Finite Group Databasebeta

← #28503096

Subgroups — #28503096

Order 15104.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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#21426269C211yesmaximal, characteristic
2#21426174C211yesmaximal, characteristic
3#21425939C211yesmaximal, characteristic
4#21422164C211yesmaximal, characteristic, complemented
5#21422157C211yesmaximal, characteristic, complemented
6#21422135C211yesmaximal, characteristic, complemented
7C59×#2245C211yesmaximal, characteristic, complemented
8#16837843—21no—
9#16837818—21no—
10#16837520C2211yescharacteristic
11#16837224—21no—
12#16837223C2211yescharacteristic
13#16837223C2211yescharacteristic
14#16837223C2211yescharacteristic
15#16837218—21no—
16#16837211—21no—
17#16837185—21no—
18C59×#499C2211yescharacteristic
19C59×#494C2211yescharacteristic
20C59×#358—21no—
21C59×#358—21no—
22C59×#346C2211yescharacteristic
23#14273749D411yescharacteristic
24#14273733—21no—
25#14273673—41no—
26#14273670—41no—
27#14273667—21no—
28#14273667—21no—
29#14273667—21no—
30#14273666—41no—
31C16×D59—41no—
32#14273663D411yescharacteristic
33#14273655—21no—
34C2×Q16×C59—21no—
35C2×Q16×C59—21no—
36C4⋊Q8×C59—21no—
37#125×C59—21no—
38C2×C16×C59D411yescharacteristic
39C2×C16×C59D411yescharacteristic
40#107×C59—21no—
41#106×C59—21no—
42C4×C8×C59C2311yescharacteristic
43C2×Q8×C59—41no—
44#11745685—21no—
45Q16×C59—41no—
46Q16×C59—41no—
47Q16×C59—41no—
48Q16×C59—41no—
49C2×C8×C59C2×D411yescharacteristic
50C2×C8×C59C2×D411yescharacteristic
51C4⋊C4×C59—41no—
52C4⋊C4×C59—81no—
53C4⋊C4×C59—81no—
54C42×C59#4111yescharacteristic
55#11745664—21no—
56Dic236SD1611yescharacteristic
57D472SD1611yescharacteristic
58#11745660—21no—
59#11745659—41no—
60C8×D59—41no—
61C16×C59—21no—
62C16×C59—21no—
63C59⋊C16—41no—
64C59⋊C16—41no—
65C59⋊C16—41no—
66Q8×C59—81no—
67Q8×C59—81no—
68C2×C4×C59—21no—
69C2×C4×C59—81no—
70C2×C4×C59—81no—
71C2×C4×C59#12111yescharacteristic
72C22⋊D59—41no—
73D236—21no—
74C4×D59—41no—
75Dic118—21no—
76C8×C59C2×SD1611yescharacteristic
77C8×C59#13511yescharacteristic
78C8×C59#13511yescharacteristic
79C8×C59AGL(1,ℤ/8)11yescharacteristic
80C59⋊C8—41no—
81#12466—591nomaximal, Sylow
82C22×C59#47211yescharacteristic
83C2×D59—41no—
84C4×C59—41no—
85C4×C59—161no—
86C4×C59—161no—
87C4×C59#46311yescharacteristic
88C4×C59#46511yescharacteristic
89Dic59—41no—
90#2245D5911yescharacteristic, complemented
91#2231—591no—
92#2214—591no—
93#2171—591no—
94#1369—591no—
95#1366—591no—
96#1358—591no—
97C2×C59—21no—
98C2×C59#167711yescharacteristic
99D59—81no—
100#510—1181no—
101#504—1181no—
102#499C2×D5911yescharacteristic
103#494C2×D5911yescharacteristic
104#443—591no—
105#365—1181no—
106#364—591no—
107#364—591no—
108#364—591no—
109#362—1181no—
110#360—1181no—
111#358—21no—
112#358—21no—
113#350—1181no—
114#346C2×D5911yescharacteristic
115C59#1246611yescharacteristic, complemented, Sylow
116#135—591no—
117C2×Q16—21no—
118C2×Q16—21no—
119#131—1181no—
120C4⋊Q8—21no—
121#125—21no—
122Q32—2361no—
123SD32—2361no—
124M32—1181no—
125M32—1181no—
126M32—1181no—
127M32—2361no—
128C2×C16—2361no—
129C2×C16C22⋊D5911yescharacteristic
130C2×C16C22⋊D5911yescharacteristic
131#108—591no—
132#107—21no—
133#106—21no—
134#104—1181no—
135C4×C8C22×D5911yescharacteristic
136#41—1181no—
137C2×Q8—41no—
138Q16—41no—
139Q16—41no—
140Q16—41no—
141Q16—41no—
142Q16—591no—
143SD16—1181no—
144D8—591no—
145M16—1181no—
146M16—2361no—
147C2×C8—2361no—
148C2×C8D4×D5911yescharacteristic
149C2×C8C2×C22⋊D5911yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C42#1174568711yescharacteristic
154C16—21no—
155C16—21no—
156C16—2361no—
157C16—2361no—
158C16—2361no—
159Q8—81no—
160Q8—81no—
161Q8—1181no—
162D4—1181no—
163D4—2361no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—2361no—
168C2×C4#1427369811yescharacteristic
169C8—2361no—
170C8SD16×D5911yescharacteristic
171C8#1427374711yescharacteristic
172C8#1427374811yescharacteristic
173C8#1427375211yescharacteristic
174C22—2361no—
175C22#1683766911yescharacteristic
176C4—41no—
177C4—161no—
178C4—161no—
179C4—2361no—
180C4#1683761411yescharacteristic
181C4#1683762811yescharacteristic
182C2—21no—
183C2—4721no—
184C2#2142350211yescharacteristic
subgroupG/H
centerC2#21423502
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2245C2
socleC2×C59#1677
prime radical#14273749D4
hypercenter#2245D59
hypocenterC59#12466
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2245

Complements

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

NH ≅ G/Ntype
4. #21422164C2semidirect
5. #21422157C2semidirect
6. #21422135C2semidirect
7. C59×#2245C2semidirect
90. #2245D59semidirect
115. C59#12466semidirect

Maximal subgroups

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

subgroupindex
#214262692
#214261742
#214259392
#214221642
#214221572
#214221352
C59×#22452
#1246659