Finite Group Databasebeta

← #28497848

Subgroups — #28497848

Order 15104.

454 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 93 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#21426721C211yesmaximal, characteristic
2#324×D59C212yesmaximal
3#21422041C212yesmaximal, complemented
4#21422040C211yesmaximal, characteristic, complemented
5C59×#1324C211yesmaximal, characteristic, complemented
6C2×#14273685C411yescharacteristic
7C2×#14273684C411yescharacteristic
8C2×C4⋊C4×D59C2211yescharacteristic
9C22⋊C8×D59—22no—
10#16837299—22no—
11#16837276—22no—
12#16837272C411yescharacteristic
13#16837271C2211yescharacteristic
14#16837267C411yescharacteristic
15#16837128C2212yes—
16#16837127—22no—
17C59×#382C2211yescharacteristic
18C59×#324C2212yes—
19C22×C4×D59C2×C411yescharacteristic
20C22×C4×D59D412yes—
21C2×C8×D59—42no—
22#14273685—42no—
23#14273684—22no—
24#14273684C814yescomplemented
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4D412yes—
31C4⋊C4×D59—42no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59C2×C411yescharacteristic
36C22⋊(C59⋊C8)—22no—
37(C2×D59)⋊C8—42no—
38C59⋊#95—22no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59C2311yescharacteristic
41C2×C42×C59C2×C411yescharacteristic
42C22⋊C8×C59—22no—
43#95×C59—22no—
44C23×D59C22⋊C411yescharacteristic
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59C22×C411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—21no—
50C22×Dic59—21no—
51C22×Dic59C22⋊C411yescharacteristic
52C22×Dic59C22⋊C412yes—
53C2×C4×D59—22no—
54C2×C4×D59—22no—
55C2×C4×D59—22no—
56C2×C4×D59—22no—
57C2×C4×D59—82no—
58C2×C4×D59C2×C812yes—
59C2×C4×D59M1612yes—
60C2×C8×C59—42no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C42×C59—21no—
64C42×C59C2×C812yescomplemented
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—42no—
67C4⋊Dic59—41no—
68C4⋊Dic59—41no—
69C4⋊Dic59—82no—
70Dic59⋊C4—24no—
71Dic59⋊C4—42no—
72Dic59⋊C4—82no—
73C4×Dic59—21no—
74C4×Dic59—42no—
75C4×Dic59C2×C812yescomplemented
76C2×C59⋊C8—42no—
77C8×D59—84no—
78C23×C59C2×C22⋊C411yescharacteristic
79C22×D59—41no—
80C22×D59—41no—
81C22×D59C22⋊C812yes—
82C22×D59#9912yes—
83C22×D59#10112yes—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59C22×C811yescharacteristic
94C2×C4×C59C2×M1611yescharacteristic
95C2×Dic59—21no—
96C2×Dic59—21no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—41no—
100C2×Dic59—41no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—41no—
104C2×Dic59C22⋊C812yes—
105C4×D59—24no—
106C4×D59—24no—
107C4×D59—44no—
108C4×D59—82no—
109C8×C59—82no—
110C59⋊C8—82no—
111#11448—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59C2×C22⋊C811yescharacteristic
115C22×C59C2×#9911yescharacteristic
116C22×C59C2×#10111yescharacteristic
117C2×D59—22no—
118C2×D59—22no—
119C2×D59—42no—
120C2×D59—42no—
121C2×D59#32414yes—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—42no—
125C4×C59—42no—
126C4×C59—81no—
127Dic59—22no—
128Dic59—42no—
129Dic59—42no—
130Dic59—81no—
131#2317—591no—
132C2×#324—592no—
133#1325—592no—
134#1324—591no—
135#1324D5911yescharacteristic, complemented
136C2×C59—21no—
137C2×C59—21no—
138C2×C59C2×#32411yescharacteristic
139C2×C59#212711yescharacteristic
140C2×C59#213311yescharacteristic
141D59—24no—
142C2×#117—591no—
143C2×#117—591no—
144C22×C4⋊C4—591no—
145C2×C22⋊C8—1182no—
146#389—1182no—
147#385—1182no—
148#384—591no—
149#384—591no—
150#382—591no—
151#382C2×D5911yescharacteristic
152#324—592no—
153#324—1182no—
154#324C2×D5912yes—
155C59#1144811yescharacteristic, complemented, Sylow
156C23×C4—591no—
157C23×C4—592no—
158C22×C8—2362no—
159#117—594no—
160#117—1182no—
161#117—2362no—
162C2×C4⋊C4—591no—
163C2×C4⋊C4—592no—
164C2×C4⋊C4—591no—
165C2×C4⋊C4—1181no—
166C2×C4⋊C4—1182no—
167C2×C4⋊C4—1182no—
168C2×C4⋊C4—2362no—
169C2×C4⋊C4C4×D5911yescharacteristic
170C2×C4⋊C4C22×D5911yescharacteristic
171C2×C22⋊C4—1181no—
172C2×C22⋊C4—1181no—
173C2×C42—591no—
174C2×C42—1181no—
175C2×C42C4×D5911yescharacteristic
176C22⋊C8—22no—
177C22⋊C8—1182no—
178C22⋊C8—2362no—
179#95—22no—
180#95—1182no—
181C24—591no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—591no—
185C22×C4—592no—
186C22×C4—592no—
187C22×C4—592no—
188C22×C4—1182no—
189C22×C4—1182no—
190C22×C4—1182no—
191C22×C4—1182no—
192C22×C4—1181no—
193C22×C4—1181no—
194C22×C4—4722no—
195C22×C4C2×C4×D5911yescharacteristic
196C22×C4D4×D5912yes—
197C2×C8—42no—
198C2×C8—4724no—
199C2×C8—2362no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—1184no—
203C4⋊C4—4722no—
204C4⋊C4—4722no—
205C4⋊C4—2362no—
206C4⋊C4—2361no—
207C4⋊C4—2361no—
208C22⋊C4—1184no—
209C22⋊C4—2362no—
210C42—21no—
211C42—592no—
212C42—1181no—
213C42—2362no—
214C42C8×D5912yescomplemented
215C23—592no—
216C23—592no—
217C23—592no—
218C23—2361no—
219C23—2361no—
220C23C22⋊C4×D5911yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—592no—
231C2×C4—1184no—
232C2×C4—1184no—
233C2×C4—1181no—
234C2×C4—1181no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—4722no—
238C2×C4—2364no—
239C2×C4—2362no—
240C2×C4—2361no—
241C2×C4—2361no—
242C2×C4—2361no—
243C2×C4—2362no—
244C2×C4C2×C8×D5911yescharacteristic
245C2×C4M16×D5911yescharacteristic
246C8—82no—
247C8—4722no—
248C22—21no—
249C22—21no—
250C22—594no—
251C22—1182no—
252C22—1182no—
253C22—2362no—
254C22—2362no—
255C22C22⋊C8×D5911yescharacteristic
256C22#99×D5911yescharacteristic
257C22#101×D5911yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—1182no—
264C4—4721no—
265C4—2362no—
266C4—2362no—
267C2—21no—
268C2—21no—
269C2—1184no—
270C2#324×D5911yescharacteristic
271C2#2142576411yescharacteristic
272C2#2142578211yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1324C2
socleC22×C59C2×#99
prime radicalC23×D59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1324D59
hypocenterC59#11448
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422040C2semidirect
5. C59×#1324C2semidirect
24. #14273684C8semidirect
64. C42×C59C2×C8semidirect
75. C4×Dic59C2×C8semidirect
135. #1324D59semidirect
155. C59#11448semidirect
214. C42C8×D59semidirect

Maximal subgroups

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

subgroupindex
#214267212
#324×D592
#214220412
#214220402
C59×#13242
#1144859