Finite Group Databasebeta

← C59×#7498 #28419607

Subgroups — C59×#7498 #28419607

Order 15104.

348 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C59×#353C212yesmaximal
2C59×#1957C211yesmaximal, characteristic
3C2×#127×C59C2211yescharacteristic
4C2×#102×C59—22no—
5C2×#99×C59—22no—
6C2×C4×C8×C59C411yescharacteristic, complemented
7C59×#384C411yescharacteristic, complemented
8C59×#353—22no—
9C59×#353C414yescomplemented
10C22×D4×C59—41no—
11C22×D4×C59D412yes—
12C22×C8×C59—21no—
13#127×C59—41no—
14#127×C59C2×C412yes—
15#127×C59D411yescharacteristic
16#127×C59Q811yescharacteristic
17C2×C4⋊C4×C59—21no—
18C2×C4⋊C4×C59—41no—
19C2×C22⋊C4×C59—42no—
20C2×C42×C59C2×C411yescharacteristic
21#102×C59—44no—
22#99×C59—24no—
23#99×C59—42no—
24C4×C8×C59—41no—
25C24×C59—22no—
26C2×D4×C59—22no—
27C2×D4×C59—42no—
28C2×D4×C59—42no—
29C2×D4×C59—42no—
30C2×D4×C59—41no—
31C2×D4×C59—41no—
32C2×D4×C59—82no—
33C2×D4×C59—82no—
34C2×D4×C59C22⋊C412yes—
35C2×D4×C59C4⋊C412yes—
36C22×C4×C59—21no—
37C22×C4×C59—41no—
38C22×C4×C59—82no—
39C22×C4×C59C22⋊C411yescharacteristic
40C2×C8×C59—22no—
41C2×C8×C59—41no—
42C2×C8×C59—41no—
43C4⋊C4×C59—42no—
44C4⋊C4×C59—82no—
45C22⋊C4×C59—44no—
46C22⋊C4×C59—82no—
47C42×C59—21no—
48C42×C59C4211yescharacteristic
49C42×C59C4⋊C411yescharacteristic
50C23×C59—22no—
51C23×C59—22no—
52C23×C59—42no—
53C23×C59—42no—
54C23×C59—82no—
55C23×C59—82no—
56C23×C59#9911yescharacteristic
57D4×C59—42no—
58D4×C59—42no—
59D4×C59—82no—
60D4×C59—84no—
61D4×C59—82no—
62D4×C59—82no—
63C2×C4×C59—22no—
64C2×C4×C59—22no—
65C2×C4×C59—21no—
66C2×C4×C59—21no—
67C2×C4×C59—21no—
68C2×C4×C59—21no—
69C2×C4×C59—82no—
70C2×C4×C59—84no—
71C2×C4×C59—81no—
72C2×C4×C59—162no—
73C2×C4×C59—162no—
74C2×C4×C59#9511yescharacteristic
75C2×C4×C59#9911yescharacteristic
76C8×C59—42no—
77#7498C5911yesmaximal, characteristic, direct product factor, Sylow
78C22×C59—21no—
79C22×C59—21no—
80C22×C59—42no—
81C22×C59—42no—
82C22×C59—82no—
83C22×C59—82no—
84C22×C59—82no—
85C22×C59—82no—
86C22×C59—162no—
87C22×C59#34211yescharacteristic
88C22×C59#35311yescharacteristic
89C22×C59#35411yescharacteristic
90C4×C59—22no—
91C4×C59—22no—
92C4×C59—21no—
93C4×C59—21no—
94C4×C59—164no—
95C4×C59—161no—
96C2×#353C2×C5912yes—
97#1957C2×C5911yescharacteristic
98C2×C59—21no—
99C2×C59—21no—
100C2×C59—82no—
101C2×C59—82no—
102C2×C59#139311yescharacteristic
103C2×C59#140811yescharacteristic
104C2×C59#140911yescharacteristic
105C2×#127C22×C5911yescharacteristic
106C2×#102—22no—
107C2×#99—22no—
108C2×C4×C8C4×C5911yescharacteristic, complemented
109#384C4×C5911yescharacteristic, complemented
110#353—22no—
111#353C4×C5914yescomplemented
112C59#749811yescharacteristic, direct product factor, Sylow
113C22×D4—41no—
114C22×D4D4×C5912yes—
115C22×C8—21no—
116#127—41no—
117#127C2×C4×C5912yes—
118#127D4×C5911yescharacteristic
119#127Q8×C5911yescharacteristic
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C42C2×C4×C5911yescharacteristic
124#102—44no—
125#99—24no—
126#99—42no—
127C4×C8—41no—
128C24—22no—
129C2×D4—22no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—82no—
136C2×D4—82no—
137C2×D4C22⋊C4×C5912yes—
138C2×D4C4⋊C4×C5912yes—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—82no—
142C22×C4C22⋊C4×C5911yescharacteristic
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C42—21no—
151C42C42×C5911yescharacteristic
152C42C4⋊C4×C5911yescharacteristic
153C23—22no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23—82no—
158C23—82no—
159C23#99×C5911yescharacteristic
160D4—42no—
161D4—42no—
162D4—82no—
163D4—84no—
164D4—82no—
165D4—82no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—82no—
173C2×C4—84no—
174C2×C4—81no—
175C2×C4—162no—
176C2×C4—162no—
177C2×C4#95×C5911yescharacteristic
178C2×C4#99×C5911yescharacteristic
179C8—42no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22—82no—
188C22—162no—
189C22C59×#34211yescharacteristic
190C22C59×#35311yescharacteristic
191C22C59×#35411yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—164no—
197C4—161no—
198C2—21no—
199C2—21no—
200C2—82no—
201C2—82no—
202C2C59×#139311yescharacteristic
203C2C59×#140811yescharacteristic
204C2C59×#140911yescharacteristic
subgroupG/H
center, socleC22×C59#353
commutatorC42C42×C59
Frattini, cosocleC2×#127C22×C59
prime radicalC2×#127×C59C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C22×C4×C59 ◁ C2×#127×C59 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×C8×C59C4semidirect
7. C59×#384C4semidirect
9. C59×#353C4semidirect
77. #7498C59direct (N × H)
108. C2×C4×C8C4×C59semidirect
109. #384C4×C59semidirect
111. #353C4×C59semidirect
112. C59#7498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3532
C59×#19572
#749859