Finite Group Databasebeta

← C2×#21437086 #29469600

Subgroups — C2×#21437086 #29469600

Order 15104.

1958 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838126C212yesmaximal, complemented
2#21437086C2116yesmaximal, direct product factor
3C2×#16837787C212yesmaximal
4C2×#16837626C214yesmaximal, complemented
5C2×#16837604C212yesmaximal, complemented
6C2×C8×Dic118C211yesmaximal, characteristic, complemented
7C2×Q16×Dic59C212yesmaximal, complemented
8C2×C4×D472C211yesmaximal, characteristic
9C2×C59×#494C211yesmaximal, characteristic, complemented
10C22×#11745671C2214yes—
11C22×D472C2211yescharacteristic
12#16838126—28no—
13#16838126C22116yescomplemented
14C2×#14273722—24no—
15C2×#14273716C2214yes—
16C2×#14273707—22no—
17C2×Q8×Dic59C2214yes—
18C2×C4×Dic118C2211yescharacteristic
19C2×C4×D236C2212yes—
20C2×#14273685—22no—
21#16837787C22116yes—
22#16837626C22132yescomplemented
23#16837604C22116yescomplemented
24C8×Dic118C2218yescomplemented
25Q16×Dic59C22116yescomplemented
26C4×D472C2218yes—
27C2×#14273661C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×Dic59⋊C8C2212yes—
30C2×#14273651C2214yes—
31C2×#14273649C2212yes—
32C2×C8×Dic59C2212yes—
33C22×Q16×C59C2212yes—
34C22×Q16×C59C2212yes—
35C2×C4⋊Q8×C59C2212yes—
36C59×#494C2218yescomplemented
37C2×C4×C8×C59C2211yescharacteristic
38C22×Dic118D411yescharacteristic
39C22×D236C2312yes—
40C22×C4×D59—22no—
41C2×#11745671—48no—
42C2×#11745671C23116yes—
43C2×D472—24no—
44C2×D472C2314yes—
45C22×C59⋊C8C2312yes—
46#14273722—232no—
47#14273716—28no—
48#14273716C23116yes—
49#14273707—216no—
50Q8×Dic59—28no—
51Q8×Dic59C23116yes—
52C4×Dic118D414yes—
53C4×Dic118C2314yes—
54C4×D236—24no—
55C4×D236C2318yes—
56#14273685—216no—
57C2×C4⋊Dic59—24no—
58C2×C4⋊Dic59D411yescharacteristic
59C2×C4⋊Dic59C2312yes—
60C2×Dic59⋊C4D412yes—
61C2×(C2×D59)⋊C4—22no—
62C2×(C2×D59)⋊C4—24no—
63C2×(C2×D59)⋊C4—24no—
64C2×C4×Dic59—24no—
65C2×C4×Dic59C2312yes—
66#14273661C2314yes—
67C4⋊(C59⋊C8)C2314yes—
68Dic59⋊C8C2318yes—
69#14273651C23116yes—
70#14273649C2318yes—
71C8×Dic59C2318yes—
72C22×Q8×C59C2314yes—
73C2×Q16×C59—28no—
74C2×Q16×C59—44no—
75C2×Q16×C59C2318yes—
76C2×Q16×C59C2318yes—
77C22×C8×C59C2311yescharacteristic
78C22×C8×C59C2311yescharacteristic
79C4⋊Q8×C59—24no—
80C4⋊Q8×C59C2318yes—
81C2×C4⋊C4×C59—22no—
82C2×C4⋊C4×C59—22no—
83C2×C42×C59C2311yescharacteristic
84C4×C8×C59C2314yes—
85C23×D59—22no—
86C2×Q8×C59—24no—
87C2×Q8×C59—28no—
88C2×Q8×C59—28no—
89C2×Q8×C59C2418yes—
90C22×C4×C59—24no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2411yescharacteristic
94C22×Dic59—22no—
95C22×Dic59C2×D412yes—
96C22×Dic59C2×D412yes—
97C2×D236—24no—
98C2×D236—24no—
99C2×D236—22no—
100C2×D236C2414yes—
101C2×C4×D59—28no—
102C2×C4×D59—48no—
103C2×Dic118—24no—
104C2×Dic118C2×D414yes—
105Q16×C59—216no—
106Q16×C59—48no—
107C2×C8×C59—22no—
108C2×C8×C59#4114yes—
109C2×C8×C59C2412yes—
110C2×C8×C59C2412yes—
111C4⋊C4×C59—28no—
112C4⋊C4×C59—28no—
113C42×C59C2×D412yes—
114C42×C59C2412yes—
115#11745671—416no—
116(C2×D59)⋊C4—28no—
117(C2×D59)⋊C4—216no—
118(C2×D59)⋊C4—216no—
119C4⋊Dic59—22no—
120C4⋊Dic59—216no—
121C4⋊Dic59C2×D414yes—
122C4⋊Dic59C2414yes—
123Dic59⋊C4C2×D418yes—
124C4×Dic59—216no—
125C4×Dic59C2×D414yes—
126C4×Dic59C2414yes—
127C2×C59⋊C8—24no—
128C2×C59⋊C8C2414yes—
129D472—28no—
130C23×C59C22×D411yescharacteristic
131C22×D59—24no—
132C22×D59—44no—
133C22×D59—44no—
134C22×D59—44no—
135Q8×C59—28no—
136Q8×C59—28no—
137Q8×C59—48no—
138C2×C4×C59—22no—
139C2×C4×C59—28no—
140C2×C4×C59—48no—
141C2×C4×C59#13514yes—
142C2×C4×C59C22×D412yes—
143C2×C4×C59C22×D411yescharacteristic
144C2×C4×C59C22×D412yes—
145C2×C4×C59C2×#4112yes—
146C2×C4×C59H(5,𝔽2)12yes—
147C2×C4×C59C2511yescharacteristic
148C2×Dic59—24no—
149C2×Dic59—24no—
150C2×Dic59—24no—
151C2×Dic59—44no—
152C2×Dic59C22×D414yes—
153C2×Dic59C22×D414yes—
154D236—24no—
155D236—24no—
156D236—44no—
157C4×D59—416no—
158Dic118—28no—
159C8×C59—22no—
160C8×C59C2×#4114yes—
161C59⋊C8—24no—
162C2×#3393—591nomaximal, Sylow
163C22×C59#55012yes—
164C22×C59C2×#13512yes—
165C22×C59#57812yes—
166C22×C59C23×D411yescharacteristic
167C2×D59—44no—
168C2×D59—44no—
169C2×D59—44no—
170C2×D59—88no—
171C4×C59—22no—
172C4×C59—48no—
173C4×C59#55012yes—
174C4×C59C2×#13514yes—
175C4×C59C22×#4111yescharacteristic
176C4×C59C2×H(5,𝔽2)11yescharacteristic
177Dic59—24no—
178Dic59—24no—
179Dic59—44no—
180C2×#556—592no—
181#3393—5916no—
182C2×#495—592no—
183C2×#494D5911yescharacteristic, complemented
184C2×#464—594no—
185C2×#461—592no—
186C2×C8×Q8—591no—
187C2×C4×Q16—592no—
188C2×C4×D8—591no—
189C2×C59#339314yescomplemented
190C2×C59C2×#55011yescharacteristic
191C2×C59C22×#13511yescharacteristic
192C2×C59C2×#57811yescharacteristic
193D59—84no—
194C22×Q16C2×D5912yes—
195C22×Q16C2×D5912yes—
196C22×SD16—594no—
197C22×D8—591no—
198#556—5916no—
199#556—1188no—
200C2×C4⋊Q8C2×D5912yes—
201C2×#126—1184no—
202C2×#124—594no—
203C2×C22⋊Q8—1182no—
204C2×C4×Q8—591no—
205C2×C4×Q8—594no—
206C2×C4×D4—592no—
207C2×#117—1182no—
208#495—5916no—
209#494C2×D5918yescomplemented
210#464—5932no—
211#461—5916no—
212C8×Q8—598no—
213C4×Q16—5916no—
214C4×D8—598no—
215C2×#107—591no—
216C2×C4⋊C8—592no—
217C2×C4⋊C8—591no—
218C2×#103—594no—
219C2×#102—592no—
220C2×C4×C8—592no—
221C2×C4×C8C2×D5911yescharacteristic
222C59C2×#339311yescharacteristic, complemented, Sylow
223C22×Q8—591no—
224C22×Q8C22×D5914yes—
225C22×D4—592no—
226C23×C4—1182no—
227C2×Q16—28no—
228C2×Q16—44no—
229C2×Q16C22×D5918yes—
230C2×Q16C22×D5918yes—
231C2×SD16—5916no—
232C2×SD16—2368no—
233C2×D8—594no—
234C2×D8—1184no—
235C22×C8—592no—
236C22×C8C22×D5911yescharacteristic
237C22×C8C22×D5911yescharacteristic
238C4⋊Q8—24no—
239C4⋊Q8C22×D5918yes—
240#126—11832no—
241#124—5916no—
242#124—1188no—
243C22⋊Q8—11816no—
244C4×Q8—594no—
245C4×Q8—594no—
246C4×Q8—5916no—
247C4×Q8—1188no—
248C4×D4—598no—
249C4×D4—1184no—
250#117—11816no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—591no—
254C2×C4⋊C4—592no—
255C2×C4⋊C4—592no—
256C2×C4⋊C4—1184no—
257C2×C22⋊C4—1184no—
258C2×C22⋊C4—1184no—
259C2×C22⋊C4—1182no—
260C2×C42—592no—
261C2×C42—1184no—
262C2×C42C22×D5911yescharacteristic
263#107—594no—
264C4⋊C8—598no—
265C4⋊C8—594no—
266#103—5916no—
267#102—598no—
268C4×C8—598no—
269C4×C8C22×D5914yes—
270C24—1182no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8—594no—
275C2×Q8—1184no—
276C2×Q8C23×D5918yes—
277C2×D4—594no—
278C2×D4—1182no—
279C2×D4—1184no—
280C2×D4—1184no—
281C22×C4—24no—
282C22×C4—592no—
283C22×C4—592no—
284C22×C4—1182no—
285C22×C4—1188no—
286C22×C4—2368no—
287C22×C4D4×D5911yescharacteristic
288C22×C4D4×D5911yescharacteristic
289C22×C4C23×D5911yescharacteristic
290Q16—216no—
291Q16—48no—
292SD16—23616no—
293D8—1188no—
294C2×C8—22no—
295C2×C8—594no—
296C2×C8—1184no—
297C2×C8#1174568714yes—
298C2×C8C23×D5912yes—
299C2×C8C23×D5912yes—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—594no—
303C4⋊C4—598no—
304C4⋊C4—594no—
305C4⋊C4—11816no—
306C4⋊C4—1182no—
307C22⋊C4—11816no—
308C22⋊C4—11816no—
309C22⋊C4—1188no—
310C42—594no—
311C42—594no—
312C42—11816no—
313C42D4×D5912yes—
314C42C23×D5912yes—
315C23—1184no—
316C23—2364no—
317C23—2364no—
318C23—2364no—
319C23C2×D4×D5911yescharacteristic
320Q8—28no—
321Q8—28no—
322Q8—48no—
323Q8—1188no—
324D4—1184no—
325D4—1184no—
326D4—2364no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—48no—
330C2×C4—594no—
331C2×C4—594no—
332C2×C4—1184no—
333C2×C4—1184no—
334C2×C4—1184no—
335C2×C4—2364no—
336C2×C4—23616no—
337C2×C4#1427375014yes—
338C2×C4C2×D4×D5912yes—
339C2×C4C2×D4×D5912yes—
340C2×C4C2×D4×D5911yescharacteristic
341C2×C4C2×#1174568712yes—
342C2×C4#1427377212yes—
343C2×C4C24×D5911yescharacteristic
344C8—22no—
345C8—1184no—
346C8C2×#1174568714yes—
347C22—4728no—
348C22—2364no—
349C22—2364no—
350C22—2364no—
351C22#1683807212yes—
352C22C2×#1427375012yes—
353C22#1683821512yes—
354C22C22×D4×D5911yescharacteristic
355C4—22no—
356C4—48no—
357C4—1184no—
358C4—1184no—
359C4—2364no—
360C4#1683807212yes—
361C4C2×#1427375014yes—
362C4C22×#1174568711yescharacteristic
363C4C2×#1427377211yescharacteristic
364C2—4724no—
365C2#2143708614yesdirect product factor
366C2C2×#1683807211yescharacteristic
367C2C22×#1427375011yescharacteristic
368C2C2×#1683821511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C59×#494C2
socleC23×C59C22×D4
prime radicalC22×D472C22
hypercenterC2×#494D59
hypocenterC59C2×#3393
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C2×#16838126C2semidirect
2. #21437086C2direct (N × H)
4. C2×#16837626C2semidirect
5. C2×#16837604C2semidirect
6. C2×C8×Dic118C2semidirect
7. C2×Q16×Dic59C2semidirect
9. C2×C59×#494C2semidirect
13. #16838126C22semidirect
22. #16837626C22semidirect
23. #16837604C22semidirect
24. C8×Dic118C22semidirect
25. Q16×Dic59C22semidirect
36. C59×#494C22semidirect
183. C2×#494D59semidirect
189. C2×C59#3393semidirect
209. #494C2×D59semidirect
222. C59C2×#3393semidirect
365. C2#21437086direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168381262
#214370862
C2×#168377872
C2×#168376262
C2×#168376042
C2×C8×Dic1182
C2×Q16×Dic592
C2×C4×D4722
C2×C59×#4942
C2×#339359