Finite Group Databasebeta

← #28484409

Subgroups — #28484409

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432435C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic
3C2×#16837138C212yesmaximal
4#21421975C212yesmaximal
5C2×C4×C4⋊C4×C59C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837308C412yescomplemented
9#16837300C412yescomplemented
10#16837286—22no—
11#16837275—21no—
12#16837265—21no—
13#16837262C2211yescharacteristic
14#16837157—22no—
15#16837138—24no—
16C22×C4⋊C4×C59—21no—
17C22×C42×C59—21no—
18C22×C42×C59C2211yescharacteristic
19C4×C4⋊C4×C59—24no—
20C2×#95×C59C2212yes—
21C22×Dic118C2×C411yescharacteristic
22C22×C59⋊C8—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C23×C4×C59—22no—
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C2×C4⋊C4×C59—24no—
42C2×C4⋊C4×C59—28no—
43C2×C42×C59—22no—
44C2×C42×C59—22no—
45C2×C42×C59—24no—
46C2×C42×C59—42no—
47C2×C42×C59—42no—
48C2×C42×C59C2×C412yes—
49C2×C42×C59D412yes—
50#95×C59—22no—
51#95×C59D414yes—
52C24×C59C2×D411yescharacteristic
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—24no—
60C22×C4×C59—28no—
61C22×C4×C59—24no—
62C22×C4×C59—42no—
63C22×C4×C59—44no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C412yes—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59SD1612yes—
69C22×C4×C59Q1612yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59#4112yes—
74C22×Dic59—22no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×Dic118—22no—
78C2×Dic118—42no—
79C4⋊C4×C59—216no—
80C42×C59—24no—
81C42×C59—24no—
82C42×C59—48no—
83C22⋊Dic59—41no—
84C22⋊Dic59—41no—
85C22⋊Dic59—41no—
86C22⋊Dic59—41no—
87C4⋊Dic59—22no—
88C4⋊Dic59—42no—
89Dic59⋊C4—44no—
90Dic59⋊C4—42no—
91C2×C59⋊C8—44no—
92C2×C59⋊C8—42no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59C2×C22⋊C411yescharacteristic
96C23×C59#12011yescharacteristic
97C23×C59#12311yescharacteristic
98C23×C59C2×SD1611yescharacteristic
99C23×C59C2×Q1611yescharacteristic
100C23×C59AGL(1,ℤ/8)11yescharacteristic
101C23×C59#13711yescharacteristic
102C2×C4×C59—24no—
103C2×C4×C59—24no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—28no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—44no—
118C2×C4×C59—48no—
119C2×C4×C59—48no—
120C2×C4×C59—44no—
121C2×C4×C59#10314yes—
122C2×C4×C59#10414yes—
123C2×C4×C59C4×D412yes—
124C2×C4×C59#12112yes—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—42no—
132Dic118—82no—
133C59⋊C8—82no—
134#9144—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—22no—
140C22×C59—22no—
141C22×C59—22no—
142C22×C59—22no—
143C22×C59#38611yescharacteristic
144C22×C59#41012yes—
145C22×C59C2×#10311yescharacteristic
146C22×C59#41711yescharacteristic
147C22×C59C2×#10412yes—
148C22×C59#45011yescharacteristic
149C22×C59#45111yescharacteristic
150C22×C59#48311yescharacteristic
151C22×C59#48411yescharacteristic
152C4×C59—24no—
153C4×C59—24no—
154C4×C59—22no—
155C4×C59—22no—
156C4×C59—48no—
157Dic59—81no—
158Dic59—81no—
159Dic59—82no—
160#2855—591no—
161C2×C4×C4⋊C4D5911yescharacteristic
162#1817—591no—
163C2×#328—592no—
164#1294—592no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59#150312yes—
170C2×C59#150912yes—
171C2×C59#187211yescharacteristic
172C2×C59#189611yescharacteristic
173C2×C59#189911yescharacteristic
174C2×C22⋊Q8—591no—
175C22×C4⋊C4—21no—
176C22×C42—21no—
177C22×C42C2×D5911yescharacteristic
178C2×C22⋊C8—592no—
179#391—592no—
180#389—592no—
181#387—1182no—
182#385—1181no—
183#381—1181no—
184#380—591no—
185C4×C4⋊C4—24no—
186C2×#95C2×D5912yes—
187#336—1182no—
188#328—1184no—
189C59#914411yescharacteristic, complemented, Sylow
190C22×Q8—591no—
191C23×C4—22no—
192C23×C4C22⋊D5911yescharacteristic
193C23×C4C22⋊D5911yescharacteristic
194C23×C4C22×D5911yescharacteristic
195C22×C8—1182no—
196C22⋊Q8—1182no—
197C22⋊Q8—1182no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—591no—
201C2×C4⋊C4—592no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—1181no—
204C2×C22⋊C4—1181no—
205C2×C22⋊C4—1181no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C4×D5912yes—
212C2×C42D23612yes—
213C22⋊C8—1182no—
214C22⋊C8—1182no—
215#95—22no—
216#95—1182no—
217#95—1182no—
218#95—1182no—
219#95—1181no—
220#95C22⋊D5914yes—
221C24C2×C22⋊D5911yescharacteristic
222C2×Q8—1182no—
223C2×Q8—2362no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1182no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4(C2×D59)⋊C412yes—
241C22×C4(C2×D59)⋊C412yes—
242C22×C4#1174567012yes—
243C22×C4#1174567212yes—
244C22×C4C2×C4×D5911yescharacteristic
245C22×C4C2×D23611yescharacteristic
246C22×C4#1174568512yes—
247C22×C4C2×C22⋊D5912yes—
248C2×C8—2364no—
249C2×C8—2362no—
250C4⋊C4—216no—
251C4⋊C4—1182no—
252C4⋊C4—2362no—
253C4⋊C4—2364no—
254C4⋊C4—2362no—
255C22⋊C4—2361no—
256C22⋊C4—2361no—
257C22⋊C4—2361no—
258C22⋊C4—2361no—
259C42—24no—
260C42—24no—
261C42—48no—
262C23—22no—
263C23—22no—
264C23C2×(C2×D59)⋊C411yescharacteristic
265C23#1427369611yescharacteristic
266C23#1427371011yescharacteristic
267C23C2×#1174567011yescharacteristic
268C23C2×#1174567211yescharacteristic
269C23#1427375311yescharacteristic
270C23#1427375811yescharacteristic
271Q8—4722no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4—2362no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4#1427365014yes—
299C2×C4#1427365514yes—
300C2×C4C4×C22⋊D5912yes—
301C2×C4#1427370112yes—
302C8—4722no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1683728111yescharacteristic
312C22#1683737512yes—
313C22C2×#1427365011yescharacteristic
314C22#1683740111yescharacteristic
315C22C2×#1427365512yes—
316C22#1683755311yescharacteristic
317C22#1683755811yescharacteristic
318C22#1683773211yescharacteristic
319C22#1683773711yescharacteristic
320C4—24no—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—48no—
325C4—4722no—
326C4—4721no—
327C4—4721no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2142259412yes—
333C2#2142261912yes—
334C2#2142440311yescharacteristic
335C2#2142453911yescharacteristic
336C2#2142455411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C4×C4⋊C4×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC2×C4×C4⋊C4D59
hypocenterC59#9144
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #16837308C4semidirect
9. #16837300C4semidirect
189. C59#9144semidirect

Maximal subgroups

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

subgroupindex
#214324352
#214241642
C2×#168371382
#214219752
C2×C4×C4⋊C4×C592
#914459