Finite Group Databasebeta

← #28503224

Subgroups — #28503224

Order 15104.

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

normal subgroups only

This group has 234 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#21426239C211yesmaximal, characteristic, complemented
2#21426133C211yesmaximal, characteristic
3#21423685C211yesmaximal, characteristic
4#21422992C211yesmaximal, characteristic, complemented
5#21422165C211yesmaximal, characteristic, complemented
6#21422152C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic
8C2×#14273671—21no—
9#16837807C2211yescharacteristic
10#16837774—21no—
11#16837748—21no—
12#16837529—21no—
13C4×Dic236C2211yescharacteristic
14#16837230—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837207—21no—
17#16837203—21no—
18#16837132—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236D411yescharacteristic
24C2×#11745670—41no—
25C2×#11745670—41no—
26#14273727—21no—
27C4×Dic118—21no—
28#14273671—42no—
29C2×C59⋊C16—21no—
30#14273661—21no—
31#14273661—21no—
32#14273661D411yescharacteristic
33Dic59⋊C8—41no—
34#14273652—21no—
35#14273648—42no—
36#14273637—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
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—21no—
49C2×Dic118—41no—
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—
58#11745670—82no—
59C4⋊Dic59—21no—
60C4⋊Dic59—41no—
61C4⋊Dic59—41no—
62Dic59⋊C4—41no—
63C4×Dic59—41no—
64C2×C59⋊C8—21no—
65C2×C59⋊C8D811yescharacteristic
66C2×C59⋊C8D811yescharacteristic
67Dic236—21no—
68Dic236—21no—
69C16×C59—41no—
70C59⋊C16—41no—
71C23×C59—81no—
72D4×C59—82no—
73D4×C59—81no—
74D4×C59—81no—
75D4×C59—81no—
76D4×C59—81no—
77C2×C4×C59#12111yescharacteristic
78C2×C4×C59C2×D811yescharacteristic
79C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
80C2×Dic59—41no—
81C2×Dic59—41no—
82C2×Dic59—81no—
83Dic118—41no—
84Dic118—41no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59—81no—
89C8×C59AGL(1,ℤ/8)11yescharacteristic
90C8×C59#13711yescharacteristic
91C59⋊C8—21no—
92C59⋊C8—41no—
93C59⋊C8SD3212yescomplemented
94#12484—591nomaximal, Sylow
95C22×C59—161no—
96C22×C59—161no—
97C22×C59—161no—
98C22×C59#45911yescharacteristic
99C4×C59—21no—
100C4×C59#46811yescharacteristic
101C4×C59#47111yescharacteristic
102C4×C59C2×SD3211yescharacteristic
103C4×C59#50911yescharacteristic
104Dic59—81no—
105Dic59—81no—
106Dic59—161no—
107#2225—591no—
108#2207—591no—
109#1706—591no—
110#1591—591no—
111#1370—591no—
112#1364—591no—
113#1361D5911yescharacteristic
114C2×C59—321no—
115C2×C59#168511yescharacteristic
116C2×C59#220711yescharacteristic
117C2×C59#221211yescharacteristic
118C2×SD32—1181no—
119#500—591no—
120#493C2×D5911yescharacteristic
121#492—1181no—
122#486—1181no—
123#446—1181no—
124C4×Q16—591no—
125#367—1181no—
126C4⋊C16—591no—
127C4⋊C16C2×D5911yescharacteristic
128#358—1181no—
129#357—1181no—
130#335C2×D5911yescharacteristic
131Q8⋊C8—1181no—
132C82—591no—
133C59#1248411yescharacteristic, complemented, Sylow
134C2×Q16—591no—
135C2×SD16—2361no—
136C2×SD16—2361no—
137C2×D8—21no—
138C2×D8—21no—
139C4⋊Q8—1181no—
140#127—21no—
141C4×Q8—1181no—
142SD32—2362no—
143C2×C16—21no—
144C2×C16—1181no—
145#107—591no—
146#107—1181no—
147#107—1181no—
148C4⋊C8—21no—
149C4⋊C8—2361no—
150#103—1181no—
151#102—2362no—
152#97—2361no—
153C4×C8—591no—
154C4×C8—591no—
155C4×C8C22×D5911yescharacteristic
156C2×Q8—1181no—
157C2×Q8—2361no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161Q16—1181no—
162Q16—1181no—
163SD16—4722no—
164SD16—4722no—
165D8—41no—
166D8—41no—
167D8—81no—
168C2×C8—41no—
169C2×C8—591no—
170C2×C8—591no—
171C2×C8—1181no—
172C2×C8#1174568511yescharacteristic
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—1181no—
175C4⋊C4—2361no—
176C4⋊C4—2361no—
177C4⋊C4—2361no—
178C42—2361no—
179C42D4×D5911yescharacteristic
180C16—41no—
181C16—2361no—
182C23—81no—
183Q8—4721no—
184Q8—4721no—
185Q8—2361no—
186Q8—2361no—
187D4—82no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192C2×C4—4721no—
193C2×C4—2361no—
194C2×C4—2361no—
195C2×C4#1427370011yescharacteristic
196C2×C4D8×D5911yescharacteristic
197C2×C4#1427375411yescharacteristic
198C8—21no—
199C8—81no—
200C8—592no—
201C8—1181no—
202C8—2361no—
203C8#1427375111yescharacteristic
204C8#1427375611yescharacteristic
205C22—161no—
206C22—161no—
207C22—161no—
208C22#1683759611yescharacteristic
209C4—21no—
210C4—9441no—
211C4—4721no—
212C4—4721no—
213C4#1683764711yescharacteristic
214C4#1683766411yescharacteristic
215C4SD32×D5911yescharacteristic
216C4#1683783611yescharacteristic
217C2—321no—
218C2#2142356011yescharacteristic
219C2#2142613511yescharacteristic
220C2#2142616411yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#459
prime radicalC59×#493C22
hypercenter#1361D59
hypocenterC59#12484
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
1. #21426239C2semidirect
4. #21422992C2semidirect
5. #21422165C2semidirect
6. #21422152C2semidirect
93. C59⋊C8SD32semidirect
133. C59#12484semidirect

Maximal subgroups

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

subgroupindex
#214262392
#214261332
#214236852
#214229922
#214221652
#214221522
C59×#13612
#1248459