Finite Group Databasebeta

← #28502973

Subgroups — #28502973

Order 15104.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426352C211yesmaximal, characteristic
2#21423466C211yesmaximal, characteristic, complemented
3#21423456C211yesmaximal, characteristic
4#21422168C211yesmaximal, characteristic, complemented
5#21422146C211yesmaximal, characteristic, complemented
6#21422143C211yesmaximal, characteristic
7C59×#2183C211yesmaximal, characteristic, complemented
8C2×#14273672—21no—
9C2×#14273672—21no—
10#16837782C2211yescharacteristic
11#16837779C2211yescharacteristic
12#16837687—21no—
13#16837602—21no—
14#16837155C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837149—21no—
17#16837145—21no—
18C8×Q8×C59C2211yescharacteristic
19C59×C4⋊C16—21no—
20C59×C4⋊C16C2211yescharacteristic
21C59×#346—21no—
22C4×C16×C59C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×#11745671—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273727—21no—
29#14273724—21no—
30#14273672—42no—
31#14273672—42no—
32#14273658—41no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)—21no—
35#14273651—41no—
36#14273647—41no—
37C4×Q8×C59D411yescharacteristic
38C2×C16×C59—21no—
39C2×C16×C59D411yescharacteristic
40C2×C16×C59D411yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59D411yescharacteristic
43C4×C8×C59—21no—
44C4×C8×C59C2311yescharacteristic
45C2×Q8×C59D811yescharacteristic
46C2×D236—41no—
47C2×D236—41no—
48C2×D236—41no—
49C2×Dic118—41no—
50C2×C8×C59—21no—
51C2×C8×C59C2×D411yescharacteristic
52C2×C8×C59#4111yescharacteristic
53C4⋊C4×C59—21no—
54C4⋊C4×C59D811yescharacteristic
55C42×C59—21no—
56C42×C59C2×D411yescharacteristic
57#11745671—82no—
58C4⋊Dic59—41no—
59C4⋊Dic59—41no—
60C2×C59⋊C8—41no—
61C2×C59⋊C8—41no—
62Dic236—41no—
63Dic236—41no—
64Dic236—81no—
65D472—41no—
66D472—41no—
67D472—81no—
68C16×C59—21no—
69C16×C59—41no—
70C16×C59SD1612yescomplemented
71C22×D59—81no—
72Q8×C59—21no—
73Q8×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59#12111yescharacteristic
77C2×C4×C59C2×D811yescharacteristic
78C2×C4×C59#13511yescharacteristic
79C2×Dic59—81no—
80D236—81no—
81D236—81no—
82D236—81no—
83D236—81no—
84D236—161no—
85Dic118—81no—
86Dic118—81no—
87C8×C59—21no—
88C8×C59—41no—
89C8×C59C2×SD1611yescharacteristic
90C8×C59AGL(1,ℤ/8)11yescharacteristic
91C59⋊C8—81no—
92C59⋊C8—81no—
93#12444—591nomaximal, Sylow
94C22×C59#46611yescharacteristic
95C2×D59—161no—
96C2×D59—161no—
97C2×D59—161no—
98C4×C59—21no—
99C4×C59—41no—
100C4×C59—41no—
101C4×C59#46011yescharacteristic
102C4×C59#47211yescharacteristic
103C4×C59#50911yescharacteristic
104C4×C59#51011yescharacteristic
105Dic59—161no—
106#2248—591no—
107#2183D5911yescharacteristic, complemented
108#1670—591no—
109#1668—591no—
110#1371—591no—
111#1362—591no—
112#1361—591no—
113C2×C59#166811yescharacteristic
114C2×C59#222011yescharacteristic
115C2×C59#222311yescharacteristic
116D59—321no—
117C2×SD32—1181no—
118C2×SD32—1181no—
119#494—591no—
120#493—591no—
121#475—1181no—
122#460—1181no—
123C8×Q8C2×D5911yescharacteristic
124C4⋊C16—21no—
125C4⋊C16C2×D5911yescharacteristic
126#346—21no—
127C4×C16C2×D5911yescharacteristic
128#335—591no—
129#335—591no—
130#332—1181no—
131#331—1181no—
132C59#1244411yescharacteristic, complemented, Sylow
133C2×Q16—1181no—
134C2×Q16—1181no—
135C2×SD16—2361no—
136C2×D8—1181no—
137C2×D8—1181no—
138C4⋊Q8—1181no—
139#127—1181no—
140C4×Q8D23611yescharacteristic
141SD32—2362no—
142SD32—2362no—
143C2×C16—21no—
144C2×C16C22⋊D5911yescharacteristic
145C2×C16C22⋊D5911yescharacteristic
146#106—2361no—
147C4⋊C8—21no—
148C4⋊C8—1181no—
149C4⋊C8—1181no—
150C4⋊C8D23611yescharacteristic
151#103—2361no—
152#102—2361no—
153C4×C8—21no—
154C4×C8C22×D5911yescharacteristic
155C2×Q8—2361no—
156C2×Q8D47211yescharacteristic
157C2×D4—2361no—
158C2×D4—2361no—
159C2×D4—2361no—
160Q16—4721no—
161Q16—2361no—
162Q16—2361no—
163SD16—4722no—
164D8—4721no—
165D8—2361no—
166D8—2361no—
167C2×C8—21no—
168C2×C8—2361no—
169C2×C8—2361no—
170C2×C8#1174568511yescharacteristic
171C2×C8C2×C22⋊D5911yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C4⋊C4D47211yescharacteristic
176C42—21no—
177C42C2×D23611yescharacteristic
178C16—21no—
179C16—41no—
180C16#1174567112yescomplemented
181C23—4721no—
182Q8—21no—
183Q8—21no—
184Q8—4721no—
185Q8—4721no—
186D4—9441no—
187D4—4721no—
188D4—4721no—
189D4—4721no—
190D4—4721no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—4721no—
194C2×C4#1427370111yescharacteristic
195C2×C4C2×D47211yescharacteristic
196C2×C4#1427374911yescharacteristic
197C8—21no—
198C8—41no—
199C8—4721no—
200C8—4721no—
201C8C2×#1174567111yescharacteristic
202C8#1427375211yescharacteristic
203C22—9441no—
204C22—9441no—
205C22—9441no—
206C22#1683763811yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—9441no—
211C4#1683760211yescharacteristic
212C4#1683767211yescharacteristic
213C4#1683784011yescharacteristic
214C4#1683784511yescharacteristic
215C2—18881no—
216C2#2142345611yescharacteristic
217C2#2142620911yescharacteristic
218C2#2142622611yescharacteristic
subgroupG/H
centerC22#16837638
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2183C2
socleC22×C59#466
prime radical#16837779C22
hypercenter#2183D59
hypocenterC59#12444
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #2183

Complements

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

NH ≅ G/Ntype
2. #21423466C2semidirect
4. #21422168C2semidirect
5. #21422146C2semidirect
7. C59×#2183C2semidirect
70. C16×C59SD16semidirect
107. #2183D59semidirect
132. C59#12444semidirect
180. C16#11745671semidirect

Maximal subgroups

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

subgroupindex
#214263522
#214234662
#214234562
#214221682
#214221462
#214221432
C59×#21832
#1244459