Finite Group Databasebeta

← C59×#2203 #21420521

Subgroups — C59×#2203 #21420521

Order 7552.

216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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×#563C211yesmaximal, characteristic, complemented
2C59×#456C212yesmaximal, complemented
3C59×#454C211yesmaximal, characteristic
4C59×#356C212yesmaximal, complemented
5C59×#355C211yesmaximal, characteristic, complemented
62−1+4×C59C2211yescharacteristic
7#137×C59—22no—
8AGL(1,ℤ/8)×C59—21no—
9C4⋊Q8×C59C2212yes—
10#126×C59—22no—
11#125×C59—21no—
12#124×C59C2211yescharacteristic
13#123×C59—21no—
14C22⋊Q8×C59—22no—
15#104×C59—21no—
16#104×C59—22no—
17#101×C59C2212yes—
18#101×C59C2211yescharacteristic
19#41×C59—21no—
20#41×C59—22no—
21#41×C59—41no—
22C2×Q8×C59—41no—
23C2×Q8×C59D412yes—
24C2×Q8×C59C2311yescharacteristic
25C2×D4×C59D411yescharacteristic
26C22×C4×C59—21no—
27Q16×C59—42no—
28SD16×C59—42no—
29SD16×C59—41no—
30D8×C59—41no—
31M16×C59—21no—
32M16×C59—22no—
33C4⋊C4×C59—21no—
34C4⋊C4×C59—22no—
35C4⋊C4×C59—21no—
36C4⋊C4×C59—22no—
37C4⋊C4×C59—22no—
38C22⋊C4×C59—22no—
39C22⋊C4×C59—21no—
40C22⋊C4×C59—22no—
41C42×C59D411yescharacteristic
42C42×C59D412yes—
43C23×C59—21no—
44Q8×C59—21no—
45Q8×C59—22no—
46Q8×C59—41no—
47Q8×C59—42no—
48D4×C59—21no—
49D4×C59—22no—
50D4×C59—41no—
51D4×C59—41no—
52C2×C4×C59—21no—
53C2×C4×C59—22no—
54C2×C4×C59—22no—
55C2×C4×C59—41no—
56C2×C4×C59—42no—
57C2×C4×C59—81no—
58C2×C4×C59C2×D412yes—
59C2×C4×C59C2×D411yescharacteristic
60C8×C59—41no—
61C8×C59—42no—
62C22×C59—42no—
63C22×C59—41no—
64C22×C59—41no—
65C22×C59#12011yescharacteristic
66C4×C59—21no—
67C4×C59—22no—
68C4×C59—41no—
69C4×C59—42no—
70C4×C59—42no—
71C4×C59—41no—
72#2203C5911yesmaximal, characteristic, direct product factor, Sylow
73C2×C59—21no—
74C2×C59—81no—
75C2×C59—81no—
76C2×C59B(4,𝔽2)11yescharacteristic
77#563C2×C5911yescharacteristic, complemented
78#456C2×C5912yescomplemented
79#454C2×C5911yescharacteristic
80#356C2×C5912yescomplemented
81#355C2×C5911yescharacteristic, complemented
82C59#220311yescharacteristic, direct product factor, Sylow
832−1+4C22×C5911yescharacteristic
84#137—22no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8C22×C5912yes—
87#126—22no—
88#125—21no—
89#124C22×C5911yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—21no—
93#104—22no—
94#101C22×C5912yes—
95#101C22×C5911yescharacteristic
96#41—21no—
97#41—22no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8D4×C5912yes—
101C2×Q8C23×C5911yescharacteristic
102C2×D4D4×C5911yescharacteristic
103C22×C4—21no—
104Q16—42no—
105SD16—42no—
106SD16—41no—
107D8—41no—
108M16—21no—
109M16—22no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42D4×C5911yescharacteristic
119C42D4×C5912yes—
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—42no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—81no—
135C2×C4C2×D4×C5912yes—
136C2×C4C2×D4×C5911yescharacteristic
137C8—41no—
138C8—42no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#120×C5911yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—41no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C59
prime radicalC59×#454C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C22×C59 ◁ C2×Q8×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#563C2semidirect
2. C59×#456C2semidirect
4. C59×#356C2semidirect
5. C59×#355C2semidirect
72. #2203C59direct (N × H)
77. #563C2×C59semidirect
78. #456C2×C59semidirect
80. #356C2×C59semidirect
81. #355C2×C59semidirect
82. C59#2203direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#5632
C59×#4562
C59×#4542
C59×#3562
C59×#3552
#220359