Finite Group Databasebeta

← #28502970

Subgroups — #28502970

Order 15104.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426004C211yesmaximal, characteristic, complemented
2#21423463C211yesmaximal, characteristic, complemented
3#21423453C211yesmaximal, characteristic
4#21422168C211yesmaximal, characteristic, complemented
5#21422145C211yesmaximal, characteristic, complemented
6#21422142C211yesmaximal, characteristic
7C59×#2248C211yesmaximal, characteristic
8#16837684—21no—
9#16837599—21no—
10C8×Dic118C2211yescharacteristic
11C4⋊(C59⋊C16)C2211yescharacteristic
12Dic59⋊C16—21no—
13#16837177—21no—
14#16837155C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837148—21no—
17#16837146—21no—
18C2×SD32×C59—21no—
19C2×SD32×C59—21no—
20C59×#494C2211yescharacteristic
21C59×#493C2211yescharacteristic
22C4×C16×C59C2211yescharacteristic
23C2×#11745670—41no—
24C4×Dic118D411yescharacteristic
25C2×C59⋊C16—21no—
26#14273658—41no—
27C4⋊(C59⋊C8)—21no—
28C4⋊(C59⋊C8)—21no—
29C4⋊(C59⋊C8)D411yescharacteristic
30Dic59⋊C8—21no—
31#14273650—41no—
32#14273648—41no—
33C8×Dic59—21no—
34C2×Q16×C59—21no—
35C2×Q16×C59—21no—
36C2×D8×C59—21no—
37C2×D8×C59—21no—
38C4⋊Q8×C59—21no—
39#127×C59—21no—
40SD32×C59—42no—
41SD32×C59—42no—
42C2×C16×C59D411yescharacteristic
43C2×C16×C59D411yescharacteristic
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×Dic118D811yescharacteristic
50Q16×C59—41no—
51Q16×C59—41no—
52Q16×C59—81no—
53D8×C59—41no—
54D8×C59—41no—
55D8×C59—81no—
56C2×C8×C59C2×D411yescharacteristic
57C2×C8×C59#4111yescharacteristic
58C4⋊C4×C59—41no—
59C4⋊C4×C59—41no—
60C42×C59C2×D411yescharacteristic
61#11745670—82no—
62C4⋊Dic59D811yescharacteristic
63Dic59⋊C4—21no—
64C4×Dic59—21no—
65C2×C59⋊C8—21no—
66C2×C59⋊C8—41no—
67C2×C59⋊C8—41no—
68C16×C59—21no—
69C16×C59SD1612yescomplemented
70C59⋊C16—41no—
71C23×C59—81no—
72Q8×C59—81no—
73Q8×C59—81no—
74D4×C59—81no—
75D4×C59—81no—
76D4×C59—81no—
77D4×C59—81no—
78D4×C59—161no—
79C2×C4×C59—81no—
80C2×C4×C59#12111yescharacteristic
81C2×C4×C59C2×D811yescharacteristic
82C2×C4×C59#13511yescharacteristic
83C2×Dic59—21no—
84C2×Dic59—21no—
85Dic118—21no—
86Dic118—21no—
87C8×C59—21no—
88C8×C59C2×SD1611yescharacteristic
89C8×C59AGL(1,ℤ/8)11yescharacteristic
90C59⋊C8—41no—
91C59⋊C8—81no—
92C59⋊C8—81no—
93#12444—591nomaximal, Sylow
94C22×C59—161no—
95C22×C59—161no—
96C22×C59—161no—
97C22×C59#46611yescharacteristic
98C4×C59—21no—
99C4×C59—161no—
100C4×C59#46011yescharacteristic
101C4×C59#47211yescharacteristic
102C4×C59#50911yescharacteristic
103C4×C59#51011yescharacteristic
104Dic59—41no—
105Dic59—41no—
106#2248D5911yescharacteristic
107#2183—591no—
108#1670—591no—
109#1668—591no—
110#1371—591no—
111#1362—591no—
112#1361—591no—
113C2×C59—321no—
114C2×C59#166811yescharacteristic
115C2×C59#222011yescharacteristic
116C2×C59#222311yescharacteristic
117C2×SD32—21no—
118C2×SD32—21no—
119#494C2×D5911yescharacteristic
120#493C2×D5911yescharacteristic
121#475—1181no—
122#460—1181no—
123C8×Q8—591no—
124C4⋊C16—591no—
125C4⋊C16—1181no—
126#346—1181no—
127C4×C16C2×D5911yescharacteristic
128#335—591no—
129#335—591no—
130#332—1181no—
131#331—1181no—
132C59#1244411yescharacteristic, complemented, Sylow
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2361no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8—591no—
141SD32—42no—
142SD32—42no—
143C2×C16—1181no—
144C2×C16C22⋊D5911yescharacteristic
145C2×C16C22⋊D5911yescharacteristic
146#106—2361no—
147C4⋊C8—591no—
148C4⋊C8—1181no—
149C4⋊C8—1181no—
150C4⋊C8—1181no—
151#103—2361no—
152#102—2361no—
153C4×C8—1181no—
154C4×C8C22×D5911yescharacteristic
155C2×Q8—41no—
156C2×Q8—591no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—4722no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—1181no—
168C2×C8—2361no—
169C2×C8—2361no—
170C2×C8#1174568711yescharacteristic
171C2×C8C2×C22⋊D5911yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—591no—
175C4⋊C4—1181no—
176C42—1181no—
177C42D4×D5911yescharacteristic
178C16—21no—
179C16—2361no—
180C16#1174567012yescomplemented
181C23—81no—
182Q8—81no—
183Q8—81no—
184Q8—1181no—
185Q8—1181no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—81no—
192C2×C4—1181no—
193C2×C4—1181no—
194C2×C4#1427369811yescharacteristic
195C2×C4D8×D5911yescharacteristic
196C2×C4#1427374611yescharacteristic
197C8—21no—
198C8—4721no—
199C8—4721no—
200C8—2361no—
201C8C2×#1174567011yescharacteristic
202C8#1427375311yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1683763511yescharacteristic
207C4—21no—
208C4—161no—
209C4—2361no—
210C4—2361no—
211C4#1683759911yescharacteristic
212C4#1683766911yescharacteristic
213C4#1683783911yescharacteristic
214C4#1683784111yescharacteristic
215C2—321no—
216C2#2142345311yescharacteristic
217C2#2142620611yescharacteristic
218C2#2142622311yescharacteristic
subgroupG/H
centerC22#16837635
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2248C2
socleC22×C59#466
prime radicalC59×#493C22
hypercenter#2248D59
hypocenterC59#12444
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #21426004C2semidirect
2. #21423463C2semidirect
4. #21422168C2semidirect
5. #21422145C2semidirect
69. C16×C59SD16semidirect
132. C59#12444semidirect
180. C16#11745670semidirect

Maximal subgroups

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

subgroupindex
#214260042
#214234632
#214234532
#214221682
#214221452
#214221422
C59×#22482
#1244459