Finite Group Databasebeta

← #28503165

Subgroups — #28503165

Order 15104.

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

normal subgroups only

This group has 220 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#21426321C211yesmaximal, characteristic
2#21425998C211yesmaximal, characteristic, complemented
3#21423487C211yesmaximal, characteristic, complemented
4#21423457C211yesmaximal, characteristic
5#21422167C211yesmaximal, characteristic, complemented
6#21422143C211yesmaximal, characteristic
7C59×#1364C211yesmaximal, characteristic, complemented
8#16837779C2211yescharacteristic
9#16837609—21no—
10#16837603—21no—
11C8×Dic118C2211yescharacteristic
12Dic59⋊C16—21no—
13#16837202—21no—
14#16837202—21no—
15C4×C59⋊C16C2211yescharacteristic
16C16×Dic59—21no—
17#16837155C2211yescharacteristic
18#16837148—21no—
19#16837145—21no—
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745672—41no—
24C2×#11745660—41no—
25C2×D472—21no—
26C2×D472—21no—
27#14273724—21no—
28C4×Dic118D411yescharacteristic
29C2×C59⋊C16D411yescharacteristic
30C2×C59⋊C16D411yescharacteristic
31C4⋊(C59⋊C8)—21no—
32C4⋊(C59⋊C8)D411yescharacteristic
33Dic59⋊C8—21no—
34#14273651—41no—
35#14273649—41no—
36C8×Dic59—21no—
37C4⋊Q8×C59—21no—
38C2×C16×C59—21no—
39#107×C59—21no—
40#107×C59—21no—
41C4⋊C8×C59—21no—
42C4×C8×C59C2311yescharacteristic
43C2×Q8×C59—41no—
44C2×D236—41no—
45C2×D236—41no—
46C2×D236—41no—
47C2×Dic118D811yescharacteristic
48C2×C8×C59—41no—
49C2×C8×C59C2×D411yescharacteristic
50C2×C8×C59#4111yescharacteristic
51C4⋊C4×C59—41no—
52C4⋊C4×C59—41no—
53C42×C59C2×D411yescharacteristic
54#11745672—82no—
55C4⋊Dic59D811yescharacteristic
56Dic59⋊C4—21no—
57C4×Dic59—21no—
58C2×C59⋊C8—21no—
59C2×C59⋊C8—41no—
60D472—41no—
61D472—41no—
62D472—81no—
63#11745660—82no—
64C16×C59—41no—
65C59⋊C16—21no—
66C59⋊C16—21no—
67C22×D59—81no—
68Q8×C59—81no—
69Q8×C59—81no—
70C2×C4×C59—81no—
71C2×C4×C59#12111yescharacteristic
72C2×C4×C59C2×D811yescharacteristic
73C2×C4×C59#13511yescharacteristic
74C2×Dic59—21no—
75C2×Dic59—21no—
76D236—81no—
77D236—81no—
78D236—81no—
79D236—81no—
80D236—161no—
81Dic118—21no—
82Dic118D1612yes—
83C8×C59—21no—
84C8×C59—81no—
85C8×C59#13511yescharacteristic
86C8×C59#13711yescharacteristic
87C59⋊C8—41no—
88C59⋊C8—81no—
89#12475—591nomaximal, Sylow
90C22×C59#46611yescharacteristic
91C2×D59—161no—
92C2×D59—161no—
93C2×D59—161no—
94C4×C59—21no—
95C4×C59—161no—
96C4×C59#46411yescharacteristic
97C4×C59#47111yescharacteristic
98C4×C59C2×D1611yescharacteristic
99C4×C59#50811yescharacteristic
100Dic59—22no—
101Dic59—41no—
102#2240—591no—
103Q8×C16—591no—
104#1674—591no—
105#1668—591no—
106#1371—591no—
107#1364D5911yescharacteristic, complemented
108#1361—591no—
109C2×C59#168211yescharacteristic
110C2×C59#221511yescharacteristic
111C2×C59#222311yescharacteristic
112D59—321no—
113#500C2×D5911yescharacteristic
114#493—591no—
115#462—1181no—
116#460—1181no—
117C8×Q8—591no—
118C4⋊C16—1181no—
119C4⋊C16C2×D5911yescharacteristic
120#357—1181no—
121#357—1181no—
122C4×C16—591no—
123C4×C16—1181no—
124#335—591no—
125#335C2×D5911yescharacteristic
126#332—1181no—
127#331—1181no—
128C59#1247511yescharacteristic, complemented, Sylow
129C2×Q16—2361no—
130C2×SD16—2361no—
131C2×D8—1181no—
132C2×D8—1181no—
133C4⋊Q8—21no—
134#127—1181no—
135C4×Q8—591no—
136C2×C16—21no—
137C2×C16—591no—
138C2×C16—591no—
139#107—21no—
140#107—21no—
141C4⋊C8—21no—
142C4⋊C8—591no—
143C4⋊C8—1181no—
144C4⋊C8—1181no—
145#103—2361no—
146#102—2361no—
147C4×C8—1181no—
148C4×C8C22×D5911yescharacteristic
149C2×Q8—41no—
150C2×Q8—591no—
151C2×D4—2361no—
152C2×D4—2361no—
153C2×D4—2361no—
154Q16—4722no—
155SD16—4722no—
156D8—4721no—
157D8—2361no—
158D8—2361no—
159C2×C8—41no—
160C2×C8—1181no—
161C2×C8—2361no—
162C2×C8#1174568511yescharacteristic
163C2×C8D4×D5911yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—591no—
167C4⋊C4—1181no—
168C42—1181no—
169C42D4×D5911yescharacteristic
170C16—41no—
171C16—1181no—
172C16—1181no—
173C23—4721no—
174Q8—81no—
175Q8—81no—
176Q8—592no—
177Q8—1181no—
178D4—9441no—
179D4—4721no—
180D4—4721no—
181D4—4721no—
182D4—4721no—
183C2×C4—81no—
184C2×C4—1181no—
185C2×C4—1181no—
186C2×C4#1427370011yescharacteristic
187C2×C4D8×D5911yescharacteristic
188C2×C4#1427374811yescharacteristic
189C8—21no—
190C8—81no—
191C8—4721no—
192C8—2361no—
193C8#1427374811yescharacteristic
194C8#1427375911yescharacteristic
195C22—9441no—
196C22—9441no—
197C22—9441no—
198C22#1683763711yescharacteristic
199C4—21no—
200C4—161no—
201C4—1182no—
202C4—2361no—
203C4#1683762711yescharacteristic
204C4#1683766811yescharacteristic
205C4D16×D5911yescharacteristic
206C4#1683783311yescharacteristic
207C2—18881no—
208C2#2142354311yescharacteristic
209C2#2142618111yescharacteristic
210C2#2142622911yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59#466
prime radical#16837779C22
hypercenter#1364D59
hypocenterC59#12475
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. #21425998C2semidirect
3. #21423487C2semidirect
5. #21422167C2semidirect
7. C59×#1364C2semidirect
107. #1364D59semidirect
128. C59#12475semidirect

Maximal subgroups

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

subgroupindex
#214263212
#214259982
#214234872
#214234572
#214221672
#214221432
C59×#13642
#1247559