Finite Group Databasebeta

← #28503089

Subgroups — #28503089

Order 15104.

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

normal subgroups only

This group has 196 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, complemented
2#21426169C211yesmaximal, characteristic
3#21425939C211yesmaximal, characteristic, complemented
4#21422164C211yesmaximal, characteristic, complemented
5#21422157C211yesmaximal, characteristic, complemented
6#21422133C211yesmaximal, characteristic, complemented
7C59×#2244C211yesmaximal, characteristic, complemented
8#16837838—21no—
9#16837818—21no—
10#16837520C2211yescharacteristic
11#16837224—21no—
12#16837223C2211yescharacteristic
13#16837223C2211yescharacteristic
14#16837223C2211yescharacteristic
15#16837215—21no—
16#16837211—21no—
17#16837185—21no—
18C59×#499C2211yescharacteristic
19C59×#493C2211yescharacteristic
20C59×#357—21no—
21C59×#357—21no—
22C59×#346C2211yescharacteristic
23#14273749D411yescharacteristic
24#14273733—21no—
25#14273671—41no—
26#14273668—41no—
27#14273667—21no—
28#14273667—21no—
29#14273667—21no—
30#14273666—41no—
31C16×D59—41no—
32#14273663D411yescharacteristic
33#14273655—21no—
34C2×D8×C59—21no—
35C2×D8×C59—21no—
36#127×C59—21no—
37#125×C59—21no—
38C2×C16×C59D411yescharacteristic
39C2×C16×C59D411yescharacteristic
40#107×C59—21no—
41#106×C59—21no—
42C4×C8×C59C2311yescharacteristic
43C2×D4×C59—41no—
44C2×D4×C59—81no—
45#11745685—21no—
46D8×C59—41no—
47D8×C59—41no—
48D8×C59—41no—
49D8×C59—41no—
50C2×C8×C59C2×D411yescharacteristic
51C2×C8×C59C2×D411yescharacteristic
52C4⋊C4×C59—41no—
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—
66C23×C59—81no—
67D4×C59—81no—
68D4×C59—81no—
69D4×C59—162no—
70C2×C4×C59—21no—
71C2×C4×C59—81no—
72C2×C4×C59#12111yescharacteristic
73C22⋊D59—41no—
74D236—21no—
75C4×D59—41no—
76Dic118—21no—
77C8×C59C2×SD1611yescharacteristic
78C8×C59#13511yescharacteristic
79C8×C59#13511yescharacteristic
80C8×C59AGL(1,ℤ/8)11yescharacteristic
81C59⋊C8—41no—
82#12465—591nomaximal, Sylow
83C22×C59—162no—
84C22×C59—161no—
85C22×C59#47211yescharacteristic
86C2×D59—41no—
87C4×C59—41no—
88C4×C59—161no—
89C4×C59#46311yescharacteristic
90C4×C59#46511yescharacteristic
91Dic59—41no—
92#2244D5911yescharacteristic, complemented
93#2231—591no—
94#2213—591no—
95#2171—591no—
96#1369—591no—
97#1366—591no—
98#1357—591no—
99C2×C59—21no—
100C2×C59—321no—
101C2×C59#167711yescharacteristic
102D59—81no—
103#509—1181no—
104#504—1181no—
105#499C2×D5911yescharacteristic
106#493C2×D5911yescharacteristic
107#443—591no—
108#365—1181no—
109#364—591no—
110#364—591no—
111#364—591no—
112#361—1181no—
113#360—1181no—
114#357—21no—
115#357—21no—
116#350—1181no—
117#346C2×D5911yescharacteristic
118C59#1246511yescharacteristic, complemented, Sylow
119#135—591no—
120C2×D8—21no—
121C2×D8—21no—
122#131—1181no—
123#127—21no—
124#125—21no—
125SD32—2361no—
126D16—2361no—
127M32—1181no—
128M32—1181no—
129M32—1181no—
130M32—2361no—
131C2×C16—2361no—
132C2×C16C22⋊D5911yescharacteristic
133C2×C16C22⋊D5911yescharacteristic
134#108—591no—
135#107—21no—
136#106—21no—
137#104—1181no—
138C4×C8C22×D5911yescharacteristic
139#41—1181no—
140C2×D4—41no—
141C2×D4—81no—
142Q16—591no—
143SD16—1181no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8—591no—
149M16—1181no—
150M16—2361no—
151C2×C8—2361no—
152C2×C8D4×D5911yescharacteristic
153C2×C8C2×C22⋊D5911yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1174568711yescharacteristic
157C16—21no—
158C16—21no—
159C16—2361no—
160C16—2361no—
161C16—2361no—
162C23—81no—
163Q8—1181no—
164D4—81no—
165D4—81no—
166D4—162no—
167D4—1181no—
168D4—2361no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—2361no—
172C2×C4#1427369811yescharacteristic
173C8—2361no—
174C8SD16×D5911yescharacteristic
175C8#1427374711yescharacteristic
176C8#1427374811yescharacteristic
177C8#1427375211yescharacteristic
178C22—162no—
179C22—161no—
180C22—2361no—
181C22#1683766911yescharacteristic
182C4—41no—
183C4—161no—
184C4—2361no—
185C4#1683761411yescharacteristic
186C4#1683762811yescharacteristic
187C2—21no—
188C2—321no—
189C2—4721no—
190C2#2142350211yescharacteristic
subgroupG/H
centerC2#21423502
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2244C2
socleC2×C59#1677
prime radical#21426169C2
hypercenter#2244D59
hypocenterC59#12465
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #21426269C2semidirect
3. #21425939C2semidirect
4. #21422164C2semidirect
5. #21422157C2semidirect
6. #21422133C2semidirect
7. C59×#2244C2semidirect
92. #2244D59semidirect
118. C59#12465semidirect

Maximal subgroups

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

subgroupindex
#214262692
#214261692
#214259392
#214221642
#214221572
#214221332
C59×#22442
#1246559