Finite Group Databasebeta

← #28503168

Subgroups — #28503168

Order 15104.

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

normal subgroups only

This group has 214 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#21426319C211yesmaximal, characteristic
2#21425998C211yesmaximal, characteristic, complemented
3#21423501C211yesmaximal, characteristic, complemented
4#21423453C211yesmaximal, characteristic
5#21422169C211yesmaximal, characteristic, complemented
6#21422153C211yesmaximal, characteristic, complemented
7C59×#1361C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837698—21no—
10#16837599—21no—
11C8×Dic118C2211yescharacteristic
12Dic59⋊C16—21no—
13#16837203—21no—
14#16837203—21no—
15C4×C59⋊C16C2211yescharacteristic
16C16×Dic59—21no—
17#16837155C2211yescharacteristic
18#16837150—21no—
19#16837146—21no—
20C59×#493C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745670—41no—
24#14273727—21no—
25C4×Dic118D411yescharacteristic
26C2×C59⋊C16D411yescharacteristic
27C2×C59⋊C16D411yescharacteristic
28#14273661—21no—
29#14273661—21no—
30#14273661—41no—
31C4⋊(C59⋊C8)—21no—
32C4⋊(C59⋊C8)D411yescharacteristic
33Dic59⋊C8—21no—
34#14273652—41no—
35#14273648—41no—
36C8×Dic59—21no—
37C2×D8×C59—21no—
38C2×D8×C59—21no—
39#127×C59—21no—
40C2×C16×C59—21no—
41C4⋊C8×C59—21no—
42C4×C8×C59C2311yescharacteristic
43C2×D4×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×Dic118—41no—
47C2×Dic118D811yescharacteristic
48D8×C59—41no—
49D8×C59—41no—
50D8×C59—81no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C42×C59C2×D411yescharacteristic
55#11745670—82no—
56C4⋊Dic59—41no—
57C4⋊Dic59—41no—
58C4⋊Dic59D811yescharacteristic
59Dic59⋊C4—21no—
60C4×Dic59—21no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8—41no—
63C16×C59—41no—
64C59⋊C16—21no—
65C59⋊C16—21no—
66C23×C59—81no—
67D4×C59—81no—
68D4×C59—81no—
69D4×C59—81no—
70D4×C59—81no—
71D4×C59—161no—
72C2×C4×C59#12111yescharacteristic
73C2×C4×C59C2×D811yescharacteristic
74C2×C4×C59#13511yescharacteristic
75C2×Dic59—21no—
76C2×Dic59—21no—
77C2×Dic59—81no—
78Dic118—21no—
79Dic118—161no—
80Dic118SD3212yes—
81C8×C59—21no—
82C8×C59—81no—
83C8×C59#13511yescharacteristic
84C8×C59#13711yescharacteristic
85C59⋊C8—41no—
86C59⋊C8—81no—
87#12476—591nomaximal, Sylow
88C22×C59—161no—
89C22×C59—161no—
90C22×C59—161no—
91C22×C59#46611yescharacteristic
92C4×C59—21no—
93C4×C59#46411yescharacteristic
94C4×C59#47111yescharacteristic
95C4×C59C2×SD3211yescharacteristic
96C4×C59#50811yescharacteristic
97Dic59—22no—
98Dic59—41no—
99Dic59—161no—
100#2240—591no—
101Q8×C16—591no—
102#1676—591no—
103#1668—591no—
104#1372—591no—
105#1365—591no—
106#1361D5911yescharacteristic
107C2×C59—321no—
108C2×C59#168211yescharacteristic
109C2×C59#221711yescharacteristic
110C2×C59#222111yescharacteristic
111#500—591no—
112#493C2×D5911yescharacteristic
113#477—1181no—
114#460—1181no—
115C8×Q8—591no—
116C4⋊C16—1181no—
117C4⋊C16C2×D5911yescharacteristic
118#357—1181no—
119#357—1181no—
120C4×C16—591no—
121C4×C16—1181no—
122#335—591no—
123#335C2×D5911yescharacteristic
124#333—1181no—
125#331—1181no—
126C59#1247611yescharacteristic, complemented, Sylow
127C2×SD16—2361no—
128C2×D8—21no—
129C2×D8—21no—
130C4⋊Q8—1181no—
131#127—21no—
132C4×Q8—591no—
133C2×C16—21no—
134C2×C16—591no—
135C2×C16—591no—
136#107—1181no—
137#107—1181no—
138#107—2361no—
139C4⋊C8—21no—
140C4⋊C8—591no—
141C4⋊C8—1181no—
142C4⋊C8—1181no—
143#103—2361no—
144#102—2361no—
145C4×C8—1181no—
146C4×C8C22×D5911yescharacteristic
147C2×Q8—591no—
148C2×Q8—2361no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152SD16—4722no—
153D8—41no—
154D8—41no—
155D8—81no—
156C2×C8—41no—
157C2×C8—1181no—
158C2×C8—2361no—
159C2×C8#1174568511yescharacteristic
160C2×C8D4×D5911yescharacteristic
161C4⋊C4—591no—
162C4⋊C4—1181no—
163C4⋊C4—2361no—
164C4⋊C4—2361no—
165C42—1181no—
166C42D4×D5911yescharacteristic
167C16—41no—
168C16—1181no—
169C16—1181no—
170C23—81no—
171Q8—592no—
172Q8—1181no—
173Q8—9441no—
174D4—81no—
175D4—81no—
176D4—81no—
177D4—81no—
178D4—161no—
179C2×C4—1181no—
180C2×C4—1181no—
181C2×C4—4721no—
182C2×C4#1427370011yescharacteristic
183C2×C4D8×D5911yescharacteristic
184C2×C4#1427374811yescharacteristic
185C8—21no—
186C8—81no—
187C8—4721no—
188C8—2361no—
189C8#1427374611yescharacteristic
190C8#1427375611yescharacteristic
191C22—161no—
192C22—161no—
193C22—161no—
194C22#1683763711yescharacteristic
195C4—21no—
196C4—1182no—
197C4—9441no—
198C4—2361no—
199C4#1683762311yescharacteristic
200C4#1683766411yescharacteristic
201C4SD32×D5911yescharacteristic
202C4#1683783111yescharacteristic
203C2—321no—
204C2#2142353911yescharacteristic
205C2#2142619111yescharacteristic
206C2#2142621511yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1361C2
socleC22×C59#466
prime radicalC59×#493C22
hypercenter#1361D59
hypocenterC59#12476
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
2. #21425998C2semidirect
3. #21423501C2semidirect
5. #21422169C2semidirect
6. #21422153C2semidirect
126. C59#12476semidirect

Maximal subgroups

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

subgroupindex
#214263192
#214259982
#214235012
#214234532
#214221692
#214221532
C59×#13612
#1247659