Finite Group Databasebeta

← #28476465

Subgroups — #28476465

Order 15104.

290 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425761C211yesmaximal, characteristic
2#21424925C211yesmaximal, characteristic
3C59×#2132C211yesmaximal, characteristic
4#16837426C411yescharacteristic, complemented
5C2×#14273654—21no—
6C2×#14273654—21no—
7#16837376—21no—
8C42×Dic59C411yescharacteristic, complemented
9#16837197—21no—
10#16837195C412yescomplemented
11C59×#532C2211yescharacteristic
12C59×#413—21no—
13C59×#356C412yescomplemented
14C59×#355—21no—
15C2×#11745664—21no—
16#14273686—41no—
17C2×C4×Dic59—21no—
18C2×C4×Dic59—41no—
19C4⋊(C59⋊C8)—41no—
20#14273654—42no—
21#14273654—42no—
22#14273642—22no—
23#14273642—41no—
24C2×#41×C59D411yescharacteristic
25C2×#41×C59D411yescharacteristic
26C2×M16×C59—41no—
27C4⋊Q8×C59C2×C411yescharacteristic
28#127×C59C2×C411yescharacteristic
29#124×C59D411yescharacteristic
30#124×C59Q811yescharacteristic
31#121×C59—41no—
32C4×D4×C59—41no—
33C2×C42×C59C2×C411yescharacteristic
34#101×C59—22no—
35#100×C59—41no—
36#41×C59—41no—
37#41×C59—41no—
38#41×C59—41no—
39#41×C59—41no—
40C2×Q8×C59C22⋊C411yescharacteristic
41C2×Q8×C59C4⋊C411yescharacteristic
42C2×D4×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×D4×C59C22⋊C411yescharacteristic
46C2×D4×C59C4⋊C411yescharacteristic
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59C22⋊C411yescharacteristic
51C22×Dic59—41no—
52M16×C59—42no—
53M16×C59—81no—
54C2×C8×C59—81no—
55C4⋊C4×C59—41no—
56C22⋊C4×C59—41no—
57C22⋊C4×C59—41no—
58C42×C59—21no—
59C42×C59C4211yescharacteristic
60C42×C59C4⋊C411yescharacteristic
61C22⋊Dic59—42no—
62C4⋊Dic59—81no—
63C4×Dic59—22no—
64C4×Dic59—41no—
65C4×Dic59—44no—
66C4×Dic59—81no—
67#11745664—41no—
68#11745664—41no—
69C2×C59⋊C8—41no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59#9911yescharacteristic
73Q8×C59—41no—
74Q8×C59—41no—
75D4×C59—41no—
76D4×C59—41no—
77D4×C59—82no—
78D4×C59—81no—
79D4×C59—81no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—42no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59#9511yescharacteristic
92C2×C4×C59#9911yescharacteristic
93C2×Dic59—42no—
94C2×Dic59—44no—
95C2×Dic59—82no—
96C8×C59—82no—
97C59⋊C8—81no—
98#7497—591nomaximal, Sylow
99C22×C59—21no—
100C22×C59—41no—
101C22×C59—41no—
102C22×C59—41no—
103C22×C59—81no—
104C22×C59—81no—
105C22×C59#34211yescharacteristic
106C4×C59—22no—
107C4×C59—21no—
108C4×C59—41no—
109C4×C59—41no—
110C4×C59—41no—
111C4×C59#35311yescharacteristic
112C4×C59#35411yescharacteristic
113Dic59—44no—
114Dic59—162no—
115#2132D5911yescharacteristic
116#2126—591no—
117#1962—591no—
118C2×C59—21no—
119C2×C59—41no—
120C2×C59—81no—
121C2×C59—81no—
122C2×C59#139311yescharacteristic
123#532C2×D5911yescharacteristic
124#423—591no—
125C2×#104—1181no—
126C2×#104—1181no—
127#413—21no—
128#410—1181no—
129C43—591no—
130#356Dic5912yescomplemented
131#355—21no—
132#354—1181no—
133#353—592no—
134C59#749711yescharacteristic, complemented, Sylow
135C2×#41D23611yescharacteristic
136C2×#41C22⋊D5911yescharacteristic
137C2×M16—41no—
138C2×M16—1181no—
139C4⋊Q8C2×Dic5911yescharacteristic
140#127C4×D5911yescharacteristic
141#124Dic11811yescharacteristic
142#124C22⋊D5911yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—2361no—
146C2×C42—1181no—
147C2×C42—2361no—
148C2×C42C4×D5911yescharacteristic
149C4⋊C8—2361no—
150#104—2362no—
151#104—2362no—
152#101—22no—
153#100—41no—
154#99—1182no—
155#99—2361no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8C4⋊Dic5911yescharacteristic
161C2×Q8C22⋊Dic5911yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4Dic59⋊C411yescharacteristic
166C2×D4(C2×D59)⋊C411yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2361no—
171C22×C4(C2×D59)⋊C411yescharacteristic
172M16—42no—
173M16—81no—
174M16—2361no—
175M16—2361no—
176C2×C8—81no—
177C2×C8—2361no—
178C4⋊C4—41no—
179C4⋊C4—4721no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—2362no—
183C42—21no—
184C42—1182no—
185C42—4721no—
186C42—2361no—
187C42—2364no—
188C42C4×Dic5911yescharacteristic
189C42Dic59⋊C411yescharacteristic
190C23—21no—
191C23—21no—
192C23#1427364111yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—4722no—
212C2×C4—2362no—
213C2×C4—2364no—
214C2×C4C59⋊#9511yescharacteristic
215C2×C4#1427364111yescharacteristic
216C8—82no—
217C8—4721no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1683716811yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—9442no—
231C4—2364no—
232C4#1683719411yescharacteristic
233C4#1683719611yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2142222011yescharacteristic
subgroupG/H
centerC4#16837194
commutatorC42×C59C42
Frattini#532C2×D59
FittingC59×#2132C2
socleC2×C59#1393
prime radical, cosocleC59×#532C22
hypercenter#2132D59
hypocenterC59#7497
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
4. #16837426C4semidirect
8. C42×Dic59C4semidirect
10. #16837195C4semidirect
13. C59×#356C4semidirect
130. #356Dic59semidirect
134. C59#7497semidirect

Maximal subgroups

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

subgroupindex
#214257612
#214249252
C59×#21322
#749759