Finite Group Databasebeta

← #28502815

Subgroups — #28502815

Order 15104.

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

normal subgroups only

This group has 240 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#21426235C211yesmaximal, characteristic, complemented
2#21426137C211yesmaximal, characteristic
3#21423689C211yesmaximal, characteristic
4#21422902C211yesmaximal, characteristic, complemented
5#21422159C211yesmaximal, characteristic, complemented
6#21422141C211yesmaximal, characteristic
7C59×#1364C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9#16837779C2211yescharacteristic
10#16837776—21no—
11#16837750—21no—
12#16837526—21no—
13C4×Dic236C2211yescharacteristic
14#16837229—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—41no—
25C2×#11745671—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273724—21no—
29C4×Dic118—21no—
30#14273672—42no—
31C2×C59⋊C16—21no—
32#14273661D411yescharacteristic
33Dic59⋊C8—41no—
34#14273652—21no—
35#14273647—42no—
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—82no—
57#11745671—82no—
58C4⋊Dic59—21no—
59Dic59⋊C4—41no—
60C4×Dic59—41no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8D811yescharacteristic
63C2×C59⋊C8D811yescharacteristic
64Dic236—21no—
65Dic236SD1612yescomplemented
66D472—41no—
67D472—41no—
68D472—81no—
69C16×C59—41no—
70C59⋊C16—41no—
71C22×D59—81no—
72Q8×C59—81no—
73Q8×C59—81no—
74C2×C4×C59—81no—
75C2×C4×C59#12111yescharacteristic
76C2×C4×C59C2×D811yescharacteristic
77C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
78C2×Dic59—41no—
79C2×Dic59—41no—
80D236—82no—
81D236—81no—
82D236—81no—
83D236—81no—
84D236—81no—
85Dic118—22no—
86Dic118—41no—
87C8×C59—21no—
88C8×C59—81no—
89C8×C59C2×SD1611yescharacteristic
90C8×C59#13511yescharacteristic
91C59⋊C8—21no—
92C59⋊C8—41no—
93C59⋊C8SD3212yescomplemented
94#12417—591nomaximal, Sylow
95C22×C59#45911yescharacteristic
96C2×D59—161no—
97C2×D59—161no—
98C2×D59—161no—
99C4×C59—21no—
100C4×C59—161no—
101C4×C59#46511yescharacteristic
102C4×C59#47111yescharacteristic
103C4×C59C2×SD3211yescharacteristic
104C4×C59#50911yescharacteristic
105Dic59—42no—
106Dic59—81no—
107#2225—591no—
108#2207—591no—
109#1706—591no—
110C8×Q16—591no—
111#1367—591no—
112#1364D5911yescharacteristic, complemented
113#1361—591no—
114C2×C59#166511yescharacteristic
115C2×C59#220711yescharacteristic
116C2×C59#221211yescharacteristic
117D59—321no—
118C2×SD32—1181no—
119#500C2×D5911yescharacteristic
120#493—591no—
121#492—1181no—
122#486—1181no—
123C8×Q8—1181no—
124C4×Q16—591no—
125#367—1181no—
126C4⋊C16—591no—
127C4⋊C16C2×D5911yescharacteristic
128#358—1181no—
129#357—1181no—
130#335C2×D5911yescharacteristic
131Q8⋊C8—1181no—
132C82—591no—
133C59#1241711yescharacteristic, complemented, Sylow
134C2×Q16—591no—
135C2×SD16—2361no—
136C2×SD16—2361no—
137C2×D8—1181no—
138C2×D8—1181no—
139C4⋊Q8—21no—
140#127—1181no—
141C4×Q8—1181no—
142SD32—2362no—
143C2×C16—21no—
144C2×C16—1181no—
145#107—21no—
146#107—21no—
147#107—591no—
148C4⋊C8—21no—
149C4⋊C8—2361no—
150#103—1181no—
151#102—2362no—
152C4×C8—591no—
153C4×C8—591no—
154C4×C8—2361no—
155C4×C8C22×D5911yescharacteristic
156C2×Q8—41no—
157C2×Q8—1181no—
158C2×D4—2361no—
159C2×D4—2361no—
160C2×D4—2361no—
161Q16—592no—
162Q16—1181no—
163SD16—4722no—
164SD16—4722no—
165D8—4721no—
166D8—2361no—
167D8—2361no—
168C2×C8—41no—
169C2×C8—591no—
170C2×C8—591no—
171C2×C8—1181no—
172C2×C8#1174568511yescharacteristic
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—1181no—
177C4⋊C4—2361no—
178C42—2361no—
179C42D4×D5911yescharacteristic
180C16—41no—
181C16—2361no—
182C23—4721no—
183Q8—81no—
184Q8—81no—
185Q8—1182no—
186Q8—2361no—
187D4—4722no—
188D4—4721no—
189D4—4721no—
190D4—4721no—
191D4—4721no—
192C2×C4—81no—
193C2×C4—2361no—
194C2×C4—2361no—
195C2×C4#1427370011yescharacteristic
196C2×C4D8×D5911yescharacteristic
197C2×C4#1427375411yescharacteristic
198C8—21no—
199C8—81no—
200C8—592no—
201C8—1181no—
202C8—2361no—
203C8SD16×D5911yescharacteristic
204C8#1427375011yescharacteristic
205C22—9441no—
206C22—9441no—
207C22—9441no—
208C22#1683759611yescharacteristic
209C4—21no—
210C4—161no—
211C4—4721no—
212C4—2362no—
213C4#1683763411yescharacteristic
214C4#1683766811yescharacteristic
215C4SD32×D5911yescharacteristic
216C4#1683783611yescharacteristic
217C2—18881no—
218C2#2142344011yescharacteristic
219C2#2142613511yescharacteristic
220C2#2142616811yescharacteristic
subgroupG/H
centerC22#16837596
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1364C2
socleC22×C59#459
prime radical#16837779C22
hypercenter#1364D59
hypocenterC59#12417
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. #21426235C2semidirect
4. #21422902C2semidirect
5. #21422159C2semidirect
7. C59×#1364C2semidirect
65. Dic236SD16semidirect
93. C59⋊C8SD32semidirect
112. #1364D59semidirect
133. C59#12417semidirect

Maximal subgroups

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

subgroupindex
#214262352
#214261372
#214236892
#214229022
#214221592
#214221412
C59×#13642
#1241759