yes—
158C23×C59C22×D411yescharacteristic
159C22×D59—24no—
160C22×D59—48no—
161C22×D59—44no—
162C22×D59—44no—
163C22×D59—44no—
164C22×D59—84no—
165C22×D59—84no—
166C22×D59—816no—
167C2×C4×C59—24no—
168C2×C4×C59—24no—
169C2×C4×C59—22no—
170C2×C4×C59—42no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—48no—
174C2×C4×C59—42no—
175C2×C4×C59—42no—
176C2×C4×C59—42no—
177C2×C4×C59—42no—
178C2×C4×C59—42no—
179C2×C4×C59—42no—
180C2×C4×C59—42no—
181C2×C4×C59—81no—
182C2×C4×C59—82no—
183C2×C4×C59—82no—
184C2×C4×C59#12012yes—
185C2×C4×C59C22×D412yes—
186C22⋊D59—416no—
187C22⋊D59—44no—
188C22⋊D59—44no—
189C22⋊D59—48no—
190C2×Dic59—24no—
191C2×Dic59—48no—
192C2×Dic59—44no—
193D236—44no—
194D236—82no—
195D236—84no—
196C59⋊C8—44no—
197C2×#1613—591nomaximal, Sylow
198C22×C59—24no—
199C22×C59—24no—
200C22×C59—21no—
201C22×C59—21no—
202C22×C59—21no—
203C22×C59—21no—
204C22×C59—24no—
205C22×C59—44no—
206C22×C59—48no—
207C22×C59—42no—
208C22×C59—42no—
209C22×C59—82no—
210C22×C59—84no—
211C22×C59#45012yes—
212C22×C59#45012yes—
213C22×C59#45312yes—
214C22×C59#45812yes—
215C22×C59C2×#12011yescharacteristic
216C22×C59C22×SD1611yescharacteristic
217C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
218C2×D59—44no—
219C2×D59—44no—
220C2×D59—44no—
221C2×D59—816no—
222C2×D59—88no—
223C4×C59—22no—
224C4×C59—22no—
225C4×C59—44no—
226C4×C59—42no—
227C4×C59—82no—
228Dic59—44no—
229C2×#534—591no—
230C2×#481—592no—
231C2×#399D5911yescharacteristic, complemented
232#1613—598no—
233C2×#327—592no—
234C2×#323—591no—
235C2×C59—21no—
236C2×C59—21no—
237C2×C59—22no—
238C2×C59—44no—
239C2×C59#161314yescomplemented
240C2×C59C2×#45011yescharacteristic
241C2×C59C2×#45311yescharacteristic
242C2×C59C2×#45811yescharacteristic
243D59—84no—
244C23×D4—1181no—
245#534—594no—
246#534—1184no—
247#534—1182no—
248C2×#123—1182no—
249C2×#121—592no—
250C2×#121—1181no—
251C2×#120—1182no—
252C22×C4⋊C4—21no—
253C22×C4⋊C4C2×D5911yescharacteristic, complemented
254C22×C22⋊C4—21no—
255C22×C22⋊C4C2×D5911yescharacteristic
256#481—598no—
257#481—1184no—
258C2×#106—1182no—
259C2×#102—1182no—
260C2×C22⋊C8—592no—
261#399—22no—
262#399C2×D5914yescomplemented
263C2×#95C2×D5911yescharacteristic, complemented
264#327—598no—
265#323—594no—
266C59C2×#161311yescharacteristic, complemented, Sylow
267C25—1181no—
268C25C22⋊D5911yescharacteristic
269C22×D4—592no—
270C22×D4—1181no—
271C22×D4—1182no—
272C22×D4—1184no—
273C22×D4—1182no—
274C22×D4—1184no—
275C22×D4—2364no—
276C22×D4—2364no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—1181no—
280C23×C4C22⋊D5911yescharacteristic
281C23×C4C22×D5911yescharacteristic
282C22×C8—1182no—
283#123—1188no—
284#123—1188no—
285#121—594no—
286#121—594no—
287#121—1184no—
288#121—1182no—
289#121—1182no—
290#121—1184no—
291#120—1188no—
292#120—1188no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—592no—
299C2×C4⋊C4—1181no—
300C2×C4⋊C4C22×D5912yescomplemented
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—1182no—
306C2×C22⋊C4—1182no—
307C2×C22⋊C4C22⋊D5912yes—
308C2×C22⋊C4C22×D5912yes—
309#106—1188no—
310#102—1188no—
311C22⋊C8—594no—
312C22⋊C8—1182no—
313#95—21no—
314#95C22×D5912yescomplemented
315C24—21no—
316C24—21no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—1184no—
321C24—1182no—
322C24—4722no—
323C24—4722no—
324C24—2364no—
325C24#1174567114yes—
326C24C2×C22⋊D5911yescharacteristic
327C24C2×C22⋊D5912yes—
328C2×D4—594no—
329C2×D4—1184no—
330C2×D4—1188no—
331C2×D4—1184no—
332C2×D4—1184no—
333C2×D4—1182no—
334C2×D4—4722no—
335C2×D4—2364no—
336C2×D4—2368no—
337C2×D4—23616no—
338C2×D4—2368no—
339C2×D4—2368no—
340C2×D4—2364no—
341C2×D4—2364no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—1184no—
356C22×C4—1182no—
357C22×C4—2364no—
358C22×C4D4×D5912yes—
359C22×C4D4×D5912yes—
360C22×C4C2×C22⋊D5912yes—
361C22×C4C2×C22⋊D5911yescharacteristic
362C22×C4C23×D5911yescharacteristic
363C2×C8—1184no—
364C2×C8—2364no—
365C4⋊C4—24no—
366C4⋊C4—48no—
367C4⋊C4—42no—
368C4⋊C4—594no—
369C4⋊C4—1184no—
370C4⋊C4—1182no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—48no—
374C22⋊C4—1188no—
375C22⋊C4—1184no—
376C22⋊C4—1184no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—24no—
383C23—42no—
384C23—48no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—84no—
389C23—81no—
390C23—81no—
391C23—1184no—
392C23—47216no—
393C23—4724no—
394C23—4724no—
395C23—2368no—
396C23—2364no—
397C23—2364no—
398C23—2364no—
399C23#1427369511yescharacteristic
400C23#1427369511yescharacteristic
401C23C2×#1174567112yes—
402C23C2×#1174567114yes—
403C23#1427375212yes—
404C23C22×C22⋊D5911yescharacteristic
405D4—4722no—
406D4—4724no—
407D4—2364no—
408D4—23616no—
409D4—2364no—
410D4—2364no—
411D4—2368no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—1184no—
430C2×C4—2368no—
431C2×C4—2364no—
432C2×C4#1427369512yes—
433C2×C4C2×D4×D5912yes—
434C8—2364no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—84no—
448C22—47216no—
449C22—4728no—
450C22—2364no—
451C22—2364no—
452C22—2364no—
453C22#1683755212yes—
454C22#1683755212yes—
455C22#1683756912yes—
456C22#1683759112yes—
457C22C2×#1427369511yescharacteristic
458C22C22×#1174567111yescharacteristic
459C22C2×#1427375211yescharacteristic
460C4—22no—
461C4—22no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—2364no—
466C2—21no—
467C2—21no—
468C2—22no—
469C2—44no—
470C2—4724no—
471C2#2142311514yesdirect product factor
472C2C2×#1683755211yescharacteristic
473C2C2×#1683756911yescharacteristic
474C2C2×#1683759111yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#399C2
socleC23×C59#120
prime radicalC2×#16837964C2
hypercenterC2×#399D59
hypocenterC59C2×#1613
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×#16837720C2semidirect
3. #21423115C2direct (N × H)
4. C2×#16837135C2semidirect
5. C2×#16837126C2semidirect
6. C2×C59×#399C2semidirect
16. #16837720C22semidirect
19. C2×C22⋊(C59⋊C8)C22semidirect
20. #16837135C22semidirect
21. #16837126C22semidirect
23. C22×C4⋊C4×C59C22semidirect
27. C59×#399C22semidirect
28. C2×#95×C59C22semidirect
57. C22⋊(C59⋊C8)C23semidirect
68. C2×C4⋊C4×C59C23semidirect
76. #95×C59C23semidirect
231. C2×#399D59semidirect
239. C2×C59#1613semidirect
253. C22×C4⋊C4C2×D59semidirect
262. #399C2×D59semidirect
263. C2×#95C2×D59semidirect
266. C59C2×#1613semidirect
300. C2×C4⋊C4C22×D59semidirect
314. #95C22×D59semidirect
471. C2#21423115direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168379642
C2×#168377202
#214231152
C2×#168371352
C2×#168371262
C2×C59×#3992
C2×#161359