Finite Group Databasebeta

← #28489386

Subgroups — #28489386

Order 15104.

698 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 87 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#21434920C211yesmaximal, characteristic
2C2×#16837558C212yesmaximal
3#21422461C212yesmaximal
4#21421983C211yesmaximal, characteristic
5C59×#1784C211yesmaximal, characteristic
6C22×#11745672—22no—
7C2×#14273706C2212yes—
8#16837733—22no—
9#16837660—22no—
10#16837560—22no—
11#16837558—24no—
12#16837558—24no—
13C2×#14273651—22no—
14C2×C22⋊(C59⋊C8)C2212yes—
15#16837127—24no—
16C23×Q8×C59C2211yescharacteristic
17C2×#101×C59—22no—
18C59×#407C2211yescharacteristic
19C59×#407C2211yescharacteristic
20C22×Dic118D412yes—
21C2×Dic236—22no—
22C2×#11745672—28no—
23C2×#11745672—48no—
24C22×C59⋊C8—22no—
25#14273706—22no—
26#14273706—22no—
27#14273706—22no—
28#14273706—22no—
29C2×C4⋊Dic59D412yes—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32#14273659—22no—
33#14273652—24no—
34#14273651—28no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)D414yes—
37C22×Q8×C59—21no—
38C22×Q8×C59—22no—
39C22×Q8×C59—22no—
40C22×Q8×C59—44no—
41C22×Q8×C59D412yes—
42C23×C4×C59—21no—
43C23×C4×C59C2311yescharacteristic
44C2×M16×C59—21no—
45C2×M16×C59D412yes—
46#101×C59—44no—
47C22⋊C8×C59—22no—
48C22⋊C8×C59—22no—
49C24×C59#4111yescharacteristic
50C2×Q8×C59—24no—
51C2×Q8×C59—28no—
52C2×Q8×C59—24no—
53C2×Q8×C59—44no—
54C2×Q8×C59—44no—
55C2×Q8×C59—42no—
56C2×Q8×C59—48no—
57C2×Q8×C59—41no—
58C2×Q8×C59—41no—
59C22×C4×C59—22no—
60C22×C4×C59—42no—
61C22×C4×C59—44no—
62C22×C4×C59—42no—
63C22×C4×C59C2×D412yes—
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D412yes—
67C22×Dic59—22no—
68C22×Dic59—22no—
69C2×Dic118—22no—
70C2×Dic118—22no—
71C2×Dic118—44no—
72M16×C59—42no—
73M16×C59—42no—
74C2×C8×C59—22no—
75C2×C8×C59—41no—
76C22⋊Dic59—42no—
77C22⋊Dic59—42no—
78#11745672—416no—
79C4⋊Dic59—22no—
80C4⋊Dic59—22no—
81Dic59⋊C4—42no—
82Dic59⋊C4—42no—
83C2×C59⋊C8—24no—
84C2×C59⋊C8—44no—
85Dic236—82no—
86C23×C59—24no—
87C23×C59—41no—
88C23×C59#12112yes—
89C23×C59#12111yescharacteristic
90Q8×C59—28no—
91Q8×C59—44no—
92Q8×C59—44no—
93Q8×C59—44no—
94C2×C4×C59—24no—
95C2×C4×C59—22no—
96C2×C4×C59—24no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—48no—
101C2×C4×C59—42no—
102C2×C4×C59#12011yescharacteristic
103C2×C4×C59#12012yes—
104C2×C4×C59#12711yescharacteristic
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59—44no—
108C2×Dic59—82no—
109Dic118—44no—
110Dic118—44no—
111C8×C59—42no—
112C8×C59—81no—
113C59⋊C8—44no—
114#9991—591nomaximal, Sylow
115C22×C59—22no—
116C22×C59—22no—
117C22×C59—22no—
118C22×C59—21no—
119C22×C59—44no—
120C22×C59#39211yescharacteristic
121C22×C59#47314yes—
122C4×C59—22no—
123C4×C59—21no—
124C4×C59—44no—
125C4×C59B(4,𝔽2)12yes—
126Dic59—82no—
127Dic59—82no—
128#3195—591no—
129C2×#451—592no—
130#1784D5911yescharacteristic
131#1472—592no—
132#1298—591no—
133C2×C59—24no—
134C2×C59—41no—
135C2×C59#202012yes—
136C2×C59#202111yescharacteristic
137C23×Q8C2×D5911yescharacteristic
138C22×Q16—1182no—
139C2×C22⋊Q8—592no—
140#483—1182no—
141#470—1182no—
142#451—1182no—
143#451—1184no—
144#451—1184no—
145C2×#103—1182no—
146C2×#101—22no—
147#407C2×D5911yescharacteristic
148#407C2×D5911yescharacteristic
149C2×C22⋊C8—592no—
150#324—1184no—
151C59#999111yescharacteristic, complemented, Sylow
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—592no—
157C22×Q8D23612yes—
158C23×C4—21no—
159C23×C4C22×D5911yescharacteristic
160C2×Q16—1182no—
161C2×Q16—1188no—
162C2×Q16—2368no—
163C2×M16—21no—
164C2×M16C22⋊D5912yes—
165C22×C8—1182no—
166C22⋊Q8—1182no—
167C22⋊Q8—1182no—
168C22⋊Q8—1182no—
169C22⋊Q8—1182no—
170C2×C4⋊C4—592no—
171C2×C4⋊C4—1182no—
172C2×C22⋊C4—1182no—
173#106—1182no—
174#103—1184no—
175#103—1188no—
176#101—44no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—594no—
180C22⋊C8—1182no—
181C24#1174568511yescharacteristic
182C2×Q8—24no—
183C2×Q8—28no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—48no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1182no—
192C2×Q8—1182no—
193C2×Q8—2364no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—1182no—
199C22×C4—1182no—
200C22×C4C2×D23611yescharacteristic
201C22×C4D4×D5912yes—
202C22×C4D4×D5912yes—
203C22×C4C2×C22⋊D5911yescharacteristic
204Q16—4722no—
205Q16—23616no—
206M16—42no—
207M16—42no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—1184no—
211C2×C8—2364no—
212C4⋊C4—1182no—
213C4⋊C4—1182no—
214C4⋊C4—2362no—
215C4⋊C4—2362no—
216C22⋊C4—2362no—
217C22⋊C4—2362no—
218C23—24no—
219C23—41no—
220C23#1427370012yes—
221C23#1427370111yescharacteristic
222Q8—28no—
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—2364no—
227Q8—2364no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—4722no—
237C2×C4—2364no—
238C2×C4—2362no—
239C2×C4—2362no—
240C2×C4#1427369511yescharacteristic
241C2×C4#1427369712yes—
242C2×C4#1427372311yescharacteristic
243C8—42no—
244C8—81no—
245C8—2364no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—44no—
251C22#1683731111yescharacteristic
252C22#1683767614yes—
253C4—21no—
254C4—22no—
255C4—44no—
256C4—4722no—
257C4—4722no—
258C4#1683758812yes—
259C2—24no—
260C2—41no—
261C2#2142521612yes—
262C2#2142522211yescharacteristic
subgroupG/H
centerC22#16837311
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1784C2
socleC22×C59#392
prime radicalC24×C59#41
hypercenter#1784D59
hypocenterC59#9991
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
151. C59#9991semidirect

Maximal subgroups

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

subgroupindex
#214349202
C2×#168375582
#214224612
#214219832
C59×#17842
#999159