Finite Group Databasebeta

← C2×#21422821 #28557225

Subgroups — C2×#21422821 #28557225

Order 15104.

1182 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273727C211yesmaximal, characteristic
2C2×#16837459C211yesmaximal, characteristic
3#21422821C218yesmaximal, direct product factor
4C2×#16837150C214yesmaximal, complemented
5C2×C59×#432C211yesmaximal, characteristic
6C23×Dic118C412yes—
7C2×#14273727—22no—
8C2×#14273727—21no—
9C2×#14273727—21no—
10C2×#14273727—22no—
11C2×#14273727C414yes—
12C2×#14273727C412yes—
13C2×#14273727C2212yes—
14C2×#14273727C2214yes—
15C22×C4⋊Dic59—21no—
16C22×C4⋊Dic59—21no—
17#16837459—22no—
18#16837459C2214yes—
19C2×C4⋊(C59⋊C8)C2212yes—
20C2×C22⋊(C59⋊C8)—21no—
21C2×#14273638—21no—
22#16837150C22116yescomplemented
23C22×C42×C59C2211yescharacteristic
24C59×#432—22no—
25C59×#432C2214yes—
26C2×C4⋊C8×C59C2212yes—
27C2×C22⋊C8×C59—21no—
28C2×#97×C59—21no—
29C22×Dic118—24no—
30C22×Dic118—22no—
31C22×Dic118—48no—
32C22×Dic118C2×C414yes—
33C22×Dic118C2×C418yes—
34C23×Dic59—22no—
35C22×C59⋊C8—22no—
36#14273727—24no—
37#14273727—24no—
38#14273727—24no—
39#14273727—24no—
40#14273727—22no—
41#14273727—24no—
42#14273727—22no—
43#14273727C2×C414yes—
44#14273727C2×C418yes—
45#14273727C2314yes—
46C2×C4⋊Dic59—24no—
47C2×C4⋊Dic59—24no—
48C2×C4⋊Dic59—22no—
49C2×C4⋊Dic59—22no—
50C2×C4⋊Dic59—24no—
51C2×C4⋊Dic59—22no—
52C2×C4⋊Dic59—24no—
53C2×C4⋊Dic59—22no—
54C4⋊(C59⋊C8)—22no—
55C4⋊(C59⋊C8)C2314yes—
56C22⋊(C59⋊C8)—24no—
57#14273638—24no—
58C23×C4×C59C2×C411yescharacteristic
59C23×C4×C59D411yescharacteristic
60C23×C4×C59D411yescharacteristic
61C22×C8×C59—22no—
62C2×C42×C59C2×C412yes—
63C2×C42×C59C2×C411yescharacteristic
64C2×C42×C59D412yes—
65C2×C42×C59D411yescharacteristic
66C2×C42×C59D411yescharacteristic
67C2×C42×C59D412yes—
68C2×C42×C59C2311yescharacteristic
69C2×C42×C59C2312yes—
70C4⋊C8×C59—22no—
71C4⋊C8×C59C2314yes—
72C22⋊C8×C59—24no—
73#97×C59—24no—
74C24×C59C22⋊C411yescharacteristic
75C22×C4×C59—24no—
76C22×C4×C59—24no—
77C22×C4×C59—24no—
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C412yes—
81C22×C4×C59C22⋊C412yes—
82C22×C4×C59C22⋊C412yes—
83C22×C4×C59C22⋊C411yescharacteristic
84C22×C4×C59C22×C412yes—
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C2×D412yes—
87C22×C4×C59C2×D412yes—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×Dic59—28no—
91C22×Dic59—24no—
92C22×Dic59—48no—
93C2×Dic118—28no—
94C2×Dic118—24no—
95C2×Dic118—416no—
96C2×Dic118—48no—
97C2×Dic118—416no—
98C2×Dic118—48no—
99C2×Dic118C22×C418yes—
100C2×C8×C59—24no—
101C2×C8×C59—44no—
102C42×C59C22⋊C412yes—
103C42×C59C22⋊C412yes—
104C42×C59C22⋊C412yes—
105C42×C59C22×C412yes—
106C42×C59C22×C411yescharacteristic
107C42×C59C2×D412yes—
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D412yes—
110C42×C59C2×D411yescharacteristic
111C42×C59C2411yescharacteristic
112C4⋊Dic59—28no—
113C4⋊Dic59—28no—
114C4⋊Dic59—24no—
115C4⋊Dic59—24no—
116C4⋊Dic59—24no—
117C4⋊Dic59—24no—
118C2×C59⋊C8—24no—
119C2×C59⋊C8—44no—
120C23×C59—22no—
121C23×C59—22no—
122C23×C59#10014yes—
123C23×C59C2×C22⋊C412yes—
124C23×C59C2×C22⋊C411yescharacteristic
125C2×C4×C59—24no—
126C2×C4×C59—22no—
127C2×C4×C59—24no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—24no—
132C2×C4×C59—24no—
133C2×C4×C59—24no—
134C2×C4×C59—24no—
135C2×C4×C59C2×C22⋊C412yes—
136C2×C4×C59C2×C22⋊C412yes—
137C2×C4×C59C2×C22⋊C412yes—
138C2×C4×C59C2×C22⋊C411yescharacteristic
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C2×C22⋊C411yescharacteristic
141C2×C4×C59#13714yes—
142C2×C4×C59#13714yes—
143C2×C4×C59C23×C411yescharacteristic
144C2×C4×C59C22×D411yescharacteristic
145C2×C4×C59C22×D411yescharacteristic
146C2×Dic59—28no—
147C2×Dic59—416no—
148C2×Dic59—48no—
149Dic118—48no—
150Dic118—416no—
151Dic118—48no—
152C8×C59—44no—
153C59⋊C8—44no—
154C2×#1558—591nomaximal, Sylow
155C22×C59—28no—
156C22×C59—22no—
157C22×C59—22no—
158C22×C59C2×#10014yes—
159C22×C59C2×#10012yes—
160C22×C59#41912yes—
161C22×C59#41912yes—
162C22×C59C22×C22⋊C411yescharacteristic
163C4×C59—21no—
164C4×C59—22no—
165C4×C59—21no—
166C4×C59—22no—
167C4×C59—22no—
168C4×C59#41912yes—
169C4×C59#41912yes—
170C4×C59C2×#13712yes—
171C4×C59C2×#13712yes—
172Dic59—48no—
173C22×C4⋊Q8—591no—
174C2×#432—591no—
175C2×#432D5911yescharacteristic
176#1558—598no—
177C2×#333—594no—
178C2×C59—24no—
179C2×C59#155814yescomplemented
180C2×C59C22×#10011yescharacteristic
181C2×C59C2×#41911yescharacteristic
182C2×C59C2×#41911yescharacteristic
183C23×Q8—592no—
184C2×C4⋊Q8—594no—
185C2×C4⋊Q8—592no—
186C2×C4⋊Q8—592no—
187C2×C4⋊Q8—594no—
188C2×C4⋊Q8—1181no—
189C2×C4⋊Q8—1182no—
190C2×C4⋊Q8—1181no—
191C2×C4⋊Q8—1182no—
192C22×C4⋊C4—1181no—
193C22×C4⋊C4—1181no—
194C22×C42C2×D5911yescharacteristic
195#432—22no—
196#432—594no—
197#432—1182no—
198#432C2×D5914yes—
199C2×C4⋊C8—592no—
200C2×C4⋊C8C2×D5912yes—
201C2×C22⋊C8—21no—
202C2×C22⋊C8—1181no—
203C2×#97—21no—
204C2×#97—1181no—
205#333—5916no—
206C59C2×#155811yescharacteristic, complemented, Sylow
207C22×Q8—598no—
208C22×Q8—594no—
209C22×Q8—1184no—
210C22×Q8—1182no—
211C22×Q8—2368no—
212C23×C4—1182no—
213C23×C4C4×D5911yescharacteristic
214C23×C4D23611yescharacteristic
215C23×C4C22⋊D5911yescharacteristic
216C22×C8—22no—
217C22×C8—1182no—
218C4⋊Q8—594no—
219C4⋊Q8—598no—
220C4⋊Q8—594no—
221C4⋊Q8—1184no—
222C4⋊Q8—1184no—
223C4⋊Q8—1184no—
224C4⋊Q8—1184no—
225C4⋊Q8—1182no—
226C4⋊Q8—1184no—
227C4⋊Q8—1182no—
228C2×C4⋊C4—1184no—
229C2×C4⋊C4—1184no—
230C2×C4⋊C4—1182no—
231C2×C4⋊C4—1182no—
232C2×C4⋊C4—1184no—
233C2×C4⋊C4—1182no—
234C2×C4⋊C4—1184no—
235C2×C4⋊C4—1182no—
236C2×C42C4×D5912yes—
237C2×C42C4×D5911yescharacteristic
238C2×C42D23611yescharacteristic
239C2×C42D23612yes—
240C2×C42C22⋊D5911yescharacteristic
241C2×C42C22⋊D5912yes—
242C2×C42C22×D5911yescharacteristic
243C2×C42C22×D5912yes—
244C4⋊C8—22no—
245C4⋊C8—594no—
246C4⋊C8—1182no—
247C4⋊C8C22×D5914yes—
248C22⋊C8—24no—
249C22⋊C8—1184no—
250#97—24no—
251#97—1184no—
252C24(C2×D59)⋊C411yescharacteristic
253C2×Q8—598no—
254C2×Q8—1188no—
255C2×Q8—1184no—
256C2×Q8—23616no—
257C2×Q8—2368no—
258C2×Q8—23616no—
259C2×Q8—2368no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—1188no—
264C22×C4—1184no—
265C22×C4—2368no—
266C22×C4(C2×D59)⋊C412yes—
267C22×C4(C2×D59)⋊C412yes—
268C22×C4(C2×D59)⋊C412yes—
269C22×C4(C2×D59)⋊C411yescharacteristic
270C22×C4(C2×D59)⋊C411yescharacteristic
271C22×C4(C2×D59)⋊C411yescharacteristic
272C22×C4C2×C4×D5912yes—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4C2×D23612yes—
275C22×C4C2×D23611yescharacteristic
276C22×C4C2×C22⋊D5912yes—
277C22×C4C2×C22⋊D5911yescharacteristic
278C2×C8—24no—
279C2×C8—44no—
280C2×C8—1184no—
281C2×C8—2364no—
282C4⋊C4—1188no—
283C4⋊C4—1188no—
284C4⋊C4—1184no—
285C4⋊C4—1184no—
286C4⋊C4—1184no—
287C4⋊C4—1184no—
288C42(C2×D59)⋊C412yes—
289C42(C2×D59)⋊C412yes—
290C42(C2×D59)⋊C412yes—
291C42C2×C4×D5912yes—
292C42C2×C4×D5911yescharacteristic
293C42C2×D23612yes—
294C42C2×D23611yescharacteristic
295C42C2×C22⋊D5912yes—
296C42C2×C22⋊D5911yescharacteristic
297C42C23×D5911yescharacteristic
298C23—22no—
299C23—22no—
300C23#1427364314yes—
301C23C2×(C2×D59)⋊C412yes—
302C23C2×(C2×D59)⋊C411yescharacteristic
303Q8—2368no—
304Q8—2368no—
305Q8—23616no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—1188no—
317C2×C4—23616no—
318C2×C4—2368no—
319C2×C4C2×(C2×D59)⋊C412yes—
320C2×C4C2×(C2×D59)⋊C412yes—
321C2×C4C2×(C2×D59)⋊C412yes—
322C2×C4C2×(C2×D59)⋊C411yescharacteristic
323C2×C4C2×(C2×D59)⋊C411yescharacteristic
324C2×C4C2×(C2×D59)⋊C411yescharacteristic
325C2×C4#1427375714yes—
326C2×C4#1427376014yes—
327C2×C4C22×C4×D5911yescharacteristic
328C2×C4C22×C22⋊D5911yescharacteristic
329C2×C4C22×D23611yescharacteristic
330C8—44no—
331C8—2364no—
332C22—22no—
333C22—22no—
334C22—28no—
335C22C2×#1427364314yes—
336C22C2×#1427364312yes—
337C22#1683741012yes—
338C22#1683741212yes—
339C22C22×(C2×D59)⋊C411yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—22no—
344C4—21no—
345C4—2368no—
346C4#1683741012yes—
347C4#1683741212yes—
348C4C2×#1427375712yes—
349C4C2×#1427376012yes—
350C2—24no—
351C2#2142282114yesdirect product factor
352C2C22×#1427364311yescharacteristic
353C2C2×#1683741011yescharacteristic
354C2C2×#1683741211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC2×C59×#432C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC42×C59C24
hypercenterC2×#432D59
hypocenterC59C2×#1558
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. #21422821C2direct (N × H)
4. C2×#16837150C2semidirect
22. #16837150C22semidirect
179. C2×C59#1558semidirect
206. C59C2×#1558semidirect
351. C2#21422821direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737272
C2×#168374592
#214228212
C2×#168371502
C2×C59×#4322
C2×#155859