Finite Group Databasebeta

← #28502829

Subgroups — #28502829

Order 15104.

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

normal subgroups only

This group has 240 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#21426245C211yesmaximal, characteristic, complemented
2#21426137C211yesmaximal, characteristic
3#21423689C211yesmaximal, characteristic
4#21422902C211yesmaximal, characteristic, complemented
5#21422159C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C59×#1365C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9#16837779C2211yescharacteristic
10#16837776—21no—
11#16837750—21no—
12#16837526—21no—
13C4×Dic236C2211yescharacteristic
14#16837227—21no—
15C4⋊(C59⋊C16)C2211yescharacteristic
16#16837205—21no—
17#16837204—21no—
18#16837132—21no—
19C8×C59⋊C8C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236D411yescharacteristic
24C2×#11745671—42no—
25C2×D472—21no—
26C2×D472—21no—
27#14273724—21no—
28C4×Dic118—21no—
29#14273672—42no—
30C2×C59⋊C16—21no—
31#14273661D411yescharacteristic
32Dic59⋊C8—41no—
33#14273652—21no—
34#14273647—41no—
35#14273647—41no—
36C4×C59⋊C8D411yescharacteristic
37C4×C59⋊C8D411yescharacteristic
38C8×Dic59—41no—
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—41no—
47C2×D236—41no—
48C2×D236—41no—
49C2×Dic118—21no—
50C2×C8×C59—41no—
51C2×C8×C59C2×D411yescharacteristic
52C2×C8×C59#4111yescharacteristic
53C4⋊C4×C59—41no—
54C4⋊C4×C59—41no—
55C42×C59C2×D411yescharacteristic
56#11745671—84no—
57C4⋊Dic59—21no—
58Dic59⋊C4—41no—
59C4×Dic59—41no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8D811yescharacteristic
62C2×C59⋊C8D811yescharacteristic
63Dic236—21no—
64Dic236SD1612yescomplemented
65D472—41no—
66D472—41no—
67D472—81no—
68C16×C59—41no—
69C59⋊C16—41no—
70C22×D59—81no—
71Q8×C59—82no—
72C2×C4×C59—81no—
73C2×C4×C59#12111yescharacteristic
74C2×C4×C59C2×D811yescharacteristic
75C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
76C2×Dic59—41no—
77C2×Dic59—41no—
78D236—82no—
79D236—81no—
80D236—81no—
81D236—81no—
82D236—81no—
83Dic118—22no—
84Dic118—41no—
85C8×C59—21no—
86C8×C59—81no—
87C8×C59C2×SD1611yescharacteristic
88C8×C59#13511yescharacteristic
89C59⋊C8—22no—
90C59⋊C8—21no—
91C59⋊C8—21no—
92#12419—591nomaximal, Sylow
93C22×C59#45911yescharacteristic
94C2×D59—161no—
95C2×D59—161no—
96C2×D59—161no—
97C4×C59—21no—
98C4×C59—161no—
99C4×C59#46511yescharacteristic
100C4×C59#47111yescharacteristic
101C4×C59#50811yescharacteristic
102C4×C59#51011yescharacteristic
103Dic59—42no—
104Dic59—81no—
105#2227—591no—
106#2207—591no—
107#1706—591no—
108C8×Q16—591no—
109#1367—591no—
110#1365D5911yescharacteristic, complemented
111#1361—591no—
112C2×C59#166511yescharacteristic
113C2×C59#221111yescharacteristic
114C2×C59#221311yescharacteristic
115D59—321no—
116C2×SD32—1181no—
117#500C2×D5911yescharacteristic
118#493—591no—
119#492—1181no—
120#486—1181no—
121C8×Q8—1181no—
122C4×Q16—591no—
123#366—1181no—
124C4⋊C16—591no—
125C4⋊C16C2×D5911yescharacteristic
126#358—1181no—
127#357—1181no—
128#335C2×D5911yescharacteristic
129Q8⋊C8—1181no—
130C82—591no—
131C59#1241911yescharacteristic, complemented, Sylow
132C2×Q16—591no—
133C2×SD16—2362no—
134C2×D8—1181no—
135C2×D8—1181no—
136C4⋊Q8—21no—
137#127—1181no—
138C4×Q8—1181no—
139SD32—2362no—
140C2×C16—21no—
141C2×C16—1181no—
142#107—21no—
143#107—21no—
144#107—591no—
145C4⋊C8—21no—
146C4⋊C8—2361no—
147#103—1181no—
148#102—2361no—
149#102—2361no—
150C4×C8—591no—
151C4×C8—591no—
152C4×C8—2361no—
153C4×C8C22×D5911yescharacteristic
154C2×Q8—41no—
155C2×Q8—1181no—
156C2×D4—2361no—
157C2×D4—2361no—
158C2×D4—2361no—
159Q16—592no—
160Q16—1181no—
161SD16—4724no—
162D8—4721no—
163D8—2361no—
164D8—2361no—
165C2×C8—41no—
166C2×C8—591no—
167C2×C8—591no—
168C2×C8—1181no—
169C2×C8#1174568511yescharacteristic
170C2×C8D4×D5911yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—1181no—
174C4⋊C4—2361no—
175C42—2361no—
176C42D4×D5911yescharacteristic
177C16—41no—
178C16—2361no—
179C23—4721no—
180Q8—82no—
181Q8—1182no—
182Q8—2361no—
183D4—4722no—
184D4—4721no—
185D4—4721no—
186D4—4721no—
187D4—4721no—
188C2×C4—81no—
189C2×C4—2361no—
190C2×C4—2361no—
191C2×C4#1427370011yescharacteristic
192C2×C4D8×D5911yescharacteristic
193C2×C4#1427375411yescharacteristic
194C8—21no—
195C8—81no—
196C8—1182no—
197C8—1181no—
198C8—1181no—
199C8SD16×D5911yescharacteristic
200C8#1427375011yescharacteristic
201C22—9441no—
202C22—9441no—
203C22—9441no—
204C22#1683759611yescharacteristic
205C4—21no—
206C4—161no—
207C4—4721no—
208C4—2362no—
209C4#1683763411yescharacteristic
210C4#1683766811yescharacteristic
211C4#1683783311yescharacteristic
212C4#1683784411yescharacteristic
213C2—18881no—
214C2#2142344011yescharacteristic
215C2#2142616111yescharacteristic
216C2#2142617111yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1365C2
socleC22×C59#459
prime radical#16837779C22
hypercenter#1365D59
hypocenterC59#12419
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #21426245C2semidirect
4. #21422902C2semidirect
5. #21422159C2semidirect
7. C59×#1365C2semidirect
64. Dic236SD16semidirect
110. #1365D59semidirect
131. C59#12419semidirect

Maximal subgroups

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

subgroupindex
#214262452
#214261372
#214236892
#214229022
#214221592
#214221412
C59×#13652
#1241959