Finite Group Databasebeta

← C2×#21421930 #28478144

Subgroups — C2×#21421930 #28478144

Order 15104.

830 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C59⋊C8)C211yesmaximal, characteristic
2C22×C4×C59⋊C8C211yesmaximal, characteristic
3#21421930C214yesmaximal, direct product factor
4C22×C4⋊C8×C59C211yesmaximal, characteristic
5C23×C59⋊C8—21no—
6C23×C59⋊C8C412yes—
7C2×C4⋊(C59⋊C8)—22no—
8C2×C4⋊(C59⋊C8)—21no—
9C2×C4⋊(C59⋊C8)C414yes—
10C2×C4⋊(C59⋊C8)C2212yes—
11C2×C4×C59⋊C8—22no—
12C2×C4×C59⋊C8—22no—
13C2×C4×C59⋊C8C412yes—
14C2×C4×C59⋊C8C2212yes—
15C23×C8×C59—21no—
16C22×C42×C59C2211yescharacteristic
17C2×C4⋊C8×C59—21no—
18C2×C4⋊C8×C59—22no—
19C2×C4⋊C8×C59C414yes—
20C2×C4⋊C8×C59C2212yes—
21C22×C59⋊C8—24no—
22C22×C59⋊C8—24no—
23C22×C59⋊C8—24no—
24C22×C59⋊C8—22no—
25C22×C59⋊C8—42no—
26C22×C59⋊C8—42no—
27C22×C59⋊C8C818yescomplemented
28C22×C59⋊C8C2×C414yes—
29C4⋊(C59⋊C8)—24no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)C2×C414yes—
32C4×C59⋊C8—24no—
33C4×C59⋊C8—22no—
34C4×C59⋊C8—22no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59D411yescharacteristic
37C23×C4×C59Q811yescharacteristic
38C22×C8×C59—24no—
39C22×C8×C59—22no—
40C22×C8×C59—42no—
41C22×C8×C59—42no—
42C2×C42×C59C2×C411yescharacteristic
43C2×C42×C59C2×C412yes—
44C2×C42×C59C2×C412yes—
45C2×C42×C59D412yes—
46C2×C42×C59D411yescharacteristic
47C2×C42×C59D412yes—
48C2×C42×C59Q811yescharacteristic
49C2×C42×C59C2311yescharacteristic
50C4⋊C8×C59—24no—
51C4⋊C8×C59—22no—
52C4⋊C8×C59C2×C414yes—
53C24×C59C4⋊C411yescharacteristic
54C22×C4×C59—24no—
55C22×C4×C59C4212yes—
56C22×C4×C59C22⋊C412yes—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22⋊C412yes—
59C22×C4×C59C4⋊C411yescharacteristic
60C22×C4×C59C4⋊C412yes—
61C22×C4×C59C4⋊C412yes—
62C22×C4×C59C4⋊C411yescharacteristic
63C22×C4×C59C4⋊C411yescharacteristic
64C22×C4×C59C2×C814yes—
65C22×C4×C59M1614yes—
66C22×C4×C59D812yes—
67C22×C4×C59SD1612yes—
68C22×C4×C59SD1612yes—
69C22×C4×C59Q1612yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×Q811yescharacteristic
73C2×C8×C59—24no—
74C2×C8×C59—48no—
75C2×C8×C59—42no—
76C2×C8×C59—42no—
77C42×C59C4212yes—
78C42×C59C22⋊C412yes—
79C42×C59C22⋊C412yes—
80C42×C59C22⋊C412yes—
81C42×C59C4⋊C412yes—
82C42×C59C4⋊C412yes—
83C42×C59C22×C411yescharacteristic
84C42×C59C22×C411yescharacteristic
85C42×C59C2×D411yescharacteristic
86C42×C59C2×D411yescharacteristic
87C2×C59⋊C8—216no—
88C2×C59⋊C8—24no—
89C2×C59⋊C8—24no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—48no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8C2×C818yescomplemented
95C23×C59#9512yes—
96C23×C59C4⋊C814yes—
97C23×C59#10612yes—
98C23×C59#10712yes—
99C23×C59#10814yes—
100C23×C59C2×C4⋊C411yescharacteristic
101C2×C4×C59—24no—
102C2×C4×C59—24no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59#9512yes—
106C2×C4×C59#9512yes—
107C2×C4×C59#9512yes—
108C2×C4×C59C4×C814yes—
109C2×C4×C59#9714yes—
110C2×C4×C59C22⋊C818yes—
111C2×C4×C59#10214yes—
112C2×C4×C59#10214yes—
113C2×C4×C59#10314yes—
114C2×C4×C59#10314yes—
115C2×C4×C59C4⋊C814yes—
116C2×C4×C59#10612yes—
117C2×C4×C59#10712yes—
118C2×C4×C59C2×C4211yescharacteristic
119C2×C4×C59C2×C22⋊C411yescharacteristic
120C2×C4×C59C2×C22⋊C411yescharacteristic
121C2×C4×C59C2×C22⋊C411yescharacteristic
122C2×C4×C59C2×C4⋊C411yescharacteristic
123C2×C4×C59C2×C4⋊C411yescharacteristic
124C2×C4×C59C22×C812yes—
125C2×C4×C59C2×M1612yes—
126C2×C4×C59C2×D811yescharacteristic
127C2×C4×C59C2×SD1611yescharacteristic
128C2×C4×C59C2×SD1611yescharacteristic
129C2×C4×C59C2×Q1611yescharacteristic
130C8×C59—44no—
131C59⋊C8—28no—
132C59⋊C8—44no—
133C2×#1277—591nomaximal, Sylow
134C22×C59#33418yes—
135C22×C59#33518yes—
136C22×C59#33614yes—
137C22×C59#34014yes—
138C22×C59#34114yes—
139C22×C59C2×#9511yescharacteristic
140C22×C59C2×C4⋊C812yes—
141C22×C59C2×#10611yescharacteristic
142C22×C59C2×#10711yescharacteristic
143C22×C59C2×#10812yes—
144C4×C59—22no—
145C4×C59—21no—
146C4×C59—21no—
147C4×C59#33614yes—
148C4×C59#34014yes—
149C4×C59C2×C4×C811yescharacteristic
150C4×C59C2×#9711yescharacteristic
151C4×C59C2×C22⋊C812yes—
152C4×C59C2×#10211yescharacteristic
153C4×C59C2×#10211yescharacteristic
154C4×C59C2×#10311yescharacteristic
155C4×C59C2×#10311yescharacteristic
156C22×C4⋊C8—591no—
157C22×C4⋊C8D5911yescharacteristic
158C22×C4×C8—591no—
159#1277—594no—
160C2×C59#127718yescomplemented
161C2×C59C2×#33412yes—
162C2×C59C2×#33512yes—
163C2×C59C2×#33611yescharacteristic
164C2×C59C2×#34011yescharacteristic
165C2×C59C2×#34111yescharacteristic
166C23×C8—21no—
167C23×C8—592no—
168C23×C8—1181no—
169C22×C42C2×D5911yescharacteristic
170C2×C4⋊C8—21no—
171C2×C4⋊C8—22no—
172C2×C4⋊C8—594no—
173C2×C4⋊C8—592no—
174C2×C4⋊C8—1182no—
175C2×C4⋊C8—1181no—
176C2×C4⋊C8Dic5914yes—
177C2×C4⋊C8C2×D5912yes—
178C2×C4×C8—592no—
179C2×C4×C8—592no—
180C2×C4×C8—1182no—
181C2×C4×C8—1182no—
182C59C2×#127711yescharacteristic, complemented, Sylow
183C23×C4Dic11811yescharacteristic
184C23×C4C4×D5911yescharacteristic
185C23×C4C22⋊D5911yescharacteristic
186C22×C8—24no—
187C22×C8—22no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—598no—
191C22×C8—594no—
192C22×C8—1184no—
193C22×C8—1184no—
194C22×C8—1184no—
195C22×C8—1182no—
196C22×C8—2362no—
197C22×C8—2362no—
198C2×C42Dic11811yescharacteristic
199C2×C42C4×D5911yescharacteristic
200C2×C42C4×D5912yes—
201C2×C42D23612yes—
202C2×C42C2×Dic5912yes—
203C2×C42C22⋊D5912yes—
204C2×C42C22⋊D5911yescharacteristic
205C2×C42C22×D5911yescharacteristic
206C4⋊C8—24no—
207C4⋊C8—22no—
208C4⋊C8—594no—
209C4⋊C8—1184no—
210C4⋊C8—1182no—
211C4⋊C8C2×Dic5914yes—
212C4×C8—1184no—
213C4×C8—1182no—
214C4×C8—1182no—
215C24Dic59⋊C411yescharacteristic
216C22×C4—24no—
217C22×C4C8×D5914yes—
218C22×C4#1174565914yes—
219C22×C4C4×Dic5912yes—
220C22×C4Dic59⋊C411yescharacteristic
221C22×C4Dic59⋊C412yes—
222C22×C4Dic59⋊C411yescharacteristic
223C22×C4Dic59⋊C411yescharacteristic
224C22×C4C4⋊Dic5912yes—
225C22×C4(C2×D59)⋊C412yes—
226C22×C4(C2×D59)⋊C412yes—
227C22×C4#1174566912yes—
228C22×C4#1174567012yes—
229C22×C4#1174567112yes—
230C22×C4#1174567212yes—
231C22×C4C22⋊Dic5912yes—
232C22×C4C2×Dic11811yescharacteristic
233C22×C4C2×C4×D5911yescharacteristic
234C22×C4C2×C22⋊D5911yescharacteristic
235C2×C8—24no—
236C2×C8—48no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—598no—
240C2×C8—11816no—
241C2×C8—1184no—
242C2×C8—1184no—
243C2×C8—1184no—
244C2×C8—2368no—
245C2×C8—2362no—
246C2×C8—2362no—
247C42C4×Dic5912yes—
248C42Dic59⋊C412yes—
249C42C4⋊Dic5912yes—
250C42(C2×D59)⋊C412yes—
251C42(C2×D59)⋊C412yes—
252C42C22⋊Dic5912yes—
253C42C2×C4×D5911yescharacteristic
254C42C2×D23611yescharacteristic
255C42C22×Dic5911yescharacteristic
256C42C2×C22⋊D5911yescharacteristic
257C23C59⋊#9512yes—
258C23Dic59⋊C814yes—
259C23#1427365812yes—
260C23#1427366012yes—
261C23#1427366214yes—
262C23C2×Dic59⋊C411yescharacteristic
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4C59⋊#9512yes—
268C2×C4C59⋊#9512yes—
269C2×C4C59⋊#9512yes—
270C2×C4C8×Dic5914yes—
271C2×C4#1427363714yes—
272C2×C4(C2×D59)⋊C818yes—
273C2×C4#1427364714yes—
274C2×C4#1427364814yes—
275C2×C4#1427365014yes—
276C2×C4#1427365114yes—
277C2×C4Dic59⋊C814yes—
278C2×C4#1427365812yes—
279C2×C4#1427366012yes—
280C2×C4C2×C4×Dic5911yescharacteristic
281C2×C4C2×(C2×D59)⋊C411yescharacteristic
282C2×C4C2×(C2×D59)⋊C411yescharacteristic
283C2×C4C2×C22⋊Dic5911yescharacteristic
284C2×C4C2×Dic59⋊C411yescharacteristic
285C2×C4C2×C4⋊Dic5911yescharacteristic
286C2×C4C2×C8×D5912yes—
287C2×C4C2×#1174565912yes—
288C2×C4C2×#1174566911yescharacteristic
289C2×C4C2×#1174567111yescharacteristic
290C2×C4C2×#1174567011yescharacteristic
291C2×C4C2×#1174567211yescharacteristic
292C8—44no—
293C8—1188no—
294C8—2364no—
295C22#1683715218yes—
296C22#1683715418yes—
297C22#1683715614yes—
298C22#1683716414yes—
299C22#1683716614yes—
300C22C2×C59⋊#9511yescharacteristic
301C22C2×Dic59⋊C812yes—
302C22C2×#1427365811yescharacteristic
303C22C2×#1427366011yescharacteristic
304C22C2×#1427366212yes—
305C4—22no—
306C4—21no—
307C4—21no—
308C4#1683715614yes—
309C4#1683716414yes—
310C4C2×C8×Dic5911yescharacteristic
311C4C2×#1427363711yescharacteristic
312C4C2×(C2×D59)⋊C812yes—
313C4C2×#1427364711yescharacteristic
314C4C2×#1427364811yescharacteristic
315C4C2×#1427365011yescharacteristic
316C4C2×#1427365111yescharacteristic
317C2#2142193018yesdirect product factor
318C2C2×#1683715212yes—
319C2C2×#1683715412yes—
320C2C2×#1683715611yescharacteristic
321C2C2×#1683716411yescharacteristic
322C2C2×#1683716611yescharacteristic
subgroupG/H
centerC23×C4C22⋊D59
commutatorC4×C59C2×C4×C8
FrattiniC2×C42C22×D59
FittingC22×C4⋊C8×C59C2
socle, prime radicalC24×C59C4⋊C4
cosocleC2×C42×C59C23
hypercenterC22×C4⋊C8D59
hypocenterC59C2×#1277
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23×C4 ◁ C22×C42 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #21421930C2direct (N × H)
27. C22×C59⋊C8C8semidirect
94. C2×C59⋊C8C2×C8semidirect
160. C2×C59#1277semidirect
182. C59C2×#1277semidirect
317. C2#21421930direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊(C59⋊C8)2
C22×C4×C59⋊C82
#214219302
C22×C4⋊C8×C592
C2×#127759