Finite Group Databasebeta

← #28499238

Subgroups — #28499238

Order 15104.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#21434755C211yesmaximal, characteristic
2#21423851C211yesmaximal, characteristic
3#21422091C212yesmaximal
4#21422085C212yesmaximal
5C59×#2166C211yesmaximal, characteristic
6#16838014—21no—
7#16837804C411yescharacteristic, complemented
8#16837782C411yescharacteristic, complemented
9#16837771—21no—
10#16837660C2212yes—
11#16837643—21no—
12#16837510C412yescomplemented
13C4×Dic236—21no—
14#16837357—21no—
15C59⋊#338C2211yescharacteristic
16#16837159C2211yescharacteristic
17#16837158—21no—
18C59×#405C2211yescharacteristic
19C59×#364—21no—
20C59×#349C2212yes—
21C59×#347—21no—
22C2×Dic236—21no—
23C2×Dic236C2×C412yes—
24C2×#11745664—21no—
25#14273727—21no—
26#14273720—42no—
27#14273719—21no—
28#14273717—42no—
29#14273706—22no—
30C4×Dic118—22no—
31#14273686—21no—
32#14273661—21no—
33#14273659C2×C412yes—
34#14273652—22no—
35#14273652—22no—
36#14273637—22no—
37C8×Dic59—22no—
38C2×M16×C59C2311yescharacteristic
39C22×C8×C59D411yescharacteristic
40#117×C59D411yescharacteristic
41M32×C59—22no—
42#97×C59C2×C411yescharacteristic
43#97×C59D411yescharacteristic
44C4×C8×C59C2×C411yescharacteristic
45C4×C8×C59D411yescharacteristic
46C22×C4×C59C2×D411yescharacteristic
47C2×Dic118—22no—
48M16×C59—41no—
49C2×C8×C59—21no—
50C2×C8×C59—21no—
51C2×C8×C59C22⋊C411yescharacteristic
52C2×C8×C59C22⋊C411yescharacteristic
53C2×C8×C59C22×C411yescharacteristic
54C2×C8×C59C2×D411yescharacteristic
55C4⋊C4×C59—21no—
56C22⋊C4×C59—21no—
57C42×C59C22⋊C411yescharacteristic
58C42×C59C22⋊C411yescharacteristic
59C22⋊Dic59—41no—
60C22⋊Dic59—42no—
61C4⋊Dic59—22no—
62C4⋊Dic59—41no—
63C4⋊Dic59—42no—
64Dic59⋊C4—42no—
65Dic59⋊C4—42no—
66Dic59⋊C4—42no—
67C4×Dic59—22no—
68C4×Dic59—42no—
69#11745664—41no—
70#11745664—41no—
71C2×C59⋊C8—22no—
72Dic236—42no—
73Dic236—41no—
74Dic236—41no—
75C16×C59—42no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59C2×C22⋊C411yescharacteristic
82C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
83C2×Dic59—42no—
84C2×Dic59—42no—
85C2×Dic59—42no—
86Dic118—42no—
87Dic118—42no—
88C8×C59—21no—
89C8×C59—21no—
90C8×C59—21no—
91C8×C59#10411yescharacteristic
92C8×C59#10411yescharacteristic
93C59⋊C8—42no—
94#11771—591nomaximal, Sylow
95C22×C59—21no—
96C22×C59—41no—
97C22×C59#41811yescharacteristic
98C4×C59—21no—
99C4×C59—41no—
100C4×C59—41no—
101C4×C59#41011yescharacteristic
102C4×C59C2×#10411yescharacteristic
103Dic59—82no—
104Dic59—82no—
105Dic59—82no—
106#3177—591no—
107#2166D5911yescharacteristic
108#1747—591no—
109#1342—592no—
110#1340—592no—
111C2×C59—21no—
112C2×C59—41no—
113C2×C59#150511yescharacteristic
114#541—1181no—
115#499—591no—
116#494—591no—
117#491—1181no—
118#470—592no—
119#467—1181no—
120#441—592no—
121C4×Q16—1181no—
122#405—1181no—
123#405C2×D5911yescharacteristic
124#364—21no—
125#349C2×D5912yes—
126#347—21no—
127#338—591no—
128H(3,ℤ/4)—591no—
129H(3,ℤ/4)—1181no—
130C59#1177111yescharacteristic, complemented, Sylow
131C2×Q16—592no—
132C2×Q16—1181no—
133C2×M16—1181no—
134C2×M16C22×D5911yescharacteristic
135C22×C8D23611yescharacteristic
136C4⋊Q8—1181no—
137#126—2362no—
138#125—1181no—
139#125—2362no—
140C22⋊Q8—1182no—
141C4×Q8—1182no—
142#117—1181no—
143#117D23611yescharacteristic
144M32—22no—
145#107—1181no—
146#106—592no—
147#103—1182no—
148#103—1182no—
149#97—1182no—
150#97C4×D5911yescharacteristic
151#97C22⋊D5911yescharacteristic
152C4×C8—1182no—
153C4×C8C4×D5911yescharacteristic
154C4×C8C22⋊D5911yescharacteristic
155C2×Q8—1182no—
156C22×C4C2×D23611yescharacteristic
157Q16—2362no—
158Q16—2361no—
159Q16—2361no—
160M16—41no—
161M16—2361no—
162M16—2361no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1182no—
166C2×C8(C2×D59)⋊C411yescharacteristic
167C2×C8(C2×D59)⋊C411yescharacteristic
168C2×C8C2×C4×D5911yescharacteristic
169C2×C8C2×C22⋊D5911yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—1182no—
172C4⋊C4—2361no—
173C4⋊C4—2362no—
174C4⋊C4—2362no—
175C4⋊C4—2362no—
176C4⋊C4—2362no—
177C22⋊C4—21no—
178C22⋊C4—2361no—
179C22⋊C4—2362no—
180C42—1182no—
181C42—2362no—
182C42(C2×D59)⋊C411yescharacteristic
183C42(C2×D59)⋊C411yescharacteristic
184C16—42no—
185C23#1427375511yescharacteristic
186Q8—2362no—
187Q8—2362no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2362no—
193C2×C4—2362no—
194C2×C4—2362no—
195C2×C4C2×(C2×D59)⋊C411yescharacteristic
196C2×C4#1427375511yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—2362no—
201C8#1427365311yescharacteristic
202C8#1427365311yescharacteristic
203C22—21no—
204C22—41no—
205C22#1683740811yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—4722no—
210C4—4722no—
211C4—4722no—
212C4#1683737511yescharacteristic
213C4C2×#1427365311yescharacteristic
214C2—21no—
215C2—41no—
216C2#2142260511yescharacteristic
subgroupG/H
centerC2#21422605
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2166C2
socleC2×C59#1505
prime radicalC23×C59AGL(1,ℤ/8)
cosocleC2×M16×C59C23
hypercenter#2166D59
hypocenterC59#11771
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
7. #16837804C4semidirect
8. #16837782C4semidirect
12. #16837510C4semidirect
130. C59#11771semidirect

Maximal subgroups

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

subgroupindex
#214347552
#214238512
#214220912
#214220852
C59×#21662
#1177159