Finite Group Databasebeta

← #28482140

Subgroups — #28482140

Order 15104.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#21426593C211yesmaximal, characteristic
2#21422408C212yesmaximal
3#21421929C211yesmaximal, characteristic, complemented
4#21421924C212yesmaximal, complemented
5C59×#1276C211yesmaximal, characteristic, complemented
6C2×#14273684C2211yescharacteristic
7#16837455C2212yes—
8#16837293—22no—
9#16837286—22no—
10C4×Dic59⋊C4C412yes—
11C4×(C2×D59)⋊C4C412yes—
12#16837248C2211yescharacteristic
13#16837127—22no—
14#16837125—22no—
15C59×#431C2212yes—
16C43×C59C2211yescharacteristic
17C22×C4×D59C2×C411yescharacteristic
18#14273684—22no—
19#14273684—21no—
20#14273684—21no—
21C2×Dic59⋊C4—22no—
22C2×Dic59⋊C4D412yes—
23C2×(C2×D59)⋊C4—22no—
24C2×(C2×D59)⋊C4D412yes—
25C2×C4×Dic59C2×C412yes—
26C2×C4×Dic59C2×C411yescharacteristic
27C4⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C4×C59⋊C8—22no—
30C59⋊#95—22no—
31C59⋊#95—22no—
32C59⋊#95—22no—
33C2×C42×C59—22no—
34C2×C42×C59C2×C412yes—
35C2×C42×C59C2311yescharacteristic
36C4⋊C8×C59—22no—
37C22⋊C8×C59—22no—
38C4×C8×C59—22no—
39C23×D59C22⋊C411yescharacteristic
40C22×C4×C59—22no—
41C22×C4×C59C22×C411yescharacteristic
42C22×C4×C59C2×D412yes—
43C22×Dic59—22no—
44C22×Dic59C22⋊C411yescharacteristic
45C22×Dic59C22⋊C412yes—
46C2×C4×D59—21no—
47C2×C4×D59—21no—
48C2×C4×D59—22no—
49C2×C4×D59—22no—
50C2×C8×C59—42no—
51C42×C59—22no—
52C42×C59—22no—
53C42×C59—44no—
54C42×C59#4112yes—
55C42×C59#4111yescharacteristic
56C42×C59#4111yescharacteristic
57(C2×D59)⋊C4—22no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—44no—
60Dic59⋊C4—22no—
61Dic59⋊C4—22no—
62Dic59⋊C4—44no—
63C4×Dic59—21no—
64C4×Dic59—21no—
65C4×Dic59—22no—
66C4×Dic59—22no—
67C2×C59⋊C8—42no—
68C23×C59C2×C22⋊C411yescharacteristic
69C22×D59—21no—
70C22×D59—22no—
71C22×D59—42no—
72C2×C4×C59—24no—
73C2×C4×C59—42no—
74C2×C4×C59—42no—
75C2×C4×C59—42no—
76C2×C4×C59#11711yescharacteristic
77C2×C4×C59#11711yescharacteristic
78C2×C4×C59#12312yes—
79C2×C4×C59#12312yes—
80C2×C4×C59#13112yes—
81C2×C4×C59#13112yes—
82C2×Dic59—22no—
83C2×Dic59—21no—
84C2×Dic59—22no—
85C2×Dic59—42no—
86C2×Dic59—42no—
87C2×Dic59—42no—
88C2×Dic59—42no—
89C4×D59—41no—
90C4×D59—42no—
91C4×D59—41no—
92C8×C59—82no—
93C59⋊C8—82no—
94#8732—591nomaximal, Sylow
95C22×C59—22no—
96C22×C59#38111yescharacteristic
97C22×C59#40812yes—
98C2×D59—41no—
99C2×D59—42no—
100C2×D59—82no—
101C4×C59—22no—
102C4×C59—22no—
103C4×C59—44no—
104C4×C59#43311yescharacteristic
105C4×C59#43312yes—
106C4×C59#43311yescharacteristic
107Dic59—41no—
108Dic59—42no—
109Dic59—82no—
110#2298—591no—
111#1458—592no—
112#1276—591no—
113#1276D5911yescharacteristic, complemented
114#1274—592no—
115C2×C59—22no—
116C2×C59#181811yescharacteristic
117C2×C59#182712yes—
118D59—81no—
119C2×#117—591no—
120#431—592no—
121#431C2×D5912yes—
122#388—1182no—
123#387—1182no—
124C4×C4⋊C4—592no—
125C4×C22⋊C4—592no—
126#376—591no—
127C43C2×D5911yescharacteristic
128#324—1182no—
129#323—1182no—
130C59#873211yescharacteristic, complemented, Sylow
131C23×C4—591no—
132#117—1182no—
133#117—1181no—
134#117—1181no—
135C2×C4⋊C4—592no—
136C2×C4⋊C4—1182no—
137C2×C22⋊C4—592no—
138C2×C22⋊C4—1182no—
139C2×C42—22no—
140C2×C42—591no—
141C2×C42—592no—
142C2×C42C4×D5912yes—
143C2×C42C22×D5911yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—1182no—
146C22⋊C8—22no—
147C22⋊C8—1182no—
148C4×C8—22no—
149C4×C8—1182no—
150#95—1182no—
151#95—1182no—
152#95—1182no—
153C24—591no—
154C22×C4—22no—
155C22×C4—591no—
156C22×C4—592no—
157C22×C4—1181no—
158C22×C4—1181no—
159C22×C4—1182no—
160C22×C4—1182no—
161C22×C4—1182no—
162C22×C4C2×C4×D5911yescharacteristic
163C22×C4D4×D5912yes—
164C2×C8—42no—
165C2×C8—2362no—
166C4⋊C4—1182no—
167C4⋊C4—1182no—
168C4⋊C4—2364no—
169C22⋊C4—1182no—
170C22⋊C4—1182no—
171C22⋊C4—2364no—
172C42—22no—
173C42—22no—
174C42—44no—
175C42—1181no—
176C42—1181no—
177C42—1182no—
178C42—1182no—
179C42#1174568512yes—
180C42#1174568711yescharacteristic
181C42#1174568911yescharacteristic
182C23—1182no—
183C23—1181no—
184C23—2362no—
185C23C22⋊C4×D5911yescharacteristic
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—1182no—
191C2×C4—1181no—
192C2×C4—1182no—
193C2×C4—2362no—
194C2×C4—2362no—
195C2×C4—2362no—
196C2×C4—2361no—
197C2×C4—2362no—
198C2×C4—2361no—
199C2×C4—2362no—
200C2×C4#1427368411yescharacteristic
201C2×C4#1427368511yescharacteristic
202C2×C4#1427370912yes—
203C2×C4#1427371212yes—
204C2×C4#1427373312yes—
205C2×C4#1427373512yes—
206C8—82no—
207C8—4722no—
208C22—22no—
209C22—4722no—
210C22—2361no—
211C22—2362no—
212C22#1683726411yescharacteristic
213C22#1683736712yes—
214C4—22no—
215C4—22no—
216C4—44no—
217C4—4722no—
218C4—2361no—
219C4—2362no—
220C4#1683746111yescharacteristic
221C4#1683746212yes—
222C4#1683746411yescharacteristic
223C2—22no—
224C2—4721no—
225C2#2142416611yescharacteristic
226C2#2142419912yes—
subgroupG/H
centerC2×C4#14273685
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1276C2
socleC22×C59#381
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1276D59
hypocenterC59#8732
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #21421929C2semidirect
4. #21421924C2semidirect
5. C59×#1276C2semidirect
113. #1276D59semidirect
130. C59#8732semidirect

Maximal subgroups

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

subgroupindex
#214265932
#214224082
#214219292
#214219242
C59×#12762
#873259