Finite Group Databasebeta

← C2×#21423668 #28571259

Subgroups — C2×#21423668 #28571259

Order 15104.

694 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic
2C2×#16837679C211yesmaximal, characteristic
3C2×#16837481C211yesmaximal, characteristic
4#21423668C218yesmaximal, direct product factor
5C2×#16837153C211yesmaximal, characteristic, complemented
6C2×#16837151C211yesmaximal, characteristic, complemented
7C2×#16837148C211yesmaximal, characteristic, complemented
8C2×C59×#487C211yesmaximal, characteristic, complemented
9C22×#11745659—21no—
10C2×C4×D236C2211yescharacteristic
11#16837689—22no—
12#16837689C2214yes—
13#16837679—22no—
14#16837679C2214yes—
15#16837481—22no—
16#16837481C2214yes—
17C2×#14273660—21no—
18C2×#14273658—21no—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273647—21no—
23C2×#14273647—21no—
24C2×(C2×D59)⋊C8—21no—
25#16837153C2214yescomplemented
26#16837151C2214yescomplemented
27#16837148C2214yescomplemented
28C2×C4⋊Q8×C59C2211yescharacteristic
29C2×C4⋊Q8×C59C2211yescharacteristic
30C59×#487—22no—
31C59×#487C2214yescomplemented
32C2×#103×C59—21no—
33C2×#103×C59—21no—
34C2×C4×C8×C59C2211yescharacteristic
35C22×D236D411yescharacteristic
36C22×C4×D59—21no—
37C2×#11745659—24no—
38C2×#11745659—44no—
39C22×C59⋊C8—21no—
40C22×C59⋊C8—21no—
41C22×C59⋊C8—21no—
42C4×D236—21no—
43C4×D236—21no—
44C4×D236D412yes—
45C4×D236C2312yes—
46C2×C4⋊Dic59D411yescharacteristic
47C2×(C2×D59)⋊C4—21no—
48#14273660—24no—
49#14273658—24no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)C2312yes—
54C4⋊(C59⋊C8)C2312yes—
55C4⋊(C59⋊C8)C2312yes—
56#14273647—24no—
57#14273647—24no—
58(C2×D59)⋊C8—24no—
59C22×Q8×C59—21no—
60C22×Q8×C59—21no—
61C22×C8×C59D411yescharacteristic
62C22×C8×C59D411yescharacteristic
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59—21no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59C2312yes—
70C4⋊Q8×C59C2312yes—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59—21no—
74C2×C4⋊C4×C59—21no—
75C2×C42×C59C2311yescharacteristic
76#103×C59—24no—
77#103×C59—24no—
78C4×C8×C59D412yes—
79C4×C8×C59C2312yes—
80C23×D59—21no—
81C2×Q8×C59—22no—
82C2×Q8×C59—22no—
83C2×Q8×C59—41no—
84C2×Q8×C59—41no—
85C2×Q8×C59—42no—
86C2×Q8×C59—42no—
87C2×Q8×C59—82no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C22×Dic59—21no—
94C2×D236—22no—
95C2×D236—22no—
96C2×D236—21no—
97C2×D236C2×D412yes—
98C2×C4×D59—22no—
99C2×C4×D59—22no—
100C2×C4×D59—42no—
101C2×C4×D59—42no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59C2×D412yes—
105C2×C8×C59C2×D412yes—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C4⋊C4×C59—41no—
114C42×C59C2×D411yescharacteristic
115C42×C59C2×D411yescharacteristic
116C42×C59#4111yescharacteristic
117C42×C59C2411yescharacteristic
118(C2×D59)⋊C4—22no—
119(C2×D59)⋊C4—22no—
120C4⋊Dic59—21no—
121C4⋊Dic59C2×D412yes—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—42no—
126C2×C59⋊C8—42no—
127C2×C59⋊C8—42no—
128#11745659—48no—
129C23×C59#12111yescharacteristic
130C22×D59—22no—
131C22×D59—42no—
132C22×D59—42no—
133C22×D59—42no—
134Q8×C59—42no—
135Q8×C59—42no—
136Q8×C59—82no—
137Q8×C59—82no—
138Q8×C59—82no—
139C2×C4×C59—22no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—82no—
143C2×C4×C59—82no—
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#13512yes—
148C2×C4×C59AGL(1,ℤ/8)12yes—
149C2×C4×C59#13712yes—
150C2×C4×C59#13712yes—
151C2×C4×C59C22×D411yescharacteristic
152C2×C4×C59C22×D411yescharacteristic
153C2×C4×C59C2×#4111yescharacteristic
154C2×Dic59—22no—
155C2×Dic59—42no—
156D236—22no—
157D236—22no—
158D236—42no—
159C4×D59—42no—
160C4×D59—42no—
161C4×D59—44no—
162C8×C59—22no—
163C8×C59—22no—
164C59⋊C8—42no—
165C59⋊C8—42no—
166C59⋊C8—42no—
167C2×#1703—591nomaximal, Sylow
168C22×C59#46312yes—
169C22×C59#47012yes—
170C22×C59#47112yes—
171C22×C59C2×#12111yescharacteristic
172C2×D59—42no—
173C2×D59—42no—
174C2×D59—42no—
175C2×D59—84no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—82no—
179C4×C59—82no—
180C4×C59#46312yes—
181C4×C59#47012yes—
182C4×C59C2×#13511yescharacteristic
183C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
184C4×C59C2×#13711yescharacteristic
185C4×C59C2×#13711yescharacteristic
186Dic59—42no—
187C2×#487D5911yescharacteristic, complemented
188C2×#476—591no—
189C2×#474—591no—
190C2×#436—591no—
191#1703—598no—
192C2×#334—591no—
193C2×#333—591no—
194C2×#332—591no—
195C2×C59#170314yescomplemented
196C2×C59C2×#46311yescharacteristic
197C2×C59C2×#47011yescharacteristic
198C2×C59C2×#47111yescharacteristic
199D59—82no—
200C22×M16—1181no—
201C2×C4⋊Q8C2×D5911yescharacteristic
202C2×C4⋊Q8C2×D5911yescharacteristic
203C2×C4×D4—591no—
204#487—22no—
205#487C2×D5914yescomplemented
206#476—594no—
207#476—1182no—
208#474—594no—
209#474—1182no—
210#436—594no—
211#436—1182no—
212C2×#107—1181no—
213C2×#106—1181no—
214C2×C4⋊C8—591no—
215C2×C4⋊C8—591no—
216C2×C4⋊C8—591no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#102—1181no—
220C2×#102—1181no—
221C2×C22⋊C8—1181no—
222C2×C4×C8C2×D5911yescharacteristic
223#334—594no—
224#333—594no—
225#332—594no—
226C59C2×#170311yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—591no—
230C23×C4—1181no—
231C2×M16—1184no—
232C2×M16—2364no—
233C22×C8—1181no—
234C22×C8—1181no—
235C22×C8—1181no—
236C22×C8C22⋊D5911yescharacteristic
237C22×C8C22⋊D5911yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22×D5912yes—
245C4⋊Q8C22×D5912yes—
246C4×D4—592no—
247C4×D4—592no—
248C4×D4—1181no—
249C4×D4—1181no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—591no—
255C2×C22⋊C4—1181no—
256C2×C42C22×D5911yescharacteristic
257#107—1184no—
258#106—1184no—
259C4⋊C8—592no—
260C4⋊C8—592no—
261C4⋊C8—592no—
262C4⋊C8—1181no—
263C4⋊C8—1181no—
264C4⋊C8—1181no—
265#103—24no—
266#103—24no—
267#102—1184no—
268#102—1184no—
269C22⋊C8—1184no—
270C4×C8C22⋊D5912yes—
271C4×C8C22×D5912yes—
272C24—1181no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×D4—592no—
281C2×D4—1181no—
282C2×D4—1182no—
283C2×D4—1182no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—1181no—
287C22×C4—1182no—
288C22×C4—1182no—
289C22×C4—2362no—
290C22×C4—2362no—
291C22×C4D4×D5911yescharacteristic
292C22×C4#1174568711yescharacteristic
293C22×C4C2×C22⋊D5911yescharacteristic
294M16—2368no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—1182no—
298C2×C8—1182no—
299C2×C8—1182no—
300C2×C8—2362no—
301C2×C8—2362no—
302C2×C8—2362no—
303C2×C8C2×C22⋊D5912yes—
304C2×C8C2×C22⋊D5912yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—592no—
314C4⋊C4—1181no—
315C22⋊C4—1182no—
316C22⋊C4—1182no—
317C42D4×D5911yescharacteristic
318C42#1174568711yescharacteristic
319C42C2×C22⋊D5911yescharacteristic
320C42C23×D5911yescharacteristic
321C23—1182no—
322C23—2362no—
323C23—2362no—
324C23—2362no—
325C23#1427369811yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—82no—
329Q8—82no—
330Q8—82no—
331D4—1182no—
332D4—1182no—
333D4—2362no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—1182no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4—2364no—
344C2×C4#1427369811yescharacteristic
345C2×C4#1427369811yescharacteristic
346C2×C4#1427369811yescharacteristic
347C2×C4#1427374712yes—
348C2×C4#1427375212yes—
349C2×C4#1427375912yes—
350C2×C4#1427375912yes—
351C2×C4C2×D4×D5911yescharacteristic
352C2×C4C22×C22⋊D5911yescharacteristic
353C2×C4C2×#1174568711yescharacteristic
354C8—22no—
355C8—22no—
356C8—2362no—
357C8—2362no—
358C8—2362no—
359C22—4724no—
360C22—2362no—
361C22—2362no—
362C22—2362no—
363C22#1683761412yes—
364C22#1683765712yes—
365C22#1683766212yes—
366C22C2×#1427369811yescharacteristic
367C4—21no—
368C4—21no—
369C4—82no—
370C4—82no—
371C4—2362no—
372C4#1683761412yes—
373C4#1683765712yes—
374C4C2×#1427374711yescharacteristic
375C4C2×#1427375211yescharacteristic
376C4C2×#1427375911yescharacteristic
377C4C2×#1427375911yescharacteristic
378C2—4722no—
379C2#2142366814yesdirect product factor
380C2C2×#1683761411yescharacteristic
381C2C2×#1683765711yescharacteristic
382C2C2×#1683766211yescharacteristic
subgroupG/H
centerC23#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#487C2
socleC23×C59#121
prime radicalC22×D236D4
hypercenterC2×#487D59
hypocenterC59C2×#1703
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #21423668C2direct (N × H)
5. C2×#16837153C2semidirect
6. C2×#16837151C2semidirect
7. C2×#16837148C2semidirect
8. C2×C59×#487C2semidirect
25. #16837153C22semidirect
26. #16837151C22semidirect
27. #16837148C22semidirect
31. C59×#487C22semidirect
187. C2×#487D59semidirect
195. C2×C59#1703semidirect
205. #487C2×D59semidirect
226. C59C2×#1703semidirect
379. C2#21423668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168376892
C2×#168376792
C2×#168374812
#214236682
C2×#168371532
C2×#168371512
C2×#168371482
C2×C59×#4872
C2×#170359