Finite Group Databasebeta

← #28503217

Subgroups — #28503217

Order 15104.

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

normal subgroups only

This group has 248 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#21426234C211yesmaximal, characteristic, complemented
2#21426128C211yesmaximal, characteristic
3#21423685C211yesmaximal, characteristic, complemented
4#21422980C211yesmaximal, characteristic, complemented
5#21422165C211yesmaximal, characteristic, complemented
6#21422152C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic, complemented
8C2×#14273668—21no—
9#16837807C2211yescharacteristic
10#16837774—21no—
11#16837748—21no—
12C4×D472C2211yescharacteristic
13#16837479—21no—
14#16837228—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837204—21no—
17#16837203—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745670—41no—
24C2×#11745670—41no—
25C2×D472D411yescharacteristic
26C2×#11745659—41no—
27#14273727—21no—
28C4×D236—21no—
29#14273668—42no—
30C2×C59⋊C16—21no—
31#14273661—21no—
32#14273661—21no—
33#14273661D411yescharacteristic
34#14273649—21no—
35#14273648—42no—
36(C2×D59)⋊C8—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×D236—21no—
49C2×C4×D59—41no—
50C2×Dic118—41no—
51D8×C59—41no—
52D8×C59—41no—
53D8×C59—81no—
54C2×C8×C59—41no—
55C2×C8×C59C2×D411yescharacteristic
56C2×C8×C59#4111yescharacteristic
57C42×C59C2×D411yescharacteristic
58#11745670—82no—
59#11745670—82no—
60(C2×D59)⋊C4—41no—
61C4⋊Dic59—21no—
62C4⋊Dic59—41no—
63C4⋊Dic59—41no—
64C2×C59⋊C8—21no—
65C2×C59⋊C8D811yescharacteristic
66C2×C59⋊C8D811yescharacteristic
67D472—21no—
68D472—21no—
69#11745659—82no—
70C16×C59—41no—
71C59⋊C16—41no—
72C23×C59—81no—
73C22×D59—41no—
74D4×C59—82no—
75D4×C59—81no—
76D4×C59—81no—
77D4×C59—81no—
78D4×C59—81no—
79C2×C4×C59#12111yescharacteristic
80C2×C4×C59C2×D811yescharacteristic
81C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
82C2×Dic59—41no—
83C2×Dic59—81no—
84D236—41no—
85D236—41no—
86C4×D59—81no—
87C4×D59—81no—
88Dic118—81no—
89Dic118—81no—
90C8×C59—21no—
91C8×C59—81no—
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C8×C59#13711yescharacteristic
94C59⋊C8—21no—
95C59⋊C8—41no—
96C59⋊C8D1612yescomplemented
97#12483—591nomaximal, Sylow
98C22×C59—161no—
99C22×C59—161no—
100C22×C59—161no—
101C22×C59#45911yescharacteristic
102C2×D59—81no—
103C2×D59—81no—
104C2×D59—81no—
105C4×C59—21no—
106C4×C59#46811yescharacteristic
107C4×C59#47111yescharacteristic
108C4×C59C2×D1611yescharacteristic
109C4×C59#50911yescharacteristic
110Dic59—81no—
111Dic59—161no—
112#2224—591no—
113#2206—591no—
114#1706—591no—
115#1589—591no—
116#1370—591no—
117#1364—591no—
118#1361D5911yescharacteristic, complemented
119C2×C59—321no—
120C2×C59#168511yescharacteristic
121C2×C59#220611yescharacteristic
122C2×C59#221211yescharacteristic
123D59—161no—
124C2×D16—1181no—
125#500—591no—
126#493C2×D5911yescharacteristic
127#492—1181no—
128#486—1181no—
129C4×D8—591no—
130#436—1181no—
131#366—1181no—
132C4⋊C16—591no—
133C4⋊C16C2×D5911yescharacteristic
134#357—1181no—
135#357—1181no—
136#335C2×D5911yescharacteristic
137D4⋊C8—1181no—
138C82—591no—
139C59#1248311yescharacteristic, complemented, Sylow
140C2×SD16—2361no—
141C2×SD16—2361no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—591no—
145C2×M16—2361no—
146C4⋊Q8—1181no—
147#127—21no—
148C4×D4—1181no—
149D16—2362no—
150C2×C16—21no—
151C2×C16—1181no—
152#107—591no—
153#107—1181no—
154#107—1181no—
155C4⋊C8—21no—
156#102—1181no—
157#102—2362no—
158C22⋊C8—2361no—
159C4×C8—591no—
160C4×C8—591no—
161C4×C8C22×D5911yescharacteristic
162C2×Q8—2361no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—1181no—
167C22×C4—2361no—
168SD16—4722no—
169SD16—4722no—
170D8—41no—
171D8—41no—
172D8—81no—
173D8—1181no—
174D8—1181no—
175M16—4722no—
176C2×C8—41no—
177C2×C8—591no—
178C2×C8—591no—
179C2×C8—1181no—
180C2×C8#1174568511yescharacteristic
181C2×C8D4×D5911yescharacteristic
182C4⋊C4—1181no—
183C4⋊C4—2361no—
184C4⋊C4—2361no—
185C22⋊C4—2361no—
186C42D4×D5911yescharacteristic
187C16—41no—
188C16—2361no—
189C23—81no—
190C23—2361no—
191Q8—4721no—
192Q8—4721no—
193D4—82no—
194D4—81no—
195D4—81no—
196D4—81no—
197D4—81no—
198D4—2361no—
199D4—2361no—
200C2×C4—4721no—
201C2×C4—4721no—
202C2×C4—4721no—
203C2×C4—2361no—
204C2×C4#1427370011yescharacteristic
205C2×C4D8×D5911yescharacteristic
206C2×C4#1427375411yescharacteristic
207C8—21no—
208C8—81no—
209C8—592no—
210C8—1181no—
211C8—2361no—
212C8#1427375111yescharacteristic
213C8#1427375611yescharacteristic
214C22—161no—
215C22—161no—
216C22—161no—
217C22—4721no—
218C22—4721no—
219C22—4721no—
220C22#1683759611yescharacteristic
221C4—21no—
222C4—9441no—
223C4—4721no—
224C4#1683764711yescharacteristic
225C4#1683766411yescharacteristic
226C4D16×D5911yescharacteristic
227C4#1683783911yescharacteristic
228C2—321no—
229C2—9441no—
230C2#2142356011yescharacteristic
231C2#2142613011yescharacteristic
232C2#2142616411yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#459
prime radical#21426128C2
hypercenter#1361D59
hypocenterC59#12483
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. #21426234C2semidirect
3. #21423685C2semidirect
4. #21422980C2semidirect
5. #21422165C2semidirect
6. #21422152C2semidirect
7. C59×#1361C2semidirect
96. C59⋊C8D16semidirect
118. #1361D59semidirect
139. C59#12483semidirect

Maximal subgroups

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

subgroupindex
#214262342
#214261282
#214236852
#214229802
#214221652
#214221522
C59×#13612
#1248359