Finite Group Databasebeta

← #28477972

Subgroups — #28477972

Order 15104.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#21430468C211yesmaximal, characteristic
2#21425660C212yesmaximal, complemented
3#21425629C211yesmaximal, characteristic
4#21421920C212yesmaximal, complemented
5C59×#1270C211yesmaximal, characteristic
6C2×#14273704—22no—
7C2×#14273658—21no—
8C2×#14273648—21no—
9#16837325—22no—
10#16837322—21no—
11#16837300C2212yes—
12#16837282—21no—
13#16837272C2211yescharacteristic
14#16837165—22no—
15#16837165—22no—
16#16837157C2211yescharacteristic
17C59×#390C2211yescharacteristic
18C59×#336C2212yes—
19C22×Dic118—21no—
20C23×Dic59—21no—
21C22×C59⋊C8—21no—
22#14273704—44no—
23#14273704—44no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59D412yes—
27C2×C4⋊Dic59D412yes—
28C2×Dic59⋊C4—22no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32#14273658—42no—
33#14273648—42no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C22×D4×C59D411yescharacteristic
37C22×C8×C59—22no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C42×C59C2311yescharacteristic
41C24×C59—21no—
42C2×D4×C59—21no—
43C2×D4×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59Q1612yes—
46C22×C4×C59—21no—
47C22×C4×C59C2×D412yes—
48C22×C4×C59C2×D411yescharacteristic
49C22×Dic59—22no—
50C22×Dic59—22no—
51C22×Dic59—44no—
52C22×Dic59—41no—
53C22×Dic59—81no—
54C2×Dic118—44no—
55C2×Dic118—81no—
56C2×C8×C59—42no—
57C2×C8×C59—42no—
58C2×C8×C59—42no—
59C4⋊C4×C59—41no—
60C22⋊C4×C59—42no—
61C22⋊C4×C59—42no—
62C42×C59—41no—
63C22⋊Dic59—44no—
64C22⋊Dic59—81no—
65C4⋊Dic59—22no—
66C4⋊Dic59—42no—
67C4⋊Dic59—41no—
68C4⋊Dic59—41no—
69C4⋊Dic59—44no—
70C4⋊Dic59SD1614yes—
71Dic59⋊C4—44no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74C2×C59⋊C8—22no—
75C2×C59⋊C8—41no—
76C2×C59⋊C8—41no—
77C23×C59—22no—
78C23×C59—42no—
79C23×C59—41no—
80C23×C59—42no—
81C23×C59—41no—
82C23×C59#12011yescharacteristic
83D4×C59—42no—
84D4×C59—81no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—42no—
90C2×C4×C59—41no—
91C2×C4×C59C2×SD1612yes—
92C2×C4×C59C2×Q1611yescharacteristic
93C2×C4×C59AGL(1,ℤ/8)12yes—
94C2×C4×C59#13711yescharacteristic
95C2×Dic59—24no—
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—84no—
102C2×Dic59—81no—
103C2×Dic59—82no—
104Dic118—84no—
105C8×C59—82no—
106C59⋊C8—42no—
107#7826—591nomaximal, Sylow
108C22×C59—42no—
109C22×C59—44no—
110C22×C59—82no—
111C22×C59—81no—
112C22×C59—81no—
113C22×C59#45012yes—
114C22×C59#45111yescharacteristic
115C22×C59#45311yescharacteristic
116C22×C59#45412yes—
117C22×C59#45811yescharacteristic
118C4×C59—21no—
119C4×C59—21no—
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—81no—
123Dic59—44no—
124Dic59—82no—
125Dic59—161no—
126#2683—591no—
127#2100—592no—
128#2094—591no—
129#1271—592no—
130#1270D5911yescharacteristic
131C2×C59—82no—
132C2×C59#161411yescharacteristic
133C2×C59#161612yes—
134C2×C59#162911yescharacteristic
135C2×C59#163712yes—
136C2×C59#165711yescharacteristic
137C2×C22⋊Q8—1182no—
138C2×#106—1181no—
139C2×#102—1181no—
140#396—1182no—
141#395—1181no—
142#390C2×D5911yescharacteristic
143#389—592no—
144#386—1181no—
145#384—591no—
146#340—1182no—
147#340—1182no—
148#336—591no—
149#336C2×D5912yes—
150C59#782611yescharacteristic, complemented, Sylow
151C22×Q8—1181no—
152C22×D4D23611yescharacteristic
153C23×C4—1181no—
154C22×C8—22no—
155C22×C8—1181no—
156C22⋊Q8—2364no—
157C22⋊Q8—2364no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—592no—
160C2×C4⋊C4—1182no—
161C2×C4⋊C4—1182no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4—1182no—
164C2×C4⋊C4D23611yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—1181no—
167C2×C22⋊C4—1182no—
168C2×C42C22×D5911yescharacteristic
169#106—2362no—
170#102—2362no—
171#95—1181no—
172#95—1182no—
173C24—21no—
174C2×Q8—4721no—
175C2×Q8—2364no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Dic23612yes—
180C22×C4—21no—
181C22×C4—1182no—
182C22×C4—1182no—
183C22×C4—4721no—
184C22×C4—2364no—
185C22×C4—2361no—
186C22×C4C2×D23611yescharacteristic
187C22×C4D4×D5912yes—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—1182no—
192C2×C8—2361no—
193C2×C8—2361no—
194C4⋊C4—41no—
195C4⋊C4—594no—
196C4⋊C4—1182no—
197C4⋊C4—2361no—
198C4⋊C4—2361no—
199C4⋊C4—2364no—
200C4⋊C4—2362no—
201C4⋊C4—2364no—
202C4⋊C4—2362no—
203C4⋊C4—2362no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—4721no—
207C22⋊C4—2364no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1427369711yescharacteristic
215Q8—4724no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—1184no—
225C2×C4—4721no—
226C2×C4—4722no—
227C2×C4—4724no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4—2362no—
231C2×C4—2362no—
232C2×C4—2362no—
233C2×C4SD16×D5912yes—
234C2×C4C2×Dic23611yescharacteristic
235C2×C4#1427375112yes—
236C2×C4#1427376011yescharacteristic
237C8—82no—
238C8—2362no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1683755412yes—
245C22#1683756011yescharacteristic
246C22#1683757211yescharacteristic
247C22#1683757612yes—
248C22#1683759311yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—9441no—
255C4—4722no—
256C4—2364no—
257C2—82no—
258C2#2142312311yescharacteristic
259C2#2142313412yes—
260C2#2142321311yescharacteristic
261C2#2142326412yes—
262C2#2142339511yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#1270D59
hypocenterC59#7826
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21425660C2semidirect
4. #21421920C2semidirect
150. C59#7826semidirect

Maximal subgroups

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

subgroupindex
#214304682
#214256602
#214256292
#214219202
C59×#12702
#782659