Finite Group Databasebeta

← #28478006

Subgroups — #28478006

Order 15104.

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

normal subgroups only

This group has 438 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#21425635C212yesmaximal, complemented
3#21421921C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#2094C211yesmaximal, characteristic
6C2×#14273660—22no—
7C2×#14273650—22no—
8#16837341—21no—
9#16837314—22no—
10#16837308C2212yes—
11#16837282—21no—
12#16837277—22no—
13#16837157C2212yes—
14C2×#106×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#14273660—44no—
27#14273650—44no—
28C59⋊#95—22no—
29C59⋊#95—21no—
30C59⋊#95—22no—
31C59⋊#95—22no—
32C22×D4×C59D411yescharacteristic
33C22×C8×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59D411yescharacteristic
37C2×C22⋊C4×C59—22no—
38C2×C42×C59C2311yescharacteristic
39#106×C59—42no—
40#102×C59—42no—
41C24×C59—21no—
42C2×D4×C59—21no—
43C2×D4×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59Q1612yes—
46C22×C4×C59—21no—
47C22×C4×C59—41no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D412yes—
50C22×Dic59—22no—
51C22×Dic59—22no—
52C22×Dic59—82no—
53C22×Dic59—81no—
54C2×Dic118—22no—
55C2×Dic118—22no—
56C2×Dic118—42no—
57C2×Dic118—42no—
58C2×C8×C59—22no—
59C2×C8×C59—41no—
60C2×C8×C59—41no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C22⋊C4×C59—42no—
68C42×C59—41no—
69C22⋊Dic59—82no—
70C22⋊Dic59—81no—
71C4⋊Dic59—42no—
72C4⋊Dic59—81no—
73C2×C59⋊C8—24no—
74C2×C59⋊C8—42no—
75C2×C59⋊C8—42no—
76C23×C59—22no—
77C23×C59—42no—
78C23×C59—42no—
79C23×C59—41no—
80C23×C59—41no—
81C23×C59#12011yescharacteristic
82D4×C59—42no—
83D4×C59—81no—
84C2×C4×C59—22no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—42no—
89C2×C4×C59—41no—
90C2×C4×C59—82no—
91C2×C4×C59—81no—
92C2×C4×C59C2×Q1611yescharacteristic
93C2×C4×C59#13512yes—
94C2×C4×C59#13712yes—
95C2×C4×C59#13711yescharacteristic
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—42no—
102C2×Dic59—84no—
103Dic118—44no—
104Dic118—82no—
105C8×C59—42no—
106C59⋊C8—44no—
107#7833—591nomaximal, Sylow
108C22×C59—42no—
109C22×C59—44no—
110C22×C59—82no—
111C22×C59—81no—
112C22×C59—81no—
113C22×C59#45111yescharacteristic
114C22×C59#45212yes—
115C22×C59#45312yes—
116C22×C59#45311yescharacteristic
117C22×C59#45811yescharacteristic
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—81no—
123C4×C59—161no—
124Dic59—82no—
125Dic59—82no—
126#2804—591no—
127#2095—592no—
128#2094D5911yescharacteristic
129#1272—592no—
130#1270—591no—
131C2×C59—82no—
132C2×C59#161411yescharacteristic
133C2×C59#161812yes—
134C2×C59#163011yescharacteristic
135C2×C59#163212yes—
136C2×C59#165511yescharacteristic
137C2×#107—1182no—
138C2×#106—21no—
139C2×#103—1182no—
140C2×#102—21no—
141#400—1181no—
142#393—1182no—
143#391—592no—
144#390C2×D5911yescharacteristic
145#386—1181no—
146#385—1182no—
147#384C2×D5911yescharacteristic
148#336—592no—
149#336C2×D5911yescharacteristic
150C59#783311yescharacteristic, complemented, Sylow
151C22×Q8—592no—
152C22×D4C22⋊D5911yescharacteristic
153C23×C4—1181no—
154C22×C8—21no—
155C22×C8—1182no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—1181no—
160C2×C4⋊C4C22⋊D5911yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1182no—
163C2×C22⋊C4—1181no—
164C2×C42C22×D5911yescharacteristic
165#107—2364no—
166#106—42no—
167#103—2364no—
168#102—42no—
169#95—1182no—
170#95—1181no—
171#95—1182no—
172#95—1182no—
173C24—21no—
174C2×Q8—1182no—
175C2×Q8—1182no—
176C2×Q8—2362no—
177C2×Q8—2362no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4#1174567212yes—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—1182no—
185C22×C4—1182no—
186C22×C4—4722no—
187C22×C4—4721no—
188C22×C4D4×D5912yes—
189C22×C4C2×C22⋊D5911yescharacteristic
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—1184no—
194C2×C8—2362no—
195C2×C8—2362no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—4721no—
202C4⋊C4—2362no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—4722no—
206C22⋊C4—4721no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1427369511yescharacteristic
214Q8—4722no—
215Q8—2364no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—4724no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4—2362no—
231C2×C4—2362no—
232C2×C4—2362no—
233C2×C4C2×#1174567211yescharacteristic
234C2×C4#1427374812yes—
235C2×C4#1427375612yes—
236C2×C4#1427375711yescharacteristic
237C8—42no—
238C8—2364no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1683755711yescharacteristic
245C22#1683756112yes—
246C22#1683756812yes—
247C22#1683756911yescharacteristic
248C22#1683759111yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—161no—
255C4—4722no—
256C4—4722no—
257C2—82no—
258C2#2142312011yescharacteristic
259C2#2142314112yes—
260C2#2142321511yescharacteristic
261C2#2142322812yes—
262C2#2142338411yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2094C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2094D59
hypocenterC59#7833
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
2. #21425635C2semidirect
3. #21421921C2semidirect
150. C59#7833semidirect

Maximal subgroups

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

subgroupindex
#214320422
#214256352
#214219212
#214219172
C59×#20942
#783359