Finite Group Databasebeta

← C59×#12304 #28424413

Subgroups — C59×#12304 #28424413

Order 15104.

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

normal subgroups only

This group has 286 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
1C59×#3259C211yesmaximal, characteristic
2C59×#2238C212yesmaximal, complemented
3C59×#2103C211yesmaximal, characteristic, complemented
4C59×#1341C212yesmaximal, complemented
5C59×#1328C211yesmaximal, characteristic, complemented
6C59×#540—21no—
7C59×#478—22no—
8C59×#469C2212yes—
9C59×#466—21no—
10C59×#459—22no—
11C59×#430C2211yescharacteristic
12C59×#424C2211yescharacteristic
13C59×#365—22no—
14C59×#361—22no—
15C59×#349C2212yes—
16C59×#344—21no—
17C59×#341—21no—
18C59×#338C2211yescharacteristic
19C2×D8×C59—21no—
20C2×D8×C59D412yes—
21C2×M16×C59—21no—
22C2×M16×C59—21no—
23C2×M16×C59C2311yescharacteristic
24C22×C8×C59D411yescharacteristic
25#127×C59—21no—
26#125×C59—21no—
27#123×C59—42no—
28#121×C59—22no—
29#121×C59—42no—
30C4×D4×C59—22no—
31#117×C59D411yescharacteristic
32M32×C59—22no—
33#108×C59—22no—
34#107×C59—21no—
35#106×C59D412yes—
36C4⋊C8×C59—22no—
37#102×C59—22no—
38#102×C59—22no—
39C2×D4×C59—22no—
40C2×D4×C59—42no—
41C2×D4×C59—42no—
42C22×C4×C59—42no—
43C22×C4×C59C2×D411yescharacteristic
44D8×C59—42no—
45D8×C59—81no—
46M16×C59—41no—
47M16×C59—41no—
48M16×C59—41no—
49M16×C59—41no—
50M16×C59—41no—
51C2×C8×C59—21no—
52C2×C8×C59—41no—
53C2×C8×C59—41no—
54C2×C8×C59—41no—
55C2×C8×C59C2×D412yes—
56C4⋊C4×C59—21no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—42no—
59C22⋊C4×C59—42no—
60C22⋊C4×C59—42no—
61C22⋊C4×C59SD1612yes—
62C42×C59—21no—
63C16×C59—42no—
64C23×C59—42no—
65C23×C59C2×SD1611yescharacteristic
66D4×C59—42no—
67D4×C59—42no—
68D4×C59—162no—
69D4×C59—162no—
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—22no—
73C2×C4×C59—42no—
74C2×C4×C59—162no—
75C2×C4×C59#12011yescharacteristic
76C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
77C8×C59—22no—
78C8×C59—41no—
79C8×C59—81no—
80C8×C59—81no—
81#12304C5911yesmaximal, characteristic, direct product factor, Sylow
82C22×C59—21no—
83C22×C59—41no—
84C22×C59—82no—
85C22×C59—82no—
86C22×C59—82no—
87C22×C59#45011yescharacteristic
88C4×C59—21no—
89C4×C59—42no—
90C4×C59—82no—
91C4×C59#45411yescharacteristic
92C4×C59#45811yescharacteristic
93#3259C2×C5911yescharacteristic
94#2238C2×C5912yescomplemented
95#2103C2×C5911yescharacteristic, complemented
96#1341C2×C5912yescomplemented
97#1328C2×C5911yescharacteristic, complemented
98C2×C59—21no—
99C2×C59—41no—
100C2×C59—162no—
101C2×C59#161611yescharacteristic
102#540—21no—
103#478—22no—
104#469C22×C5912yes—
105#466—21no—
106#459—22no—
107#430C22×C5911yescharacteristic
108#424C22×C5911yescharacteristic
109#365—22no—
110#361—22no—
111#349C22×C5912yes—
112#344—21no—
113#341—21no—
114#338C22×C5911yescharacteristic
115C59#1230411yescharacteristic, direct product factor, Sylow
116C2×D8—21no—
117C2×D8D4×C5912yes—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16C23×C5911yescharacteristic
121C22×C8D4×C5911yescharacteristic
122#127—21no—
123#125—21no—
124#123—42no—
125#121—22no—
126#121—42no—
127C4×D4—22no—
128#117D4×C5911yescharacteristic
129M32—22no—
130#108—22no—
131#107—21no—
132#106D4×C5912yes—
133C4⋊C8—22no—
134#102—22no—
135#102—22no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C22×C4—42no—
140C22×C4C2×D4×C5911yescharacteristic
141D8—42no—
142D8—81no—
143M16—41no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—41no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8C2×D4×C5912yes—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4SD16×C5912yes—
159C42—21no—
160C16—42no—
161C23—42no—
162C23C2×SD16×C5911yescharacteristic
163D4—42no—
164D4—42no—
165D4—162no—
166D4—162no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—162no—
172C2×C4#120×C5911yescharacteristic
173C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
174C8—22no—
175C8—41no—
176C8—81no—
177C8—81no—
178C22—21no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22C59×#45011yescharacteristic
184C4—21no—
185C4—42no—
186C4—82no—
187C4C59×#45411yescharacteristic
188C4C59×#45811yescharacteristic
189C2—21no—
190C2—41no—
191C2—162no—
192C2C59×#161611yescharacteristic
subgroupG/H
center, socleC2×C59#1616
commutator, Frattini, cosocleC2×M16C23×C59
prime radicalC59×#3259C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C2×M16×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2238C2semidirect
3. C59×#2103C2semidirect
4. C59×#1341C2semidirect
5. C59×#1328C2semidirect
81. #12304C59direct (N × H)
94. #2238C2×C59semidirect
95. #2103C2×C59semidirect
96. #1341C2×C59semidirect
97. #1328C2×C59semidirect
115. C59#12304direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32592
C59×#22382
C59×#21032
C59×#13412
C59×#13282
#1230459