Finite Group Databasebeta

← #28503223

Subgroups — #28503223

Order 15104.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#21426238C211yesmaximal, characteristic, complemented
2#21423685C211yesmaximal, characteristic
3#21422993C211yesmaximal, characteristic, complemented
4#21422165C211yesmaximal, characteristic, complemented
5#21422150C211yesmaximal, characteristic, complemented
6#21422142C211yesmaximal, characteristic
7C59×#2207C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837774—21no—
10#16837748—21no—
11#16837533—21no—
12#16837229—21no—
13C4⋊(C59⋊C16)C2211yescharacteristic
14#16837206—21no—
15#16837155C2211yescharacteristic
16Q8⋊(C59⋊C8)—21no—
17C8×C59⋊C8C2211yescharacteristic
18C2×SD32×C59—21no—
19C59×#493C2211yescharacteristic
20C4×Q16×C59C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#357—21no—
23C2×#11745670—41no—
24C2×#11745670—41no—
25#14273727—21no—
26C2×C59⋊C16—21no—
27#14273661—21no—
28#14273661—21no—
29C4⋊(C59⋊C8)—21no—
30C4⋊(C59⋊C8)—41no—
31#14273648—42no—
32#14273638—41no—
33C4×C59⋊C8D411yescharacteristic
34C4×C59⋊C8D411yescharacteristic
35C2×Q16×C59D411yescharacteristic
36C2×D8×C59—21no—
37C2×D8×C59—21no—
38#127×C59—21no—
39C4×Q8×C59—21no—
40SD32×C59—42no—
41C2×C16×C59—21no—
42#107×C59D411yescharacteristic
43#103×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—21no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×Dic118—41no—
50Q16×C59—21no—
51Q16×C59—21no—
52D8×C59—41no—
53D8×C59—41no—
54D8×C59—81no—
55C2×C8×C59C2×D411yescharacteristic
56C2×C8×C59#4111yescharacteristic
57C4⋊C4×C59—21no—
58C4⋊C4×C59—41no—
59C42×C59—41no—
60C42×C59C2×D411yescharacteristic
61#11745670—82no—
62#11745670—82no—
63C4⋊Dic59—41no—
64C4⋊Dic59—41no—
65C2×C59⋊C8—21no—
66C2×C59⋊C8—41no—
67C2×C59⋊C8D811yescharacteristic
68C2×C59⋊C8D811yescharacteristic
69C16×C59—41no—
70C59⋊C16—41no—
71C23×C59—81no—
72Q8×C59—41no—
73Q8×C59—41no—
74D4×C59—82no—
75D4×C59—81no—
76D4×C59—81no—
77D4×C59—81no—
78D4×C59—81no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59#12111yescharacteristic
82C2×C4×C59C2×D811yescharacteristic
83C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
84C2×Dic59—81no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59AGL(1,ℤ/8)11yescharacteristic
89C8×C59#13711yescharacteristic
90C59⋊C8—21no—
91C59⋊C8—41no—
92C59⋊C8—81no—
93C59⋊C8SD3212yescomplemented
94#12484—591nomaximal, Sylow
95C22×C59—161no—
96C22×C59—161no—
97C22×C59—161no—
98C22×C59#45911yescharacteristic
99C4×C59—21no—
100C4×C59—81no—
101C4×C59—81no—
102C4×C59#46811yescharacteristic
103C4×C59#47111yescharacteristic
104C4×C59C2×SD3211yescharacteristic
105C4×C59#50911yescharacteristic
106Dic59—161no—
107#2225—591no—
108#2207D5911yescharacteristic
109#1706—591no—
110#1591—591no—
111#1370—591no—
112#1364—591no—
113#1361—591no—
114C2×C59—321no—
115C2×C59#168511yescharacteristic
116C2×C59#220711yescharacteristic
117C2×C59#221211yescharacteristic
118C2×SD32—21no—
119#500—591no—
120#493C2×D5911yescharacteristic
121#492—1181no—
122#486—1181no—
123#446—1181no—
124C4×Q16C2×D5911yescharacteristic
125#367—1181no—
126C4⋊C16—591no—
127C4⋊C16C2×D5911yescharacteristic
128#358—1181no—
129#357—21no—
130#335—591no—
131Q8⋊C8—1181no—
132C82—591no—
133C59#1248411yescharacteristic, complemented, Sylow
134C2×Q16C22⋊D5911yescharacteristic
135C2×SD16—2361no—
136C2×SD16—2361no—
137C2×D8—21no—
138C2×D8—21no—
139C4⋊Q8—1181no—
140#127—21no—
141C4×Q8—21no—
142SD32—42no—
143C2×C16—21no—
144C2×C16—1181no—
145#107—1181no—
146#107—1181no—
147#107C22⋊D5911yescharacteristic
148C4⋊C8—1181no—
149C4⋊C8—2361no—
150#103—21no—
151#102—2362no—
152#97—2361no—
153C4×C8—591no—
154C4×C8—591no—
155C4×C8C22×D5911yescharacteristic
156C2×Q8—21no—
157C2×Q8—2361no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161Q16—21no—
162Q16—21no—
163SD16—4722no—
164SD16—4722no—
165D8—41no—
166D8—41no—
167D8—81no—
168C2×C8—591no—
169C2×C8—591no—
170C2×C8—1181no—
171C2×C8—2361no—
172C2×C8#1174568711yescharacteristic
173C2×C8C2×C22⋊D5911yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—2361no—
177C4⋊C4—2361no—
178C42—41no—
179C42D4×D5911yescharacteristic
180C16—41no—
181C16—2361no—
182C23—81no—
183Q8—41no—
184Q8—41no—
185Q8—4721no—
186Q8—4721no—
187D4—82no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—4721no—
195C2×C4#1427369911yescharacteristic
196C2×C4D8×D5911yescharacteristic
197C2×C4#1427375111yescharacteristic
198C8—21no—
199C8—592no—
200C8—1181no—
201C8—4721no—
202C8—2361no—
203C8#1427375311yescharacteristic
204C8#1427375711yescharacteristic
205C22—161no—
206C22—161no—
207C22—161no—
208C22#1683759511yescharacteristic
209C4—21no—
210C4—81no—
211C4—81no—
212C4—9441no—
213C4#1683764611yescharacteristic
214C4#1683766311yescharacteristic
215C4SD32×D5911yescharacteristic
216C4#1683783911yescharacteristic
217C2—321no—
218C2#2142355911yescharacteristic
219C2#2142613411yescharacteristic
220C2#2142616311yescharacteristic
subgroupG/H
centerC22#16837595
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2207C2
socleC22×C59#459
prime radicalC59×#493C22
hypercenter#2207D59
hypocenterC59#12484
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21426238C2semidirect
3. #21422993C2semidirect
4. #21422165C2semidirect
5. #21422150C2semidirect
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
#214262382
#214236852
#214229932
#214221652
#214221502
#214221422
C59×#22072
#1248459