Finite Group Databasebeta

← #28502936

Subgroups — #28502936

Order 15104.

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

normal subgroups only

This group has 224 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#21426174C211yesmaximal, characteristic, complemented
2#21426169C211yesmaximal, characteristic
3#21425934C211yesmaximal, characteristic, complemented
4#21422172C211yesmaximal, characteristic, complemented
5#21422135C211yesmaximal, characteristic, complemented
6#21422133C211yesmaximal, characteristic, complemented
7C59×#2248C211yesmaximal, characteristic, complemented
8#16837843—21no—
9#16837838—21no—
10#16837818—21no—
11#16837520C2211yescharacteristic
12#16837223C2211yescharacteristic
13#16837223C2211yescharacteristic
14#16837223C2211yescharacteristic
15#16837218—21no—
16#16837215—21no—
17#16837185—21no—
18C2×SD32×C59—21no—
19C2×SD32×C59—21no—
20C59×#494C2211yescharacteristic
21C59×#493C2211yescharacteristic
22C4×C16×C59C2211yescharacteristic
23#14273749D411yescharacteristic
24#14273733—21no—
25#14273673—41no—
26#14273671—41no—
27#14273670—41no—
28#14273668—41no—
29#14273667—21no—
30#14273667—21no—
31#14273667—21no—
32#14273666—41no—
33C16×D59—41no—
34#14273663D411yescharacteristic
35#14273655—21no—
36C2×Q16×C59—21no—
37C2×Q16×C59—21no—
38C2×D8×C59—21no—
39C2×D8×C59—21no—
40C4⋊Q8×C59—21no—
41#127×C59—21no—
42SD32×C59—42no—
43SD32×C59—41no—
44SD32×C59—41no—
45C2×C16×C59D411yescharacteristic
46C2×C16×C59D411yescharacteristic
47C4×C8×C59C2311yescharacteristic
48C2×Q8×C59—41no—
49C2×D4×C59—41no—
50C2×D4×C59—81no—
51#11745685—21no—
52Q16×C59—41no—
53Q16×C59—41no—
54Q16×C59—41no—
55Q16×C59—41no—
56D8×C59—41no—
57D8×C59—41no—
58D8×C59—41no—
59D8×C59—41no—
60C2×C8×C59C2×D411yescharacteristic
61C2×C8×C59C2×D411yescharacteristic
62C4⋊C4×C59—81no—
63C42×C59#4111yescharacteristic
64#11745664—21no—
65Dic236D811yescharacteristic
66D472D811yescharacteristic
67#11745660—21no—
68#11745659—41no—
69C8×D59—41no—
70C16×C59—21no—
71C16×C59D811yescharacteristic
72C16×C59D811yescharacteristic
73C59⋊C16—41no—
74C59⋊C16—41no—
75C59⋊C16—41no—
76C23×C59—81no—
77Q8×C59—81no—
78Q8×C59—81no—
79D4×C59—81no—
80D4×C59—81no—
81D4×C59—162no—
82C2×C4×C59—21no—
83C2×C4×C59—81no—
84C2×C4×C59#12111yescharacteristic
85C22⋊D59—41no—
86D236—21no—
87C4×D59—41no—
88Dic118—21no—
89C8×C59C2×D811yescharacteristic
90C8×C59C2×D811yescharacteristic
91C8×C59#13511yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C59⋊C8—41no—
94#12438—591nomaximal, Sylow
95C22×C59—162no—
96C22×C59—161no—
97C22×C59#47211yescharacteristic
98C2×D59—41no—
99C4×C59—22no—
100C4×C59—161no—
101C4×C59#45911yescharacteristic
102C4×C59#46611yescharacteristic
103Dic59—41no—
104#2248D5911yescharacteristic, complemented
105#2214—591no—
106#2213—591no—
107#2170—591no—
108#1373—591no—
109#1358—591no—
110#1357—591no—
111C2×C59—21no—
112C2×C59—321no—
113C2×C59#166711yescharacteristic
114D59—81no—
115#510—1181no—
116#509—1181no—
117C2×SD32—21no—
118C2×SD32—21no—
119#504—1181no—
120#494C2×D5911yescharacteristic
121#493C2×D5911yescharacteristic
122#443—591no—
123#364—591no—
124#364—591no—
125#364—591no—
126#362—1181no—
127#361—1181no—
128#350—1181no—
129C4×C16C2×D5911yescharacteristic
130C59#1243811yescharacteristic, complemented, Sylow
131#135—591no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—21no—
135C2×D8—21no—
136#131—1181no—
137C4⋊Q8—21no—
138#127—21no—
139Q32—2361no—
140SD32—42no—
141SD32—41no—
142SD32—41no—
143SD32—2361no—
144SD32—2361no—
145D16—2361no—
146M32—1181no—
147M32—1181no—
148M32—1181no—
149M32—2361no—
150C2×C16—2361no—
151C2×C16C22⋊D5911yescharacteristic
152C2×C16C22⋊D5911yescharacteristic
153#108—591no—
154#104—1181no—
155C4×C8C22×D5911yescharacteristic
156#41—1181no—
157C2×Q8—41no—
158C2×D4—41no—
159C2×D4—81no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16—591no—
165SD16—1181no—
166D8—41no—
167D8—41no—
168D8—41no—
169D8—41no—
170D8—591no—
171M16—1181no—
172M16—2361no—
173C2×C8—2361no—
174C2×C8D4×D5911yescharacteristic
175C2×C8C2×C22⋊D5911yescharacteristic
176C4⋊C4—81no—
177C42#1174568711yescharacteristic
178C16—21no—
179C16—2361no—
180C16—2361no—
181C16—2361no—
182C16#1174566911yescharacteristic
183C16#1174566911yescharacteristic
184C23—81no—
185Q8—81no—
186Q8—81no—
187Q8—1181no—
188D4—81no—
189D4—81no—
190D4—162no—
191D4—1181no—
192D4—2361no—
193C2×C4—21no—
194C2×C4—81no—
195C2×C4—2361no—
196C2×C4#1427369811yescharacteristic
197C8—2361no—
198C8D8×D5911yescharacteristic
199C8C2×#1174566911yescharacteristic
200C8#1427374611yescharacteristic
201C8#1427375311yescharacteristic
202C22—162no—
203C22—161no—
204C22—2361no—
205C22#1683766911yescharacteristic
206C4—22no—
207C4—161no—
208C4—2361no—
209C4#1683759411yescharacteristic
210C4#1683763511yescharacteristic
211C2—21no—
212C2—321no—
213C2—4721no—
214C2#2142344811yescharacteristic
subgroupG/H
centerC2#21423448
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2248C2
socleC2×C59#1667
prime radical#21426169C2
hypercenter#2248D59
hypocenterC59#12438
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #21426174C2semidirect
3. #21425934C2semidirect
4. #21422172C2semidirect
5. #21422135C2semidirect
6. #21422133C2semidirect
7. C59×#2248C2semidirect
104. #2248D59semidirect
130. C59#12438semidirect

Maximal subgroups

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

subgroupindex
#214261742
#214261692
#214259342
#214221722
#214221352
#214221332
C59×#22482
#1243859