Finite Group Databasebeta

← #28476418

Subgroups — #28476418

Order 15104.

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

normal subgroups only

This group has 278 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#21424480C211yesmaximal, characteristic
2#21422831C211yesmaximal, characteristic
3C59×#2126C211yesmaximal, characteristic
4#16838169C411yescharacteristic, complemented
5#16837426—21no—
6#16837419—21no—
7#16837376C411yescharacteristic, complemented
8#16837162—21no—
9#16837151C412yescomplemented
10#16837146—21no—
11C59×#532C2211yescharacteristic
12C59×#410—21no—
13C59×#354C412yescomplemented
14C59×#353—21no—
15#14273734—42no—
16C2×#11745664—21no—
17C2×#11745664—41no—
18C2×#11745664—41no—
19#14273686—21no—
20C4⋊(C59⋊C8)—22no—
21C4⋊(C59⋊C8)—41no—
22#14273654—42no—
23#14273642—41no—
24C2×#41×C59C2×C411yescharacteristic
25C2×#41×C59D411yescharacteristic
26C4⋊Q8×C59C2×C411yescharacteristic
27#127×C59D411yescharacteristic
28#124×C59C2×C411yescharacteristic
29#124×C59Q811yescharacteristic
30#121×C59—41no—
31C4×D4×C59—41no—
32#117×C59—41no—
33C2×C42×C59D411yescharacteristic
34#99×C59—22no—
35#99×C59—41no—
36#41×C59—41no—
37#41×C59—41no—
38#41×C59—41no—
39#41×C59—41no—
40C2×Q8×C59C4211yescharacteristic
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
51C4⋊C4×C59—41no—
52C4⋊C4×C59—81no—
53C22⋊C4×C59—41no—
54C22⋊C4×C59—41no—
55C22⋊C4×C59—42no—
56C42×C59—21no—
57C42×C59—81no—
58C42×C59C22⋊C411yescharacteristic
59C42×C59C4⋊C411yescharacteristic
60C22⋊Dic59—41no—
61C4⋊Dic59—41no—
62C4×Dic59—41no—
63#11745664—22no—
64#11745664—42no—
65#11745664—41no—
66#11745664—82no—
67C2×C59⋊C8—41no—
68C2×C59⋊C8—42no—
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—81no—
79D4×C59—81no—
80D4×C59—81no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—82no—
94C2×C4×C59#9511yescharacteristic
95C2×C4×C59#9911yescharacteristic
96C2×Dic59—81no—
97C59⋊C8—42no—
98C59⋊C8—82no—
99#7482—591nomaximal, Sylow
100C22×C59—21no—
101C22×C59—41no—
102C22×C59—41no—
103C22×C59—41no—
104C22×C59—81no—
105C22×C59—81no—
106C22×C59#34211yescharacteristic
107C4×C59—21no—
108C4×C59—21no—
109C4×C59—21no—
110C4×C59—41no—
111C4×C59—41no—
112C4×C59—41no—
113C4×C59—162no—
114C4×C59AΣL(1,𝔽16)11yescharacteristic
115C4×C59#35211yescharacteristic
116Dic59—161no—
117#2126D5911yescharacteristic
118#1884—591no—
119#1560—591no—
120C2×C59—21no—
121C2×C59—41no—
122C2×C59—81no—
123C2×C59—81no—
124C2×C59#139011yescharacteristic
125#568—591no—
126#532C2×D5911yescharacteristic
127#423—1181no—
128#421—1181no—
129#410—21no—
130#410—591no—
131#354Dic5912yescomplemented
132#353—21no—
133#339—1181no—
134#333—592no—
135#331—1181no—
136C59#748211yescharacteristic, complemented, Sylow
137C2×#41C4×D5911yescharacteristic
138C2×#41C22⋊D5911yescharacteristic
139#131—2362no—
140C2×M16—1181no—
141C2×M16—2361no—
142C2×M16—2361no—
143C4⋊Q8C4×D5911yescharacteristic
144#127C22⋊D5911yescharacteristic
145#124Dic11811yescharacteristic
146#124C2×Dic5911yescharacteristic
147#121—41no—
148C4×D4—41no—
149#117—41no—
150#117—1181no—
151C2×C42D23611yescharacteristic
152C4⋊C8—1182no—
153C4⋊C8—2361no—
154#104—2362no—
155#99—22no—
156#99—41no—
157#99—2361no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162C2×Q8C4×Dic5911yescharacteristic
163C2×Q8Dic59⋊C411yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4Dic59⋊C411yescharacteristic
168C2×D4C22⋊Dic5911yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4(C2×D59)⋊C411yescharacteristic
173M16—1182no—
174M16—4722no—
175M16—2362no—
176M16—2361no—
177C2×C8—2362no—
178C2×C8—2361no—
179C2×C8—2362no—
180C4⋊C4—41no—
181C4⋊C4—81no—
182C4⋊C4—2361no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—2361no—
187C42—21no—
188C42—81no—
189C42—2361no—
190C42C4⋊Dic5911yescharacteristic
191C42(C2×D59)⋊C411yescharacteristic
192C23—21no—
193C23—21no—
194C23#1427364111yescharacteristic
195Q8—41no—
196Q8—41no—
197D4—41no—
198D4—41no—
199D4—42no—
200D4—81no—
201D4—81no—
202D4—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4—4721no—
217C2×C4C59⋊#9511yescharacteristic
218C2×C4#1427364111yescharacteristic
219C8—4722no—
220C8—2362no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22#1683716811yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—162no—
235C4—9441no—
236C4#1683718811yescharacteristic
237C4#1683719111yescharacteristic
238C2—21no—
239C2—41no—
240C2—81no—
241C2—81no—
242C2#2142221311yescharacteristic
subgroupG/H
centerC2#21422213
commutatorC2×Q8×C59C42
Frattini#532C2×D59
FittingC59×#2126C2
socleC2×C59#1390
prime radical, cosocleC59×#532C22
hypercenter#2126D59
hypocenterC59#7482
LCSG ▷ C2×Q8×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #16838169C4semidirect
7. #16837376C4semidirect
9. #16837151C4semidirect
13. C59×#354C4semidirect
131. #354Dic59semidirect
136. C59#7482semidirect

Maximal subgroups

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

subgroupindex
#214244802
#214228312
C59×#21262
#748259