Finite Group Databasebeta

← #28497975

Subgroups — #28497975

Order 15104.

524 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 119 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#21422414C212yesmaximal, complemented
2#21422040C212yesmaximal, complemented
3#21422028C212yesmaximal, complemented
4C59×#2328C211yesmaximal, characteristic
5C2×C22⋊(C59⋊C8)—22no—
6#16837128C2214yes—
7#16837126—22no—
8C2×#127×C59C411yescharacteristic
9C2×C4×D4×C59C411yescharacteristic
10C2×C4×D4×C59C2211yescharacteristic
11C59×#390C411yescharacteristic
12C59×#390C2211yescharacteristic
13C59×#386—22no—
14C59×#386—22no—
15C59×#382C2211yescharacteristic
16C59×#376C411yescharacteristic
17C22×C59⋊C8—42no—
18C22⋊(C59⋊C8)—24no—
19C22⋊(C59⋊C8)—42no—
20C22×D4×C59—22no—
21C22×D4×C59C2×C411yescharacteristic
22C22×D4×C59C2×C411yescharacteristic
23C23×C4×C59—21no—
24C23×C4×C59D412yes—
25#127×C59—22no—
26#127×C59C814yescomplemented
27C4×D4×C59—42no—
28C4×D4×C59—42no—
29C2×C4⋊C4×C59C2×C411yescharacteristic
30C2×C4⋊C4×C59C2311yescharacteristic
31C2×C22⋊C4×C59—21no—
32C2×C22⋊C4×C59—22no—
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59D412yes—
35C2×C42×C59C2×C411yescharacteristic
36C2×C42×C59C2×C411yescharacteristic
37C2×C42×C59C2×C411yescharacteristic
38#95×C59—21no—
39#95×C59—21no—
40#95×C59—22no—
41C24×C59—21no—
42C24×C59C22⋊C412yes—
43C2×D4×C59—28no—
44C2×D4×C59—21no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D4×C59—82no—
50C2×D4×C59—82no—
51C2×D4×C59C2×C812yes—
52C2×D4×C59C2×C812yescomplemented
53C2×D4×C59M1612yes—
54C22×C4×C59—21no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—42no—
60C22×C4×C59—82no—
61C22×C4×C59—82no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D412yes—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C22⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—82no—
71C22⋊C4×C59—82no—
72C42×C59—22no—
73C42×C59—21no—
74C42×C59—41no—
75C42×C59C2×C812yescomplemented
76C2×C59⋊C8—44no—
77C2×C59⋊C8—84no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—42no—
82C23×C59—42no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59—41no—
86C23×C59—41no—
87C23×C59C22⋊C814yes—
88C23×C59C2×C22⋊C411yescharacteristic
89D4×C59—42no—
90D4×C59—42no—
91D4×C59—42no—
92D4×C59—42no—
93D4×C59—88no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—41no—
100C2×C4×C59—44no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—82no—
111C2×C4×C59—84no—
112C2×C4×C59#9912yes—
113C2×C4×C59C22×C811yescharacteristic
114C2×C4×C59C2×M1611yescharacteristic
115C59⋊C8—84no—
116#11479—591nomaximal, Sylow
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—24no—
120C22×C59—44no—
121C22×C59—44no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—82no—
126C22×C59—82no—
127C22×C59—81no—
128C22×C59—81no—
129C22×C59C2×C22⋊C811yescharacteristic
130C22×C59C2×#9911yescharacteristic
131C22×C59#41311yescharacteristic
132C4×C59—22no—
133C4×C59—21no—
134C4×C59—21no—
135C4×C59—42no—
136C4×C59—42no—
137C4×C59—41no—
138C4×C59—81no—
139C4×C59—81no—
140#2328D5911yescharacteristic
141#1459—592no—
142#1324—592no—
143#1318—592no—
144C2×C59—21no—
145C2×C59—21no—
146C2×C59—44no—
147C2×C59—82no—
148C2×C59#145911yescharacteristic
149C2×C59#212711yescharacteristic
150C2×C59#213411yescharacteristic
151C2×#127Dic5911yescharacteristic
152C2×C4×D4Dic5911yescharacteristic
153C2×C4×D4C2×D5911yescharacteristic
154C2×C22⋊C8—1182no—
155#390Dic5911yescharacteristic
156#390C2×D5911yescharacteristic
157#386—22no—
158#386—22no—
159#382C2×D5911yescharacteristic
160#376Dic5911yescharacteristic
161#324—594no—
162#323—1182no—
163C59#1147911yescharacteristic, complemented, Sylow
164C22×D4—22no—
165C22×D4C2×Dic5911yescharacteristic
166C22×D4C2×Dic5911yescharacteristic
167C23×C4—21no—
168C23×C4C22⋊D5912yes—
169C22×C8—2362no—
170#127—22no—
171#127C59⋊C814yescomplemented
172C4×D4—42no—
173C4×D4—42no—
174C2×C4⋊C4C2×Dic5911yescharacteristic
175C2×C4⋊C4C22×D5911yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊D5912yes—
180C2×C42C2×Dic5911yescharacteristic
181C2×C42C2×Dic5911yescharacteristic
182C2×C42C2×Dic5911yescharacteristic
183C22⋊C8—1184no—
184C22⋊C8—2362no—
185#95—21no—
186#95—21no—
187#95—22no—
188C24—21no—
189C24C22⋊Dic5912yes—
190C2×D4—28no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—82no—
197C2×D4—82no—
198C2×D4C2×C59⋊C812yes—
199C2×D4C2×C59⋊C812yescomplemented
200C2×D4#1174566412yes—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4C22⋊Dic5911yescharacteristic
210C22×C4C22⋊Dic5911yescharacteristic
211C22×C4C22×Dic5911yescharacteristic
212C22×C4C2×C22⋊D5912yes—
213C2×C8—4724no—
214C2×C8—2364no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—22no—
222C42—21no—
223C42—41no—
224C42C2×C59⋊C812yescomplemented
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23C22⋊(C59⋊C8)14yes—
235C23C2×C22⋊Dic5911yescharacteristic
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—88no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4#1427364212yes—
260C2×C4C22×C59⋊C811yescharacteristic
261C2×C4C2×#1174566411yescharacteristic
262C8—4724no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22C2×C22⋊(C59⋊C8)11yescharacteristic
276C22C2×#1427364211yescharacteristic
277C22#1683738611yescharacteristic
278C4—21no—
279C4—21no—
280C4—22no—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—81no—
285C4—81no—
286C2—21no—
287C2—21no—
288C2—44no—
289C2—82no—
290C2#2142241411yescharacteristic
291C2#2142576511yescharacteristic
292C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2328C2
socleC22×C59C2×#99
prime radicalC2×#127×C59C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2328D59
hypocenterC59#11479
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #21422414C2semidirect
2. #21422040C2semidirect
3. #21422028C2semidirect
26. #127×C59C8semidirect
52. C2×D4×C59C2×C8semidirect
75. C42×C59C2×C8semidirect
163. C59#11479semidirect
171. #127C59⋊C8semidirect
199. C2×D4C2×C59⋊C8semidirect
224. C42C2×C59⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214224142
#214220402
#214220282
C59×#23282
#1147959