Finite Group Databasebeta

← #28503123

Subgroups — #28503123

Order 15104.

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

normal subgroups only

This group has 218 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#21426325C211yesmaximal, characteristic, complemented
2#21425915C211yesmaximal, characteristic
3#21423492C211yesmaximal, characteristic
4#21423462C211yesmaximal, characteristic
5#21422167C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#1365C211yesmaximal, characteristic, complemented
8C2×C16×D59—21no—
9#16837782C2211yescharacteristic
10#16837689—21no—
11#16837683—21no—
12C8×D236C2211yescharacteristic
13#16837205—21no—
14#16837205—21no—
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837151—21no—
19#16837147—21no—
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×C8×D59—21no—
26#14273727—21no—
27C4×D236D411yescharacteristic
28C2×C59⋊C16D411yescharacteristic
29C2×C59⋊C16D411yescharacteristic
30C16×D59—42no—
31#14273661—41no—
32#14273658—41no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)D411yescharacteristic
35#14273649—41no—
36#14273647—41no—
37(C2×D59)⋊C8—21no—
38C4⋊Q8×C59—21no—
39C2×C16×C59—21no—
40#107×C59—21no—
41#107×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×Q8×C59—41no—
45C2×D236D811yescharacteristic
46C2×C4×D59—21no—
47C2×Dic118—41no—
48C2×C8×C59—41no—
49C2×C8×C59C2×D411yescharacteristic
50C2×C8×C59#4111yescharacteristic
51C4⋊C4×C59—41no—
52C4⋊C4×C59—41no—
53C42×C59C2×D411yescharacteristic
54(C2×D59)⋊C4—21no—
55C4⋊Dic59—41no—
56C4⋊Dic59—41no—
57C4⋊Dic59D811yescharacteristic
58C2×C59⋊C8—21no—
59C2×C59⋊C8—41no—
60Dic236—41no—
61Dic236—41no—
62Dic236—81no—
63C8×D59—41no—
64C8×D59—41no—
65C16×C59—41no—
66C59⋊C16—21no—
67C59⋊C16—21no—
68C22×D59—21no—
69Q8×C59—161no—
70C2×C4×C59—81no—
71C2×C4×C59#12111yescharacteristic
72C2×C4×C59C2×D811yescharacteristic
73C2×C4×C59#13511yescharacteristic
74C2×Dic59—21no—
75C2×Dic59—81no—
76D236—21no—
77D236Q3212yes—
78C4×D59—41no—
79C4×D59—41no—
80Dic118—81no—
81Dic118—81no—
82C8×C59—21no—
83C8×C59—81no—
84C8×C59#13511yescharacteristic
85C8×C59AGL(1,ℤ/8)11yescharacteristic
86C59⋊C8—41no—
87C59⋊C8—81no—
88#12469—591nomaximal, Sylow
89C22×C59#46611yescharacteristic
90C2×D59—22no—
91C2×D59—41no—
92C2×D59—41no—
93C4×C59—21no—
94C4×C59—161no—
95C4×C59#46311yescharacteristic
96C4×C59#47111yescharacteristic
97C4×C59C2×Q3211yescharacteristic
98C4×C59#50811yescharacteristic
99Dic59—41no—
100Dic59—161no—
101#2241—591no—
102D4×C16—591no—
103#1675—591no—
104#1669—591no—
105#1371—591no—
106#1365D5911yescharacteristic, complemented
107#1362—591no—
108C2×C59#168111yescharacteristic
109C2×C59#221611yescharacteristic
110C2×C59#222311yescharacteristic
111D59—42no—
112C22×C16—1181no—
113#500C2×D5911yescharacteristic
114#494—591no—
115#476—1181no—
116#474—1181no—
117C8×D4—591no—
118C4⋊C16C2×D5911yescharacteristic
119#358—1181no—
120#358—1181no—
121C22⋊C16—1181no—
122C4×C16—591no—
123#335—591no—
124#335C2×D5911yescharacteristic
125#333—1181no—
126#332—1181no—
127C59#1246911yescharacteristic, complemented, Sylow
128C2×Q16—1181no—
129C2×Q16—1181no—
130C22×C8—1181no—
131C4⋊Q8—21no—
132C4⋊Q8—1181no—
133C4×D4—591no—
134C2×C16—21no—
135C2×C16—591no—
136C2×C16—591no—
137C2×C16—2362no—
138#107—21no—
139#107—21no—
140#107—2361no—
141#106—2361no—
142C4⋊C8—21no—
143C4⋊C8—591no—
144C4⋊C8—1181no—
145#102—2361no—
146#102—2361no—
147C22⋊C8—1181no—
148C4×C8C22×D5911yescharacteristic
149C2×Q8—41no—
150C2×Q8—2361no—
151C2×D4—591no—
152C22×C4—1181no—
153Q16—4721no—
154Q16—2361no—
155Q16—2361no—
156C2×C8—41no—
157C2×C8—1181no—
158C2×C8—2361no—
159C2×C8—2361no—
160C2×C8—2361no—
161C2×C8#1174568511yescharacteristic
162C2×C8D4×D5911yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—591no—
166C4⋊C4—2361no—
167C4⋊C4—2361no—
168C22⋊C4—1181no—
169C42D4×D5911yescharacteristic
170C16—41no—
171C16—1181no—
172C16—1181no—
173C23—1181no—
174Q8—161no—
175Q8—4721no—
176Q8—4721no—
177D4—592no—
178D4—1181no—
179C2×C4—81no—
180C2×C4—1181no—
181C2×C4—4721no—
182C2×C4—2361no—
183C2×C4—2361no—
184C2×C4#1427370011yescharacteristic
185C2×C4D8×D5911yescharacteristic
186C2×C4#1427374811yescharacteristic
187C8—21no—
188C8—81no—
189C8—4721no—
190C8—2361no—
191C8#1427375011yescharacteristic
192C8#1427375411yescharacteristic
193C22—1182no—
194C22—2361no—
195C22—2361no—
196C22#1683763711yescharacteristic
197C4—21no—
198C4—161no—
199C4—9441no—
200C4—2361no—
201C4#1683762011yescharacteristic
202C4#1683766811yescharacteristic
203C4Q32×D5911yescharacteristic
204C4#1683783311yescharacteristic
205C2—2362no—
206C2#2142353611yescharacteristic
207C2#2142618611yescharacteristic
208C2#2142622511yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1365C2
socleC22×C59#466
prime radicalC2×D236D8
hypercenter#1365D59
hypocenterC59#12469
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #21426325C2semidirect
5. #21422167C2semidirect
6. #21422146C2semidirect
7. C59×#1365C2semidirect
106. #1365D59semidirect
127. C59#12469semidirect

Maximal subgroups

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

subgroupindex
#214263252
#214259152
#214234922
#214234622
#214221672
#214221462
C59×#13652
#1246959