Finite Group Databasebeta

← #28607103

Subgroups — #28607103

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D472C212yesmaximal
2C2×C4×D472C211yesmaximal, characteristic
3#21427007C212yesmaximal, complemented
4#21424829C212yesmaximal, complemented
5#21424360C214yesmaximal, complemented
6#21424260C211yesmaximal, characteristic, complemented
7#21424107C212yesmaximal, complemented
8C8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
9C22×D472C2211yescharacteristic
10C2×C4×D236C2214yes—
11C2×C4×D236C2212yes—
12C4×D472—28no—
13C4×D472C418yescomplemented
14C2×#14273661C2212yes—
15C2×#14273661C2211yescharacteristic
16C2×#14273659C2212yes—
17C2×#14273649C2214yes—
18C2×#14273649C2212yes—
19#16837300C2214yes—
20#16837273—24no—
21#16837268C2212yes—
22#16837261—22no—
23#95×D59—22no—
24#16837165C2214yes—
25C2×C4⋊C8×C59C2211yescharacteristic
26C2×C4×C8×C59C2212yes—
27C2×C4×C8×C59C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C59×#336C2212yes—
30C22×D236C2312yes—
31C22×C4×D59—22no—
32C22×C4×D59—24no—
33C2×D472C2×C414yes—
34C2×D472D414yes—
35C2×D472Q814yes—
36C4×D236—24no—
37C4×D236—28no—
38C4×D236—44no—
39C4×D236C2×C418yescomplemented
40C2×C4⋊Dic59C2314yes—
41C2×C4⋊Dic59C2312yes—
42C2×Dic59⋊C4—24no—
43C2×(C2×D59)⋊C4—22no—
44C2×(C2×D59)⋊C4—24no—
45#14273661—24no—
46#14273661C2×C414yescomplemented
47#14273659—42no—
48#14273649—28no—
49#14273649C2×C418yescomplemented
50C59⋊#95—22no—
51C59⋊#95—24no—
52C59⋊#95—22no—
53C22×C8×C59C2312yes—
54C22×C8×C59C2311yescharacteristic
55C22×C8×C59C2311yescharacteristic
56C2×C4⋊C4×C59D412yes—
57C2×C42×C59C2312yes—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—22no—
60C4×C8×C59—24no—
61C4×C8×C59C2×C414yescomplemented
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64C23×D59—22no—
65C22×C4×C59C2×D412yes—
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D412yes—
69C22×C4×C59C2411yescharacteristic
70C22×Dic59—22no—
71C22×Dic59—24no—
72C2×D236—22no—
73C2×D236—22no—
74C2×D236—22no—
75C2×D236C22×C414yes—
76C2×D236C2×D414yes—
77C2×D236C2×Q814yes—
78C2×C4×D59—28no—
79C2×C4×D59—48no—
80C2×C4×D59—48no—
81C2×C4×D59—84no—
82C2×C4×D59—84no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—22no—
86C2×C8×C59—22no—
87C2×C8×C59—22no—
88C2×C8×C59C22×C412yes—
89C2×C8×C59C22×C412yes—
90C2×C8×C59C2×D412yes—
91C2×C8×C59C2×Q812yes—
92C4⋊C4×C59—24no—
93C42×C59—22no—
94C42×C59—22no—
95C42×C59C22×C412yes—
96C42×C59#4112yes—
97(C2×D59)⋊C4—28no—
98(C2×D59)⋊C4—48no—
99C4⋊Dic59—22no—
100C4⋊Dic59—24no—
101C4⋊Dic59—24no—
102C4⋊Dic59C22×C414yes—
103Dic59⋊C4—84no—
104D472C4⋊C4116yescomplemented
105C23×C59C22×D411yescharacteristic
106C22×D59—24no—
107C22×D59—24no—
108C22×D59—24no—
109C22×D59—44no—
110C22×D59—44no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59C4×D412yes—
119C2×C4×C59C4×D412yes—
120C2×C4×C59#13514yes—
121C2×C4×C59C23×C411yescharacteristic
122C2×C4×C59C22×D411yescharacteristic
123C2×C4×C59C22×Q811yescharacteristic
124C2×C4×C59C2×#4111yescharacteristic
125C2×C4×C59H(5,𝔽2)11yescharacteristic
126C2×C4×C592−1+411yescharacteristic
127C2×Dic59—24no—
128C2×Dic59—44no—
129C2×Dic59—44no—
130C2×Dic59—44no—
131C2×Dic59—44no—
132D236—24no—
133D236—24no—
134D236—24no—
135D236C2×C4⋊C418yes—
136C4×D59—88no—
137C4×D59—816no—
138C8×C59—22no—
139C8×C59—42no—
140C8×C59C2×C4⋊C414yes—
141#22682—591nomaximal, Sylow
142C22×C59C2×C4×D411yescharacteristic
143C22×C59D4211yescharacteristic
144C22×C59D4×Q811yescharacteristic
145C22×C59C2×#13512yes—
146C22×C59#57612yes—
147C2×D59—28no—
148C2×D59—48no—
149C2×D59—44no—
150C2×D59—44no—
151C2×D59—48no—
152C4×C59—22no—
153C4×C59—22no—
154C4×C59—24no—
155C4×C59—42no—
156C4×C59C22×C4⋊C411yescharacteristic
157C4×C59#52011yescharacteristic
158C4×C59#54711yescharacteristic
159C4×C59#55411yescharacteristic
160Dic59—44no—
161Dic59—84no—
162C2×C4×D8—591no—
163C2×C4×D8—592no—
164#2349—592no—
165#1942—592no—
166#1864—594no—
167#1846—591no—
168#1804—592no—
169C8×C4⋊C4D5911yescharacteristic, complemented
170C2×C59D4×C4⋊C411yescharacteristic
171C2×C59C4×#13511yescharacteristic
172C2×C59#294811yescharacteristic
173C2×C59#328012yes—
174C2×C59#338012yes—
175D59—48no—
176C22×D8—591no—
177C2×C4×D4—592no—
178C2×C4×D4—594no—
179C4×D8—598no—
180C4×D8—1188no—
181C2×#107—591no—
182C2×#107—592no—
183C2×#106—592no—
184C2×C4⋊C8C2×D5911yescharacteristic
185C2×#102—592no—
186C2×#102—594no—
187C2×C4×C8C2×D5912yes—
188C2×C4×C8C2×D5911yescharacteristic
189#389—594no—
190#385—1184no—
191#382—592no—
192#380—1182no—
193C4×C4⋊C4C2×D5911yescharacteristic
194C2×#95—1182no—
195#340—594no—
196#336C2×D5912yes—
197C59#2268211yescharacteristic, complemented, Sylow
198C22×D4—592no—
199C23×C4—1182no—
200C23×C4—1184no—
201C2×D8—594no—
202C2×D8—594no—
203C2×D8—594no—
204C22×C8C22×D5912yes—
205C22×C8C22×D5911yescharacteristic
206C22×C8C22×D5911yescharacteristic
207C4×D4—598no—
208C4×D4—1188no—
209C4×D4—1184no—
210C4×D4—2364no—
211C2×C4⋊C4—592no—
212C2×C4⋊C4—594no—
213C2×C4⋊C4—1184no—
214C2×C4⋊C4D23612yes—
215C2×C22⋊C4—1182no—
216C2×C22⋊C4—1184no—
217C2×C42C22×D5912yes—
218C2×C42C22×D5911yescharacteristic
219#107—594no—
220#107—1184no—
221#106—2362no—
222C4⋊C8—22no—
223#102—598no—
224#102—1188no—
225C4×C8—24no—
226C4×C8C4×D5914yescomplemented
227#95—1182no—
228#95—1184no—
229#95—1182no—
230#95D23611yescharacteristic
231#95D23611yescharacteristic
232C24—1182no—
233C2×D4—594no—
234C2×D4—594no—
235C2×D4—594no—
236C2×D4—1182no—
237C2×D4—1182no—
238C2×D4—1182no—
239C22×C4—1182no—
240C22×C4—1188no—
241C22×C4—1184no—
242C22×C4—4724no—
243C22×C4—4724no—
244C22×C4—2368no—
245C22×C4—2368no—
246C22×C4C2×D23612yes—
247C22×C4C2×D23611yescharacteristic
248C22×C4C2×D23611yescharacteristic
249C22×C4C2×D23612yes—
250C22×C4C23×D5911yescharacteristic
251D8—5916no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8C2×C4×D5912yes—
258C2×C8C2×C4×D5912yes—
259C2×C8D4×D5912yes—
260C2×C8Q8×D5912yes—
261C4⋊C4—24no—
262C4⋊C4—594no—
263C4⋊C4—1182no—
264C4⋊C4—1184no—
265C4⋊C4—1184no—
266C4⋊C4—4724no—
267C22⋊C4—1188no—
268C22⋊C4—2368no—
269C42—22no—
270C42—22no—
271C42C2×C4×D5912yes—
272C42#1174568512yes—
273C23—1184no—
274C23—1184no—
275C23—1184no—
276C23—2364no—
277C23—2364no—
278C23C22×D23611yescharacteristic
279D4—598no—
280D4—1184no—
281D4—1184no—
282D4—1184no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—1184no—
291C2×C4—47216no—
292C2×C4—4728no—
293C2×C4—2364no—
294C2×C4—2364no—
295C2×C4—2364no—
296C2×C4—2364no—
297C2×C4C4×D23612yes—
298C2×C4C4×D23612yes—
299C2×C4#1427374914yes—
300C2×C4C22×C4×D5911yescharacteristic
301C2×C4C2×D4×D5911yescharacteristic
302C2×C4C2×Q8×D5911yescharacteristic
303C2×C4C2×#1174568511yescharacteristic
304C2×C4#1427377311yescharacteristic
305C2×C4#1427377411yescharacteristic
306C8—22no—
307C8—42no—
308C8C4⋊C4×D5914yes—
309C22—1188no—
310C22—2368no—
311C22—2364no—
312C22—2364no—
313C22—2368no—
314C22C2×C4×D23611yescharacteristic
315C22D4×D23611yescharacteristic
316C22Q8×D23611yescharacteristic
317C22C2×#1427374912yes—
318C22#1683820712yes—
319C4—22no—
320C4—22no—
321C4—24no—
322C4—42no—
323C4—4724no—
324C4—2364no—
325C4C2×C4⋊C4×D5911yescharacteristic
326C4#1683788211yescharacteristic
327C4#1683805711yescharacteristic
328C4#1683811211yescharacteristic
329C2—2368no—
330C2C4⋊C4×D23611yescharacteristic
331C2C4×#1427374911yescharacteristic
332C2#2143294911yescharacteristic
333C2#2143578312yes—
334C2#2143696412yes—
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC8×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×D472C22
cosocleC22×C4×C59C24
hypercenterC8×C4⋊C4D59
hypocenterC59#22682
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21427007C2semidirect
4. #21424829C2semidirect
5. #21424360C2semidirect
6. #21424260C2semidirect
7. #21424107C2semidirect
8. C8×C4⋊C4×C59C2semidirect
13. C4×D472C4semidirect
39. C4×D236C2×C4semidirect
46. #14273661C2×C4semidirect
49. #14273649C2×C4semidirect
61. C4×C8×C59C2×C4semidirect
104. D472C4⋊C4semidirect
169. C8×C4⋊C4D59semidirect
197. C59#22682semidirect
226. C4×C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4722
#214270072
#214248292
#214243602
#214242602
#214241072
C8×C4⋊C4×C592
#2268259