Finite Group Databasebeta

← #28502972

Subgroups — #28502972

Order 15104.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426351C211yesmaximal, characteristic
2#21426001C211yesmaximal, characteristic, complemented
3#21423467C211yesmaximal, characteristic, complemented
4#21423453C211yesmaximal, characteristic
5#21422167C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic
8C2×#14273671—21no—
9C2×#14273671—21no—
10#16837782C2211yescharacteristic
11#16837688—21no—
12#16837599—21no—
13C8×Dic118C2211yescharacteristic
14Dic59⋊C16—21no—
15#16837177—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837147—21no—
19#16837146—21no—
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×#11745670—41no—
26#14273727—21no—
27C4×Dic118D411yescharacteristic
28#14273671—42no—
29#14273671—42no—
30C2×C59⋊C16D411yescharacteristic
31C2×C59⋊C16D411yescharacteristic
32#14273659—41no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)D411yescharacteristic
35Dic59⋊C8—21no—
36#14273652—41no—
37#14273648—41no—
38C8×Dic59—21no—
39C2×D8×C59—21no—
40C2×D8×C59—21no—
41#127×C59—21no—
42C2×C16×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×Dic118—41no—
49C2×Dic118D811yescharacteristic
50D8×C59—41no—
51D8×C59—41no—
52D8×C59—81no—
53C2×C8×C59—41no—
54C2×C8×C59C2×D411yescharacteristic
55C2×C8×C59#4111yescharacteristic
56C42×C59C2×D411yescharacteristic
57#11745670—82no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C4⋊Dic59D811yescharacteristic
61Dic59⋊C4—21no—
62C4×Dic59—21no—
63C2×C59⋊C8—21no—
64C2×C59⋊C8—41no—
65Dic236—41no—
66Dic236—41no—
67Dic236—81no—
68C16×C59—41no—
69C59⋊C16—21no—
70C59⋊C16SD1612yescomplemented
71C23×C59—81no—
72D4×C59—81no—
73D4×C59—81no—
74D4×C59—81no—
75D4×C59—81no—
76D4×C59—161no—
77C2×C4×C59#12111yescharacteristic
78C2×C4×C59C2×D811yescharacteristic
79C2×C4×C59#13511yescharacteristic
80C2×Dic59—21no—
81C2×Dic59—21no—
82C2×Dic59—81no—
83Dic118—21no—
84Dic118—21no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59—81no—
89C8×C59C2×SD1611yescharacteristic
90C8×C59AGL(1,ℤ/8)11yescharacteristic
91C59⋊C8—41no—
92C59⋊C8—81no—
93#12444—591nomaximal, Sylow
94C22×C59—161no—
95C22×C59—161no—
96C22×C59—161no—
97C22×C59#46611yescharacteristic
98C4×C59—21no—
99C4×C59#46011yescharacteristic
100C4×C59#47211yescharacteristic
101C4×C59#50911yescharacteristic
102C4×C59#51011yescharacteristic
103Dic59—41no—
104Dic59—41no—
105Dic59—161no—
106#2248—591no—
107#2183—591no—
108#1670—591no—
109#1668—591no—
110#1371—591no—
111#1362—591no—
112#1361D5911yescharacteristic
113C2×C59—321no—
114C2×C59#166811yescharacteristic
115C2×C59#222011yescharacteristic
116C2×C59#222311yescharacteristic
117C2×SD32—1181no—
118C2×SD32—1181no—
119#494—591no—
120#493C2×D5911yescharacteristic
121#475—1181no—
122#460—1181no—
123C8×Q8—591no—
124C4⋊C16—1181no—
125C4⋊C16C2×D5911yescharacteristic
126#346—1181no—
127C4×C16—591no—
128#335—591no—
129#335C2×D5911yescharacteristic
130#332—1181no—
131#331—1181no—
132C59#1244411yescharacteristic, complemented, Sylow
133C2×Q16—1181no—
134C2×Q16—1181no—
135C2×SD16—2361no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1181no—
139#127—21no—
140C4×Q8—591no—
141SD32—2362no—
142SD32—2362no—
143C2×C16—21no—
144C2×C16—591no—
145C2×C16—591no—
146#106—2361no—
147C4⋊C8—21no—
148C4⋊C8—591no—
149C4⋊C8—1181no—
150C4⋊C8—1181no—
151#103—2361no—
152#102—2361no—
153C4×C8—1181no—
154C4×C8C22×D5911yescharacteristic
155C2×Q8—591no—
156C2×Q8—2361no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—4721no—
161Q16—2361no—
162Q16—2361no—
163SD16—4722no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—41no—
168C2×C8—1181no—
169C2×C8—2361no—
170C2×C8#1174568511yescharacteristic
171C2×C8D4×D5911yescharacteristic
172C4⋊C4—591no—
173C4⋊C4—1181no—
174C4⋊C4—2361no—
175C4⋊C4—2361no—
176C42—1181no—
177C42D4×D5911yescharacteristic
178C16—41no—
179C16—592no—
180C16—1181no—
181C23—81no—
182Q8—1181no—
183Q8—1181no—
184Q8—4721no—
185Q8—4721no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—1181no—
192C2×C4—1181no—
193C2×C4—4721no—
194C2×C4#1427370011yescharacteristic
195C2×C4D8×D5911yescharacteristic
196C2×C4#1427374811yescharacteristic
197C8—21no—
198C8—81no—
199C8—4721no—
200C8—2361no—
201C8SD16×D5911yescharacteristic
202C8#1427375111yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1683763711yescharacteristic
207C4—21no—
208C4—9441no—
209C4—2361no—
210C4—2361no—
211C4#1683760111yescharacteristic
212C4#1683767111yescharacteristic
213C4#1683783611yescharacteristic
214C4#1683784111yescharacteristic
215C2—321no—
216C2#2142345511yescharacteristic
217C2#2142620811yescharacteristic
218C2#2142622511yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#466
prime radicalC59×#493C22
hypercenter#1361D59
hypocenterC59#12444
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
2. #21426001C2semidirect
3. #21423467C2semidirect
5. #21422167C2semidirect
6. #21422146C2semidirect
70. C59⋊C16SD16semidirect
132. C59#12444semidirect

Maximal subgroups

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

subgroupindex
#214263512
#214260012
#214234672
#214234532
#214221672
#214221462
C59×#13612
#1244459