Finite Group Databasebeta

← #28502827

Subgroups — #28502827

Order 15104.

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

normal subgroups only

This group has 240 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#21426249C211yesmaximal, characteristic, complemented
2#21426137C211yesmaximal, characteristic
3#21423688C211yesmaximal, characteristic
4C8×Dic236C211yesmaximal, characteristic, complemented
5#21422153C211yesmaximal, characteristic, complemented
6#21422143C211yesmaximal, characteristic
7C59×#1367C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9#16837807C2211yescharacteristic
10#16837779C2211yescharacteristic
11#16837775—21no—
12#16837749—21no—
13C8×Dic118—21no—
14C4×Dic236C2211yescharacteristic
15#16837228—21no—
16#16837207—21no—
17#16837204—21no—
18#16837155C2211yescharacteristic
19#16837134—21no—
20C59×C4⋊C16C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C82×C59C2211yescharacteristic
23C2×Dic236D411yescharacteristic
24C2×#11745660—42no—
25C2×D472—21no—
26C2×D472—21no—
27#14273727—21no—
28#14273724—21no—
29C4×Dic118—21no—
30#14273672—42no—
31#14273661—21no—
32#14273661—21no—
33#14273661D411yescharacteristic
34C4⋊(C59⋊C8)—21no—
35Dic59⋊C8—41no—
36#14273652—21no—
37#14273649—41no—
38#14273649—41no—
39C8×Dic59—41no—
40C2×C16×C59—21no—
41C2×C16×C59—21no—
42C4×C8×C59D411yescharacteristic
43C4×C8×C59D411yescharacteristic
44C4×C8×C59C2311yescharacteristic
45C2×D236—41no—
46C2×D236—41no—
47C2×D236—41no—
48C2×Dic118—21no—
49C2×Dic118—41no—
50C2×C8×C59—21no—
51C2×C8×C59D811yescharacteristic
52C2×C8×C59D811yescharacteristic
53C2×C8×C59C2×D411yescharacteristic
54C2×C8×C59#4111yescharacteristic
55C42×C59C2×D411yescharacteristic
56C4⋊Dic59—21no—
57C4⋊Dic59—41no—
58C4⋊Dic59—41no—
59Dic59⋊C4—41no—
60C4×Dic59—41no—
61C2×C59⋊C8—41no—
62Dic236—21no—
63Dic236SD1612yescomplemented
64D472—41no—
65D472—41no—
66D472—81no—
67#11745660—84no—
68C16×C59—41no—
69C16×C59—41no—
70C22×D59—81no—
71C2×C4×C59#12111yescharacteristic
72C2×C4×C59C2×D811yescharacteristic
73C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
74C2×Dic59—41no—
75C2×Dic59—41no—
76C2×Dic59—81no—
77D236—82no—
78D236—81no—
79D236—81no—
80D236—81no—
81D236—81no—
82Dic118—22no—
83Dic118—41no—
84Dic118—82no—
85C8×C59—22no—
86C8×C59—21no—
87C8×C59—21no—
88C8×C59—21no—
89C8×C59C2×SD1611yescharacteristic
90C8×C59#13511yescharacteristic
91C59⋊C8—81no—
92#12419—591nomaximal, Sylow
93C22×C59#45911yescharacteristic
94C2×D59—161no—
95C2×D59—161no—
96C2×D59—161no—
97C4×C59—21no—
98C4×C59#46511yescharacteristic
99C4×C59#47111yescharacteristic
100C4×C59#50811yescharacteristic
101C4×C59#51011yescharacteristic
102Dic59—42no—
103Dic59—81no—
104Dic59—161no—
105#2227—591no—
106#2207—591no—
107#1706—591no—
108C8×Q16—591no—
109#1367D5911yescharacteristic, complemented
110#1365—591no—
111#1361—591no—
112C2×C59#166511yescharacteristic
113C2×C59#221111yescharacteristic
114C2×C59#221311yescharacteristic
115D59—321no—
116C2×SD32—1181no—
117#500—591no—
118#493—591no—
119#492—1181no—
120#486—1181no—
121C8×Q8—1181no—
122C4×Q16—591no—
123#366—1181no—
124C4⋊C16C2×D5911yescharacteristic
125C4⋊C16C2×D5911yescharacteristic
126#358—1181no—
127#357—1181no—
128#335—591no—
129Q8⋊C8—1181no—
130C82C2×D5911yescharacteristic
131C59#1241911yescharacteristic, complemented, Sylow
132C2×Q16—591no—
133C2×SD16—2362no—
134C2×D8—1181no—
135C2×D8—1181no—
136C4⋊Q8—1181no—
137#127—1181no—
138C4×Q8—1181no—
139SD32—2362no—
140C2×C16—21no—
141C2×C16—21no—
142#107—591no—
143#107—1181no—
144#107—1181no—
145C4⋊C8—1181no—
146C4⋊C8—2361no—
147#103—1181no—
148#102—2361no—
149#102—2361no—
150C4×C8—2361no—
151C4×C8D23611yescharacteristic
152C4×C8D23611yescharacteristic
153C4×C8C22×D5911yescharacteristic
154C2×Q8—1181no—
155C2×Q8—2361no—
156C2×D4—2361no—
157C2×D4—2361no—
158C2×D4—2361no—
159Q16—592no—
160Q16—1181no—
161SD16—4724no—
162D8—4721no—
163D8—2361no—
164D8—2361no—
165C2×C8—21no—
166C2×C8—2361no—
167C2×C8D47211yescharacteristic
168C2×C8D47211yescharacteristic
169C2×C8D4×D5911yescharacteristic
170C2×C8#1174568911yescharacteristic
171C4⋊C4—1181no—
172C4⋊C4—2361no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C42—2361no—
176C42C2×D23611yescharacteristic
177C16—41no—
178C16—41no—
179C23—4721no—
180Q8—1182no—
181Q8—4722no—
182Q8—2361no—
183D4—4722no—
184D4—4721no—
185D4—4721no—
186D4—4721no—
187D4—4721no—
188C2×C4—4721no—
189C2×C4—2361no—
190C2×C4—2361no—
191C2×C4#1427370211yescharacteristic
192C2×C4C2×D47211yescharacteristic
193C2×C4#1427375511yescharacteristic
194C8—22no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—4721no—
199C8SD16×D5911yescharacteristic
200C8#1427374811yescharacteristic
201C22—9441no—
202C22—9441no—
203C22—9441no—
204C22#1683759811yescharacteristic
205C4—21no—
206C4—9441no—
207C4—4721no—
208C4—2362no—
209C4#1683763211yescharacteristic
210C4#1683766611yescharacteristic
211C4#1683783411yescharacteristic
212C4#1683784511yescharacteristic
213C2—18881no—
214C2#2142343811yescharacteristic
215C2#2142615911yescharacteristic
216C2#2142617311yescharacteristic
subgroupG/H
centerC22#16837598
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1367C2
socleC22×C59#459
prime radical#16837779C22
hypercenter#1367D59
hypocenterC59#12419
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1367

Complements

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

NH ≅ G/Ntype
1. #21426249C2semidirect
4. C8×Dic236C2semidirect
5. #21422153C2semidirect
7. C59×#1367C2semidirect
63. Dic236SD16semidirect
109. #1367D59semidirect
131. C59#12419semidirect

Maximal subgroups

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

subgroupindex
#214262492
#214261372
#214236882
C8×Dic2362
#214221532
#214221432
C59×#13672
#1241959