Finite Group Databasebeta

← #28499882

Subgroups — #28499882

Order 15104.

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

normal subgroups only

This group has 326 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#21434471C211yesmaximal, characteristic
2#21425908C211yesmaximal, characteristic, complemented
3#21422128C214yesmaximal, complemented
4C59×#2165C211yesmaximal, characteristic, complemented
5#16837959C411yescharacteristic
6C2×#14273724C411yescharacteristic
7#16837767C412yes—
8#16837767C2212yes—
9#16837749—21no—
10#16837749—21no—
11#16837408—21no—
12#16837408—21no—
13#16837223—21no—
14#16837184—21no—
15#16837180C2212yescomplemented
16C4×M16×C59C2211yescharacteristic
17C59×#364—21no—
18C59×#349—21no—
19C59×#347C2212yescomplemented
20C22×D236—21no—
21C22×D236—41no—
22#14273727—21no—
23#14273724—21no—
24#14273724—41no—
25#14273724C2×C412yes—
26#14273719C2×C412yes—
27#14273706—41no—
28C4×Dic118—41no—
29#14273686—21no—
30#14273667—22no—
31#14273649—22no—
32#14273649—22no—
33#14273649—22no—
34#14273649—22no—
35C2×M16×C59D411yescharacteristic
36C2×M16×C59D411yescharacteristic
37C2×C42×C59C2×C411yescharacteristic
38M32×C59—22no—
39#97×C59C2×C411yescharacteristic
40#97×C59C2311yescharacteristic
41C4×C8×C59D411yescharacteristic
42C4×C8×C59D411yescharacteristic
43C23×D59—41no—
44C22×C4×C59—21no—
45C22×C4×C59C22⋊C411yescharacteristic
46C2×D236—22no—
47C2×D236—42no—
48C2×D236—42no—
49C2×D236—42no—
50C2×D236—42no—
51C2×D236—41no—
52C2×D236—84no—
53C2×Dic118—41no—
54M16×C59—41no—
55M16×C59—41no—
56C2×C8×C59C22⋊C411yescharacteristic
57C2×C8×C59C22⋊C411yescharacteristic
58C2×C8×C59C2×D411yescharacteristic
59C2×C8×C59C2×D411yescharacteristic
60C42×C59—21no—
61C42×C59C22⋊C411yescharacteristic
62C42×C59C22×C411yescharacteristic
63C22⋊Dic59—41no—
64C4⋊Dic59—22no—
65C4⋊Dic59—41no—
66Dic59⋊C4—42no—
67C4×Dic59—41no—
68C16×C59—42no—
69C59⋊C16—42no—
70C23×C59#10011yescharacteristic
71C22×D59—42no—
72C22×D59—82no—
73C22×D59—84no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—42no—
79C2×C4×C59#10011yescharacteristic
80C2×C4×C59C2×C22⋊C411yescharacteristic
81C2×Dic59—42no—
82D236—42no—
83D236—42no—
84D236—82no—
85D236—84no—
86D236—84no—
87Dic118—82no—
88C8×C59—21no—
89C8×C59—21no—
90C8×C59—21no—
91C8×C59—21no—
92#11915—591nomaximal, Sylow
93C22×C59—21no—
94C22×C59—41no—
95C22×C59C2×#10011yescharacteristic
96C2×D59—84no—
97C2×D59—82no—
98C2×D59—164no—
99C4×C59—22no—
100C4×C59—21no—
101C4×C59—41no—
102C4×C59#42111yescharacteristic
103C4×C59#42111yescharacteristic
104Dic59—82no—
105#3138—591no—
106#2165—591no—
107#2165D5911yescharacteristic, complemented
108#1355—594no—
109C2×C59—21no—
110C2×C59—41no—
111C2×C59#153311yescharacteristic
112D59—162no—
113#533—591no—
114C2×#127—591no—
115#490—592no—
116#490—592no—
117#486—1181no—
118#486—1181no—
119#418—1181no—
120#418—1181no—
121C4×M16C2×D5911yescharacteristic
122#364—21no—
123#364—1181no—
124#349—21no—
125#349—1181no—
126#347—592no—
127#347C2×D5912yescomplemented
128C59#1191511yescharacteristic, complemented, Sylow
129C22×D4—1181no—
130C22×D4—2361no—
131C2×M16D23611yescharacteristic
132C2×M16C22⋊D5911yescharacteristic
133C4⋊Q8—1181no—
134#127—592no—
135#127—1181no—
136#127—2361no—
137#125—592no—
138C22⋊Q8—2361no—
139C4×Q8—2361no—
140#117—1181no—
141C2×C42C4×D5911yescharacteristic
142M32—22no—
143M32—1182no—
144#102—1182no—
145#102—1182no—
146#102—1182no—
147#102—1182no—
148#97C4×D5911yescharacteristic
149#97C22×D5911yescharacteristic
150C4×C8D23611yescharacteristic
151C4×C8C22⋊D5911yescharacteristic
152C24—2361no—
153C2×Q8—2361no—
154C2×D4—1182no—
155C2×D4—4724no—
156C2×D4—2362no—
157C2×D4—2362no—
158C2×D4—2362no—
159C2×D4—2362no—
160C2×D4—2361no—
161C22×C4—21no—
162C22×C4(C2×D59)⋊C411yescharacteristic
163M16—41no—
164M16—41no—
165C2×C8(C2×D59)⋊C411yescharacteristic
166C2×C8(C2×D59)⋊C411yescharacteristic
167C2×C8C2×D23611yescharacteristic
168C2×C8C2×C22⋊D5911yescharacteristic
169C4⋊C4—1182no—
170C4⋊C4—2362no—
171C4⋊C4—2361no—
172C22⋊C4—2361no—
173C42—21no—
174C42—2361no—
175C42(C2×D59)⋊C411yescharacteristic
176C42C2×C4×D5911yescharacteristic
177C16—42no—
178C16—2362no—
179C23—4722no—
180C23—4724no—
181C23—2362no—
182C23#1427364311yescharacteristic
183Q8—4722no—
184D4—4724no—
185D4—4722no—
186D4—4724no—
187D4—2362no—
188D4—2362no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—2362no—
195C2×C4#1427364311yescharacteristic
196C2×C4C2×(C2×D59)⋊C411yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C22—21no—
202C22—41no—
203C22—9444no—
204C22—4724no—
205C22—4722no—
206C22C2×#1427364311yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—4722no—
211C4#1683741811yescharacteristic
212C4#1683742011yescharacteristic
213C2—21no—
214C2—41no—
215C2—9442no—
216C2#2142271911yescharacteristic
subgroupG/H
centerC2#21422719
commutatorC42×C59C22×C4
Frattini#97C22×D59
FittingC59×#2165C2
socleC2×C59#1533
prime radicalC2×#14273724C4
cosocle#97×C59C23
hypercenter#2165D59
hypocenterC59#11915
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21425908C2semidirect
3. #21422128C2semidirect
4. C59×#2165C2semidirect
15. #16837180C22semidirect
19. C59×#347C22semidirect
107. #2165D59semidirect
127. #347C2×D59semidirect
128. C59#11915semidirect

Maximal subgroups

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

subgroupindex
#214344712
#214259082
#214221282
C59×#21652
#1191559