Finite Group Databasebeta

← C2×#21423086 #28560815

Subgroups — C2×#21423086 #28560815

Order 15104.

1136 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1136 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×#16837732C212yesmaximal, complemented
3#21423086C218yesmaximal, direct product factor
4C2×#16837138C212yesmaximal, complemented
5C2×#16837126C211yesmaximal, characteristic
6C2×C59×#397C211yesmaximal, characteristic
7#16838028—24no—
8#16838028—22no—
9#16838028C2214yes—
10C2×#14273706—21no—
11C2×#14273706C2212yes—
12C2×#14273704—22no—
13C2×#14273704—22no—
14C22×C22⋊Dic59—21no—
15#16837732—24no—
16#16837732C2218yescomplemented
17C2×#14273658—22no—
18C2×#14273650—22no—
19C2×C22⋊(C59⋊C8)C2212yes—
20#16837138C2218yescomplemented
21#16837126C2214yes—
22C22×C4⋊C4×C59C2211yescharacteristic
23C22×C22⋊C4×C59—21no—
24C22×C22⋊C4×C59C2211yescharacteristic
25C59×#397—22no—
26C59×#397C2214yes—
27C2×#95×C59—21no—
28C2×#95×C59C2211yescharacteristic
29C22×Dic118—21no—
30C22×Dic118D412yes—
31C23×Dic59—21no—
32C23×Dic59—21no—
33C22×C59⋊C8—22no—
34#14273706—24no—
35#14273706—24no—
36#14273706—22no—
37#14273706—22no—
38#14273706D414yes—
39#14273706C2314yes—
40#14273704—28no—
41#14273704—28no—
42#14273704—28no—
43#14273704—28no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59D412yes—
46C2×Dic59⋊C4—22no—
47C2×Dic59⋊C4—22no—
48C2×C22⋊Dic59—22no—
49C2×C22⋊Dic59—24no—
50C2×C22⋊Dic59—22no—
51C2×C22⋊Dic59—24no—
52#14273658—28no—
53#14273650—28no—
54C22⋊(C59⋊C8)—22no—
55C22⋊(C59⋊C8)C2314yes—
56C25×C59D411yescharacteristic
57C23×C4×C59—21no—
58C23×C4×C59—21no—
59C23×C4×C59D411yescharacteristic
60C23×C4×C59C2311yescharacteristic
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59C2312yes—
65C2×C22⋊C4×C59—22no—
66C2×C22⋊C4×C59—22no—
67C2×C22⋊C4×C59—24no—
68C2×C22⋊C4×C59—44no—
69C2×C22⋊C4×C59D412yes—
70C2×C22⋊C4×C59C2312yes—
71#95×C59—24no—
72#95×C59—21no—
73#95×C59C2312yes—
74C24×C59—21no—
75C24×C59—21no—
76C24×C59—22no—
77C24×C59—42no—
78C24×C59—42no—
79C24×C59SD1614yes—
80C24×C59C2×D411yescharacteristic
81C24×C59C2×D412yes—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—22no—
86C22×C4×C59—42no—
87C22×C4×C59—42no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—42no—
93C22×C4×C59—42no—
94C22×C4×C59—42no—
95C22×C4×C59C2×D412yes—
96C22×C4×C59C2×D412yes—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D412yes—
99C22×C4×C59C2411yescharacteristic
100C22×Dic59—22no—
101C22×Dic59—22no—
102C22×Dic59—24no—
103C22×Dic59—24no—
104C22×Dic59—82no—
105C22×Dic59—82no—
106C2×Dic118—24no—
107C2×Dic118—22no—
108C2×Dic118—44no—
109C2×Dic118—82no—
110C2×Dic118C2×D414yes—
111C4⋊C4×C59—24no—
112C4⋊C4×C59—42no—
113C22⋊C4×C59—24no—
114C22⋊C4×C59—24no—
115C22⋊C4×C59—48no—
116C22⋊Dic59—28no—
117C22⋊Dic59—24no—
118C22⋊Dic59—24no—
119C4⋊Dic59—24no—
120C4⋊Dic59—22no—
121C4⋊Dic59C2×D414yes—
122Dic59⋊C4—28no—
123Dic59⋊C4—24no—
124Dic59⋊C4—24no—
125C2×C59⋊C8—24no—
126C2×C59⋊C8—44no—
127C23×C59—22no—
128C23×C59—22no—
129C23×C59—22no—
130C23×C59—24no—
131C23×C59—22no—
132C23×C59—24no—
133C23×C59—48no—
134C23×C59—42no—
135C23×C59—42no—
136C23×C59—42no—
137C23×C59—42no—
138C23×C59—84no—
139C23×C59—81no—
140C23×C59—81no—
141C23×C59#12011yescharacteristic
142C23×C59#12011yescharacteristic
143C23×C59C2×SD1612yes—
144C23×C59C2×SD1614yes—
145C23×C59#13712yes—
146C23×C59C22×D411yescharacteristic
147C2×C4×C59—24no—
148C2×C4×C59—22no—
149C2×C4×C59—42no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59—42no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—81no—
161C2×C4×C59—84no—
162C2×C4×C59—82no—
163C2×C4×C59—82no—
164C2×C4×C59#12012yes—
165C2×C4×C59C22×D412yes—
166C2×Dic59—24no—
167C2×Dic59—24no—
168C2×Dic59—44no—
169C2×Dic59—44no—
170C2×Dic59—84no—
171C2×Dic59—84no—
172Dic118—44no—
173Dic118—82no—
174Dic118—84no—
175C59⋊C8—44no—
176C2×#1608—591nomaximal, Sylow
177C22×C59—24no—
178C22×C59—24no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—24no—
184C22×C59—44no—
185C22×C59—48no—
186C22×C59—42no—
187C22×C59—42no—
188C22×C59—82no—
189C22×C59—84no—
190C22×C59#44812yes—
191C22×C59#44812yes—
192C22×C59#45512yes—
193C22×C59#45812yes—
194C22×C59C2×#12011yescharacteristic
195C22×C59C22×SD1611yescharacteristic
196C22×C59C2×#13711yescharacteristic
197C4×C59—22no—
198C4×C59—22no—
199C4×C59—42no—
200C4×C59—82no—
201C4×C59—82no—
202Dic59—44no—
203Dic59—44no—
204C2×#543—591no—
205C2×#483—592no—
206C2×#397D5911yescharacteristic
207#1608—598no—
208C2×#328—592no—
209C2×#323—591no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—22no—
213C2×C59—44no—
214C2×C59#160814yescomplemented
215C2×C59C2×#44811yescharacteristic
216C2×C59C2×#45511yescharacteristic
217C2×C59C2×#45811yescharacteristic
218#543—594no—
219#543—1184no—
220#543—1182no—
221C2×C22⋊Q8—592no—
222C2×C22⋊Q8—1182no—
223C2×C22⋊Q8—1182no—
224C2×C22⋊Q8—1181no—
225C22×C4⋊C4C2×D5911yescharacteristic
226C22×C22⋊C4—21no—
227C22×C22⋊C4—1181no—
228C22×C22⋊C4C2×D5911yescharacteristic
229#483—598no—
230#483—1184no—
231C2×#106—1182no—
232C2×#103—1182no—
233C2×C22⋊C8—592no—
234#397—22no—
235#397C2×D5914yes—
236C2×#95—21no—
237C2×#95C2×D5911yescharacteristic
238#328—598no—
239#323—594no—
240C59C2×#160811yescharacteristic, complemented, Sylow
241C25C22⋊D5911yescharacteristic
242C22×Q8—592no—
243C22×Q8—1181no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—1181no—
247C23×C4—1181no—
248C23×C4C22⋊D5911yescharacteristic
249C23×C4C22×D5911yescharacteristic
250C22×C8—1182no—
251C22⋊Q8—594no—
252C22⋊Q8—594no—
253C22⋊Q8—1188no—
254C22⋊Q8—1188no—
255C22⋊Q8—1188no—
256C22⋊Q8—1188no—
257C22⋊Q8—1184no—
258C22⋊Q8—1182no—
259C22⋊Q8—1182no—
260C22⋊Q8—1184no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—592no—
265C2×C4⋊C4—1182no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4—1182no—
268C2×C4⋊C4C22×D5912yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—1182no—
274C2×C22⋊C4—1184no—
275C2×C22⋊C4—1182no—
276C2×C22⋊C4—1184no—
277C2×C22⋊C4C22⋊D5912yes—
278C2×C22⋊C4C22×D5912yes—
279#106—1188no—
280#103—1188no—
281C22⋊C8—594no—
282C22⋊C8—1182no—
283#95—24no—
284#95—21no—
285#95C22×D5912yes—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24#1174567014yes—
292C24C2×C22⋊D5911yescharacteristic
293C24C2×C22⋊D5912yes—
294C2×Q8—594no—
295C2×Q8—1184no—
296C2×Q8—1182no—
297C2×Q8—4722no—
298C2×Q8—2364no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—1184no—
313C22×C4—1184no—
314C22×C4—1182no—
315C22×C4—1182no—
316C22×C4—4722no—
317C22×C4—4722no—
318C22×C4D4×D5912yes—
319C22×C4D4×D5912yes—
320C22×C4C2×C22⋊D5912yes—
321C22×C4C2×C22⋊D5911yescharacteristic
322C22×C4C23×D5911yescharacteristic
323C2×C8—1184no—
324C2×C8—2364no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—594no—
328C4⋊C4—1188no—
329C4⋊C4—1184no—
330C4⋊C4—1182no—
331C4⋊C4—1184no—
332C4⋊C4—1184no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—48no—
336C22⋊C4—1188no—
337C22⋊C4—1184no—
338C22⋊C4—1184no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—48no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—84no—
353C23#1427369511yescharacteristic
354C23#1427369511yescharacteristic
355C23C2×#1174567012yes—
356C23C2×#1174567014yes—
357C23#1427375712yes—
358C23C22×C22⋊D5911yescharacteristic
359Q8—4722no—
360Q8—4724no—
361Q8—2364no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—1184no—
380C2×C4—1184no—
381C2×C4—4724no—
382C2×C4—4724no—
383C2×C4—2364no—
384C2×C4—2364no—
385C2×C4#1427369512yes—
386C2×C4C2×D4×D5912yes—
387C8—2364no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—48no—
399C22—82no—
400C22—84no—
401C22#1683754012yes—
402C22#1683754012yes—
403C22#1683757912yes—
404C22#1683759112yes—
405C22C2×#1427369511yescharacteristic
406C22C22×#1174567011yescharacteristic
407C22C2×#1427375711yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—2364no—
414C4—2364no—
415C2—21no—
416C2—21no—
417C2—22no—
418C2—44no—
419C2#2142308614yesdirect product factor
420C2C2×#1683754011yescharacteristic
421C2C2×#1683757911yescharacteristic
422C2C2×#1683759111yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#397C2
socleC23×C59#120
prime radicalC25×C59D4
hypercenterC2×#397D59
hypocenterC59C2×#1608
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×#16837732C2semidirect
3. #21423086C2direct (N × H)
4. C2×#16837138C2semidirect
16. #16837732C22semidirect
20. #16837138C22semidirect
214. C2×C59#1608semidirect
240. C59C2×#1608semidirect
419. C2#21423086direct (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×#168377322
#214230862
C2×#168371382
C2×#168371262
C2×C59×#3972
C2×#160859