Finite Group Databasebeta

← C59×#12307 #28424416

Subgroups — C59×#12307 #28424416

Order 15104.

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

normal subgroups only

This group has 318 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×#3253C211yesmaximal, characteristic
2C59×#2238C212yesmaximal, complemented
3C59×#2102C211yesmaximal, characteristic, complemented
4C59×#1339C212yesmaximal, complemented
5C59×#1327C211yesmaximal, characteristic, complemented
6C59×#532—21no—
7C59×#476—22no—
8C59×#469—21no—
9C59×#469C2212yes—
10C59×#459—22no—
11C59×#430C2211yescharacteristic
12C59×#423C2211yescharacteristic
13C59×#365—22no—
14C59×#361—22no—
15C59×#349C2212yes—
16C59×#344—21no—
17C59×#339—21no—
18C59×H(3,ℤ/4)C2211yescharacteristic
19C2×#41×C59—41no—
20C2×D8×C59—21no—
21C2×D8×C59D412yes—
22C2×M16×C59—21no—
23C2×M16×C59D411yescharacteristic
24C2×M16×C59C2311yescharacteristic
25C4⋊Q8×C59—21no—
26#127×C59—21no—
27#124×C59—41no—
28#121×C59—22no—
29#121×C59—41no—
30C4×D4×C59—22no—
31C4×D4×C59—41no—
32#117×C59—21no—
33C2×C42×C59D411yescharacteristic
34M32×C59—22no—
35#108×C59—22no—
36#106×C59—21no—
37#106×C59D412yes—
38C4⋊C8×C59—22no—
39#102×C59—22no—
40#102×C59—22no—
41#41×C59—84no—
42C2×Q8×C59—41no—
43C2×D4×C59—22no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59—42no—
47C22×C4×C59—21no—
48C22×C4×C59—42no—
49C22×C4×C59C2×D411yescharacteristic
50D8×C59—42no—
51D8×C59—81no—
52M16×C59—41no—
53M16×C59—41no—
54M16×C59—41no—
55M16×C59—41no—
56C2×C8×C59—21no—
57C2×C8×C59—41no—
58C2×C8×C59C2×D412yes—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—41no—
62C22⋊C4×C59—42no—
63C22⋊C4×C59—42no—
64C22⋊C4×C59—41no—
65C42×C59—21no—
66C42×C59—41no—
67C42×C59SD1611yescharacteristic
68C42×C59SD1611yescharacteristic
69C16×C59—42no—
70C23×C59—42no—
71C23×C59AGL(1,ℤ/8)11yescharacteristic
72Q8×C59—82no—
73D4×C59—42no—
74D4×C59—42no—
75D4×C59—82no—
76D4×C59—82no—
77D4×C59—84no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—84no—
85C2×C4×C59—81no—
86C2×C4×C59#12011yescharacteristic
87C2×C4×C59C2×SD1611yescharacteristic
88C8×C59—22no—
89C8×C59—41no—
90C8×C59—81no—
91#12307C5911yesmaximal, characteristic, direct product factor, Sylow
92C22×C59—21no—
93C22×C59—41no—
94C22×C59—82no—
95C22×C59—82no—
96C22×C59—82no—
97C22×C59#45011yescharacteristic
98C4×C59—22no—
99C4×C59—21no—
100C4×C59—41no—
101C4×C59—82no—
102C4×C59—161no—
103C4×C59#45411yescharacteristic
104C4×C59#45811yescharacteristic
105#3253C2×C5911yescharacteristic
106#2238C2×C5912yescomplemented
107#2102C2×C5911yescharacteristic, complemented
108#1339C2×C5912yescomplemented
109#1327C2×C5911yescharacteristic, complemented
110C2×C59—21no—
111C2×C59—41no—
112C2×C59—162no—
113C2×C59#161611yescharacteristic
114#532—21no—
115#476—22no—
116#469—21no—
117#469C22×C5912yes—
118#459—22no—
119#430C22×C5911yescharacteristic
120#423C22×C5911yescharacteristic
121#365—22no—
122#361—22no—
123#349C22×C5912yes—
124#344—21no—
125#339—21no—
126H(3,ℤ/4)C22×C5911yescharacteristic
127C59#1230711yescharacteristic, direct product factor, Sylow
128C2×#41—41no—
129C2×D8—21no—
130C2×D8D4×C5912yes—
131C2×M16—21no—
132C2×M16D4×C5911yescharacteristic
133C2×M16C23×C5911yescharacteristic
134C4⋊Q8—21no—
135#127—21no—
136#124—41no—
137#121—22no—
138#121—41no—
139C4×D4—22no—
140C4×D4—41no—
141#117—21no—
142C2×C42D4×C5911yescharacteristic
143M32—22no—
144#108—22no—
145#106—21no—
146#106D4×C5912yes—
147C4⋊C8—22no—
148#102—22no—
149#102—22no—
150#41—84no—
151C2×Q8—41no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4C2×D4×C5911yescharacteristic
159D8—42no—
160D8—81no—
161M16—41no—
162M16—41no—
163M16—41no—
164M16—41no—
165C2×C8—21no—
166C2×C8—41no—
167C2×C8C2×D4×C5912yes—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—41no—
174C42—21no—
175C42—41no—
176C42SD16×C5911yescharacteristic
177C42SD16×C5911yescharacteristic
178C16—42no—
179C23—42no—
180C23AGL(1,ℤ/8)×C5911yescharacteristic
181Q8—82no—
182D4—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—81no—
195C2×C4#120×C5911yescharacteristic
196C2×C4C2×SD16×C5911yescharacteristic
197C8—22no—
198C8—41no—
199C8—81no—
200C22—21no—
201C22—41no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22C59×#45011yescharacteristic
206C4—22no—
207C4—21no—
208C4—41no—
209C4—82no—
210C4—161no—
211C4C59×#45411yescharacteristic
212C4C59×#45811yescharacteristic
213C2—21no—
214C2—41no—
215C2—162no—
216C2C59×#161611yescharacteristic
subgroupG/H
center, socleC2×C59#1616
commutator, Frattini, cosocleC2×M16C23×C59
prime radicalC59×#3253C2
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×#2102C2semidirect
4. C59×#1339C2semidirect
5. C59×#1327C2semidirect
91. #12307C59direct (N × H)
106. #2238C2×C59semidirect
107. #2102C2×C59semidirect
108. #1339C2×C59semidirect
109. #1327C2×C59semidirect
127. C59#12307direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32532
C59×#22382
C59×#21022
C59×#13392
C59×#13272
#1230759