Finite Group Databasebeta

← C2×#21422875 #28557755

Subgroups — C2×#21422875 #28557755

Order 15104.

670 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837458C211yesmaximal, characteristic
2C2×#16837454C211yesmaximal, characteristic
3C2×#16837453C211yesmaximal, characteristic
4#21422875C218yesmaximal, direct product factor
5C2×#16837155C211yesmaximal, characteristic, complemented
6C2×#16837152C211yesmaximal, characteristic, complemented
7C2×#16837152C211yesmaximal, characteristic, complemented
8C2×C59×#335C211yesmaximal, characteristic, complemented
9C2×C42×D59C2211yescharacteristic
10#16837458—22no—
11#16837458C2214yes—
12#16837454—22no—
13#16837454C2214yes—
14#16837453C414yes—
15#16837453C2214yes—
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17C2×C4⋊(C59⋊C8)C2211yescharacteristic
18C2×Dic59⋊C8C411yescharacteristic
19C2×Dic59⋊C8C411yescharacteristic
20C2×(C2×D59)⋊C8—21no—
21C2×(C2×D59)⋊C8—21no—
22C2×(C2×D59)⋊C8C411yescharacteristic
23C2×(C2×D59)⋊C8C411yescharacteristic
24C2×#14273637—21no—
25C2×C4×C59⋊C8C2211yescharacteristic
26C2×C8×Dic59—21no—
27#16837155C2214yescomplemented
28#16837152C2214yescomplemented
29#16837152C2214yescomplemented
30C2×C4⋊C8×C59C2211yescharacteristic
31C2×C4⋊C8×C59C2211yescharacteristic
32C2×C4×C8×C59C2211yescharacteristic
33C59×#335C2214yescomplemented
34C22×C4×D59C2×C411yescharacteristic
35C22×C4×D59D411yescharacteristic
36C22×C4×D59Q811yescharacteristic
37C22×C59⋊C8—21no—
38C22×C59⋊C8—21no—
39C22×C59⋊C8C2×C411yescharacteristic
40C22×C59⋊C8C2×C411yescharacteristic
41C2×C4×Dic59C2×C411yescharacteristic
42C2×C4×Dic59D411yescharacteristic
43C2×C4×Dic59Q811yescharacteristic
44C42×D59C2×C412yes—
45C42×D59D412yes—
46C42×D59Q812yes—
47C42×D59C2312yes—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)C2312yes—
51C4⋊(C59⋊C8)C2312yes—
52Dic59⋊C8C2×C414yes—
53Dic59⋊C8C2×C414yes—
54(C2×D59)⋊C8—24no—
55(C2×D59)⋊C8—24no—
56(C2×D59)⋊C8C2×C414yes—
57(C2×D59)⋊C8C2×C414yes—
58#14273637—24no—
59C4×C59⋊C8C2×C412yes—
60C4×C59⋊C8C2312yes—
61C8×Dic59—24no—
62C22×C8×C59—21no—
63C22×C8×C59—21no—
64C22×C8×C59C2×C411yescharacteristic
65C22×C8×C59C2×C411yescharacteristic
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59C2312yes—
70C4⋊C8×C59C2312yes—
71C4×C8×C59C2×C412yes—
72C4×C8×C59C2312yes—
73C23×D59C4⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×Q811yescharacteristic
77C22×Dic59C4⋊C411yescharacteristic
78C22×Dic59C4⋊C411yescharacteristic
79C22×Dic59C4⋊C411yescharacteristic
80C2×C4×D59—22no—
81C2×C4×D59—22no—
82C2×C4×D59—22no—
83C2×C4×D59—22no—
84C2×C4×D59—42no—
85C2×C4×D59C4⋊C412yes—
86C2×C4×D59C4⋊C412yes—
87C2×C4×D59C4⋊C412yes—
88C2×C4×D59C22×C412yes—
89C2×C4×D59C2×D412yes—
90C2×C4×D59C2×Q812yes—
91C2×C8×C59—22no—
92C2×C8×C59—22no—
93C2×C8×C59—22no—
94C2×C8×C59—22no—
95C2×C8×C59—42no—
96C2×C8×C59—42no—
97C2×C8×C59C22×C412yes—
98C2×C8×C59C22×C412yes—
99C42×C59C22×C411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59C2×Q811yescharacteristic
102C42×C59C2411yescharacteristic
103C4×Dic59C4⋊C412yes—
104C4×Dic59C4⋊C412yes—
105C4×Dic59C4⋊C412yes—
106C4×Dic59C22×C412yes—
107C4×Dic59C2×D412yes—
108C4×Dic59C2×Q812yes—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—42no—
114C2×C59⋊C8—42no—
115C2×C59⋊C8C22×C412yes—
116C2×C59⋊C8C22×C412yes—
117C23×C59C2×C4⋊C411yescharacteristic
118C22×D59—22no—
119C22×D59—22no—
120C22×D59—22no—
121C22×D59C2×C4⋊C412yes—
122C2×C4×C59—22no—
123C2×C4×C59C2×C4⋊C411yescharacteristic
124C2×C4×C59C2×C4⋊C411yescharacteristic
125C2×C4×C59C2×C4⋊C411yescharacteristic
126C2×C4×C59#13112yes—
127C2×C4×C59#13112yes—
128C2×C4×C59#13512yes—
129C2×C4×C59#13512yes—
130C2×C4×C59C23×C411yescharacteristic
131C2×C4×C59C22×D411yescharacteristic
132C2×C4×C59C22×Q811yescharacteristic
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59C2×C4⋊C412yes—
137C2×Dic59C2×C4⋊C412yes—
138C2×Dic59C2×C4⋊C412yes—
139C4×D59—22no—
140C4×D59—22no—
141C4×D59—22no—
142C4×D59—22no—
143C4×D59—24no—
144C4×D59—24no—
145C4×D59—42no—
146C4×D59—42no—
147C8×C59—22no—
148C8×C59—22no—
149C8×C59—42no—
150C8×C59—42no—
151C59⋊C8—22no—
152C59⋊C8—22no—
153C59⋊C8—42no—
154C59⋊C8—42no—
155C2×#1572—591nomaximal, Sylow
156C22×C59#42412yes—
157C22×C59#42712yes—
158C22×C59#43012yes—
159C22×C59C22×C4⋊C411yescharacteristic
160C2×D59—22no—
161C2×D59—22no—
162C2×D59—22no—
163C2×D59—44no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59#42412yes—
167C4×C59#42712yes—
168C4×C59C2×#13111yescharacteristic
169C4×C59C2×#13111yescharacteristic
170C4×C59C2×#13511yescharacteristic
171C4×C59C2×#13511yescharacteristic
172Dic59—22no—
173Dic59—22no—
174Dic59—22no—
175C2×#432—591no—
176C2×#431—591no—
177C2×#431—591no—
178#1572—598no—
179C2×#335—591no—
180C2×#335D5911yescharacteristic, complemented
181C2×#334—591no—
182C2×#334—591no—
183C2×C59#157214yescomplemented
184C2×C59C2×#42411yescharacteristic
185C2×C59C2×#42711yescharacteristic
186C2×C59C2×#43011yescharacteristic
187D59—42no—
188C22×C42—591no—
189#432—594no—
190#432—1182no—
191#431—594no—
192#431—594no—
193#431—594no—
194#431—1182no—
195C2×C4⋊C8—591no—
196C2×C4⋊C8—591no—
197C2×C4⋊C8—591no—
198C2×C4⋊C8—591no—
199C2×C4⋊C8C2×D5911yescharacteristic
200C2×C4⋊C8C2×D5911yescharacteristic
201C2×C22⋊C8—591no—
202C2×C22⋊C8—591no—
203C2×C22⋊C8—1181no—
204C2×C22⋊C8—1181no—
205C2×#97—1181no—
206C2×C4×C8—591no—
207C2×C4×C8—1181no—
208C2×C4×C8C2×D5911yescharacteristic
209#335—594no—
210#335C2×D5914yescomplemented
211#334—594no—
212#334—594no—
213C59C2×#157211yescharacteristic, complemented, Sylow
214C23×C4—591no—
215C23×C4—591no—
216C23×C4—591no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—591no—
220C22×C8—591no—
221C22×C8—1181no—
222C22×C8—1181no—
223C22×C8C4×D5911yescharacteristic
224C22×C8C4×D5911yescharacteristic
225C2×C42—591no—
226C2×C42—591no—
227C2×C42—591no—
228C2×C42—592no—
229C2×C42—592no—
230C2×C42—592no—
231C2×C42—592no—
232C2×C42C22×D5911yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—594no—
236C4⋊C8—594no—
237C4⋊C8—592no—
238C4⋊C8—592no—
239C4⋊C8—1181no—
240C4⋊C8—1181no—
241C4⋊C8C22×D5912yes—
242C4⋊C8C22×D5912yes—
243C22⋊C8—594no—
244C22⋊C8—594no—
245C22⋊C8—1184no—
246C22⋊C8—1184no—
247#97—1184no—
248C4×C8—592no—
249C4×C8—592no—
250C4×C8—1184no—
251C4×C8C4×D5912yes—
252C4×C8C22×D5912yes—
253C24—591no—
254C22×C4—591no—
255C22×C4—591no—
256C22×C4—591no—
257C22×C4—592no—
258C22×C4—592no—
259C22×C4—592no—
260C22×C4—592no—
261C22×C4—592no—
262C22×C4—592no—
263C22×C4—1182no—
264C22×C4—1182no—
265C22×C4—1182no—
266C22×C4—1182no—
267C22×C4—2362no—
268C22×C4C2×C4×D5911yescharacteristic
269C22×C4D4×D5911yescharacteristic
270C22×C4Q8×D5911yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—592no—
278C2×C8—592no—
279C2×C8—1182no—
280C2×C8—1182no—
281C2×C8—1182no—
282C2×C8—1182no—
283C2×C8—2362no—
284C2×C8—2362no—
285C2×C8C2×C4×D5912yes—
286C2×C8C2×C4×D5912yes—
287C42—592no—
288C42—592no—
289C42—592no—
290C42—592no—
291C42—592no—
292C42—592no—
293C42C2×C4×D5911yescharacteristic
294C42D4×D5911yescharacteristic
295C42Q8×D5911yescharacteristic
296C42C23×D5911yescharacteristic
297C23—592no—
298C23—1182no—
299C23—1182no—
300C23—1182no—
301C23C4⋊C4×D5911yescharacteristic
302C2×C4—22no—
303C2×C4—592no—
304C2×C4—592no—
305C2×C4—592no—
306C2×C4—1182no—
307C2×C4—1182no—
308C2×C4—1182no—
309C2×C4—1182no—
310C2×C4—1182no—
311C2×C4—1182no—
312C2×C4—1182no—
313C2×C4—1184no—
314C2×C4—1184no—
315C2×C4—2362no—
316C2×C4—2362no—
317C2×C4C4⋊C4×D5911yescharacteristic
318C2×C4C4⋊C4×D5911yescharacteristic
319C2×C4C4⋊C4×D5911yescharacteristic
320C2×C4#1427373312yes—
321C2×C4#1427373512yes—
322C2×C4#1427374612yes—
323C2×C4#1427375012yes—
324C2×C4C22×C4×D5911yescharacteristic
325C2×C4C2×D4×D5911yescharacteristic
326C2×C4C2×Q8×D5911yescharacteristic
327C8—22no—
328C8—22no—
329C8—42no—
330C8—42no—
331C8—1182no—
332C8—1182no—
333C8—2362no—
334C8—2362no—
335C22—1182no—
336C22—1182no—
337C22—1182no—
338C22—2364no—
339C22#1683742712yes—
340C22#1683743912yes—
341C22#1683744912yes—
342C22C2×C4⋊C4×D5911yescharacteristic
343C4—21no—
344C4—21no—
345C4—1182no—
346C4—1182no—
347C4—1182no—
348C4#1683742712yes—
349C4#1683743912yes—
350C4C2×#1427373311yescharacteristic
351C4C2×#1427373511yescharacteristic
352C4C2×#1427374611yescharacteristic
353C4C2×#1427375011yescharacteristic
354C2—2362no—
355C2#2142287514yesdirect product factor
356C2C2×#1683742711yescharacteristic
357C2C2×#1683743911yescharacteristic
358C2C2×#1683744911yescharacteristic
subgroupG/H
centerC22×C4D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC2×C59×#335C2
socleC23×C59C2×C4⋊C4
prime radicalC23×D59C4⋊C4
cosocleC42×C59C24
hypercenterC2×#335D59
hypocenterC59C2×#1572
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
4. #21422875C2direct (N × H)
5. C2×#16837155C2semidirect
6. C2×#16837152C2semidirect
7. C2×#16837152C2semidirect
8. C2×C59×#335C2semidirect
27. #16837155C22semidirect
28. #16837152C22semidirect
29. #16837152C22semidirect
33. C59×#335C22semidirect
180. C2×#335D59semidirect
183. C2×C59#1572semidirect
210. #335C2×D59semidirect
213. C59C2×#1572semidirect
355. C2#21422875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168374582
C2×#168374542
C2×#168374532
#214228752
C2×#168371552
C2×#168371522
C2×C59×#3352
C2×#157259