Finite Group Databasebeta

← #28501496

Subgroups — #28501496

Order 15104.

456 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425260C211yesmaximal, characteristic
2#21423120C212yesmaximal, complemented
3#21422120C212yesmaximal, complemented
4#21422024C211yesmaximal, characteristic
5C59×#2842C211yesmaximal, characteristic
6#16838028C2211yescharacteristic
7#16837737—22no—
8#16837258—21no—
9#16837169C2211yescharacteristic
10#16837168—21no—
11#16837138—22no—
12#16837126C2212yes—
13C59×#399—21no—
14C59×#399C2212yes—
15C59×#396—22no—
16C59×#396—22no—
17C59×#379—21no—
18C59×#379C2211yescharacteristic
19#14273706—22no—
20#14273704—42no—
21C2×C22⋊Dic59—21no—
22C2×C22⋊Dic59—21no—
23C2×C22⋊Dic59—42no—
24#14273660—42no—
25#14273650—42no—
26C22⋊(C59⋊C8)—22no—
27C25×C59D411yescharacteristic
28C2×C4⋊C4×C59—22no—
29C2×C4⋊C4×C59—21no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59D412yes—
32C2×C22⋊C4×C59—21no—
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59—21no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59D411yescharacteristic
38C2×C22⋊C4×C59C2311yescharacteristic
39#95×C59—21no—
40#95×C59D412yes—
41C24×C59—21no—
42C24×C59—21no—
43C24×C59—41no—
44C24×C59—41no—
45C24×C59—82no—
46C24×C59Q1612yes—
47C24×C59C2×D411yescharacteristic
48C22×C4×C59—22no—
49C22×C4×C59—21no—
50C22×C4×C59—22no—
51C22×C4×C59C2×D412yes—
52C22×Dic59—22no—
53C22×Dic59—41no—
54C2×Dic118—22no—
55C4⋊C4×C59—42no—
56C4⋊C4×C59—82no—
57C4⋊C4×C59—82no—
58C4⋊C4×C59—81no—
59C22⋊C4×C59—41no—
60C22⋊C4×C59—41no—
61C22⋊C4×C59—82no—
62C22⋊C4×C59—82no—
63C22⋊C4×C59—82no—
64C22⋊C4×C59—81no—
65C22⋊C4×C59—81no—
66C22⋊Dic59—41no—
67C22⋊Dic59—41no—
68C22⋊Dic59—41no—
69C22⋊Dic59—41no—
70C22⋊Dic59—84no—
71C4⋊Dic59—22no—
72Dic59⋊C4—42no—
73C2×C59⋊C8—42no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—22no—
77C23×C59—42no—
78C23×C59—42no—
79C23×C59—42no—
80C23×C59—42no—
81C23×C59—44no—
82C23×C59—41no—
83C23×C59—41no—
84C23×C59—81no—
85C23×C59—82no—
86C23×C59—82no—
87C23×C59—82no—
88C23×C59—84no—
89C23×C59#12011yescharacteristic
90C23×C59C2×Q1611yescharacteristic
91C23×C59#13711yescharacteristic
92C2×C4×C59—22no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×Dic59—42no—
103C2×Dic59—82no—
104C2×Dic59—81no—
105C2×Dic59—82no—
106Dic118—82no—
107C59⋊C8—82no—
108#12218—591nomaximal, Sylow
109C22×C59—22no—
110C22×C59—21no—
111C22×C59—21no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—42no—
115C22×C59—42no—
116C22×C59—42no—
117C22×C59—44no—
118C22×C59—42no—
119C22×C59—81no—
120C22×C59—82no—
121C22×C59—82no—
122C22×C59—84no—
123C22×C59—82no—
124C22×C59#45111yescharacteristic
125C22×C59#45511yescharacteristic
126C22×C59B(4,𝔽2)11yescharacteristic
127C4×C59—42no—
128C4×C59—81no—
129C4×C59—82no—
130C4×C59—82no—
131Dic59—82no—
132Dic59—161no—
133#2842D5911yescharacteristic
134#2029—591no—
135#1614—592no—
136#1352—592no—
137#1316—591no—
138C2×C59—21no—
139C2×C59—21no—
140C2×C59—41no—
141C2×C59—41no—
142C2×C59—82no—
143C2×C59#160211yescharacteristic
144C2×C59#220012yes—
145#543—591no—
146#484—1182no—
147#399—21no—
148#399C2×D5912yes—
149#396—22no—
150#396—22no—
151#379—21no—
152#379—1181no—
153#379C2×D5911yescharacteristic
154#342—591no—
155#342—1181no—
156#328—1182no—
157#323—592no—
158C59#1221811yescharacteristic, complemented, Sylow
159C25C22⋊D5911yescharacteristic
160C22⋊Q8—1182no—
161C22⋊Q8—2362no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊D5912yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—1181no—
172C2×C22⋊C4—1181no—
173C2×C22⋊C4—2362no—
174C2×C22⋊C4C22⋊D5911yescharacteristic
175C2×C22⋊C4C22×D5911yescharacteristic
176#107—2362no—
177#103—2362no—
178C22⋊C8—1182no—
179#95—21no—
180#95C22⋊D5912yes—
181C24—21no—
182C24—21no—
183C24—41no—
184C24—41no—
185C24—82no—
186C24#1174567212yes—
187C24C2×C22⋊D5911yescharacteristic
188C2×Q8—1182no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—1182no—
193C22×C4—2361no—
194C22×C4C2×C22⋊D5912yes—
195C2×C8—2362no—
196C4⋊C4—42no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—81no—
200C4⋊C4—1182no—
201C4⋊C4—2362no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—4724no—
210C22⋊C4—2361no—
211C22⋊C4—2361no—
212C22⋊C4—2361no—
213C22⋊C4—2361no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—44no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—84no—
229C23#1427369611yescharacteristic
230C23C2×#1174567211yescharacteristic
231C23#1427375811yescharacteristic
232Q8—4722no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—4721no—
244C2×C4—4722no—
245C2×C4—4722no—
246C2×C4—2362no—
247C8—4722no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#1683755811yescharacteristic
264C22#1683758011yescharacteristic
265C22#1683758911yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—9441no—
271C4—4722no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2#2142305311yescharacteristic
278C2#2142610312yes—
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2842C2
socleC22×C59B(4,𝔽2)
prime radicalC25×C59D4
hypercenter#2842D59
hypocenterC59#12218
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2842

Complements

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

NH ≅ G/Ntype
2. #21423120C2semidirect
3. #21422120C2semidirect
158. C59#12218semidirect

Maximal subgroups

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

subgroupindex
#214252602
#214231202
#214221202
#214220242
C59×#28422
#1221859