Finite Group Databasebeta

← #28492205

Subgroups — #28492205

Order 15104.

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

normal subgroups only

This group has 630 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
1#21429012C211yesmaximal, characteristic
2#21424093C212yesmaximal, complemented
3C2×#16837131C212yesmaximal, complemented
4#21421979C211yesmaximal, characteristic, complemented
5C59×#1844C211yesmaximal, characteristic, complemented
6C2×#14273702—22no—
7C2×C4×D236C2212yes—
8C2×C4⋊(C59⋊C8)C2212yes—
9C2×#14273647—22no—
10#16837319—22no—
11#16837259—22no—
12#16837164—22no—
13#16837131—24no—
14C2×#106×C59—22no—
15C2×C4×C8×C59C2211yescharacteristic
16C59×#384C2211yescharacteristic
17C59×#384C2211yescharacteristic
18C22×D236—22no—
19C22×D236D412yes—
20C22×C4×D59—22no—
21C22×C59⋊C8—22no—
22#14273702—44no—
23#14273702—44no—
24C4×D236—22no—
25C4×D236—22no—
26C4×D236—22no—
27C4×D236—22no—
28C2×C4⋊Dic59D412yes—
29C2×(C2×D59)⋊C4—24no—
30C2×(C2×D59)⋊C4—22no—
31C4⋊(C59⋊C8)—22no—
32C4⋊(C59⋊C8)—22no—
33#14273647—44no—
34C59⋊#95—22no—
35C22×C8×C59—21no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59D412yes—
40C2×C42×C59C2311yescharacteristic
41#106×C59—44no—
42C4×C8×C59—21no—
43C4×C8×C59D412yescomplemented
44C23×D59—22no—
45C22×C4×C59—22no—
46C22×C4×C59—41no—
47C22×C4×C59C2×D411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—22no—
50C2×D236—22no—
51C2×D236—22no—
52C2×D236—22no—
53C2×D236—22no—
54C2×D236—22no—
55C2×D236—48no—
56C2×D236—82no—
57C2×D236SD1614yes—
58C2×C4×D59—42no—
59C2×C4×D59—42no—
60C2×C4×D59—42no—
61C2×C4×D59—42no—
62C2×C4×D59—42no—
63C2×C4×D59—42no—
64C2×C8×C59—22no—
65C2×C8×C59—22no—
66C2×C8×C59—22no—
67C4⋊C4×C59—42no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—42no—
73C42×C59C2×D411yescharacteristic
74C42×C59#4111yescharacteristic
75C42×C59#4112yes—
76(C2×D59)⋊C4—48no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—22no—
80C4⋊Dic59—22no—
81C2×C59⋊C8—42no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—42no—
84C23×C59#12011yescharacteristic
85C22×D59—24no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59—42no—
89C22×D59—42no—
90C22×D59—42no—
91C22×D59—42no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—44no—
95C2×C4×C59—42no—
96C2×C4×C59—82no—
97C2×C4×C59—81no—
98C2×C4×C59#12111yescharacteristic
99C2×C4×C59#12112yes—
100C2×C4×C59#12312yes—
101C2×C4×C59#12411yescharacteristic
102C2×C4×C59C2×SD1612yes—
103C2×C4×C59#13512yes—
104C2×C4×C59AGL(1,ℤ/8)12yes—
105C2×C4×C59AGL(1,ℤ/8)12yes—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—42no—
109D236—24no—
110D236—24no—
111D236—42no—
112D236—42no—
113D236—88no—
114C4×D59—82no—
115C4×D59—82no—
116C4×D59—82no—
117C4×D59—82no—
118C8×C59—24no—
119C59⋊C8—82no—
120#10432—591nomaximal, Sylow
121C22×C59#39411yescharacteristic
122C22×C59#44912yes—
123C22×C59#45012yes—
124C22×C59#45311yescharacteristic
125C22×C59#45611yescharacteristic
126C2×D59—44no—
127C2×D59—44no—
128C2×D59—44no—
129C2×D59—82no—
130C2×D59—82no—
131C2×D59—82no—
132C2×D59—82no—
133C4×C59—21no—
134C4×C59—21no—
135C4×C59—82no—
136C4×C59—161no—
137C4×C59#46512yes—
138C4×C59#46912yes—
139C4×C59#48112yes—
140C4×C59#48212yes—
141Dic59—82no—
142#2537—591no—
143#1844D5911yescharacteristic, complemented
144#1802—592no—
145C2×D4⋊C8—592no—
146#1296—591no—
147C2×C59#162411yescharacteristic
148C2×C59#162911yescharacteristic
149C2×C59#165412yes—
150C2×C59#203412yes—
151C2×C59#203911yescharacteristic
152D59—84no—
153C2×#121—1182no—
154C2×C4×D4—592no—
155C2×#106—22no—
156C2×C4⋊C8—592no—
157C2×#102—1182no—
158C2×C4×C8C2×D5911yescharacteristic
159#394—1182no—
160#384C2×D5911yescharacteristic
161#384C2×D5911yescharacteristic
162#380—1182no—
163#340—1182no—
164D4⋊C8—1184no—
165C59#1043211yescharacteristic, complemented, Sylow
166C22×D4—592no—
167C22×D4—1182no—
168C23×C4—1182no—
169C22×C8—21no—
170C22×C8—1182no—
171#121—2364no—
172#121—2364no—
173C4×D4—1182no—
174C4×D4—1182no—
175C4×D4—1182no—
176C4×D4—1182no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—592no—
181C2×C4⋊C4C22⋊D5912yes—
182C2×C22⋊C4—1184no—
183C2×C22⋊C4—1182no—
184C2×C42C22×D5911yescharacteristic
185#106—44no—
186C4⋊C8—1182no—
187C4⋊C8—1182no—
188#102—2364no—
189C4×C8—21no—
190C4×C8D23612yescomplemented
191#95—1182no—
192C24—1182no—
193C2×D4—594no—
194C2×D4—1182no—
195C2×D4—1182no—
196C2×D4—1182no—
197C2×D4—1182no—
198C2×D4—1182no—
199C2×D4—4722no—
200C2×D4—2368no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—1182no—
204C22×C4—2362no—
205C22×C4—2362no—
206C22×C4—2362no—
207C22×C4—2362no—
208C22×C4—2362no—
209C22×C4—2362no—
210C22×C4D4×D5912yes—
211C22×C4C2×C22⋊D5911yescharacteristic
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—2362no—
216C2×C8—2362no—
217C2×C8—2362no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—1182no—
225C4⋊C4—1182no—
226C22⋊C4—2368no—
227C22⋊C4—2362no—
228C22⋊C4—2362no—
229C42C2×D23611yescharacteristic
230C42#1174568511yescharacteristic
231C42#1174568912yes—
232C23—1184no—
233C23—2362no—
234C23—2362no—
235C23—2362no—
236C23—2362no—
237C23—2362no—
238C23—2362no—
239C23#1427369511yescharacteristic
240D4—1184no—
241D4—1184no—
242D4—4728no—
243D4—2362no—
244D4—2362no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—4722no—
252C2×C4—4722no—
253C2×C4—4722no—
254C2×C4—4722no—
255C2×C4—2362no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4#1427370111yescharacteristic
259C2×C4#1427370212yes—
260C2×C4#1427371212yes—
261C2×C4#1427371611yescharacteristic
262C2×C4SD16×D5912yes—
263C2×C4#1427374812yes—
264C2×C4#1427375412yes—
265C2×C4#1427375412yes—
266C8—24no—
267C8—4722no—
268C22—4722no—
269C22—4722no—
270C22—4722no—
271C22—4722no—
272C22—2364no—
273C22—2364no—
274C22—2364no—
275C22#1683731911yescharacteristic
276C22#1683754612yes—
277C22#1683755112yes—
278C22#1683756911yescharacteristic
279C22#1683758411yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—161no—
284C4—4722no—
285C4#1683763212yes—
286C4#1683765612yes—
287C4#1683772212yes—
288C4#1683772812yes—
289C2—4724no—
290C2#2142317711yescharacteristic
291C2#2142321011yescharacteristic
292C2#2142337712yes—
293C2#2142528712yes—
294C2#2142532011yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1844C2
socleC23×C59#120
prime radical#21429012C2
hypercenter#1844D59
hypocenterC59#10432
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. #21424093C2semidirect
3. C2×#16837131C2semidirect
4. #21421979C2semidirect
5. C59×#1844C2semidirect
43. C4×C8×C59D4semidirect
143. #1844D59semidirect
165. C59#10432semidirect
190. C4×C8D236semidirect

Maximal subgroups

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

subgroupindex
#214290122
#214240932
C2×#168371312
#214219792
C59×#18442
#1043259