Finite Group Databasebeta

← #28503247

Subgroups — #28503247

Order 15104.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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#21426244C211yesmaximal, characteristic
2#21426142C211yesmaximal, characteristic
3#21423693C211yesmaximal, characteristic, complemented
4#21422982C211yesmaximal, characteristic
5#21422153C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#1370C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9#16837807C2211yescharacteristic
10#16837800—21no—
11#16837782C2211yescharacteristic
12#16837753—21no—
13C4×D472C2211yescharacteristic
14#16837481—21no—
15#16837230—21no—
16#16837207—21no—
17#16837204—21no—
18#16837155C2211yescharacteristic
19#16837131—21no—
20C59×C4⋊C16C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C82×C59C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×D472D411yescharacteristic
26C2×#11745659—41no—
27#14273727—21no—
28#14273727—21no—
29C4×D236—21no—
30#14273672—42no—
31#14273661—21no—
32#14273661—21no—
33#14273661D411yescharacteristic
34#14273659—41no—
35#14273659—41no—
36C4⋊(C59⋊C8)—21no—
37#14273652—42no—
38#14273649—21no—
39(C2×D59)⋊C8—41no—
40C2×C16×C59—21no—
41C2×C16×C59—21no—
42C4×C8×C59D411yescharacteristic
43C4×C8×C59D411yescharacteristic
44C4×C8×C59C2311yescharacteristic
45C2×D236—21no—
46C2×C4×D59—41no—
47C2×Dic118—41no—
48C2×Dic118—41no—
49C2×C8×C59—21no—
50C2×C8×C59D811yescharacteristic
51C2×C8×C59D811yescharacteristic
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C42×C59C2×D411yescharacteristic
55(C2×D59)⋊C4—41no—
56C4⋊Dic59—21no—
57C4⋊Dic59—41no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C4⋊Dic59—41no—
61C2×C59⋊C8—41no—
62Dic236—41no—
63Dic236—41no—
64Dic236—81no—
65D472—21no—
66D472—21no—
67#11745659—82no—
68C16×C59—41no—
69C16×C59—41no—
70C22×D59—41no—
71C2×C4×C59#12111yescharacteristic
72C2×C4×C59C2×D811yescharacteristic
73C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
74C2×Dic59—41no—
75C2×Dic59—81no—
76C2×Dic59—81no—
77D236—41no—
78D236—41no—
79C4×D59—81no—
80C4×D59—81no—
81Dic118—82no—
82Dic118—81no—
83Dic118—81no—
84C8×C59—21no—
85C8×C59—21no—
86C8×C59—41no—
87C8×C59SD3212yescomplemented
88C8×C59AGL(1,ℤ/8)11yescharacteristic
89C8×C59#13711yescharacteristic
90C59⋊C8—81no—
91#12487—591nomaximal, Sylow
92C22×C59#45911yescharacteristic
93C2×D59—81no—
94C2×D59—81no—
95C2×D59—81no—
96C4×C59—21no—
97C4×C59#46811yescharacteristic
98C4×C59#47111yescharacteristic
99C4×C59C2×SD3211yescharacteristic
100C4×C59#51011yescharacteristic
101Dic59—81no—
102Dic59—161no—
103Dic59—161no—
104#2226—591no—
105#2208—591no—
106#1707—591no—
107#1589—591no—
108#1370D5911yescharacteristic, complemented
109#1365—591no—
110#1362—591no—
111C2×C59#168511yescharacteristic
112C2×C59#220811yescharacteristic
113C2×C59#221211yescharacteristic
114D59—161no—
115C2×SD32—1181no—
116#500—591no—
117#498—1181no—
118#494—591no—
119#487—1181no—
120C4×D8—591no—
121#436—1181no—
122#367—1181no—
123C4⋊C16C2×D5911yescharacteristic
124C4⋊C16C2×D5911yescharacteristic
125#358—1181no—
126#357—1181no—
127#335—591no—
128D4⋊C8—1181no—
129C82C2×D5911yescharacteristic
130C59#1248711yescharacteristic, complemented, Sylow
131C2×Q16—1181no—
132C2×Q16—1181no—
133C2×D8—591no—
134C2×M16—2361no—
135C4⋊Q8—1181no—
136C4⋊Q8—1181no—
137C4×D4—1181no—
138SD32—2362no—
139C2×C16—21no—
140C2×C16—21no—
141#107—591no—
142#107—1181no—
143#107—1181no—
144#106—2361no—
145#106—2361no—
146C4⋊C8—1181no—
147#103—2362no—
148#102—1181no—
149C22⋊C8—2361no—
150C4×C8D23611yescharacteristic
151C4×C8D23611yescharacteristic
152C4×C8C22×D5911yescharacteristic
153C2×Q8—2361no—
154C2×Q8—2361no—
155C2×D4—1181no—
156C22×C4—2361no—
157Q16—4721no—
158Q16—2361no—
159Q16—2361no—
160D8—1181no—
161D8—1181no—
162M16—4722no—
163C2×C8—21no—
164C2×C8—2361no—
165C2×C8D47211yescharacteristic
166C2×C8D47211yescharacteristic
167C2×C8D4×D5911yescharacteristic
168C2×C8#1174568911yescharacteristic
169C4⋊C4—1181no—
170C4⋊C4—2361no—
171C4⋊C4—2361no—
172C4⋊C4—2361no—
173C4⋊C4—2361no—
174C22⋊C4—2361no—
175C42C2×D23611yescharacteristic
176C16—41no—
177C16—41no—
178C23—2361no—
179Q8—4722no—
180Q8—4721no—
181Q8—4721no—
182D4—2361no—
183D4—2361no—
184C2×C4—4721no—
185C2×C4—4721no—
186C2×C4—4721no—
187C2×C4—4721no—
188C2×C4—2361no—
189C2×C4#1427370211yescharacteristic
190C2×C4C2×D47211yescharacteristic
191C2×C4#1427375511yescharacteristic
192C8—21no—
193C8—21no—
194C8—41no—
195C8—4721no—
196C8#1427367212yescomplemented
197C8#1427375111yescharacteristic
198C8#1427375911yescharacteristic
199C22—4721no—
200C22—4721no—
201C22—4721no—
202C22#1683759811yescharacteristic
203C4—21no—
204C4—9441no—
205C4—9441no—
206C4—4721no—
207C4#1683764911yescharacteristic
208C4#1683766611yescharacteristic
209C4C2×#1427367211yescharacteristic
210C4#1683784511yescharacteristic
211C2—9441no—
212C2#2142356211yescharacteristic
213C2#2142614211yescharacteristic
214C2#2142616611yescharacteristic
subgroupG/H
centerC22#16837598
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1370C2
socleC22×C59#459
prime radicalC2×D472D4
hypercenter#1370D59
hypocenterC59#12487
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1370

Complements

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

NH ≅ G/Ntype
3. #21423693C2semidirect
5. #21422153C2semidirect
6. #21422146C2semidirect
7. C59×#1370C2semidirect
87. C8×C59SD32semidirect
108. #1370D59semidirect
130. C59#12487semidirect
196. C8#14273672semidirect

Maximal subgroups

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

subgroupindex
#214262442
#214261422
#214236932
#214229822
#214221532
#214221462
C59×#13702
#1248759