Finite Group Databasebeta

← #28477942

Subgroups — #28477942

Order 15104.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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#21432042C211yesmaximal, characteristic
2#21425505C212yesmaximal, complemented
3#21421921C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#2093C211yesmaximal, characteristic
6C2×#14273650—22no—
7#16837341—21no—
8#16837314—22no—
9#16837308C2212yes—
10#16837282—21no—
11#16837277—22no—
12#16837164—22no—
13#16837157C2212yes—
14C2×#107×C59—21no—
15C2×#102×C59—21no—
16C59×#390C2211yescharacteristic
17C59×#384C2211yescharacteristic
18C59×#336C2211yescharacteristic
19C22×Dic118D412yes—
20C23×Dic59—21no—
21C22×C59⋊C8—22no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59D412yes—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—21no—
26#14273650—44no—
27C59⋊#95—22no—
28C59⋊#95—21no—
29C59⋊#95—22no—
30C59⋊#95—22no—
31C22×D4×C59D411yescharacteristic
32C22×C8×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59D411yescharacteristic
36C2×C22⋊C4×C59—22no—
37C2×C42×C59C2311yescharacteristic
38#107×C59—42no—
39#102×C59—42no—
40C24×C59—21no—
41C2×D4×C59—21no—
42C2×D4×C59—41no—
43C2×D4×C59—41no—
44C2×D4×C59Q1612yes—
45C22×C4×C59—21no—
46C22×C4×C59—41no—
47C22×C4×C59C2×D411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—22no—
50C22×Dic59—22no—
51C22×Dic59—81no—
52C22×Dic59—82no—
53C2×Dic118—22no—
54C2×Dic118—22no—
55C2×Dic118—42no—
56C2×Dic118—42no—
57C2×C8×C59—22no—
58C2×C8×C59—41no—
59C2×C8×C59—41no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—42no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—41no—
68C22⋊Dic59—82no—
69C22⋊Dic59—81no—
70C4⋊Dic59—42no—
71C4⋊Dic59—81no—
72C2×C59⋊C8—42no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8—42no—
75C23×C59—22no—
76C23×C59—42no—
77C23×C59—41no—
78C23×C59—42no—
79C23×C59—41no—
80C23×C59#12011yescharacteristic
81D4×C59—42no—
82D4×C59—81no—
83C2×C4×C59—21no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—42no—
88C2×C4×C59—41no—
89C2×C4×C59—82no—
90C2×C4×C59—81no—
91C2×C4×C59C2×Q1611yescharacteristic
92C2×C4×C59#13512yes—
93C2×C4×C59AGL(1,ℤ/8)12yes—
94C2×C4×C59#13711yescharacteristic
95C2×Dic59—42no—
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—84no—
102Dic118—44no—
103Dic118—82no—
104C8×C59—42no—
105C59⋊C8—82no—
106#7821—591nomaximal, Sylow
107C22×C59—42no—
108C22×C59—44no—
109C22×C59—82no—
110C22×C59—81no—
111C22×C59—81no—
112C22×C59#44912yes—
113C22×C59#45111yescharacteristic
114C22×C59#45311yescharacteristic
115C22×C59#45512yes—
116C22×C59#45811yescharacteristic
117C4×C59—22no—
118C4×C59—22no—
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—81no—
122C4×C59—161no—
123Dic59—82no—
124Dic59—82no—
125#2804—591no—
126#2093D5911yescharacteristic
127#2072—592no—
128#1272—592no—
129#1270—591no—
130C2×C59—82no—
131C2×C59#161212yes—
132C2×C59#161411yescharacteristic
133C2×C59#162011yescharacteristic
134C2×C59#164412yes—
135C2×C59#165911yescharacteristic
136C2×#107—21no—
137C2×#103—1182no—
138C2×#102—21no—
139#400—1181no—
140#393—1182no—
141#391—592no—
142#390C2×D5911yescharacteristic
143#386—1181no—
144#385—1182no—
145#384C2×D5911yescharacteristic
146#340—1182no—
147#336—592no—
148#336C2×D5911yescharacteristic
149C59#782111yescharacteristic, complemented, Sylow
150C22×Q8—592no—
151C22×D4C22⋊D5911yescharacteristic
152C23×C4—1181no—
153C22×C8—21no—
154C22×C8—1182no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4—1181no—
159C2×C4⋊C4C22⋊D5911yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—1182no—
162C2×C22⋊C4—1181no—
163C2×C42C22×D5911yescharacteristic
164#107—42no—
165#103—2364no—
166#102—42no—
167#95—1182no—
168#95—1181no—
169#95—1182no—
170#95—1182no—
171C24—21no—
172C2×Q8—1182no—
173C2×Q8—1182no—
174C2×Q8—2362no—
175C2×Q8—2362no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1174567212yes—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—1182no—
183C22×C4—1182no—
184C22×C4—4721no—
185C22×C4—4722no—
186C22×C4D4×D5912yes—
187C22×C4C2×C22⋊D5911yescharacteristic
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—2362no—
192C2×C8—2362no—
193C2×C8—2362no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—4721no—
200C4⋊C4—2362no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—4722no—
204C22⋊C4—4721no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1427369511yescharacteristic
212Q8—4722no—
213Q8—2364no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—4724no—
225C2×C4—2362no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4—2362no—
231C2×C4C2×#1174567211yescharacteristic
232C2×C4#1427374612yes—
233C2×C4#1427375112yes—
234C2×C4#1427375711yescharacteristic
235C8—42no—
236C8—4722no—
237C22—42no—
238C22—44no—
239C22—82no—
240C22—81no—
241C22—81no—
242C22#1683754412yes—
243C22#1683755711yescharacteristic
244C22#1683756911yescharacteristic
245C22#1683757812yes—
246C22#1683759111yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—161no—
253C4—4722no—
254C4—4722no—
255C2—82no—
256C2#2142310712yes—
257C2#2142312011yescharacteristic
258C2#2142315311yescharacteristic
259C2#2142330912yes—
260C2#2142340011yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2093C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2093D59
hypocenterC59#7821
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #21425505C2semidirect
3. #21421921C2semidirect
149. C59#7821semidirect

Maximal subgroups

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

subgroupindex
#214320422
#214255052
#214219212
#214219172
C59×#20932
#782159