Finite Group Databasebeta

← #28476455

Subgroups — #28476455

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#21425761C211yesmaximal, characteristic
2#21422731C211yesmaximal, characteristic
3C59×#1884C211yesmaximal, characteristic
4#16837459—21no—
5#16837376—21no—
6#16837197—21no—
7#16837195C412yescomplemented
8#16837142—21no—
9#16837142C412yescomplemented
10C59×#568C411yescharacteristic, complemented
11C59×#532C2211yescharacteristic
12C59×#421—21no—
13C59×#410C411yescharacteristic, complemented
14C59×#339—21no—
15#14273686—41no—
16C4⋊(C59⋊C8)—41no—
17#14273642—22no—
18#14273642—41no—
19C22⋊(C59⋊C8)—41no—
20#14273638—22no—
21C2×#41×C59C2×C411yescharacteristic
22C2×#41×C59D411yescharacteristic
23#131×C59—42no—
24C2×M16×C59—21no—
25C2×M16×C59—41no—
26C4⋊Q8×C59Q811yescharacteristic
27#127×C59C2×C411yescharacteristic
28#124×C59C2×C411yescharacteristic
29#124×C59D411yescharacteristic
30#121×C59—41no—
31C4×D4×C59—41no—
32#117×C59—21no—
33C2×C42×C59D411yescharacteristic
34#104×C59—42no—
35#99×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—22no—
52M16×C59—41no—
53M16×C59—42no—
54C2×C8×C59—41no—
55C2×C8×C59—42no—
56C4⋊C4×C59—41no—
57C4⋊C4×C59—41no—
58C22⋊C4×C59—41no—
59C22⋊C4×C59—41no—
60C22⋊C4×C59—41no—
61C42×C59—21no—
62C42×C59—41no—
63C42×C59C22⋊C411yescharacteristic
64C42×C59C4⋊C411yescharacteristic
65C22⋊Dic59—42no—
66C4⋊Dic59—81no—
67C4×Dic59—81no—
68C2×C59⋊C8—42no—
69C23×C59—21no—
70C23×C59—21no—
71C23×C59#9911yescharacteristic
72Q8×C59—41no—
73Q8×C59—41no—
74D4×C59—41no—
75D4×C59—41no—
76D4×C59—42no—
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—81no—
90C2×C4×C59—81no—
91C2×C4×C59#9511yescharacteristic
92C2×C4×C59#9911yescharacteristic
93C2×Dic59—82no—
94C8×C59—42no—
95C59⋊C8—82no—
96#7494—591nomaximal, Sylow
97C22×C59—21no—
98C22×C59—41no—
99C22×C59—41no—
100C22×C59—41no—
101C22×C59—81no—
102C22×C59—81no—
103C22×C59#34211yescharacteristic
104C4×C59—21no—
105C4×C59—41no—
106C4×C59—41no—
107C4×C59—41no—
108C4×C59—41no—
109C4×C59—161no—
110C4×C59AΣL(1,𝔽16)11yescharacteristic
111C4×C59#35211yescharacteristic
112Dic59—162no—
113#2126—591no—
114#1884D5911yescharacteristic
115#1536—591no—
116C2×C59—21no—
117C2×C59—41no—
118C2×C59—81no—
119C2×C59—81no—
120C2×C59#139011yescharacteristic
121#568Dic5911yescharacteristic, complemented
122#532C2×D5911yescharacteristic
123#432—1181no—
124#421—21no—
125#410—1181no—
126#410Dic5911yescharacteristic, complemented
127#354—1181no—
128#353—592no—
129#339—21no—
130#329—592no—
131#329—1181no—
132C59#749411yescharacteristic, complemented, Sylow
133C2×#41C2×Dic5911yescharacteristic
134C2×#41C22⋊D5911yescharacteristic
135#131—42no—
136C2×M16—21no—
137C2×M16—41no—
138C4⋊Q8Dic11811yescharacteristic
139#127C4×D5911yescharacteristic
140#124C4×D5911yescharacteristic
141#124D23611yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145#117—2361no—
146C2×C42C22⋊D5911yescharacteristic
147C4⋊C8—2361no—
148#104—42no—
149#99—41no—
150#99—1182no—
151#99—2361no—
152C22⋊C8—2361no—
153#97—1182no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8Dic59⋊C411yescharacteristic
159C2×Q8C4⋊Dic5911yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C4×Dic5911yescharacteristic
164C2×D4(C2×D59)⋊C411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C22⋊Dic5911yescharacteristic
169M16—22no—
170M16—41no—
171M16—42no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—2362no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—4721no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2362no—
182C42—21no—
183C42—41no—
184C42—4721no—
185C42Dic59⋊C411yescharacteristic
186C42(C2×D59)⋊C411yescharacteristic
187C23—21no—
188C23—21no—
189C23#1427364211yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—4722no—
210C2×C4C59⋊#9511yescharacteristic
211C2×C4#1427364211yescharacteristic
212C8—42no—
213C8—4722no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#1683716911yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—161no—
227C4—9442no—
228C4#1683718911yescharacteristic
229C4#1683719211yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2142221411yescharacteristic
subgroupG/H
centerC2#21422214
commutatorC2×D4×C59C42
Frattini#532C2×D59
FittingC59×#1884C2
socleC2×C59#1390
prime radical, cosocleC59×#532C22
hypercenter#1884D59
hypocenterC59#7494
LCSG ▷ C2×D4×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
7. #16837195C4semidirect
9. #16837142C4semidirect
10. C59×#568C4semidirect
13. C59×#410C4semidirect
121. #568Dic59semidirect
126. #410Dic59semidirect
132. C59#7494semidirect

Maximal subgroups

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

subgroupindex
#214257612
#214227312
C59×#18842
#749459