Finite Group Databasebeta

← #28746837

Subgroups — #28746837

Order 15104.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#21437234C212yesmaximal, complemented
2#21426410C214yesmaximal, complemented
3#21426408C212yesmaximal, complemented
4#21426406C212yesmaximal, complemented
5C2×D1888C211yesmaximal, characteristic, complemented
6D32×D59C212yesmaximal, complemented
7#21426389C211yesmaximal, characteristic, complemented
8C59×#2262C211yesmaximal, characteristic, complemented
9#16838204—22no—
10#16837839—24no—
11#16837837—22no—
12#16837835C2214yescomplemented
13C2×D944C2212yescomplemented
14D16×D59C2214yescomplemented
15#16837818C2211yescharacteristic
16#16837239C2214yescomplemented
17D1888C2212yescomplemented
18D1888C2212yescomplemented
19#16837237C2214yescomplemented
20#16837236C2211yescharacteristic, complemented
21#16837235C2212yescomplemented
22C32×D59C2212yescomplemented
23C59×#508C2212yescomplemented
24C59×Q64C2211yescharacteristic, complemented
25C59×SD64C2212yescomplemented
26C59×D32C2211yescharacteristic, complemented
27C2×C32×C59C2211yescharacteristic, complemented
28#14273773—42no—
29#14273754—44no—
30#14273752—42no—
31#14273750—24no—
32C2×D472—22no—
33D8×D59—24no—
34#14273733D411yescharacteristic
35#14273670—24no—
36D944—22no—
37D944C2314yes—
38#14273668—24no—
39#14273667D411yescharacteristic
40#14273666D412yes—
41C16×D59C2312yes—
42C59⋊C32C2312yes—
43#135×C59—22no—
44Q32×C59C2312yes—
45SD32×C59—22no—
46D16×C59C2312yes—
47C2×C16×C59C2311yescharacteristic
48C32×C59C2311yescharacteristic
49C32×C59C2311yescharacteristic
50#41×C59—42no—
51#11745689—44no—
52D4×D59—44no—
53D4×D59—84no—
54#11745685—82no—
55#11745685D811yescharacteristic
56C2×D236—42no—
57C2×D236—82no—
58Q16×C59—22no—
59SD16×C59—42no—
60D8×C59—22no—
61C2×C8×C59C2×D411yescharacteristic
62#11745671—44no—
63#11745669—44no—
64#11745664D811yescharacteristic
65D472—24no—
66D472—42no—
67#11745659D812yes—
68C8×D59C2×D412yes—
69C16×C59C2×D411yescharacteristic
70C16×C59C2411yescharacteristic
71C59⋊C16C2×D412yes—
72C22×D59—82no—
73C22×D59—84no—
74Q8×C59—42no—
75D4×C59—42no—
76D4×C59—82no—
77C2×C4×C59—82no—
78C2×C4×C59C2×D811yescharacteristic
79C22⋊D59—84no—
80C22⋊D59D1612yes—
81D236—44no—
82D236—84no—
83D236—82no—
84D236D1611yescharacteristic
85C4×D59—84no—
86C4×D59C2×D812yes—
87Dic118D1611yescharacteristic
88C8×C59C2×D811yescharacteristic
89C8×C59C22×D411yescharacteristic
90C59⋊C8C2×D812yes—
91#33982—591nomaximal, Sylow
92C22×C59—82no—
93C22×C59C2×D1611yescharacteristic
94C2×D59—84no—
95C2×D59—324no—
96C2×D59—322no—
97C2×D59C2×D1612yes—
98C4×C59—82no—
99C4×C59C2×D1611yescharacteristic
100C4×C59C22×D811yescharacteristic
101Dic59C2×D1612yes—
102#3415—592no—
103#2263—592no—
104#2263—594no—
105#2262—592no—
106#2262D5911yescharacteristic, complemented
107C2×D32—592no—
108C2×D32—591no—
109#2258—591no—
110C2×C59—21no—
111C2×C59—162no—
112C2×C59C22×D1611yescharacteristic
113D59—22no—
114D59—164no—
115#576—1182no—
116#509—1184no—
117#509—1182no—
118#508—594no—
119#508C2×D5912yescomplemented
120C2×D16—594no—
121C2×D16—592no—
122#504—591no—
123Q64C2×D5911yescharacteristic, complemented
124SD64—594no—
125SD64C2×D5912yescomplemented
126D32—594no—
127D32—592no—
128D32—592no—
129D32C2×D5911yescharacteristic, complemented
130M64—591no—
131M64—592no—
132C2×C32—592no—
133C2×C32C2×D5911yescharacteristic, complemented
134C59#3398211yescharacteristic, complemented, Sylow
135H(5,𝔽2)—2362no—
136AGL(1,ℤ/8)—2364no—
137AGL(1,ℤ/8)—2362no—
138#135—22no—
139#135—1184no—
140C2×D8—1184no—
141C2×D8—1182no—
142#131—591no—
143Q32C22×D5912yes—
144SD32—22no—
145SD32—1184no—
146D16—594no—
147D16—1184no—
148D16—1182no—
149D16C22×D5912yes—
150M32—591no—
151M32—592no—
152C2×C16—592no—
153C2×C16C22×D5911yescharacteristic
154C32—592no—
155C32C22×D5911yescharacteristic
156C32C22×D5911yescharacteristic
157#41—42no—
158#41—591no—
159#41—4722no—
160#41—2364no—
161C2×D4—4722no—
162C2×D4—4724no—
163C2×D4—2364no—
164C2×D4—2362no—
165Q16—22no—
166SD16—42no—
167SD16—2364no—
168D8—22no—
169D8—1184no—
170D8—2364no—
171D8—2362no—
172M16—591no—
173M16—592no—
174C2×C8—592no—
175C2×C8D4×D5911yescharacteristic
176C16—592no—
177C16D4×D5911yescharacteristic
178C16C23×D5911yescharacteristic
179C23—4724no—
180C23—4722no—
181Q8—42no—
182Q8—591no—
183D4—42no—
184D4—82no—
185D4—592no—
186D4—591no—
187D4—4724no—
188D4—4724no—
189D4—4722no—
190D4—2364no—
191C2×C4—82no—
192C2×C4—592no—
193C2×C4—4724no—
194C2×C4D8×D5911yescharacteristic
195C8—592no—
196C8D8×D5911yescharacteristic
197C8C2×D4×D5911yescharacteristic
198C22—82no—
199C22—592no—
200C22—18882no—
201C22—18884no—
202C22—4724no—
203C22D16×D5911yescharacteristic
204C4—82no—
205C4—592no—
206C4D16×D5911yescharacteristic
207C4C2×D8×D5911yescharacteristic
208C2—21no—
209C2—162no—
210C2—1182no—
211C2—9444no—
212C2C2×D16×D5911yescharacteristic
subgroupG/H
centerC2C2×D16×D59
commutator, cosocleC16×C59C24
FrattiniC16C23×D59
FittingC59×#2262C2
socleC2×C59C22×D16
hypercenter#2262D59
hypocenterC59#33982
LCSG ▷ C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #21437234C2semidirect
2. #21426410C2semidirect
3. #21426408C2semidirect
4. #21426406C2semidirect
5. C2×D1888C2semidirect
6. D32×D59C2semidirect
7. #21426389C2semidirect
8. C59×#2262C2semidirect
12. #16837835C22semidirect
13. C2×D944C22semidirect
14. D16×D59C22semidirect
16. #16837239C22semidirect
17. D1888C22semidirect
18. D1888C22semidirect
19. #16837237C22semidirect
20. #16837236C22semidirect
21. #16837235C22semidirect
22. C32×D59C22semidirect
23. C59×#508C22semidirect
24. C59×Q64C22semidirect
25. C59×SD64C22semidirect
26. C59×D32C22semidirect
27. C2×C32×C59C22semidirect
106. #2262D59semidirect
119. #508C2×D59semidirect
123. Q64C2×D59semidirect
125. SD64C2×D59semidirect
129. D32C2×D59semidirect
133. C2×C32C2×D59semidirect
134. C59#33982semidirect

Maximal subgroups

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

subgroupindex
#214372342
#214264102
#214264082
#214264062
C2×D18882
D32×D592
#214263892
C59×#22622
#3398259