Finite Group Databasebeta

← #28502932

Subgroups — #28502932

Order 15104.

214 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 59 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#21426348C211yesmaximal, characteristic, complemented
2#21426177C211yesmaximal, characteristic
3#21425933C211yesmaximal, characteristic
4#21422171C211yesmaximal, characteristic, complemented
5#21422135C211yesmaximal, characteristic, complemented
6#21422134C211yesmaximal, characteristic, complemented
7C59×#2214C211yesmaximal, characteristic
8#16837842—21no—
9C2×#14273673—21no—
10C2×#14273673—21no—
11#16837819—21no—
12#16837782C2211yescharacteristic
13#16837223C2211yescharacteristic
14#16837223C2211yescharacteristic
15#16837217—21no—
16#16837186—21no—
17C4×C59⋊C16C2211yescharacteristic
18C59×#510—21no—
19C59×#494C2211yescharacteristic
20C59×#443C2211yescharacteristic
21C59×#364C2211yescharacteristic
22C59×#362—21no—
23C2×Dic236—21no—
24C2×Dic236—21no—
25#14273727—21no—
26#14273673—42no—
27#14273673—41no—
28#14273673—41no—
29#14273673—41no—
30#14273671—41no—
31#14273667—21no—
32#14273667—21no—
33#14273667—41no—
34C2×C59⋊C16—41no—
35C2×C59⋊C16D411yescharacteristic
36C2×C59⋊C16D411yescharacteristic
37#135×C59D411yescharacteristic
38C2×Q16×C59—21no—
39C2×Q16×C59—21no—
40#131×C59—21no—
41C4⋊Q8×C59—21no—
42Q32×C59—41no—
43SD32×C59—41no—
44M32×C59—21no—
45#108×C59D411yescharacteristic
46#104×C59—21no—
47C4×C8×C59C2311yescharacteristic
48#41×C59—21no—
49C2×Q8×C59—41no—
50C2×Dic118—41no—
51Q16×C59—41no—
52Q16×C59—41no—
53Q16×C59—41no—
54Q16×C59—41no—
55Q16×C59D811yescharacteristic
56SD16×C59—21no—
57D8×C59D811yescharacteristic
58M16×C59—21no—
59M16×C59—41no—
60C2×C8×C59—41no—
61C2×C8×C59C2×D411yescharacteristic
62C2×C8×C59C2×D411yescharacteristic
63C4⋊C4×C59—81no—
64C42×C59#4111yescharacteristic
65C4⋊Dic59—81no—
66Dic236—41no—
67Dic236—41no—
68Dic236—41no—
69Dic236—41no—
70C16×C59—41no—
71C59⋊C16—21no—
72C59⋊C16—41no—
73C59⋊C16—41no—
74C59⋊C16D811yescharacteristic
75C59⋊C16D811yescharacteristic
76Q8×C59—21no—
77Q8×C59—81no—
78Q8×C59—81no—
79D4×C59—21no—
80D4×C59—41no—
81C2×C4×C59—21no—
82C2×C4×C59—41no—
83C2×C4×C59—81no—
84C2×C4×C59#12111yescharacteristic
85C2×Dic59—81no—
86Dic118—81no—
87Dic118—81no—
88C8×C59—41no—
89C8×C59C2×D811yescharacteristic
90C8×C59C2×D811yescharacteristic
91C8×C59#13511yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93#12437—591nomaximal, Sylow
94C22×C59—41no—
95C22×C59#47211yescharacteristic
96C4×C59—22no—
97C4×C59—41no—
98C4×C59—161no—
99C4×C59#45911yescharacteristic
100C4×C59#46611yescharacteristic
101Dic59—161no—
102#2247—591no—
103#2214—591no—
104#2214D5911yescharacteristic
105#2170—591no—
106#1373—591no—
107#1358—591no—
108#1358—591no—
109C2×C59—21no—
110C2×C59—81no—
111C2×C59#166711yescharacteristic
112#510—21no—
113#510—1181no—
114C2×Q32—1181no—
115C2×Q32—1181no—
116#504—1181no—
117#494—591no—
118#494C2×D5911yescharacteristic
119#443C2×D5911yescharacteristic
120#364—591no—
121#364—591no—
122#364C2×D5911yescharacteristic
123#362—21no—
124#362—1181no—
125#350—1181no—
126C4×C16—591no—
127C59#1243711yescharacteristic, complemented, Sylow
128#135C22⋊D5911yescharacteristic
129C2×Q16—21no—
130C2×Q16—21no—
131C2×Q16—1181no—
132C2×Q16—1181no—
133#131—21no—
134C4⋊Q8—21no—
135C4⋊Q8—1181no—
136Q32—41no—
137Q32—2361no—
138Q32—2361no—
139Q32—2362no—
140Q32—2361no—
141SD32—41no—
142SD32—2361no—
143M32—21no—
144M32—1181no—
145M32—1181no—
146M32—2361no—
147C2×C16—591no—
148C2×C16—591no—
149C2×C16—2361no—
150#108C22⋊D5911yescharacteristic
151#104—21no—
152C4×C8C22×D5911yescharacteristic
153#41—21no—
154C2×Q8—41no—
155C2×Q8—2361no—
156Q16—41no—
157Q16—41no—
158Q16—41no—
159Q16—41no—
160Q16—2361no—
161Q16—2361no—
162Q16—2361no—
163Q16—2361no—
164Q16#1174566911yescharacteristic
165SD16—21no—
166D8#1174566911yescharacteristic
167M16—21no—
168M16—41no—
169C2×C8—41no—
170C2×C8D4×D5911yescharacteristic
171C2×C8C2×C22⋊D5911yescharacteristic
172C4⋊C4—81no—
173C4⋊C4—4721no—
174C42#1174568711yescharacteristic
175C16—41no—
176C16—591no—
177C16—591no—
178C16—1181no—
179C16—2361no—
180C16—2361no—
181Q8—21no—
182Q8—81no—
183Q8—81no—
184Q8—4721no—
185Q8—4721no—
186D4—21no—
187D4—41no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—4721no—
192C2×C4#1427369911yescharacteristic
193C8—41no—
194C8D8×D5911yescharacteristic
195C8C2×#1174566911yescharacteristic
196C8#1427374711yescharacteristic
197C8#1427375111yescharacteristic
198C22—41no—
199C22#1683767011yescharacteristic
200C4—22no—
201C4—41no—
202C4—161no—
203C4—9441no—
204C4#1683759511yescharacteristic
205C4#1683763611yescharacteristic
206C2—21no—
207C2—81no—
208C2#2142344911yescharacteristic
subgroupG/H
centerC2#21423449
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2214C2
socleC2×C59#1667
prime radical#135×C59D4
hypercenter#2214D59
hypocenterC59#12437
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #21426348C2semidirect
4. #21422171C2semidirect
5. #21422135C2semidirect
6. #21422134C2semidirect
127. C59#12437semidirect

Maximal subgroups

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

subgroupindex
#214263482
#214261772
#214259332
#214221712
#214221352
#214221342
C59×#22142
#1243759