Finite Group Databasebeta

← #28500209

Subgroups — #28500209

Order 15104.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434755C211yesmaximal, characteristic
2#21425912C211yesmaximal, characteristic
3#21422158C212yesmaximal, complemented
4#21422134C212yesmaximal, complemented
5C59×#2153C211yesmaximal, characteristic
6#16838014C412yes—
7#16837804C2211yescharacteristic
8#16837782C2211yescharacteristic
9#16837771C412yes—
10#16837660—21no—
11#16837643—21no—
12#16837510—21no—
13C4×Dic236—21no—
14#16837223C2212yes—
15#16837184—21no—
16#16837180—21no—
17C2×M32×C59—21no—
18C59×#405C2211yescharacteristic
19C59×#364—21no—
20C59×#364C2212yes—
21C2×Dic236—21no—
22C2×Dic236—21no—
23#14273727C2×C412yes—
24#14273720—42no—
25#14273719C2×C412yes—
26#14273717—42no—
27#14273706—22no—
28C4×Dic118—22no—
29#14273667—22no—
30#14273661—21no—
31#14273659—21no—
32#14273652—22no—
33#14273652—22no—
34C2×M16×C59D411yescharacteristic
35C22×C8×C59D411yescharacteristic
36#117×C59C2×C411yescharacteristic
37M32×C59—22no—
38M32×C59—41no—
39C2×C16×C59—41no—
40#97×C59C2×C411yescharacteristic
41#97×C59D411yescharacteristic
42C4×C8×C59D411yescharacteristic
43C4×C8×C59C2311yescharacteristic
44C22×C4×C59C22⋊C411yescharacteristic
45C2×Dic118—22no—
46M16×C59—22no—
47C2×C8×C59—21no—
48C2×C8×C59—21no—
49C2×C8×C59C22⋊C411yescharacteristic
50C2×C8×C59C22⋊C411yescharacteristic
51C2×C8×C59C2×D411yescharacteristic
52C2×C8×C59C2×D411yescharacteristic
53C4⋊C4×C59—21no—
54C22⋊C4×C59—21no—
55C42×C59C22⋊C411yescharacteristic
56C42×C59C22×C411yescharacteristic
57C22⋊Dic59—42no—
58C4⋊Dic59—22no—
59C4⋊Dic59—42no—
60Dic59⋊C4—42no—
61Dic59⋊C4—42no—
62Dic59⋊C4—42no—
63C4×Dic59—42no—
64Dic236—42no—
65Dic236—41no—
66Dic236—41no—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59#10011yescharacteristic
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59#10011yescharacteristic
75C2×C4×C59C2×C22⋊C411yescharacteristic
76C2×Dic59—42no—
77C2×Dic59—42no—
78Dic118—42no—
79Dic118—42no—
80C8×C59—21no—
81C8×C59—21no—
82C8×C59#13512yes—
83C8×C59AGL(1,ℤ/8)11yescharacteristic
84C8×C59#13711yescharacteristic
85#11990—591nomaximal, Sylow
86C22×C59—21no—
87C22×C59—41no—
88C22×C59C2×#10011yescharacteristic
89C4×C59—21no—
90C4×C59—41no—
91C4×C59—41no—
92C4×C59#41611yescharacteristic
93C4×C59#41711yescharacteristic
94Dic59—82no—
95Dic59—82no—
96#3177—591no—
97#2166—591no—
98#2153D5911yescharacteristic
99#1366—592no—
100#1358—592no—
101C2×C59—21no—
102C2×C59—41no—
103C2×C59#155111yescharacteristic
104#541—592no—
105C2×M32—21no—
106#499—591no—
107#494—591no—
108#491—592no—
109#470—1181no—
110#467—1181no—
111#441—1181no—
112C4×Q16—1181no—
113#405C2×D5911yescharacteristic
114#364—21no—
115#364—592no—
116#364C2×D5912yes—
117#349—1181no—
118#347—1181no—
119C59#1199011yescharacteristic, complemented, Sylow
120C2×Q16—1181no—
121C2×Q16—1181no—
122C2×M16D23611yescharacteristic
123C22×C8C22⋊D5911yescharacteristic
124C4⋊Q8—592no—
125#126—2362no—
126#125—592no—
127#125—2362no—
128C22⋊Q8—1182no—
129C4×Q8—1182no—
130#117C4×D5911yescharacteristic
131M32—22no—
132M32—41no—
133M32—1182no—
134C2×C16—41no—
135#107—1181no—
136#106—1181no—
137#103—1182no—
138#103—1182no—
139#97C4×D5911yescharacteristic
140#97D23611yescharacteristic
141C4×C8C22⋊D5911yescharacteristic
142C4×C8C22×D5911yescharacteristic
143C2×Q8—1182no—
144C22×C4(C2×D59)⋊C411yescharacteristic
145Q16—2362no—
146Q16—2361no—
147Q16—2361no—
148M16—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8(C2×D59)⋊C411yescharacteristic
152C2×C8(C2×D59)⋊C411yescharacteristic
153C2×C8C2×D23611yescharacteristic
154C2×C8C2×C22⋊D5911yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—1182no—
157C4⋊C4—2362no—
158C4⋊C4—2362no—
159C4⋊C4—2362no—
160C4⋊C4—2362no—
161C22⋊C4—21no—
162C22⋊C4—2362no—
163C42—2362no—
164C42(C2×D59)⋊C411yescharacteristic
165C42C2×C4×D5911yescharacteristic
166C16—42no—
167C16—2362no—
168C23#1427364311yescharacteristic
169Q8—2362no—
170Q8—2362no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—2362no—
176C2×C4—2362no—
177C2×C4#1427364311yescharacteristic
178C2×C4C2×(C2×D59)⋊C411yescharacteristic
179C8—21no—
180C8—21no—
181C8#1427374912yes—
182C8#1427375311yescharacteristic
183C8#1427375811yescharacteristic
184C22—21no—
185C22—41no—
186C22C2×#1427364311yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—4722no—
191C4—4722no—
192C4#1683739911yescharacteristic
193C4#1683740111yescharacteristic
194C2—21no—
195C2—41no—
196C2#2142278911yescharacteristic
subgroupG/H
centerC2#21422789
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#2153C2
socleC2×C59#1551
prime radicalC23×C59#100
cosocleC4×C8×C59C23
hypercenter#2153D59
hypocenterC59#11990
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
3. #21422158C2semidirect
4. #21422134C2semidirect
119. C59#11990semidirect

Maximal subgroups

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

subgroupindex
#214347552
#214259122
#214221582
#214221342
C59×#21532
#1199059