Finite Group Databasebeta

← C2×#21423547 #28570990

Subgroups — C2×#21423547 #28570990

Order 15104.

726 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×#16837753C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic
3C2×#16837682C211yesmaximal, characteristic
4#21423547C218yesmaximal, direct product factor
5C2×#16837155C211yesmaximal, characteristic, complemented
6C2×#16837150C211yesmaximal, characteristic, complemented
7C2×#16837149C211yesmaximal, characteristic, complemented
8C2×C8×D4×C59C211yesmaximal, characteristic
9C2×#14273727C2211yescharacteristic
10C2×#14273727C2211yescharacteristic
11#16837753—22no—
12#16837753C2214yescomplemented
13#16837692—22no—
14#16837692C2214yes—
15#16837682—22no—
16#16837682C2214yes—
17C2×#14273660—21no—
18C2×#14273658—21no—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273652—21no—
22C2×#14273652—21no—
23C2×#14273648—21no—
24C2×#14273648—21no—
25#16837155C2214yescomplemented
26#16837150C2214yescomplemented
27#16837149C2214yescomplemented
28C23×C8×C59—21no—
29C2×C4×D4×C59C2211yescharacteristic
30C8×D4×C59—22no—
31C8×D4×C59C2214yes—
32C2×C4⋊C8×C59C2211yescharacteristic
33C2×C22⋊C8×C59—21no—
34C2×C4×C8×C59C2211yescharacteristic
35C22×Dic118—21no—
36C22×Dic118—21no—
37C22×C59⋊C8—21no—
38C22×C59⋊C8—21no—
39#14273727—21no—
40#14273727—21no—
41#14273727—21no—
42#14273727—21no—
43#14273727—21no—
44#14273727—21no—
45#14273727C2312yes—
46#14273727C2312yes—
47C2×C4⋊Dic59—21no—
48C2×C4⋊Dic59—21no—
49C2×C4⋊Dic59—21no—
50C2×C4⋊Dic59—21no—
51#14273660—24no—
52#14273658—24no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)—21no—
55C4⋊(C59⋊C8)C2312yes—
56C4⋊(C59⋊C8)C2312yes—
57#14273652—24no—
58#14273652—24no—
59#14273648—24no—
60#14273648—24no—
61C22×D4×C59D411yescharacteristic
62C23×C4×C59—21no—
63C22×C8×C59—21no—
64C22×C8×C59—24no—
65C22×C8×C59—44no—
66C22×C8×C59D411yescharacteristic
67C22×C8×C59D411yescharacteristic
68C4×D4×C59—21no—
69C4×D4×C59—21no—
70C4×D4×C59D412yes—
71C4×D4×C59C2312yes—
72C2×C4⋊C4×C59D411yescharacteristic
73C2×C22⋊C4×C59—21no—
74C2×C42×C59C2311yescharacteristic
75C4⋊C8×C59—21no—
76C4⋊C8×C59C2312yes—
77C22⋊C8×C59—24no—
78C4×C8×C59D412yes—
79C4×C8×C59C2312yes—
80C24×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—22no—
83C2×D4×C59Q1614yes—
84C2×D4×C59C2×D412yes—
85C22×C4×C59—21no—
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C22×Dic59—41no—
94C22×Dic59—41no—
95C2×Dic118—22no—
96C2×Dic118—22no—
97C2×Dic118—42no—
98C2×Dic118—42no—
99C2×Dic118—41no—
100C2×Dic118—41no—
101C2×Dic118—82no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59—22no—
105C2×C8×C59—48no—
106C2×C8×C59—42no—
107C2×C8×C59C2×D412yes—
108C2×C8×C59C2×D412yes—
109C4⋊C4×C59—21no—
110C4⋊C4×C59C2×D412yes—
111C22⋊C4×C59—22no—
112C22⋊C4×C59—22no—
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C42×C59#4111yescharacteristic
116C42×C59C2411yescharacteristic
117C4⋊Dic59—22no—
118C4⋊Dic59—22no—
119C4⋊Dic59—22no—
120C4⋊Dic59—22no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123C4⋊Dic59—41no—
124C4⋊Dic59—41no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—42no—
128C2×C59⋊C8—42no—
129C23×C59—22no—
130C23×C59—24no—
131C23×C59—42no—
132C23×C59—42no—
133C23×C59#12111yescharacteristic
134D4×C59—22no—
135D4×C59—24no—
136D4×C59C2×Q1614yes—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—44no—
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59C2×Q1612yes—
147C2×C4×C59#13512yes—
148C2×C4×C59#13512yes—
149C2×C4×C59AGL(1,ℤ/8)12yes—
150C2×C4×C59C22×D411yescharacteristic
151C2×C4×C59C22×D411yescharacteristic
152C2×C4×C59C2×#4111yescharacteristic
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—82no—
156C2×Dic59—82no—
157Dic118—42no—
158Dic118—42no—
159Dic118—82no—
160Dic118—82no—
161Dic118—82no—
162C8×C59—22no—
163C8×C59—22no—
164C8×C59—42no—
165C59⋊C8—42no—
166C59⋊C8—42no—
167C2×#1683—591nomaximal, Sylow
168C22×C59—24no—
169C22×C59—48no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59#46312yes—
173C22×C59#46712yes—
174C22×C59#47112yes—
175C22×C59C2×#12111yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—42no—
179C4×C59#46312yes—
180C4×C59#46712yes—
181C4×C59C22×Q1611yescharacteristic
182C4×C59C2×#13511yescharacteristic
183C4×C59C2×#13511yescharacteristic
184C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
185Dic59—82no—
186Dic59—82no—
187C2×#487—591no—
188C2×#476—591no—
189C2×#474—591no—
190C2×C8×D4D5911yescharacteristic
191#1683—598no—
192C2×#335—591no—
193C2×#333—591no—
194C2×#332—591no—
195C2×C59—44no—
196C2×C59#168314yescomplemented
197C2×C59C2×#46311yescharacteristic
198C2×C59C2×#46711yescharacteristic
199C2×C59C2×#47111yescharacteristic
200C23×C8—21no—
201C2×C4⋊Q8—591no—
202C2×C4⋊Q8—591no—
203C2×C4×D4C2×D5911yescharacteristic
204#487—594no—
205#487—1182no—
206#476—594no—
207#476—1182no—
208#474—594no—
209#474—1182no—
210C8×D4—22no—
211C8×D4C2×D5914yes—
212C2×#107—1181no—
213C2×#106—1181no—
214C2×C4⋊C8—591no—
215C2×C4⋊C8—591no—
216C2×C4⋊C8C2×D5911yescharacteristic
217C2×#103—1181no—
218C2×#103—1181no—
219C2×#102—1181no—
220C2×#102—1181no—
221C2×C22⋊C8—21no—
222C2×C4×C8C2×D5911yescharacteristic
223#335—594no—
224#333—594no—
225#332—594no—
226C59C2×#168311yescharacteristic, complemented, Sylow
227C22×Q8—1181no—
228C22×Q8—1181no—
229C22×D4D23611yescharacteristic
230C23×C4—21no—
231C22×C8—21no—
232C22×C8—24no—
233C22×C8—44no—
234C22×C8—1181no—
235C22×C8—1181no—
236C22×C8C22⋊D5911yescharacteristic
237C22×C8C22⋊D5911yescharacteristic
238C4⋊Q8—592no—
239C4⋊Q8—592no—
240C4⋊Q8—1181no—
241C4⋊Q8—1181no—
242C4⋊Q8—1181no—
243C4⋊Q8—1181no—
244C4⋊Q8—1181no—
245C4⋊Q8—1181no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4D23612yes—
249C4×D4C22×D5912yes—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4D23611yescharacteristic
255C2×C22⋊C4—21no—
256C2×C42C22×D5911yescharacteristic
257#107—1184no—
258#106—1184no—
259C4⋊C8—21no—
260C4⋊C8—592no—
261C4⋊C8—592no—
262C4⋊C8—1181no—
263C4⋊C8—1181no—
264C4⋊C8C22×D5912yes—
265#103—1184no—
266#103—1184no—
267#102—1184no—
268#102—1184no—
269C22⋊C8—24no—
270C4×C8C22⋊D5912yes—
271C4×C8C22×D5912yes—
272C24—21no—
273C2×Q8—1182no—
274C2×Q8—1182no—
275C2×Q8—4722no—
276C2×Q8—2362no—
277C2×Q8—2362no—
278C2×Q8—2361no—
279C2×Q8—2361no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4Dic23614yes—
283C2×D4C2×D23612yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—2361no—
290C22×C4—2361no—
291C22×C4C2×D23611yescharacteristic
292C22×C4#1174568511yescharacteristic
293C22×C4C2×C22⋊D5911yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—48no—
298C2×C8—42no—
299C2×C8—1182no—
300C2×C8—1182no—
301C2×C8—2362no—
302C2×C8—2362no—
303C2×C8C2×C22⋊D5912yes—
304C2×C8C2×C22⋊D5912yes—
305C4⋊C4—21no—
306C4⋊C4—1182no—
307C4⋊C4—1182no—
308C4⋊C4—1182no—
309C4⋊C4—1182no—
310C4⋊C4—2361no—
311C4⋊C4—2361no—
312C4⋊C4—2361no—
313C4⋊C4—2361no—
314C4⋊C4C2×D23612yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C42C2×D23611yescharacteristic
318C42#1174568511yescharacteristic
319C42C2×C22⋊D5911yescharacteristic
320C42C23×D5911yescharacteristic
321C23—22no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23#1427370111yescharacteristic
326Q8—4722no—
327Q8—4722no—
328Q8—4722no—
329Q8—2362no—
330Q8—2362no—
331D4—24no—
332D4—22no—
333D4C2×Dic23614yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—4722no—
341C2×C4—4722no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4#1427370111yescharacteristic
345C2×C4#1427370111yescharacteristic
346C2×C4#1427370111yescharacteristic
347C2×C4C2×Dic23612yes—
348C2×C4#1427374712yes—
349C2×C4#1427374912yes—
350C2×C4#1427375312yes—
351C2×C4C22×C22⋊D5911yescharacteristic
352C2×C4C22×D23611yescharacteristic
353C2×C4C2×#1174568511yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—2362no—
358C8—2362no—
359C22—24no—
360C22—42no—
361C22—48no—
362C22—42no—
363C22#1683761712yes—
364C22#1683764312yes—
365C22#1683766512yes—
366C22C2×#1427370111yescharacteristic
367C4—21no—
368C4—21no—
369C4—42no—
370C4—4722no—
371C4—4722no—
372C4#1683761712yes—
373C4#1683764312yes—
374C4C22×Dic23611yescharacteristic
375C4C2×#1427374711yescharacteristic
376C4C2×#1427374911yescharacteristic
377C4C2×#1427375311yescharacteristic
378C2—44no—
379C2#2142354714yesdirect product factor
380C2C2×#1683761711yescharacteristic
381C2C2×#1683764311yescharacteristic
382C2C2×#1683766511yescharacteristic
subgroupG/H
centerC23#14273701
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C8×D4×C59C2
socleC23×C59#121
prime radicalC22×D4×C59D4
hypercenterC2×C8×D4D59
hypocenterC59C2×#1683
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16837753C2semidirect
4. #21423547C2direct (N × H)
5. C2×#16837155C2semidirect
6. C2×#16837150C2semidirect
7. C2×#16837149C2semidirect
12. #16837753C22semidirect
25. #16837155C22semidirect
26. #16837150C22semidirect
27. #16837149C22semidirect
196. C2×C59#1683semidirect
226. C59C2×#1683semidirect
379. C2#21423547direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377532
C2×#168376922
C2×#168376822
#214235472
C2×#168371552
C2×#168371502
C2×#168371492
C2×C8×D4×C592
C2×#168359