Finite Group Databasebeta

← #28476432

Subgroups — #28476432

Order 15104.

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

normal subgroups only

This group has 286 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#21424480C211yesmaximal, characteristic
3C59×#1525C211yesmaximal, characteristic
4#16838169C411yescharacteristic, complemented
5#16837419—21no—
6#16837376—21no—
7#16837376C411yescharacteristic, complemented
8#16837197—21no—
9#16837195C412yescomplemented
10#16837162—21no—
11C59×#532C2211yescharacteristic
12C4×M16×C59—21no—
13C59×#329—21no—
14C59×#329C412yescomplemented
15#14273734—42no—
16C2×#11745664—21no—
17C2×#11745664—41no—
18#14273686—21no—
19#14273686—41no—
20#14273654—42no—
21#14273642—22no—
22#14273642—41no—
23#14273642—41no—
24C2×#41×C59C2×C411yescharacteristic
25C2×#41×C59D411yescharacteristic
26C2×M16×C59—41no—
27C4⋊Q8×C59Q811yescharacteristic
28#127×C59C2×C411yescharacteristic
29#124×C59C2×C411yescharacteristic
30#124×C59D411yescharacteristic
31#121×C59—41no—
32C4×D4×C59—41no—
33C2×C42×C59D411yescharacteristic
34#97×C59—22no—
35C4×C8×C59—41no—
36#41×C59—41no—
37#41×C59—41no—
38#41×C59—41no—
39#41×C59—41no—
40C2×Q8×C59C4⋊C411yescharacteristic
41C2×Q8×C59C4⋊C411yescharacteristic
42C2×D4×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×D4×C59C4211yescharacteristic
46C2×D4×C59C22⋊C411yescharacteristic
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59C22⋊C411yescharacteristic
51M16×C59—44no—
52C2×C8×C59—42no—
53C4⋊C4×C59—41no—
54C22⋊C4×C59—41no—
55C22⋊C4×C59—41no—
56C42×C59—21no—
57C42×C59C22⋊C411yescharacteristic
58C42×C59C4⋊C411yescharacteristic
59C22⋊Dic59—42no—
60C22⋊Dic59—41no—
61C4⋊Dic59—41no—
62C4⋊Dic59—81no—
63C4×Dic59—41no—
64C4×Dic59—81no—
65#11745664—22no—
66#11745664—42no—
67#11745664—41no—
68C2×C59⋊C8—41no—
69C2×C59⋊C8—42no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59#9911yescharacteristic
73Q8×C59—41no—
74Q8×C59—41no—
75D4×C59—41no—
76D4×C59—41no—
77D4×C59—42no—
78D4×C59—82no—
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—81no—
94C2×Dic59—82no—
95C8×C59—44no—
96C59⋊C8—42no—
97#7486—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—22no—
106C4×C59—21no—
107C4×C59—41no—
108C4×C59—41no—
109C4×C59—41no—
110C4×C59AΣL(1,𝔽16)11yescharacteristic
111C4×C59#35211yescharacteristic
112Dic59—161no—
113Dic59—162no—
114#2126—591no—
115#1884—591no—
116#1525D5911yescharacteristic
117C2×C59—21no—
118C2×C59—41no—
119C2×C59—81no—
120C2×C59—81no—
121C2×C59#139011yescharacteristic
122#568—591no—
123#532C2×D5911yescharacteristic
124#421—1181no—
125#410—591no—
126#410—1181no—
127C4×M16—21no—
128#354—1181no—
129#353—592no—
130#339—1181no—
131#329—21no—
132#329Dic5912yescomplemented
133C59#748611yescharacteristic, complemented, Sylow
134C2×#41C4×D5911yescharacteristic
135C2×#41D23611yescharacteristic
136#131—2362no—
137C2×M16—41no—
138C2×M16—1181no—
139C2×M16—2361no—
140C4⋊Q8Dic11811yescharacteristic
141#127C4×D5911yescharacteristic
142#124C2×Dic5911yescharacteristic
143#124C22⋊D5911yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—1181no—
147#117—2361no—
148C2×C42C22⋊D5911yescharacteristic
149#104—2362no—
150#99—1182no—
151#99—2361no—
152#99—2361no—
153#97—22no—
154C4×C8—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8Dic59⋊C411yescharacteristic
160C2×Q8C4⋊Dic5911yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C4×Dic5911yescharacteristic
165C2×D4(C2×D59)⋊C411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4(C2×D59)⋊C411yescharacteristic
170M16—44no—
171M16—1182no—
172M16—2362no—
173M16—2361no—
174C2×C8—42no—
175C2×C8—2361no—
176C2×C8—2362no—
177C4⋊C4—41no—
178C4⋊C4—4721no—
179C4⋊C4—2361no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—2362no—
183C22⋊C4—2361no—
184C42—21no—
185C42—4721no—
186C42—2361no—
187C42Dic59⋊C411yescharacteristic
188C42C22⋊Dic5911yescharacteristic
189C23—21no—
190C23—21no—
191C23#1427364111yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—82no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—4721no—
211C2×C4—4722no—
212C2×C4C59⋊#9511yescharacteristic
213C2×C4#1427364111yescharacteristic
214C8—44no—
215C8—2362no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1683716811yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—9441no—
229C4—9442no—
230C4#1683719011yescharacteristic
231C4#1683719311yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2142221511yescharacteristic
subgroupG/H
centerC4#16837190
commutatorC2×D4×C59C42
Frattini#532C2×D59
FittingC59×#1525C2
socleC2×C59#1390
prime radical, cosocleC59×#532C22
hypercenter#1525D59
hypocenterC59#7486
LCSG ▷ C2×D4×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1525

Complements

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

NH ≅ G/Ntype
4. #16838169C4semidirect
7. #16837376C4semidirect
9. #16837195C4semidirect
14. C59×#329C4semidirect
132. #329Dic59semidirect
133. C59#7486semidirect

Maximal subgroups

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

subgroupindex
#214257612
#214244802
C59×#15252
#748659