Finite Group Databasebeta

← #28502921

Subgroups — #28502921

Order 15104.

246 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426351C211yesmaximal, characteristic, complemented
2#21425927C211yesmaximal, characteristic, complemented
3#21423462C211yesmaximal, characteristic, complemented
4#21423448C211yesmaximal, characteristic
5#21422167C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic, complemented
8C2×#14273671—21no—
9C2×#14273671—21no—
10C2×#14273666—21no—
11#16837782C2211yescharacteristic
12#16837683—21no—
13#16837594—21no—
14C8×D236C2211yescharacteristic
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837147—21no—
19#16837146—21no—
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×#11745669—41no—
26C2×C8×D59—21no—
27#14273727—21no—
28C4×D236D411yescharacteristic
29#14273671—42no—
30#14273671—42no—
31#14273666—42no—
32C2×C59⋊C16D411yescharacteristic
33C2×C59⋊C16D411yescharacteristic
34#14273661—41no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)D411yescharacteristic
37#14273649—41no—
38#14273648—41no—
39(C2×D59)⋊C8—21no—
40C2×D8×C59—21no—
41C2×D8×C59—21no—
42#127×C59—21no—
43C2×C16×C59—21no—
44C4⋊C8×C59—21no—
45C4×C8×C59C2311yescharacteristic
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D236D811yescharacteristic
50C2×C4×D59—21no—
51C2×Dic118—41no—
52D8×C59—41no—
53D8×C59—41no—
54D8×C59—81no—
55C2×C8×C59—41no—
56C2×C8×C59C2×D411yescharacteristic
57C2×C8×C59#4111yescharacteristic
58C42×C59C2×D411yescharacteristic
59#11745669—82no—
60(C2×D59)⋊C4—21no—
61C4⋊Dic59—41no—
62C4⋊Dic59—41no—
63C4⋊Dic59D811yescharacteristic
64C2×C59⋊C8—21no—
65C2×C59⋊C8—41no—
66Dic236—41no—
67Dic236—41no—
68Dic236—81no—
69C8×D59—41no—
70C8×D59—41no—
71C16×C59—41no—
72C59⋊C16—21no—
73C59⋊C16D812yescomplemented
74C23×C59—81no—
75C22×D59—21no—
76D4×C59—81no—
77D4×C59—81no—
78D4×C59—81no—
79D4×C59—81no—
80D4×C59—161no—
81C2×C4×C59#12111yescharacteristic
82C2×C4×C59C2×D811yescharacteristic
83C2×C4×C59#13511yescharacteristic
84C2×Dic59—21no—
85C2×Dic59—81no—
86D236—21no—
87D236—21no—
88C4×D59—41no—
89C4×D59—41no—
90Dic118—81no—
91Dic118—81no—
92C8×C59—21no—
93C8×C59—81no—
94C8×C59C2×D811yescharacteristic
95C8×C59AGL(1,ℤ/8)11yescharacteristic
96C59⋊C8—41no—
97C59⋊C8—81no—
98#12435—591nomaximal, Sylow
99C22×C59—161no—
100C22×C59—161no—
101C22×C59—161no—
102C22×C59#46611yescharacteristic
103C2×D59—41no—
104C2×D59—41no—
105C2×D59—41no—
106C4×C59—21no—
107C4×C59#45911yescharacteristic
108C4×C59#47211yescharacteristic
109C4×C59#50911yescharacteristic
110C4×C59#51011yescharacteristic
111Dic59—41no—
112Dic59—161no—
113#2248—591no—
114#2169—591no—
115#1669—591no—
116#1667—591no—
117#1371—591no—
118#1362—591no—
119#1361D5911yescharacteristic, complemented
120C2×C59—321no—
121C2×C59#166711yescharacteristic
122C2×C59#222011yescharacteristic
123C2×C59#222311yescharacteristic
124D59—81no—
125C2×SD32—1181no—
126C2×SD32—1181no—
127C2×M32—1181no—
128#494—591no—
129#493C2×D5911yescharacteristic
130#474—1181no—
131#459—1181no—
132C8×D4—591no—
133C4⋊C16C2×D5911yescharacteristic
134C22⋊C16—1181no—
135C4×C16—591no—
136#335—591no—
137#335C2×D5911yescharacteristic
138#332—1181no—
139#331—1181no—
140C59#1243511yescharacteristic, complemented, Sylow
141C2×Q16—1181no—
142C2×Q16—1181no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—2361no—
146C22×C8—1181no—
147C4⋊Q8—1181no—
148#127—21no—
149C4×D4—591no—
150SD32—2362no—
151SD32—2362no—
152M32—2362no—
153C2×C16—21no—
154C2×C16—591no—
155C2×C16—591no—
156#107—2361no—
157C4⋊C8—21no—
158C4⋊C8—591no—
159C4⋊C8—1181no—
160#102—2361no—
161#102—2361no—
162C22⋊C8—1181no—
163C4×C8C22×D5911yescharacteristic
164C2×Q8—2361no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—591no—
169C22×C4—1181no—
170Q16—4721no—
171Q16—2361no—
172Q16—2361no—
173D8—41no—
174D8—41no—
175D8—81no—
176D8—4722no—
177C2×C8—41no—
178C2×C8—1181no—
179C2×C8—2361no—
180C2×C8—2361no—
181C2×C8—2361no—
182C2×C8#1174568511yescharacteristic
183C2×C8D4×D5911yescharacteristic
184C4⋊C4—591no—
185C4⋊C4—2361no—
186C4⋊C4—2361no—
187C22⋊C4—1181no—
188C42D4×D5911yescharacteristic
189C16—41no—
190C16—592no—
191C16—1181no—
192C23—81no—
193C23—1181no—
194Q8—4721no—
195Q8—4721no—
196D4—81no—
197D4—81no—
198D4—81no—
199D4—81no—
200D4—161no—
201D4—1181no—
202D4—1181no—
203C2×C4—1181no—
204C2×C4—4721no—
205C2×C4—2361no—
206C2×C4—2361no—
207C2×C4#1427370011yescharacteristic
208C2×C4D8×D5911yescharacteristic
209C2×C4#1427374811yescharacteristic
210C8—21no—
211C8—81no—
212C8—4721no—
213C8—2361no—
214C8D8×D5911yescharacteristic
215C8#1427375411yescharacteristic
216C22—161no—
217C22—161no—
218C22—161no—
219C22—2361no—
220C22—2361no—
221C22—2361no—
222C22#1683763711yescharacteristic
223C4—21no—
224C4—9441no—
225C4—2361no—
226C4#1683759611yescharacteristic
227C4#1683767111yescharacteristic
228C4#1683783611yescharacteristic
229C4#1683784111yescharacteristic
230C2—321no—
231C2—4721no—
232C2#2142345011yescharacteristic
233C2#2142620811yescharacteristic
234C2#2142622511yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#466
prime radical#21423448C2
hypercenter#1361D59
hypocenterC59#12435
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
1. #21426351C2semidirect
2. #21425927C2semidirect
3. #21423462C2semidirect
5. #21422167C2semidirect
6. #21422146C2semidirect
7. C59×#1361C2semidirect
73. C59⋊C16D8semidirect
119. #1361D59semidirect
140. C59#12435semidirect

Maximal subgroups

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

subgroupindex
#214263512
#214259272
#214234622
#214234482
#214221672
#214221462
C59×#13612
#1243559