Finite Group Databasebeta

← C2×#21423052 #28560770

Subgroups — C2×#21423052 #28560770

Order 15104.

1308 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1308 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×#16838028C211yesmaximal, characteristic
2C2×#16837558C212yesmaximal, complemented
3#21423052C218yesmaximal, direct product factor
4C2×#16837138C212yesmaximal, complemented
5C2×#16837126C211yesmaximal, characteristic
6C2×C59×#393C211yesmaximal, characteristic
7C22×#11745672—22no—
8#16838028—24no—
9#16838028—22no—
10#16838028C2214yes—
11C2×#14273706—21no—
12C2×#14273706C2212yes—
13C2×#14273704—22no—
14C2×#14273704—22no—
15C22×C22⋊Dic59—21no—
16#16837558—24no—
17#16837558C2218yescomplemented
18C2×#14273651—22no—
19C2×C22⋊(C59⋊C8)C2212yes—
20#16837138C2218yescomplemented
21#16837126C2214yes—
22C23×Q8×C59C2211yescharacteristic
23C22×C22⋊C4×C59—21no—
24C22×C22⋊C4×C59C2211yescharacteristic
25C59×#393—22no—
26C59×#393C2214yes—
27C2×#95×C59—21no—
28C2×#95×C59C2211yescharacteristic
29C22×Dic118—21no—
30C22×Dic118D412yes—
31C23×Dic59—21no—
32C23×Dic59—21no—
33C2×#11745672—28no—
34C2×#11745672—48no—
35C22×C59⋊C8—22no—
36#14273706—24no—
37#14273706—24no—
38#14273706—22no—
39#14273706—22no—
40#14273706D414yes—
41#14273706C2314yes—
42#14273704—28no—
43#14273704—28no—
44#14273704—28no—
45#14273704—28no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59D412yes—
48C2×Dic59⋊C4—22no—
49C2×Dic59⋊C4—22no—
50C2×C22⋊Dic59—22no—
51C2×C22⋊Dic59—24no—
52C2×C22⋊Dic59—22no—
53C2×C22⋊Dic59—24no—
54#14273651—28no—
55C22⋊(C59⋊C8)—22no—
56C22⋊(C59⋊C8)C2314yes—
57C25×C59D411yescharacteristic
58C22×Q8×C59—21no—
59C22×Q8×C59—22no—
60C22×Q8×C59—22no—
61C22×Q8×C59—44no—
62C22×Q8×C59C2312yes—
63C23×C4×C59—21no—
64C23×C4×C59—21no—
65C23×C4×C59D411yescharacteristic
66C23×C4×C59C2311yescharacteristic
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59—22no—
69C2×C22⋊C4×C59—24no—
70C2×C22⋊C4×C59—44no—
71C2×C22⋊C4×C59D412yes—
72C2×C22⋊C4×C59C2312yes—
73#95×C59—24no—
74#95×C59—21no—
75#95×C59C2312yes—
76C24×C59—21no—
77C24×C59—21no—
78C24×C59—22no—
79C24×C59—42no—
80C24×C59—42no—
81C24×C59Q1614yescomplemented
82C24×C59C2×D411yescharacteristic
83C24×C59C2×D412yes—
84C2×Q8×C59—28no—
85C2×Q8×C59—24no—
86C2×Q8×C59—44no—
87C2×Q8×C59—44no—
88C2×Q8×C59—44no—
89C2×Q8×C59—48no—
90C2×Q8×C59—42no—
91C22×C4×C59—24no—
92C22×C4×C59—21no—
93C22×C4×C59—22no—
94C22×C4×C59—21no—
95C22×C4×C59—22no—
96C22×C4×C59—42no—
97C22×C4×C59—42no—
98C22×C4×C59—41no—
99C22×C4×C59—41no—
100C22×C4×C59—42no—
101C22×C4×C59—42no—
102C22×C4×C59—42no—
103C22×C4×C59—42no—
104C22×C4×C59C2×D412yes—
105C22×C4×C59C2×D412yes—
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D412yes—
108C22×C4×C59C2411yescharacteristic
109C22×Dic59—22no—
110C22×Dic59—22no—
111C22×Dic59—24no—
112C22×Dic59—24no—
113C22×Dic59—82no—
114C22×Dic59—82no—
115C2×Dic118—24no—
116C2×Dic118—22no—
117C2×Dic118—44no—
118C2×Dic118—82no—
119C2×Dic118C2×D414yes—
120C22⋊C4×C59—24no—
121C22⋊C4×C59—24no—
122C22⋊C4×C59—48no—
123C22⋊Dic59—28no—
124C22⋊Dic59—24no—
125C22⋊Dic59—24no—
126#11745672—416no—
127C4⋊Dic59—24no—
128C4⋊Dic59—22no—
129C4⋊Dic59C2×D414yes—
130Dic59⋊C4—28no—
131Dic59⋊C4—24no—
132Dic59⋊C4—24no—
133C2×C59⋊C8—24no—
134C2×C59⋊C8—44no—
135C23×C59—22no—
136C23×C59—22no—
137C23×C59—22no—
138C23×C59—24no—
139C23×C59—22no—
140C23×C59—24no—
141C23×C59—42no—
142C23×C59—48no—
143C23×C59—42no—
144C23×C59—42no—
145C23×C59—42no—
146C23×C59—84no—
147C23×C59—81no—
148C23×C59—81no—
149C23×C59#12011yescharacteristic
150C23×C59#12011yescharacteristic
151C23×C59C2×Q1612yes—
152C23×C59C2×Q1614yescomplemented
153C23×C59#13712yes—
154C23×C59C22×D411yescharacteristic
155Q8×C59—28no—
156Q8×C59—44no—
157Q8×C59—48no—
158C2×C4×C59—24no—
159C2×C4×C59—24no—
160C2×C4×C59—22no—
161C2×C4×C59—42no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—42no—
165C2×C4×C59—44no—
166C2×C4×C59—42no—
167C2×C4×C59—42no—
168C2×C4×C59—44no—
169C2×C4×C59—42no—
170C2×C4×C59—42no—
171C2×C4×C59—42no—
172C2×C4×C59—42no—
173C2×C4×C59—81no—
174C2×C4×C59—84no—
175C2×C4×C59#12012yes—
176C2×C4×C59C22×D412yes—
177C2×Dic59—24no—
178C2×Dic59—24no—
179C2×Dic59—44no—
180C2×Dic59—44no—
181C2×Dic59—84no—
182C2×Dic59—84no—
183Dic118—44no—
184Dic118—82no—
185Dic118—84no—
186C59⋊C8—44no—
187C2×#1602—591nomaximal, Sylow
188C22×C59—24no—
189C22×C59—24no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—24no—
195C22×C59—44no—
196C22×C59—48no—
197C22×C59—42no—
198C22×C59—42no—
199C22×C59—82no—
200C22×C59—84no—
201C22×C59#45112yes—
202C22×C59#45112yes—
203C22×C59#45512yes—
204C22×C59B(4,𝔽2)12yes—
205C22×C59C2×#12011yescharacteristic
206C22×C59C22×Q1611yescharacteristic
207C22×C59C2×#13711yescharacteristic
208C4×C59—22no—
209C4×C59—22no—
210C4×C59—42no—
211C4×C59—44no—
212C4×C59—82no—
213Dic59—44no—
214Dic59—44no—
215C2×#543—591no—
216C2×#451—592no—
217C2×#393D5911yescharacteristic
218#1602—598no—
219C2×#328—592no—
220C2×#323—591no—
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—22no—
224C2×C59—44no—
225C2×C59#160214yescomplemented
226C2×C59C2×#45111yescharacteristic
227C2×C59C2×#45511yescharacteristic
228C2×C59C2×B(4,𝔽2)11yescharacteristic
229C23×Q8C2×D5911yescharacteristic
230C22×Q16—1182no—
231#543—594no—
232#543—1184no—
233#543—1182no—
234C2×C22⋊Q8—592no—
235C2×C22⋊Q8—1182no—
236C2×C22⋊Q8—1182no—
237C2×C22⋊Q8—1181no—
238C22×C22⋊C4—21no—
239C22×C22⋊C4—1181no—
240C22×C22⋊C4C2×D5911yescharacteristic
241#451—598no—
242#451—1184no—
243C2×#103—1182no—
244C2×C22⋊C8—592no—
245#393—22no—
246#393C2×D5914yes—
247C2×#95—21no—
248C2×#95C2×D5911yescharacteristic
249#328—598no—
250#323—594no—
251C59C2×#160211yescharacteristic, complemented, Sylow
252C25C22⋊D5911yescharacteristic
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—44no—
257C22×Q8—592no—
258C22×Q8—1181no—
259C22×Q8C22×D5912yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—1181no—
263C23×C4—1181no—
264C23×C4C22⋊D5911yescharacteristic
265C23×C4C22×D5911yescharacteristic
266C2×Q16—1188no—
267C2×Q16—2368no—
268C22×C8—1182no—
269C22⋊Q8—594no—
270C22⋊Q8—594no—
271C22⋊Q8—1188no—
272C22⋊Q8—1188no—
273C22⋊Q8—1188no—
274C22⋊Q8—1188no—
275C22⋊Q8—1184no—
276C22⋊Q8—1182no—
277C22⋊Q8—1182no—
278C22⋊Q8—1184no—
279C2×C4⋊C4—592no—
280C2×C4⋊C4—1182no—
281C2×C4⋊C4—1181no—
282C2×C4⋊C4—1182no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—1182no—
288C2×C22⋊C4—1184no—
289C2×C22⋊C4—1182no—
290C2×C22⋊C4—1184no—
291C2×C22⋊C4C22⋊D5912yes—
292C2×C22⋊C4C22×D5912yes—
293#103—1188no—
294C22⋊C8—594no—
295C22⋊C8—1182no—
296#95—24no—
297#95—21no—
298#95C22×D5912yes—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—42no—
303C24—42no—
304C24#1174567214yescomplemented
305C24C2×C22⋊D5911yescharacteristic
306C24C2×C22⋊D5912yes—
307C2×Q8—28no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—594no—
315C2×Q8—1184no—
316C2×Q8—1182no—
317C2×Q8—4722no—
318C2×Q8—2364no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—1184no—
333C22×C4—1184no—
334C22×C4—1182no—
335C22×C4—1182no—
336C22×C4—4722no—
337C22×C4—4722no—
338C22×C4D4×D5912yes—
339C22×C4D4×D5912yes—
340C22×C4C2×C22⋊D5912yes—
341C22×C4C2×C22⋊D5911yescharacteristic
342C22×C4C23×D5911yescharacteristic
343Q16—23616no—
344C2×C8—1184no—
345C2×C8—2364no—
346C4⋊C4—594no—
347C4⋊C4—1188no—
348C4⋊C4—1184no—
349C4⋊C4—1182no—
350C4⋊C4—1184no—
351C4⋊C4—1184no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—48no—
355C22⋊C4—1188no—
356C22⋊C4—1184no—
357C22⋊C4—1184no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—42no—
365C23—48no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23#1427369511yescharacteristic
373C23#1427369511yescharacteristic
374C23C2×#1174567212yes—
375C23C2×#1174567214yescomplemented
376C23#1427375712yes—
377C23C22×C22⋊D5911yescharacteristic
378Q8—28no—
379Q8—44no—
380Q8—48no—
381Q8—4722no—
382Q8—4724no—
383Q8—2364no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—84no—
401C2×C4—1184no—
402C2×C4—1184no—
403C2×C4—4724no—
404C2×C4—4724no—
405C2×C4—2364no—
406C2×C4—2364no—
407C2×C4#1427369512yes—
408C2×C4C2×D4×D5912yes—
409C8—2364no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—44no—
418C22—42no—
419C22—42no—
420C22—48no—
421C22—82no—
422C22—84no—
423C22#1683755712yes—
424C22#1683755712yes—
425C22#1683757912yes—
426C22#1683758812yes—
427C22C2×#1427369511yescharacteristic
428C22C22×#1174567211yescharacteristic
429C22C2×#1427375711yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—44no—
434C4—82no—
435C4—2364no—
436C4—2364no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—44no—
441C2#2142305214yesdirect product factor
442C2C2×#1683755711yescharacteristic
443C2C2×#1683757911yescharacteristic
444C2C2×#1683758811yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#393C2
socleC23×C59#120
prime radicalC25×C59D4
hypercenterC2×#393D59
hypocenterC59C2×#1602
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#16837558C2semidirect
3. #21423052C2direct (N × H)
4. C2×#16837138C2semidirect
17. #16837558C22semidirect
20. #16837138C22semidirect
81. C24×C59Q16semidirect
152. C23×C59C2×Q16semidirect
225. C2×C59#1602semidirect
251. C59C2×#1602semidirect
304. C24#11745672semidirect
375. C23C2×#11745672semidirect
441. C2#21423052direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168380282
C2×#168375582
#214230522
C2×#168371382
C2×#168371262
C2×C59×#3932
C2×#160259