Finite Group Databasebeta

← C2×#21423121 #28560822

Subgroups — C2×#21423121 #28560822

Order 15104.

1180 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837735C212yesmaximal, complemented
2C2×#16837338C211yesmaximal, characteristic
3#21423121C218yesmaximal, direct product factor
4C2×#16837139C212yesmaximal, complemented
5C2×#16837126C211yesmaximal, characteristic
6C2×C59×#543C211yesmaximal, characteristic
7C22×C4⋊Dic59C2211yescharacteristic
8C22×Dic59⋊C4—21no—
9C22×C22⋊Dic59—21no—
10#16837735—24no—
11#16837735C2218yescomplemented
12C2×#14273660—22no—
13C2×#14273651—22no—
14C2×C22⋊(C59⋊C8)C2212yes—
15#16837338—22no—
16#16837338C2214yes—
17C2×C59⋊#95C2211yescharacteristic
18#16837139C2218yescomplemented
19#16837126C2214yes—
20C59×#543—24no—
21C59×#543—22no—
22C59×#543C2214yes—
23C2×C22⋊Q8×C59—22no—
24C2×C22⋊Q8×C59—21no—
25C2×C22⋊Q8×C59—22no—
26C2×C22⋊Q8×C59C2212yes—
27C22×C22⋊C4×C59—21no—
28C22×C22⋊C4×C59C2211yescharacteristic
29C23×Dic59—21no—
30C23×Dic59—21no—
31C22×C59⋊C8—22no—
32C2×C4⋊Dic59—22no—
33C2×C4⋊Dic59—22no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59C2312yes—
36C2×Dic59⋊C4—24no—
37C2×Dic59⋊C4—44no—
38C2×C22⋊Dic59—24no—
39C2×C22⋊Dic59—44no—
40#14273660—28no—
41#14273651—28no—
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)C2314yes—
44C59⋊#95—21no—
45C59⋊#95C2312yes—
46C25×C59D411yescharacteristic
47C22×Q8×C59—21no—
48C22×Q8×C59D412yes—
49C23×C4×C59—21no—
50C23×C4×C59—21no—
51C23×C4×C59D411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C22⋊Q8×C59—28no—
54C22⋊Q8×C59—28no—
55C22⋊Q8×C59—24no—
56C22⋊Q8×C59—28no—
57C22⋊Q8×C59—24no—
58C22⋊Q8×C59—22no—
59C22⋊Q8×C59—22no—
60C22⋊Q8×C59—28no—
61C22⋊Q8×C59D414yes—
62C22⋊Q8×C59C2314yes—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—22no—
65C2×C4⋊C4×C59—22no—
66C2×C4⋊C4×C59D412yes—
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59—24no—
69C2×C22⋊C4×C59—22no—
70C2×C22⋊C4×C59—24no—
71C2×C22⋊C4×C59—22no—
72C2×C22⋊C4×C59—22no—
73C2×C22⋊C4×C59D412yes—
74C2×C22⋊C4×C59C2312yes—
75C24×C59—21no—
76C24×C59—21no—
77C24×C59—22no—
78C24×C59—42no—
79C24×C59—42no—
80C24×C59Q1614yes—
81C24×C59C2×D411yescharacteristic
82C24×C59C2×D412yes—
83C2×Q8×C59—24no—
84C2×Q8×C59—22no—
85C2×Q8×C59—44no—
86C2×Q8×C59—82no—
87C2×Q8×C59C2×D414yes—
88C22×C4×C59—22no—
89C22×C4×C59—21no—
90C22×C4×C59—22no—
91C22×C4×C59—24no—
92C22×C4×C59—24no—
93C22×C4×C59—21no—
94C22×C4×C59—42no—
95C22×C4×C59—42no—
96C22×C4×C59—82no—
97C22×C4×C59—82no—
98C22×C4×C59C2×D412yes—
99C22×C4×C59C2×D412yes—
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D412yes—
102C22×C4×C59C2411yescharacteristic
103C22×Dic59—24no—
104C22×Dic59—22no—
105C22×Dic59—22no—
106C22×Dic59—42no—
107C22×Dic59—42no—
108C22×Dic59—42no—
109C22×Dic59—42no—
110C22×Dic59—41no—
111C22×Dic59—41no—
112C4⋊C4×C59—24no—
113C4⋊C4×C59—28no—
114C4⋊C4×C59—24no—
115C4⋊C4×C59—22no—
116C4⋊C4×C59—24no—
117C4⋊C4×C59C2×D414yes—
118C22⋊C4×C59—28no—
119C22⋊C4×C59—24no—
120C22⋊C4×C59—24no—
121C22⋊C4×C59—24no—
122C22⋊C4×C59—24no—
123C22⋊Dic59—48no—
124C4⋊Dic59—24no—
125C4⋊Dic59—42no—
126Dic59⋊C4—48no—
127C2×C59⋊C8—24no—
128C2×C59⋊C8—44no—
129C23×C59—22no—
130C23×C59—22no—
131C23×C59—22no—
132C23×C59—24no—
133C23×C59—22no—
134C23×C59—24no—
135C23×C59—42no—
136C23×C59—48no—
137C23×C59—42no—
138C23×C59—42no—
139C23×C59—42no—
140C23×C59—84no—
141C23×C59—81no—
142C23×C59—81no—
143C23×C59#12011yescharacteristic
144C23×C59#12011yescharacteristic
145C23×C59C2×Q1612yes—
146C23×C59C2×Q1614yes—
147C23×C59#13712yes—
148C23×C59C22×D411yescharacteristic
149Q8×C59—44no—
150Q8×C59—82no—
151Q8×C59—84no—
152C2×C4×C59—24no—
153C2×C4×C59—24no—
154C2×C4×C59—22no—
155C2×C4×C59—24no—
156C2×C4×C59—42no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—42no—
160C2×C4×C59—44no—
161C2×C4×C59—44no—
162C2×C4×C59—42no—
163C2×C4×C59—81no—
164C2×C4×C59—84no—
165C2×C4×C59—84no—
166C2×C4×C59#12012yes—
167C2×C4×C59C22×D412yes—
168C2×Dic59—24no—
169C2×Dic59—48no—
170C2×Dic59—42no—
171C2×Dic59—42no—
172C2×Dic59—42no—
173C2×Dic59—42no—
174C2×Dic59—42no—
175C2×Dic59—82no—
176C2×Dic59—82no—
177C59⋊C8—44no—
178C2×#1614—591nomaximal, Sylow
179C22×C59—24no—
180C22×C59—24no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—24no—
186C22×C59—44no—
187C22×C59—48no—
188C22×C59—42no—
189C22×C59—42no—
190C22×C59—82no—
191C22×C59—84no—
192C22×C59#45112yes—
193C22×C59#45112yes—
194C22×C59#45312yes—
195C22×C59#45812yes—
196C22×C59C2×#12011yescharacteristic
197C22×C59C22×Q1611yescharacteristic
198C22×C59C2×#13711yescharacteristic
199C4×C59—22no—
200C4×C59—22no—
201C4×C59—42no—
202C4×C59—44no—
203C4×C59—44no—
204Dic59—44no—
205Dic59—82no—
206C2×#543D5911yescharacteristic
207C2×#484—592no—
208C2×#399—591no—
209#1614—598no—
210C2×#328—592no—
211C2×#323—591no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—22no—
215C2×C59—44no—
216C2×C59#161414yescomplemented
217C2×C59C2×#45111yescharacteristic
218C2×C59C2×#45311yescharacteristic
219C2×C59C2×#45811yescharacteristic
220#543—24no—
221#543—22no—
222#543C2×D5914yes—
223C2×C22⋊Q8—22no—
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8C2×D5912yes—
227C22×C4⋊C4—591no—
228C22×C4⋊C4—1181no—
229C22×C22⋊C4—21no—
230C22×C22⋊C4—1181no—
231C22×C22⋊C4C2×D5911yescharacteristic
232#484—598no—
233#484—1184no—
234C2×#107—1182no—
235C2×#103—1182no—
236C2×C22⋊C8—592no—
237#399—594no—
238#399—1182no—
239C2×#95—591no—
240#328—598no—
241#323—594no—
242C59C2×#161411yescharacteristic, complemented, Sylow
243C25C22⋊D5911yescharacteristic
244C22×Q8—21no—
245C22×Q8C22⋊D5912yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—1181no—
249C23×C4—1181no—
250C23×C4C22⋊D5911yescharacteristic
251C23×C4C22×D5911yescharacteristic
252C22×C8—1182no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—28no—
261C22⋊Q8C22⋊D5914yes—
262C22⋊Q8C22×D5914yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—592no—
267C2×C4⋊C4—1182no—
268C2×C4⋊C4—1182no—
269C2×C4⋊C4—1181no—
270C2×C4⋊C4—1184no—
271C2×C4⋊C4—2364no—
272C2×C4⋊C4C22⋊D5912yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—1184no—
280C2×C22⋊C4—2364no—
281C2×C22⋊C4C22⋊D5912yes—
282C2×C22⋊C4C22×D5912yes—
283#107—1188no—
284#103—1188no—
285C22⋊C8—594no—
286C22⋊C8—1182no—
287#95—592no—
288#95—1181no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24#1174567214yes—
295C24C2×C22⋊D5911yescharacteristic
296C24C2×C22⋊D5912yes—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—44no—
300C2×Q8—82no—
301C2×Q8C2×C22⋊D5914yes—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—1184no—
313C22×C4—1182no—
314C22×C4—1182no—
315C22×C4—2362no—
316C22×C4—2362no—
317C22×C4—2361no—
318C22×C4—2361no—
319C22×C4—2362no—
320C22×C4—2362no—
321C22×C4C2×C22⋊D5912yes—
322C22×C4C2×C22⋊D5912yes—
323C22×C4C2×C22⋊D5911yescharacteristic
324C22×C4C2×C22⋊D5912yes—
325C22×C4C23×D5911yescharacteristic
326C2×C8—1184no—
327C2×C8—2364no—
328C4⋊C4—24no—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—1184no—
334C4⋊C4—2368no—
335C4⋊C4—2362no—
336C4⋊C4C2×C22⋊D5914yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—2368no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—24no—
349C23—42no—
350C23—48no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—84no—
355C23—81no—
356C23—81no—
357C23#1427369611yescharacteristic
358C23#1427369611yescharacteristic
359C23C2×#1174567212yes—
360C23C2×#1174567214yes—
361C23#1427375812yes—
362C23C22×C22⋊D5911yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—1184no—
381C2×C4—4722no—
382C2×C4—4722no—
383C2×C4—2362no—
384C2×C4—2362no—
385C2×C4—2362no—
386C2×C4—2368no—
387C2×C4—2362no—
388C2×C4—2362no—
389C2×C4#1427369612yes—
390C2×C4C22×C22⋊D5912yes—
391C8—2364no—
392C22—24no—
393C22—24no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1683755812yes—
406C22#1683755812yes—
407C22#1683757012yes—
408C22#1683759212yes—
409C22C2×#1427369611yescharacteristic
410C22C22×#1174567211yescharacteristic
411C22C2×#1427375811yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—44no—
416C4—44no—
417C4—4722no—
418C4—2364no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—44no—
423C2#2142312114yesdirect product factor
424C2C2×#1683755811yescharacteristic
425C2C2×#1683757011yescharacteristic
426C2C2×#1683759211yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#543C2
socleC23×C59#120
prime radicalC25×C59D4
hypercenterC2×#543D59
hypocenterC59C2×#1614
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. C2×#16837735C2semidirect
3. #21423121C2direct (N × H)
4. C2×#16837139C2semidirect
11. #16837735C22semidirect
18. #16837139C22semidirect
216. C2×C59#1614semidirect
242. C59C2×#1614semidirect
423. C2#21423121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377352
C2×#168373382
#214231212
C2×#168371392
C2×#168371262
C2×C59×#5432
C2×#161459