Finite Group Databasebeta

← #28480423

Subgroups — #28480423

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×#16838122C211yesmaximal, characteristic
2#21422769C212yesmaximal, complemented
3#21422524C212yesmaximal, complemented
4#21422417C211yesmaximal, characteristic
5C59×#1800C211yesmaximal, characteristic
6#16838122—24no—
7#16838122—24no—
8C2×#14273727C2211yescharacteristic
9C2×#14273720—22no—
10C2×#14273714—21no—
11C2×#14273706C412yes—
12C2×C4×Dic118C2211yescharacteristic
13C2×#14273692—21no—
14C2×#14273686C412yes—
15#16837455C2212yes—
16#16837149—24no—
17#16837132—24no—
18#16837126—22no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#423C2212yes—
21C2×#100×C59—21no—
22C59×#343—21no—
23C22×Dic118—21no—
24C22×Dic118C2×C411yescharacteristic
25#14273727—22no—
26#14273727D414yes—
27#14273720—216no—
28#14273714—28no—
29#14273706—24no—
30#14273706—24no—
31C4×Dic118—22no—
32C4×Dic118D414yes—
33#14273692—28no—
34#14273686—24no—
35#14273686—24no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59C2×C412yes—
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×Dic59⋊C4—22no—
40C2×Dic59⋊C4—21no—
41C2×C22⋊Dic59—22no—
42C2×C22⋊Dic59—22no—
43C2×C4×Dic59—22no—
44C2×C4×Dic59—21no—
45C4⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—22no—
47C4×C59⋊C8—22no—
48C22×D4×C59D411yescharacteristic
49C23×C4×C59C2×C412yes—
50C2×M16×C59—22no—
51C4×D4×C59—22no—
52C4×D4×C59—22no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D412yes—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—22no—
57C4⋊C8×C59—22no—
58#100×C59—42no—
59C24×C59C22⋊C412yes—
60C2×D4×C59—22no—
61C2×D4×C59—22no—
62C2×D4×C59—21no—
63C2×D4×C59—21no—
64C22×C4×C59—22no—
65C22×C4×C59—22no—
66C22×C4×C59—44no—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—22no—
74C2×Dic118—21no—
75C2×Dic118—24no—
76C2×Dic118—42no—
77C2×Dic118—42no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118C22⋊C412yes—
81M16×C59—42no—
82M16×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—22no—
85C22⋊C4×C59—24no—
86C42×C59C2×D412yes—
87C42×C59C2×D412yes—
88C22⋊Dic59—28no—
89C22⋊Dic59—28no—
90C4⋊Dic59—21no—
91C4⋊Dic59—22no—
92C4⋊Dic59—24no—
93C4⋊Dic59C22⋊C412yes—
94C4⋊Dic59C22⋊C414yes—
95Dic59⋊C4—28no—
96Dic59⋊C4—24no—
97C4×Dic59—28no—
98C4×Dic59—24no—
99C2×C59⋊C8—42no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—22no—
103C23×C59—42no—
104C23×C59—42no—
105C23×C59C2×C22⋊C411yescharacteristic
106D4×C59—42no—
107D4×C59—42no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—41no—
111C2×C4×C59—44no—
112C2×C4×C59—44no—
113C2×C4×C59—42no—
114C2×C4×C59C2×C22⋊C411yescharacteristic
115C2×C4×C59C2×C22⋊C411yescharacteristic
116C2×C4×C59#12012yes—
117C2×C4×C59#12012yes—
118C2×C4×C59#13512yes—
119C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C59#13711yescharacteristic
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—24no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128Dic118—24no—
129Dic118—42no—
130Dic118—44no—
131Dic118—44no—
132C8×C59—82no—
133C59⋊C8—82no—
134#8389—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—82no—
141C22×C59—82no—
142C22×C59#37911yescharacteristic
143C22×C59#41611yescharacteristic
144C22×C59#41711yescharacteristic
145C4×C59—22no—
146C4×C59—21no—
147C4×C59—42no—
148C4×C59#41312yes—
149C4×C59#44912yes—
150C4×C59#45212yes—
151Dic59—42no—
152Dic59—42no—
153Dic59—44no—
154C2×#556—591no—
155#1800D5911yescharacteristic
156#1546—592no—
157#1487—592no—
158#1460—591no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59—82no—
162C2×C59#178511yescharacteristic
163C2×C59#178811yescharacteristic
164C2×C59#179611yescharacteristic
165#556—1184no—
166#556—1184no—
167C2×C4⋊Q8—591no—
168C2×#126—1182no—
169C2×#124—1181no—
170C2×C22⋊Q8—592no—
171C2×C4×Q8—591no—
172C2×C4×Q8—1181no—
173C2×C4×D4C2×D5911yescharacteristic
174C2×#117—592no—
175#431—592no—
176#423C2×D5912yes—
177C2×#100—21no—
178#343—21no—
179#332—1184no—
180Q8⋊C8—1184no—
181#323—1182no—
182C59#838911yescharacteristic, complemented, Sylow
183C22×Q8—591no—
184C22×Q8—1181no—
185C22×D4C22⋊D5911yescharacteristic
186C23×C4C4×D5912yes—
187C2×M16—22no—
188C4⋊Q8—594no—
189C4⋊Q8—1182no—
190#126—11816no—
191#124—1188no—
192C22⋊Q8—1184no—
193C22⋊Q8—1184no—
194C4×Q8—594no—
195C4×Q8—1188no—
196C4×Q8—1182no—
197C4×D4—22no—
198C4×D4—22no—
199#117—1184no—
200#117—1184no—
201C2×C4⋊C4—591no—
202C2×C4⋊C4—592no—
203C2×C4⋊C4—1182no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C22⋊C4—1182no—
208C2×C22⋊C4—1182no—
209C2×C22⋊C4D23612yes—
210C2×C42—1182no—
211C2×C42—1181no—
212C2×C42C22×D5911yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—1182no—
216#100—42no—
217C22⋊C8—1182no—
218C4×C8—1182no—
219C24(C2×D59)⋊C412yes—
220C2×Q8—592no—
221C2×Q8—1181no—
222C2×Q8—1184no—
223C2×Q8—2362no—
224C2×Q8—2362no—
225C2×Q8—2361no—
226C2×Q8—2361no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—1181no—
235C22×C4—1182no—
236C22×C4—1181no—
237C22×C4(C2×D59)⋊C412yes—
238C22×C4C2×C4×D5911yescharacteristic
239C22×C4C2×D23611yescharacteristic
240C22×C4C2×C22⋊D5911yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—2362no—
245C4⋊C4—22no—
246C4⋊C4—592no—
247C4⋊C4—594no—
248C4⋊C4—1188no—
249C4⋊C4—1181no—
250C4⋊C4—1182no—
251C4⋊C4—1184no—
252C4⋊C4—1184no—
253C22⋊C4—24no—
254C22⋊C4—1188no—
255C22⋊C4—1188no—
256C42—1188no—
257C42—1184no—
258C42D4×D5912yes—
259C42D4×D5912yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×(C2×D59)⋊C411yescharacteristic
266Q8—1184no—
267Q8—2362no—
268Q8—2364no—
269Q8—2364no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—1182no—
279C2×C4—1184no—
280C2×C4—1182no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2362no—
285C2×C4C22⋊C4×D5911yescharacteristic
286C2×C4C22⋊C4×D5911yescharacteristic
287C2×C4#1427369512yes—
288C2×C4#1427369712yes—
289C2×C4#1427374712yes—
290C2×C4#1427375311yescharacteristic
291C2×C4#1427375811yescharacteristic
292C8—82no—
293C8—4722no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1683725711yescharacteristic
302C22#1683739611yescharacteristic
303C22#1683740111yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—2362no—
308C4—2364no—
309C4—2362no—
310C4#1683738312yes—
311C4#1683754512yes—
312C4#1683756212yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2142402111yescharacteristic
317C2#2142403411yescharacteristic
318C2#2142407211yescharacteristic
subgroupG/H
centerC22#16837257
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#379
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#1800D59
hypocenterC59#8389
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21422769C2semidirect
3. #21422524C2semidirect
182. C59#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#168381222
#214227692
#214225242
#214224172
C59×#18002
#838959