Finite Group Databasebeta

← #28502808

Subgroups — #28502808

Order 15104.

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

normal subgroups only

This group has 256 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#21426230C211yesmaximal, characteristic, complemented
2#21426132C211yesmaximal, characteristic
3#21423689C211yesmaximal, characteristic, complemented
4#21422890C211yesmaximal, characteristic, complemented
5#21422159C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic, complemented
7C59×#1364C211yesmaximal, characteristic, complemented
8C2×D944—21no—
9#16837779C2211yescharacteristic
10#16837776—21no—
11#16837750—21no—
12C4×D472C2211yescharacteristic
13#16837467—21no—
14#16837227—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837204—21no—
17#16837202—21no—
18#16837129—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×#11745671—41no—
24C2×#11745671—41no—
25C2×D472—21no—
26C2×D472—21no—
27C2×D472D411yescharacteristic
28C2×C8×D59—41no—
29#14273724—21no—
30C4×D236—21no—
31D944—42no—
32C2×C59⋊C16—21no—
33#14273661D411yescharacteristic
34#14273649—21no—
35#14273647—42no—
36(C2×D59)⋊C8—41no—
37C4×C59⋊C8D411yescharacteristic
38C4×C59⋊C8D411yescharacteristic
39C4⋊Q8×C59—21no—
40C2×C16×C59—21no—
41#107×C59—21no—
42#107×C59—21no—
43C4⋊C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59—41no—
46C2×D236—21no—
47C2×D236—41no—
48C2×D236—41no—
49C2×D236—41no—
50C2×C4×D59—41no—
51C2×C8×C59—41no—
52C2×C8×C59C2×D411yescharacteristic
53C2×C8×C59#4111yescharacteristic
54C4⋊C4×C59—41no—
55C4⋊C4×C59—41no—
56C42×C59C2×D411yescharacteristic
57#11745671—82no—
58#11745671—82no—
59(C2×D59)⋊C4—41no—
60C4⋊Dic59—21no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8D811yescharacteristic
63C2×C59⋊C8D811yescharacteristic
64D472—21no—
65D472—41no—
66D472—41no—
67D472—81no—
68D472SD1612yescomplemented
69C8×D59—82no—
70C16×C59—41no—
71C59⋊C16—41no—
72C22×D59—41no—
73C22×D59—81no—
74Q8×C59—81no—
75Q8×C59—81no—
76C2×C4×C59—81no—
77C2×C4×C59#12111yescharacteristic
78C2×C4×C59C2×D811yescharacteristic
79C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
80C2×Dic59—41no—
81D236—22no—
82D236—41no—
83D236—82no—
84D236—81no—
85D236—81no—
86D236—81no—
87D236—81no—
88C4×D59—81no—
89C4×D59—81no—
90C8×C59—21no—
91C8×C59—81no—
92C8×C59C2×SD1611yescharacteristic
93C8×C59#13511yescharacteristic
94C59⋊C8—21no—
95C59⋊C8—41no—
96C59⋊C8D1612yescomplemented
97#12416—591nomaximal, Sylow
98C22×C59#45911yescharacteristic
99C2×D59—42no—
100C2×D59—81no—
101C2×D59—81no—
102C2×D59—161no—
103C2×D59—161no—
104C2×D59—161no—
105C4×C59—21no—
106C4×C59—161no—
107C4×C59#46511yescharacteristic
108C4×C59#47111yescharacteristic
109C4×C59C2×D1611yescharacteristic
110C4×C59#50911yescharacteristic
111Dic59—81no—
112#2224—591no—
113#2206—591no—
114#1706—591no—
115C8×D8—591no—
116#1367—591no—
117#1364D5911yescharacteristic, complemented
118#1361—591no—
119C2×C59#166511yescharacteristic
120C2×C59#220611yescharacteristic
121C2×C59#221211yescharacteristic
122D59—82no—
123D59—321no—
124C2×D16—1181no—
125#500C2×D5911yescharacteristic
126#493—591no—
127#492—1181no—
128#486—1181no—
129C4×D8—591no—
130C8×D4—1181no—
131#366—1181no—
132C4⋊C16—591no—
133C4⋊C16C2×D5911yescharacteristic
134#357—1181no—
135#357—1181no—
136#335C2×D5911yescharacteristic
137D4⋊C8—1181no—
138C82—591no—
139C59#1241611yescharacteristic, complemented, Sylow
140C2×SD16—2361no—
141C2×SD16—2361no—
142C2×D8—591no—
143C2×D8—1181no—
144C2×D8—1181no—
145C22×C8—2361no—
146C4⋊Q8—21no—
147#127—1181no—
148C4×D4—1181no—
149D16—2362no—
150C2×C16—21no—
151C2×C16—1181no—
152#107—21no—
153#107—21no—
154#107—591no—
155C4⋊C8—21no—
156#102—1181no—
157#102—2362no—
158C22⋊C8—2361no—
159C4×C8—591no—
160C4×C8—591no—
161C4×C8C22×D5911yescharacteristic
162C2×Q8—41no—
163C2×D4—1181no—
164C2×D4—2361no—
165C2×D4—2361no—
166C2×D4—2361no—
167C22×C4—2361no—
168SD16—4722no—
169SD16—4722no—
170D8—592no—
171D8—1181no—
172D8—4721no—
173D8—2361no—
174D8—2361no—
175C2×C8—41no—
176C2×C8—591no—
177C2×C8—591no—
178C2×C8—1181no—
179C2×C8—4722no—
180C2×C8#1174568511yescharacteristic
181C2×C8D4×D5911yescharacteristic
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—1181no—
185C22⋊C4—2361no—
186C42D4×D5911yescharacteristic
187C16—41no—
188C16—2361no—
189C23—4721no—
190C23—2361no—
191Q8—81no—
192Q8—81no—
193D4—1182no—
194D4—4722no—
195D4—4721no—
196D4—4721no—
197D4—4721no—
198D4—4721no—
199D4—2361no—
200C2×C4—81no—
201C2×C4—4721no—
202C2×C4—4721no—
203C2×C4—2361no—
204C2×C4#1427370011yescharacteristic
205C2×C4D8×D5911yescharacteristic
206C2×C4#1427375411yescharacteristic
207C8—21no—
208C8—81no—
209C8—592no—
210C8—1181no—
211C8—2361no—
212C8SD16×D5911yescharacteristic
213C8#1427375011yescharacteristic
214C22—9441no—
215C22—9441no—
216C22—9441no—
217C22—4721no—
218C22—4721no—
219C22—2362no—
220C22#1683759611yescharacteristic
221C4—21no—
222C4—161no—
223C4—4721no—
224C4#1683763411yescharacteristic
225C4#1683766811yescharacteristic
226C4D16×D5911yescharacteristic
227C4#1683783911yescharacteristic
228C2—18881no—
229C2—4722no—
230C2#2142344011yescharacteristic
231C2#2142613011yescharacteristic
232C2#2142616811yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59#459
prime radical#21426132C2
hypercenter#1364D59
hypocenterC59#12416
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #21426230C2semidirect
3. #21423689C2semidirect
4. #21422890C2semidirect
5. #21422159C2semidirect
6. #21422141C2semidirect
7. C59×#1364C2semidirect
68. D472SD16semidirect
96. C59⋊C8D16semidirect
117. #1364D59semidirect
139. C59#12416semidirect

Maximal subgroups

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

subgroupindex
#214262302
#214261322
#214236892
#214228902
#214221592
#214221412
C59×#13642
#1241659