Finite Group Databasebeta

← #28502974

Subgroups — #28502974

Order 15104.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426352C211yesmaximal, characteristic
2#21426004C211yesmaximal, characteristic, complemented
3#21423467C211yesmaximal, characteristic, complemented
4#21423457C211yesmaximal, characteristic
5#21422144C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C59×#1371C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9C2×#14273672—21no—
10#16837782C2211yescharacteristic
11#16837779C2211yescharacteristic
12#16837688—21no—
13#16837603—21no—
14C8×Dic118C2211yescharacteristic
15C4⋊(C59⋊C16)C2211yescharacteristic
16Dic59⋊C16—21no—
17#16837177—21no—
18#16837147—21no—
19#16837145—21no—
20C4×C16×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×#11745660—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273727—21no—
29#14273724—21no—
30C4×Dic118D411yescharacteristic
31#14273672—42no—
32#14273672—42no—
33C2×C59⋊C16—21no—
34#14273659—41no—
35C4⋊(C59⋊C8)D411yescharacteristic
36Dic59⋊C8—21no—
37#14273652—41no—
38#14273649—41no—
39C8×Dic59—21no—
40C2×C16×C59D411yescharacteristic
41C2×C16×C59D411yescharacteristic
42C4⋊C8×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×D236—41no—
46C2×D236—41no—
47C2×D236—41no—
48C2×Dic118—41no—
49C2×Dic118D811yescharacteristic
50C2×C8×C59—41no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C42×C59C2×D411yescharacteristic
55C4⋊Dic59—41no—
56C4⋊Dic59—41no—
57C4⋊Dic59D811yescharacteristic
58Dic59⋊C4—21no—
59C4×Dic59—21no—
60C2×C59⋊C8—21no—
61Dic236—41no—
62Dic236—41no—
63Dic236—81no—
64D472—41no—
65D472—41no—
66D472—81no—
67#11745660—82no—
68C16×C59—21no—
69C16×C59SD1612yescomplemented
70C59⋊C16—41no—
71C22×D59—81no—
72C2×C4×C59#12111yescharacteristic
73C2×C4×C59C2×D811yescharacteristic
74C2×C4×C59#13511yescharacteristic
75C2×Dic59—21no—
76C2×Dic59—21no—
77C2×Dic59—81no—
78D236—81no—
79D236—81no—
80D236—81no—
81D236—81no—
82D236—161no—
83Dic118—21no—
84Dic118—21no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59—81no—
89C8×C59—81no—
90C8×C59C2×SD1611yescharacteristic
91C8×C59AGL(1,ℤ/8)11yescharacteristic
92C59⋊C8—41no—
93#12444—591nomaximal, Sylow
94C22×C59#46611yescharacteristic
95C2×D59—161no—
96C2×D59—161no—
97C2×D59—161no—
98C4×C59—21no—
99C4×C59#46011yescharacteristic
100C4×C59#47211yescharacteristic
101C4×C59#50911yescharacteristic
102C4×C59#51011yescharacteristic
103Dic59—41no—
104Dic59—41no—
105Dic59—161no—
106#2248—591no—
107#2183—591no—
108#1670—591no—
109#1668—591no—
110#1371D5911yescharacteristic, complemented
111#1362—591no—
112#1361—591no—
113C2×C59#166811yescharacteristic
114C2×C59#222011yescharacteristic
115C2×C59#222311yescharacteristic
116D59—321no—
117C2×SD32—1181no—
118C2×SD32—1181no—
119#494—591no—
120#493—591no—
121#475—1181no—
122#460—1181no—
123C8×Q8—591no—
124C4⋊C16—591no—
125C4⋊C16—1181no—
126#346—1181no—
127C4×C16C2×D5911yescharacteristic
128#335C2×D5911yescharacteristic
129#335C2×D5911yescharacteristic
130#332—1181no—
131#331—1181no—
132C59#1244411yescharacteristic, complemented, Sylow
133C2×Q16—1181no—
134C2×Q16—1181no—
135C2×SD16—2361no—
136C2×D8—1181no—
137C2×D8—1181no—
138C4⋊Q8—1181no—
139#127—1181no—
140C4×Q8—591no—
141SD32—2362no—
142SD32—2362no—
143C2×C16—1181no—
144C2×C16D23611yescharacteristic
145C2×C16D23611yescharacteristic
146#106—2361no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—591no—
150C4⋊C8—1181no—
151#103—2361no—
152#102—2361no—
153C4×C8—1181no—
154C4×C8C22×D5911yescharacteristic
155C2×Q8—591no—
156C2×Q8—2361no—
157C2×D4—2361no—
158C2×D4—2361no—
159C2×D4—2361no—
160Q16—4721no—
161Q16—2361no—
162Q16—2361no—
163SD16—4722no—
164D8—4721no—
165D8—2361no—
166D8—2361no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1181no—
170C2×C8C2×D23611yescharacteristic
171C2×C8#1174568911yescharacteristic
172C4⋊C4—591no—
173C4⋊C4—1181no—
174C4⋊C4—2361no—
175C4⋊C4—2361no—
176C42—1181no—
177C42D4×D5911yescharacteristic
178C16—21no—
179C16—2361no—
180C16#1174566012yescomplemented
181C23—4721no—
182Q8—1181no—
183Q8—1181no—
184Q8—4721no—
185Q8—4721no—
186D4—9441no—
187D4—4721no—
188D4—4721no—
189D4—4721no—
190D4—4721no—
191C2×C4—1181no—
192C2×C4—1181no—
193C2×C4—4721no—
194C2×C4#1427370211yescharacteristic
195C2×C4D8×D5911yescharacteristic
196C2×C4#1427375011yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—2361no—
201C8C2×#1174566011yescharacteristic
202C8#1427375511yescharacteristic
203C22—9441no—
204C22—9441no—
205C22—9441no—
206C22#1683763911yescharacteristic
207C4—21no—
208C4—9441no—
209C4—2361no—
210C4—2361no—
211C4#1683760311yescharacteristic
212C4#1683767311yescharacteristic
213C4#1683783611yescharacteristic
214C4#1683784411yescharacteristic
215C2—18881no—
216C2#2142345711yescharacteristic
217C2#2142621011yescharacteristic
218C2#2142622711yescharacteristic
subgroupG/H
centerC22#16837639
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1371C2
socleC22×C59#466
prime radical#16837779C22
hypercenter#1371D59
hypocenterC59#12444
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1371

Complements

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

NH ≅ G/Ntype
2. #21426004C2semidirect
3. #21423467C2semidirect
5. #21422144C2semidirect
7. C59×#1371C2semidirect
69. C16×C59SD16semidirect
110. #1371D59semidirect
132. C59#12444semidirect
180. C16#11745660semidirect

Maximal subgroups

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

subgroupindex
#214263522
#214260042
#214234672
#214234572
#214221442
#214221412
C59×#13712
#1244459