Finite Group Databasebeta

← #28503128

Subgroups — #28503128

Order 15104.

224 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 65 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#21426324C211yesmaximal, characteristic, complemented
2C16×D236C211yesmaximal, characteristic
3#21423486C211yesmaximal, characteristic
4#21423462C211yesmaximal, characteristic
5#21422152C211yesmaximal, characteristic, complemented
6#21422144C211yesmaximal, characteristic, complemented
7C59×#1372C211yesmaximal, characteristic, complemented
8C2×C16×D59—21no—
9#16837807C2211yescharacteristic
10#16837782C2211yescharacteristic
11#16837683—21no—
12#16837608—21no—
13C8×D236C2211yescharacteristic
14C4⋊(C59⋊C16)C2211yescharacteristic
15#16837207—21no—
16#16837207—21no—
17(C2×D59)⋊C16—21no—
18#16837149—21no—
19#16837147—21no—
20C4×C16×C59C2211yescharacteristic
21C59×#335C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×#11745660—41no—
26C2×C8×D59—21no—
27#14273727—21no—
28#14273727—21no—
29C4×D236D411yescharacteristic
30C2×C59⋊C16—21no—
31C16×D59—42no—
32#14273661—21no—
33#14273661—21no—
34#14273661—41no—
35C4⋊(C59⋊C8)D411yescharacteristic
36#14273652—41no—
37#14273649—41no—
38(C2×D59)⋊C8—21no—
39C2×C16×C59D411yescharacteristic
40C2×C16×C59D411yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59—21no—
43C4×C8×C59C2311yescharacteristic
44C2×D236D811yescharacteristic
45C2×C4×D59—21no—
46C2×Dic118—41no—
47C2×Dic118—41no—
48C2×C8×C59—41no—
49C2×C8×C59—41no—
50C2×C8×C59C2×D411yescharacteristic
51C2×C8×C59#4111yescharacteristic
52C42×C59C2×D411yescharacteristic
53(C2×D59)⋊C4—21no—
54C4⋊Dic59—41no—
55C4⋊Dic59—41no—
56C4⋊Dic59—41no—
57C4⋊Dic59—41no—
58C4⋊Dic59D811yescharacteristic
59C2×C59⋊C8—21no—
60Dic236—41no—
61Dic236—41no—
62Dic236—81no—
63#11745660—82no—
64C8×D59—41no—
65C8×D59—41no—
66C16×C59—21no—
67C16×C59—21no—
68C59⋊C16—41no—
69C22×D59—21no—
70C2×C4×C59#12111yescharacteristic
71C2×C4×C59C2×D811yescharacteristic
72C2×C4×C59#13511yescharacteristic
73C2×Dic59—21no—
74C2×Dic59—81no—
75C2×Dic59—81no—
76D236—21no—
77D236SD3212yes—
78C4×D59—41no—
79C4×D59—41no—
80Dic118—81no—
81Dic118—81no—
82Dic118—81no—
83Dic118—81no—
84C8×C59—21no—
85C8×C59—81no—
86C8×C59—81no—
87C8×C59#13511yescharacteristic
88C8×C59AGL(1,ℤ/8)11yescharacteristic
89C59⋊C8—41no—
90#12470—591nomaximal, Sylow
91C22×C59#46611yescharacteristic
92C2×D59—22no—
93C2×D59—41no—
94C2×D59—41no—
95C4×C59—21no—
96C4×C59#46311yescharacteristic
97C4×C59#47111yescharacteristic
98C4×C59C2×SD3211yescharacteristic
99C4×C59#50811yescharacteristic
100Dic59—41no—
101Dic59—161no—
102Dic59—161no—
103#2241—591no—
104D4×C16—591no—
105#1673—591no—
106#1669—591no—
107#1372D5911yescharacteristic, complemented
108#1364—591no—
109#1362—591no—
110C2×C59#168111yescharacteristic
111C2×C59#221711yescharacteristic
112C2×C59#222211yescharacteristic
113D59—42no—
114C22×C16—1181no—
115#500—591no—
116#494—591no—
117#474—1181no—
118#461—1181no—
119C8×D4—591no—
120C4⋊C16—591no—
121#358—1181no—
122#358—1181no—
123C22⋊C16—1181no—
124C4×C16C2×D5911yescharacteristic
125#335C2×D5911yescharacteristic
126#335C2×D5911yescharacteristic
127#332—1181no—
128#332—1181no—
129C59#1247011yescharacteristic, complemented, Sylow
130C2×Q16—1181no—
131C2×Q16—1181no—
132C2×SD16—2361no—
133C22×C8—1181no—
134C4⋊Q8—1181no—
135C4⋊Q8—1181no—
136C4×D4—591no—
137C2×C16—1181no—
138C2×C16—2362no—
139C2×C16D23611yescharacteristic
140C2×C16D23611yescharacteristic
141#107—1181no—
142#107—1181no—
143#107—2361no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—591no—
147#103—2361no—
148#102—2361no—
149C22⋊C8—1181no—
150C4×C8C22×D5911yescharacteristic
151C2×Q8—2361no—
152C2×Q8—2361no—
153C2×D4—591no—
154C22×C4—1181no—
155Q16—4721no—
156Q16—2361no—
157Q16—2361no—
158SD16—4722no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—1181no—
162C2×C8—2361no—
163C2×C8—2361no—
164C2×C8C2×D23611yescharacteristic
165C2×C8#1174568911yescharacteristic
166C4⋊C4—591no—
167C4⋊C4—2361no—
168C4⋊C4—2361no—
169C4⋊C4—2361no—
170C4⋊C4—2361no—
171C22⋊C4—1181no—
172C42D4×D5911yescharacteristic
173C16—21no—
174C16—21no—
175C16—2361no—
176C23—1181no—
177Q8—4721no—
178Q8—4721no—
179Q8—4721no—
180Q8—4721no—
181D4—592no—
182D4—1181no—
183C2×C4—1181no—
184C2×C4—4721no—
185C2×C4—4721no—
186C2×C4—2361no—
187C2×C4—2361no—
188C2×C4#1427370211yescharacteristic
189C2×C4D8×D5911yescharacteristic
190C2×C4#1427375011yescharacteristic
191C8—21no—
192C8—81no—
193C8—81no—
194C8—2361no—
195C8#1427374911yescharacteristic
196C8#1427375511yescharacteristic
197C22—1182no—
198C22—2361no—
199C22—2361no—
200C22#1683763911yescharacteristic
201C4—21no—
202C4—9441no—
203C4—9441no—
204C4—2361no—
205C4#1683761811yescharacteristic
206C4#1683766611yescharacteristic
207C4SD32×D5911yescharacteristic
208C4#1683783311yescharacteristic
209C2—2362no—
210C2#2142353411yescharacteristic
211C2#2142619311yescharacteristic
212C2#2142622211yescharacteristic
subgroupG/H
centerC22#16837639
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1372C2
socleC22×C59#466
prime radicalC2×D236D8
hypercenter#1372D59
hypocenterC59#12470
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #21426324C2semidirect
5. #21422152C2semidirect
6. #21422144C2semidirect
7. C59×#1372C2semidirect
107. #1372D59semidirect
129. C59#12470semidirect

Maximal subgroups

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

subgroupindex
#214263242
C16×D2362
#214234862
#214234622
#214221522
#214221442
C59×#13722
#1247059