Finite Group Databasebeta

← #28502916

Subgroups — #28502916

Order 15104.

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

normal subgroups only

This group has 230 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#21426348C211yesmaximal, characteristic, complemented
2#21425927C211yesmaximal, characteristic
3#21423462C211yesmaximal, characteristic
4#21423458C211yesmaximal, characteristic
5#21422169C211yesmaximal, characteristic, complemented
6#21422146C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8C2×#14273673—21no—
9C2×#14273673—21no—
10C2×#14273666—21no—
11#16837782C2211yescharacteristic
12#16837683—21no—
13#16837679—21no—
14C8×D236C2211yescharacteristic
15(C2×D59)⋊C16—21no—
16C4×C59⋊C16C2211yescharacteristic
17#16837155C2211yescharacteristic
18#16837148—21no—
19#16837147—21no—
20C59×#494C2211yescharacteristic
21C59×C4⋊C16C2211yescharacteristic
22C59×#335C2211yescharacteristic
23C2×Dic236—21no—
24C2×Dic236—21no—
25C2×C8×D59—21no—
26#14273727—21no—
27C4×D236D411yescharacteristic
28#14273673—42no—
29#14273673—42no—
30#14273666—42no—
31C2×C59⋊C16D411yescharacteristic
32C2×C59⋊C16D411yescharacteristic
33#14273661—41no—
34#14273660—41no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)D411yescharacteristic
37#14273649—41no—
38#14273647—41no—
39(C2×D59)⋊C8—21no—
40C2×Q16×C59—21no—
41C2×Q16×C59—21no—
42C4⋊Q8×C59—21no—
43C2×C16×C59—21no—
44C4⋊C8×C59—21no—
45C4×C8×C59C2311yescharacteristic
46C2×Q8×C59—41no—
47C2×D236D811yescharacteristic
48C2×C4×D59—21no—
49C2×Dic118—41no—
50Q16×C59—41no—
51Q16×C59—41no—
52Q16×C59—81no—
53C2×C8×C59—41no—
54C2×C8×C59C2×D411yescharacteristic
55C2×C8×C59#4111yescharacteristic
56C4⋊C4×C59—41no—
57C4⋊C4×C59—41no—
58C42×C59C2×D411yescharacteristic
59(C2×D59)⋊C4—21no—
60C4⋊Dic59—41no—
61C4⋊Dic59—41no—
62C4⋊Dic59D811yescharacteristic
63C2×C59⋊C8—21no—
64C2×C59⋊C8—41no—
65Dic236—41no—
66Dic236—41no—
67Dic236—81no—
68C8×D59—41no—
69C8×D59—41no—
70C16×C59—41no—
71C59⋊C16—21no—
72C59⋊C16D812yes—
73C22×D59—21no—
74Q8×C59—81no—
75Q8×C59—81no—
76C2×C4×C59—81no—
77C2×C4×C59#12111yescharacteristic
78C2×C4×C59C2×D811yescharacteristic
79C2×C4×C59#13511yescharacteristic
80C2×Dic59—21no—
81C2×Dic59—81no—
82D236—21no—
83D236—21no—
84C4×D59—41no—
85C4×D59—41no—
86Dic118—81no—
87Dic118—81no—
88C8×C59—21no—
89C8×C59—81no—
90C8×C59C2×D811yescharacteristic
91C8×C59AGL(1,ℤ/8)11yescharacteristic
92C59⋊C8—41no—
93C59⋊C8—81no—
94#12434—591nomaximal, Sylow
95C22×C59#46611yescharacteristic
96C2×D59—41no—
97C2×D59—41no—
98C2×D59—41no—
99C4×C59—21no—
100C4×C59—161no—
101C4×C59#45911yescharacteristic
102C4×C59#47211yescharacteristic
103C4×C59#51011yescharacteristic
104C4×C59#51011yescharacteristic
105Dic59—41no—
106Dic59—161no—
107#2247—591no—
108#2169—591no—
109#1669—591no—
110#1669—591no—
111#1372—591no—
112#1362—591no—
113#1362D5911yescharacteristic, complemented
114C2×C59#166711yescharacteristic
115C2×C59#221911yescharacteristic
116C2×C59#222211yescharacteristic
117D59—81no—
118C2×Q32—1181no—
119C2×Q32—1181no—
120C2×M32—1181no—
121#494—591no—
122#494C2×D5911yescharacteristic
123#474—1181no—
124#474—1181no—
125C8×D4—591no—
126C4⋊C16C2×D5911yescharacteristic
127C22⋊C16—1181no—
128C4×C16—591no—
129#335—591no—
130#335C2×D5911yescharacteristic
131#332—1181no—
132#332—1181no—
133C59#1243411yescharacteristic, complemented, Sylow
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—1181no—
137C2×Q16—1181no—
138C22×C8—1181no—
139C4⋊Q8—21no—
140C4⋊Q8—1181no—
141C4×D4—591no—
142Q32—2362no—
143Q32—2362no—
144M32—2362no—
145C2×C16—21no—
146C2×C16—591no—
147C2×C16—591no—
148#107—2361no—
149#107—2361no—
150C4⋊C8—21no—
151C4⋊C8—591no—
152C4⋊C8—1181no—
153#102—2361no—
154#102—2361no—
155C22⋊C8—1181no—
156C4×C8C22×D5911yescharacteristic
157C2×Q8—41no—
158C2×Q8—2361no—
159C2×D4—591no—
160C22×C4—1181no—
161Q16—41no—
162Q16—41no—
163Q16—81no—
164Q16—4721no—
165Q16—2361no—
166Q16—2361no—
167C2×C8—41no—
168C2×C8—1181no—
169C2×C8—2361no—
170C2×C8—2361no—
171C2×C8—2361no—
172C2×C8#1174568511yescharacteristic
173C2×C8D4×D5911yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—591no—
177C4⋊C4—2361no—
178C4⋊C4—2361no—
179C22⋊C4—1181no—
180C42D4×D5911yescharacteristic
181C16—41no—
182C16—592no—
183C16—1181no—
184C23—1181no—
185Q8—81no—
186Q8—81no—
187Q8—4721no—
188Q8—4721no—
189D4—1181no—
190D4—1181no—
191C2×C4—81no—
192C2×C4—1181no—
193C2×C4—4721no—
194C2×C4—2361no—
195C2×C4—2361no—
196C2×C4#1427370011yescharacteristic
197C2×C4D8×D5911yescharacteristic
198C2×C4#1427374811yescharacteristic
199C8—21no—
200C8—81no—
201C8—4721no—
202C8—2361no—
203C8D8×D5911yescharacteristic
204C8#1427375411yescharacteristic
205C22—2361no—
206C22—2361no—
207C22—2361no—
208C22#1683763711yescharacteristic
209C4—21no—
210C4—161no—
211C4—9441no—
212C4—2361no—
213C4#1683759611yescharacteristic
214C4#1683767111yescharacteristic
215C4#1683784111yescharacteristic
216C4#1683784411yescharacteristic
217C2—4721no—
218C2#2142345011yescharacteristic
219C2#2142620311yescharacteristic
220C2#2142622011yescharacteristic
subgroupG/H
centerC22#16837637
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59#466
prime radicalC2×D236D8
hypercenter#1362D59
hypocenterC59#12434
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #21426348C2semidirect
5. #21422169C2semidirect
6. #21422146C2semidirect
7. C59×#1362C2semidirect
113. #1362D59semidirect
133. C59#12434semidirect

Maximal subgroups

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

subgroupindex
#214263482
#214259272
#214234622
#214234582
#214221692
#214221462
C59×#13622
#1243459