Finite Group Databasebeta

← #28480448

Subgroups — #28480448

Order 15104.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838125C211yesmaximal, characteristic
2#21422778C212yesmaximal, complemented
3#21422520C212yesmaximal
4#21422420C211yesmaximal, characteristic, complemented
5C59×#1801C211yesmaximal, characteristic, complemented
6#16838125—24no—
7#16838125—24no—
8C2×#14273727C2211yescharacteristic
9C2×#14273722—22no—
10C2×#14273715C412yes—
11C2×#14273703—21no—
12C2×C4×Dic118C412yes—
13C2×C4×D236C2211yescharacteristic
14C2×#14273685—21no—
15#16837455C2212yes—
16#16837150—24no—
17#16837129—24no—
18#16837128—22no—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#423C2212yes—
21C2×#101×C59—21no—
22C59×#343—21no—
23C22×Dic118C2×C412yes—
24C22×D236C2×C411yescharacteristic
25C22×C4×D59—21no—
26#14273727—22no—
27#14273727D414yes—
28#14273722—216no—
29#14273715—24no—
30#14273715—24no—
31#14273703—28no—
32C4×Dic118—24no—
33C4×Dic118—24no—
34C4×D236—22no—
35C4×D236D414yes—
36#14273685—28no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×Dic59⋊C4—22no—
41C2×(C2×D59)⋊C4—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—21no—
44C2×C4×Dic59—22no—
45C4⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—22no—
47C4×C59⋊C8—22no—
48C22×Q8×C59D411yescharacteristic
49C2×M16×C59—22no—
50C4×Q8×C59—22no—
51C4×Q8×C59—22no—
52C2×C4⋊C4×C59D412yes—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C42×C59C2×C412yes—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—22no—
57C4⋊C8×C59—22no—
58#101×C59—42no—
59C23×D59—21no—
60C2×Q8×C59—22no—
61C2×Q8×C59—22no—
62C2×Q8×C59—21no—
63C2×Q8×C59—21no—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C22⋊C412yes—
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—22no—
71C2×D236—21no—
72C2×D236—42no—
73C2×D236C22⋊C412yes—
74C2×C4×D59—24no—
75C2×C4×D59—42no—
76C2×C4×D59—41no—
77C2×C4×D59—41no—
78C2×Dic118—22no—
79C2×Dic118—44no—
80C2×Dic118C22⋊C414yes—
81M16×C59—42no—
82M16×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—24no—
86C42×C59—22no—
87C42×C59—22no—
88C42×C59C2×D412yes—
89C42×C59C2×D412yes—
90(C2×D59)⋊C4—28no—
91(C2×D59)⋊C4—28no—
92(C2×D59)⋊C4—24no—
93C4⋊Dic59—21no—
94C4⋊Dic59—24no—
95C4⋊Dic59—24no—
96C4⋊Dic59C22⋊C412yes—
97Dic59⋊C4—28no—
98C4×Dic59—28no—
99C2×C59⋊C8—42no—
100C23×C59C2×C22⋊C411yescharacteristic
101C22×D59—22no—
102C22×D59—42no—
103C22×D59—42no—
104C22×D59—41no—
105C22×D59—41no—
106Q8×C59—42no—
107Q8×C59—42no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59C2×C22⋊C411yescharacteristic
116C2×C4×C59C2×C22⋊C411yescharacteristic
117C2×C4×C59#12012yes—
118C2×C4×C59#12012yes—
119C2×C4×C59#13512yes—
120C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C59#13711yescharacteristic
122C2×Dic59—22no—
123C2×Dic59—24no—
124C2×Dic59—44no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127D236—24no—
128D236—42no—
129C4×D59—44no—
130C4×D59—44no—
131Dic118—44no—
132Dic118—44no—
133C8×C59—82no—
134C59⋊C8—82no—
135#8394—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59#37911yescharacteristic
139C22×C59#41611yescharacteristic
140C22×C59#41711yescharacteristic
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—42no—
144C2×D59—82no—
145C2×D59—82no—
146C4×C59—22no—
147C4×C59—21no—
148C4×C59—42no—
149C4×C59—42no—
150C4×C59#41312yes—
151C4×C59#44912yes—
152C4×C59#45212yes—
153Dic59—44no—
154Dic59—42no—
155C2×#556—591no—
156#1801D5911yescharacteristic, complemented
157#1548—592no—
158#1486—592no—
159#1461—591no—
160C2×C59—21no—
161C2×C59—21no—
162C2×C59#178511yescharacteristic
163C2×C59#178811yescharacteristic
164C2×C59#179611yescharacteristic
165D59—82no—
166#556—1184no—
167#556—1184no—
168C2×C4⋊Q8—591no—
169C2×#126—1182no—
170C2×#124—592no—
171C2×C22⋊Q8—1181no—
172C2×C4×Q8—592no—
173C2×C4×Q8C2×D5911yescharacteristic
174C2×C4×D4—591no—
175C2×#117—1181no—
176#431—592no—
177#423C2×D5912yes—
178C2×#101—21no—
179#343—21no—
180#333—1184no—
181D4⋊C8—1184no—
182#324—1182no—
183C59#839411yescharacteristic, complemented, Sylow
184C22×Q8—592no—
185C22×Q8C22⋊D5911yescharacteristic
186C22×D4—591no—
187C23×C4—1181no—
188C2×M16—22no—
189C4⋊Q8—594no—
190C4⋊Q8—1182no—
191#126—11816no—
192#124—1184no—
193#124—1184no—
194C22⋊Q8—1188no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—1184no—
198C4×Q8—1184no—
199C4×D4—594no—
200C4×D4—1182no—
201#117—1188no—
202C2×C4⋊C4—591no—
203C2×C4⋊C4—1182no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4D23612yes—
207C2×C4⋊C4C22⋊D5911yescharacteristic
208C2×C22⋊C4—1182no—
209C2×C22⋊C4—1182no—
210C2×C22⋊C4—1181no—
211C2×C42—1182no—
212C2×C42C4×D5912yes—
213C2×C42C22×D5911yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—1182no—
217#101—42no—
218C22⋊C8—1182no—
219C4×C8—1182no—
220C24—1181no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—594no—
226C2×Q8—1182no—
227C2×Q8—2364no—
228C2×D4—592no—
229C2×D4—1181no—
230C2×D4—2362no—
231C22×C4—1181no—
232C22×C4—1182no—
233C22×C4—1184no—
234C22×C4—2362no—
235C22×C4—2361no—
236C22×C4—2361no—
237C22×C4(C2×D59)⋊C412yes—
238C22×C4(C2×D59)⋊C412yes—
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C2×D23611yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—2362no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—592no—
249C4⋊C4—1188no—
250C4⋊C4—1181no—
251C4⋊C4—1184no—
252C4⋊C4—1184no—
253C22⋊C4—1188no—
254C22⋊C4—1188no—
255C22⋊C4—1184no—
256C42—22no—
257C42—22no—
258C42—1188no—
259C42D4×D5912yes—
260C42D4×D5912yes—
261C23—1182no—
262C23—2362no—
263C23—2362no—
264C23—2361no—
265C23—2361no—
266C23C2×(C2×D59)⋊C411yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—2364no—
270Q8—2364no—
271D4—1184no—
272D4—2362no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—1182no—
281C2×C4—1184no—
282C2×C4—2364no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2364no—
286C2×C4—2364no—
287C2×C4C22⋊C4×D5911yescharacteristic
288C2×C4C22⋊C4×D5911yescharacteristic
289C2×C4#1427369512yes—
290C2×C4#1427369712yes—
291C2×C4#1427374712yes—
292C2×C4#1427375311yescharacteristic
293C2×C4#1427375811yescharacteristic
294C8—82no—
295C8—4722no—
296C22—21no—
297C22—21no—
298C22—4722no—
299C22—4722no—
300C22—2362no—
301C22—2362no—
302C22—2362no—
303C22#1683725711yescharacteristic
304C22#1683739611yescharacteristic
305C22#1683740111yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—2364no—
311C4—2362no—
312C4#1683738512yes—
313C4#1683754512yes—
314C4#1683756212yes—
315C2—21no—
316C2—21no—
317C2—4722no—
318C2#2142402311yescharacteristic
319C2#2142403411yescharacteristic
320C2#2142407211yescharacteristic
subgroupG/H
centerC22#16837257
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#379
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1801D59
hypocenterC59#8394
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #21422778C2semidirect
4. #21422420C2semidirect
5. C59×#1801C2semidirect
156. #1801D59semidirect
183. C59#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#168381252
#214227782
#214225202
#214224202
C59×#18012
#839459