Finite Group Databasebeta

← #28476478

Subgroups — #28476478

Order 15104.

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

normal subgroups only

This group has 270 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#21425781C211yesmaximal, characteristic
2#21425761C211yesmaximal, characteristic
3C59×#1963C211yesmaximal, characteristic
4#16837386—21no—
5#16837376—21no—
6#16837201C412yescomplemented
7#16837199—21no—
8#16837197—21no—
9#16837195C412yescomplemented
10C59×#532C2211yescharacteristic
11C59×#423C411yescharacteristic, complemented
12C2×#104×C59—21no—
13C2×#104×C59—21no—
14C59×#376C411yescharacteristic, complemented
15C2×#11745664—41no—
16#14273686—41no—
17#14273646—22no—
18#14273644—41no—
19#14273642—22no—
20#14273642—41no—
21C2×#41×C59D411yescharacteristic
22C2×#41×C59D411yescharacteristic
23C2×M16×C59—21no—
24C4⋊Q8×C59C2×C411yescharacteristic
25#127×C59C2×C411yescharacteristic
26#124×C59D411yescharacteristic
27#124×C59Q811yescharacteristic
28#121×C59—41no—
29C4×D4×C59—41no—
30C2×C42×C59—21no—
31C2×C42×C59C2×C411yescharacteristic
32C4⋊C8×C59—41no—
33#104×C59—42no—
34#104×C59—42no—
35#95×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×C59—41no—
51C22×C4×C59C22⋊C411yescharacteristic
52M16×C59—41no—
53M16×C59—41no—
54C2×C8×C59—41no—
55C4⋊C4×C59—41no—
56C22⋊C4×C59—41no—
57C22⋊C4×C59—41no—
58C42×C59—21no—
59C42×C59—22no—
60C42×C59—41no—
61C42×C59C4211yescharacteristic
62C42×C59C4⋊C411yescharacteristic
63C22⋊Dic59—42no—
64C4⋊Dic59—81no—
65C4×Dic59—81no—
66#11745664—42no—
67#11745664—81no—
68C2×C59⋊C8—81no—
69C23×C59—21no—
70C23×C59—21no—
71C23×C59#9911yescharacteristic
72Q8×C59—41no—
73Q8×C59—41no—
74D4×C59—41no—
75D4×C59—41no—
76D4×C59—81no—
77D4×C59—81no—
78D4×C59—82no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59—81no—
91C2×C4×C59—82no—
92C2×C4×C59#9511yescharacteristic
93C2×C4×C59#9911yescharacteristic
94C2×Dic59—82no—
95C8×C59—81no—
96C59⋊C8—82no—
97#7503—591nomaximal, Sylow
98C22×C59—21no—
99C22×C59—41no—
100C22×C59—41no—
101C22×C59—41no—
102C22×C59—81no—
103C22×C59—81no—
104C22×C59#34211yescharacteristic
105C4×C59—21no—
106C4×C59—41no—
107C4×C59—41no—
108C4×C59—41no—
109C4×C59—41no—
110C4×C59—82no—
111C4×C59#35311yescharacteristic
112C4×C59#35411yescharacteristic
113Dic59—162no—
114#2132—591no—
115#2126—591no—
116#1963D5911yescharacteristic
117C2×C59—21no—
118C2×C59—41no—
119C2×C59—81no—
120C2×C59—81no—
121C2×C59#139311yescharacteristic
122#532C2×D5911yescharacteristic
123#423Dic5911yescharacteristic, complemented
124C2×#104—21no—
125C2×#104—21no—
126#413—1181no—
127#410—1181no—
128#376Dic5911yescharacteristic, complemented
129#356—592no—
130#355—1181no—
131#354—1181no—
132#353—592no—
133C59#750311yescharacteristic, complemented, Sylow
134C2×#41C22⋊D5911yescharacteristic
135C2×#41C22⋊D5911yescharacteristic
136C2×M16—21no—
137C2×M16—2361no—
138C4⋊Q8C4×D5911yescharacteristic
139#127C4×D5911yescharacteristic
140#124Dic11811yescharacteristic
141#124D23611yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—2361no—
145C2×C42—21no—
146C2×C42C2×Dic5911yescharacteristic
147C4⋊C8—41no—
148#104—42no—
149#104—42no—
150#101—1182no—
151#100—2361no—
152#99—1182no—
153#99—2361no—
154#95—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8Dic59⋊C411yescharacteristic
160C2×Q8(C2×D59)⋊C411yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4Dic59⋊C411yescharacteristic
165C2×D4(C2×D59)⋊C411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4C22⋊Dic5911yescharacteristic
171M16—41no—
172M16—41no—
173M16—4721no—
174M16—2362no—
175C2×C8—41no—
176C2×C8—4721no—
177C4⋊C4—41no—
178C4⋊C4—4721no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2362no—
182C42—21no—
183C42—22no—
184C42—41no—
185C42—4721no—
186C42C4×Dic5911yescharacteristic
187C42C4⋊Dic5911yescharacteristic
188C23—21no—
189C23—21no—
190C23#1427364211yescharacteristic
191Q8—41no—
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—81no—
196D4—81no—
197D4—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—4722no—
212C2×C4C59⋊#9511yescharacteristic
213C2×C4#1427364211yescharacteristic
214C8—81no—
215C8—4722no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1683716911yescharacteristic
223C4—21no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4—9442no—
230C4#1683719511yescharacteristic
231C4#1683719711yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2142222111yescharacteristic
subgroupG/H
centerC4#16837197
commutatorC42×C59C42
Frattini#532C2×D59
FittingC59×#1963C2
socleC2×C59#1393
prime radical, cosocleC59×#532C22
hypercenter#1963D59
hypocenterC59#7503
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1963

Complements

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

NH ≅ G/Ntype
6. #16837201C4semidirect
9. #16837195C4semidirect
11. C59×#423C4semidirect
14. C59×#376C4semidirect
123. #423Dic59semidirect
128. #376Dic59semidirect
133. C59#7503semidirect

Maximal subgroups

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

subgroupindex
#214257812
#214257612
C59×#19632
#750359