Finite Group Databasebeta

← #28501491

Subgroups — #28501491

Order 15104.

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

normal subgroups only

This group has 484 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#21423052C212yesmaximal, complemented
3#21422117C212yesmaximal, complemented
4#21422024C211yesmaximal, characteristic
5C59×#2840C211yesmaximal, characteristic
6#16838028C2211yescharacteristic
7#16837558—22no—
8#16837258—21no—
9#16837169C2211yescharacteristic
10#16837168—21no—
11#16837138—22no—
12#16837126C2212yes—
13C59×#393—22no—
14C59×#393—21no—
15C59×#393—22no—
16C59×#393C2212yes—
17C59×#379—21no—
18C59×#379C2211yescharacteristic
19C2×#11745672—42no—
20#14273706—22no—
21#14273704—42no—
22C2×C22⋊Dic59—21no—
23C2×C22⋊Dic59—21no—
24C2×C22⋊Dic59—42no—
25#14273651—42no—
26C22⋊(C59⋊C8)—22no—
27C25×C59D411yescharacteristic
28C22×Q8×C59—21no—
29C22×Q8×C59D412yes—
30C2×C22⋊C4×C59—21no—
31C2×C22⋊C4×C59—22no—
32C2×C22⋊C4×C59—21no—
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59D411yescharacteristic
36C2×C22⋊C4×C59C2311yescharacteristic
37#95×C59—21no—
38#95×C59—22no—
39#95×C59—22no—
40#95×C59D412yes—
41C24×C59—21no—
42C24×C59—21no—
43C24×C59—41no—
44C24×C59—41no—
45C24×C59—82no—
46C24×C59Q1612yescomplemented
47C24×C59C2×D411yescharacteristic
48C2×Q8×C59—42no—
49C2×Q8×C59—82no—
50C2×Q8×C59—82no—
51C2×Q8×C59—81no—
52C22×C4×C59—22no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59C2×D412yes—
56C22×Dic59—22no—
57C22×Dic59—41no—
58C2×Dic118—22no—
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—
71#11745672—84no—
72C4⋊Dic59—22no—
73Dic59⋊C4—42no—
74C2×C59⋊C8—42no—
75C23×C59—21no—
76C23×C59—21no—
77C23×C59—22no—
78C23×C59—42no—
79C23×C59—42no—
80C23×C59—42no—
81C23×C59—42no—
82C23×C59—44no—
83C23×C59—41no—
84C23×C59—41no—
85C23×C59—81no—
86C23×C59—82no—
87C23×C59—82no—
88C23×C59—82no—
89C23×C59—84no—
90C23×C59#12011yescharacteristic
91C23×C59C2×Q1611yescharacteristic
92C23×C59#13711yescharacteristic
93Q8×C59—44no—
94Q8×C59—82no—
95Q8×C59—84no—
96C2×C4×C59—22no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×Dic59—42no—
107C2×Dic59—82no—
108C2×Dic59—81no—
109C2×Dic59—82no—
110Dic118—82no—
111C59⋊C8—82no—
112#12217—591nomaximal, Sylow
113C22×C59—22no—
114C22×C59—21no—
115C22×C59—21no—
116C22×C59—41no—
117C22×C59—41no—
118C22×C59—42no—
119C22×C59—42no—
120C22×C59—42no—
121C22×C59—44no—
122C22×C59—42no—
123C22×C59—81no—
124C22×C59—82no—
125C22×C59—82no—
126C22×C59—84no—
127C22×C59—82no—
128C22×C59#45111yescharacteristic
129C22×C59#45511yescharacteristic
130C22×C59B(4,𝔽2)11yescharacteristic
131C4×C59—42no—
132C4×C59—81no—
133C4×C59—82no—
134C4×C59—82no—
135Dic59—82no—
136Dic59—161no—
137#2840D5911yescharacteristic
138#2029—591no—
139#1602—592no—
140#1351—592no—
141#1316—591no—
142C2×C59—21no—
143C2×C59—21no—
144C2×C59—41no—
145C2×C59—41no—
146C2×C59—82no—
147C2×C59#160211yescharacteristic
148C2×C59#219912yes—
149#543—591no—
150#451—1182no—
151#393—22no—
152#393—21no—
153#393—22no—
154#393C2×D5912yes—
155#379—21no—
156#379—1181no—
157#379C2×D5911yescharacteristic
158#342—591no—
159#342—1181no—
160#328—1182no—
161#323—592no—
162C59#1221711yescharacteristic, complemented, Sylow
163C25C22⋊D5911yescharacteristic
164C22×Q8—21no—
165C22×Q8C22⋊D5912yes—
166C2×Q16—2362no—
167C22⋊Q8—1182no—
168C22⋊Q8—2362no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1181no—
175C2×C22⋊C4—1181no—
176C2×C22⋊C4—2362no—
177C2×C22⋊C4C22⋊D5911yescharacteristic
178C2×C22⋊C4C22×D5911yescharacteristic
179#103—2362no—
180C22⋊C8—1182no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95C22⋊D5912yes—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24#1174567212yescomplemented
191C24C2×C22⋊D5911yescharacteristic
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—81no—
196C2×Q8—1182no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—1182no—
201C22×C4—2361no—
202C22×C4C2×C22⋊D5912yes—
203Q16—4724no—
204C2×C8—2362no—
205C4⋊C4—1182no—
206C4⋊C4—2362no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—4724no—
215C22⋊C4—2361no—
216C22⋊C4—2361no—
217C22⋊C4—2361no—
218C22⋊C4—2361no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23#1427369611yescharacteristic
235C23C2×#1174567211yescharacteristic
236C23#1427375811yescharacteristic
237Q8—44no—
238Q8—82no—
239Q8—84no—
240Q8—4722no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—4721no—
252C2×C4—4722no—
253C2×C4—4722no—
254C2×C4—2362no—
255C8—4722no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22#1683755811yescharacteristic
272C22#1683758011yescharacteristic
273C22#1683758911yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—9441no—
279C4—4722no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2#2142305311yescharacteristic
286C2#2142609712yes—
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2840C2
socleC22×C59B(4,𝔽2)
prime radicalC25×C59D4
hypercenter#2840D59
hypocenterC59#12217
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #21423052C2semidirect
3. #21422117C2semidirect
46. C24×C59Q16semidirect
162. C59#12217semidirect
190. C24#11745672semidirect

Maximal subgroups

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

subgroupindex
#214252602
#214230522
#214221172
#214220242
C59×#28402
#1221759