Finite Group Databasebeta

← C31×#7521 #21464280

Subgroups — C31×#7521 #21464280

Order 7936.

148 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C31×#1634C211yesmaximal, characteristic
2C31×#1411C211yesmaximal, characteristic, complemented
3C31×#1358C211yesmaximal, characteristic, complemented
4C31×#555C411yescharacteristic
5C31×#494C2211yescharacteristic
6C31×#476—21no—
7C31×#364—21no—
8C31×#356—21no—
9C31×#333C411yescharacteristic
10C31×D4⋊C8—21no—
11C2×Q16×C31—41no—
12C31×C4⋊Q8C2×C411yescharacteristic
13C31×C4⋊Q8D411yescharacteristic
14C31×#124—41no—
15C31×#121—41no—
16C4×D4×C31—21no—
17C31×#117—41no—
18C31×M32—41no—
19C31×#106—41no—
20C31×C4⋊C8—21no—
21C31×#102—41no—
22C31×#101—41no—
23C4×C8×C31D411yescharacteristic
24C2×Q8×C31—21no—
25C2×Q8×C31—21no—
26C2×D4×C31—21no—
27C2×D4×C31—41no—
28C22×C4×C31—41no—
29Q16×C31—81no—
30Q16×C31—81no—
31M16×C31—81no—
32C2×C8×C31—21no—
33C2×C8×C31—41no—
34C4⋊C4×C31—21no—
35C4⋊C4×C31—41no—
36C4⋊C4×C31—41no—
37C22⋊C4×C31—42no—
38C22⋊C4×C31—41no—
39C42×C31—41no—
40C42×C31C22⋊C411yescharacteristic
41C16×C31—81no—
42#7521C3111yesmaximal, characteristic, direct product factor, Sylow
43C23×C31—41no—
44Q8×C31—81no—
45Q8×C31—81no—
46D4×C31—41no—
47D4×C31—41no—
48D4×C31—161no—
49C2×C4×C31—21no—
50C2×C4×C31—41no—
51C2×C4×C31—41no—
52C2×C4×C31—41no—
53C2×C4×C31—81no—
54C2×C4×C31—81no—
55C2×C4×C31#9911yescharacteristic
56C8×C31—21no—
57C8×C31—21no—
58C8×C31—81no—
59C8×C31—161no—
60#1634C2×C3111yescharacteristic
61#1411C2×C3111yescharacteristic, complemented
62#1358C2×C3111yescharacteristic, complemented
63C22×C31—81no—
64C22×C31—81no—
65C22×C31—81no—
66C22×C31AΣL(1,𝔽16)11yescharacteristic
67C4×C31—21no—
68C4×C31—21no—
69C4×C31—21no—
70C4×C31—81no—
71C4×C31—81no—
72C4×C31—81no—
73#555C4×C3111yescharacteristic
74#494C22×C3111yescharacteristic
75#476—21no—
76#364—21no—
77#356—21no—
78#333C4×C3111yescharacteristic
79D4⋊C8—21no—
80C2×C31—21no—
81C2×C31—161no—
82C2×C31#140211yescharacteristic
83C2×Q16—41no—
84C4⋊Q8C2×C4×C3111yescharacteristic
85C4⋊Q8D4×C3111yescharacteristic
86#124—41no—
87#121—41no—
88C4×D4—21no—
89#117—41no—
90M32—41no—
91#106—41no—
92C4⋊C8—21no—
93#102—41no—
94#101—41no—
95C4×C8D4×C3111yescharacteristic
96C31#752111yescharacteristic, direct product factor, Sylow
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—21no—
100C2×D4—41no—
101C22×C4—41no—
102Q16—81no—
103Q16—81no—
104M16—81no—
105C2×C8—21no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C42—41no—
113C42C22⋊C4×C3111yescharacteristic
114C16—81no—
115C23—41no—
116Q8—81no—
117Q8—81no—
118D4—41no—
119D4—41no—
120D4—161no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4C31×#9911yescharacteristic
128C8—21no—
129C8—21no—
130C8—81no—
131C8—161no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22C31×AΣL(1,𝔽16)11yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—81no—
142C2—21no—
143C2—161no—
144C2C31×#140211yescharacteristic
subgroupG/H
center, socleC2×C31#1402
commutatorC4⋊Q8C2×C4×C31
Frattini, cosocle#494C22×C31
prime radicalC31×#555C4
LCSG ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C31 ◁ C2×C4×C31 ◁ C42×C31 ◁ C31×#494 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1411C2semidirect
3. C31×#1358C2semidirect
42. #7521C31direct (N × H)
61. #1411C2×C31semidirect
62. #1358C2×C31semidirect
96. C31#7521direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#16342
C31×#14112
C31×#13582
#752131