Finite Group Databasebeta

← C31×#7822 #21464581

Subgroups — C31×#7822 #21464581

Order 7936.

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2679C211yesmaximal, characteristic
2C31×#2094C213yesmaximal, complemented
3C31×#1270C213yesmaximal, complemented
4C2×C31×#120—23no—
5C2×C31×#106—23no—
6C2×C31×#102—23no—
7C31×#394—23no—
8C31×#390C2213yes—
9C31×#384C2211yescharacteristic, complemented
10C31×#336C2213yescomplemented
11C25×C31—21no—
12C22×D4×C31—23no—
13C22×D4×C31D413yes—
14C22×C8×C31—23no—
15C31×#120—46no—
16C31×#120—46no—
17C2×C4⋊C4×C31—23no—
18C2×C4⋊C4×C31D413yes—
19C2×C22⋊C4×C31—23no—
20C2×C22⋊C4×C31—26no—
21C2×C42×C31C2311yescharacteristic
22C31×#106—46no—
23C31×#102—46no—
24C31×#95—21no—
25C24×C31—23no—
26C24×C31—46no—
27C24×C31—83no—
28C24×C31—81no—
29C2×D4×C31—23no—
30C2×D4×C31—43no—
31C2×D4×C31—43no—
32C2×D4×C31—412no—
33C2×D4×C31—83no—
34C2×D4×C31SD1616yes—
35C22×C4×C31—23no—
36C22×C4×C31—41no—
37C22×C4×C31C2×D413yes—
38C2×C8×C31—26no—
39C2×C8×C31—43no—
40C2×C8×C31—43no—
41C4⋊C4×C31—43no—
42C4⋊C4×C31—43no—
43C4⋊C4×C31—43no—
44C22⋊C4×C31—46no—
45C22⋊C4×C31—46no—
46C22⋊C4×C31—46no—
47C42×C31—41no—
48#7822C3111yesmaximal, characteristic, direct product factor, Sylow
49C23×C31—26no—
50C23×C31—46no—
51C23×C31—43no—
52C23×C31—46no—
53C23×C31—43no—
54C23×C31—812no—
55C23×C31—812no—
56C23×C31—84no—
57C23×C31#12011yescharacteristic
58D4×C31—46no—
59D4×C31—83no—
60D4×C31—812no—
61C2×C4×C31—23no—
62C2×C4×C31—23no—
63C2×C4×C31—46no—
64C2×C4×C31—43no—
65C2×C4×C31—83no—
66C2×C4×C31C2×SD1613yes—
67C2×C4×C31AGL(1,ℤ/8)13yes—
68C8×C31—46no—
69#2679C2×C3111yescharacteristic
70#2094C2×C3113yescomplemented
71#1270C2×C3113yescomplemented
72C22×C31—46no—
73C22×C31—412no—
74C22×C31—816no—
75C22×C31—86no—
76C22×C31—83no—
77C22×C31—83no—
78C22×C31#45013yes—
79C22×C31#45313yes—
80C22×C31#45811yescharacteristic
81C4×C31—23no—
82C4×C31—23no—
83C4×C31—83no—
84C4×C31—161no—
85C2×#120—23no—
86C2×#106—23no—
87C2×#102—23no—
88#394—23no—
89#390C22×C3113yes—
90#384C22×C3111yescharacteristic, complemented
91#336C22×C3113yescomplemented
92C2×C31—86no—
93C2×C31#161313yes—
94C2×C31#162913yes—
95C2×C31#165511yescharacteristic
96C25—21no—
97C22×D4—23no—
98C22×D4D4×C3113yes—
99C22×C8—23no—
100#120—46no—
101#120—46no—
102C2×C4⋊C4—23no—
103C2×C4⋊C4D4×C3113yes—
104C2×C22⋊C4—23no—
105C2×C22⋊C4—26no—
106C2×C42C23×C3111yescharacteristic
107#106—46no—
108#102—46no—
109#95—21no—
110C31#782211yescharacteristic, direct product factor, Sylow
111C24—23no—
112C24—46no—
113C24—83no—
114C24—81no—
115C2×D4—23no—
116C2×D4—43no—
117C2×D4—43no—
118C2×D4—412no—
119C2×D4—83no—
120C2×D4SD16×C3116yes—
121C22×C4—23no—
122C22×C4—41no—
123C22×C4C2×D4×C3113yes—
124C2×C8—26no—
125C2×C8—43no—
126C2×C8—43no—
127C4⋊C4—43no—
128C4⋊C4—43no—
129C4⋊C4—43no—
130C22⋊C4—46no—
131C22⋊C4—46no—
132C22⋊C4—46no—
133C42—41no—
134C23—26no—
135C23—46no—
136C23—43no—
137C23—46no—
138C23—43no—
139C23—812no—
140C23—812no—
141C23—84no—
142C23C31×#12011yescharacteristic
143D4—46no—
144D4—83no—
145D4—812no—
146C2×C4—23no—
147C2×C4—23no—
148C2×C4—46no—
149C2×C4—43no—
150C2×C4—83no—
151C2×C4C2×SD16×C3113yes—
152C2×C4C31×AGL(1,ℤ/8)13yes—
153C8—46no—
154C22—46no—
155C22—412no—
156C22—816no—
157C22—86no—
158C22—83no—
159C22—83no—
160C22C31×#45013yes—
161C22C31×#45313yes—
162C22C31×#45811yescharacteristic
163C4—23no—
164C4—23no—
165C4—83no—
166C4—161no—
167C2—86no—
168C2C31×#161313yes—
169C2C31×#162913yes—
170C2C31×#165511yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2679C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2094C2semidirect
3. C31×#1270C2semidirect
9. C31×#384C22semidirect
10. C31×#336C22semidirect
48. #7822C31direct (N × H)
70. #2094C2×C31semidirect
71. #1270C2×C31semidirect
90. #384C22×C31semidirect
91. #336C22×C31semidirect
110. C31#7822direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26792
C31×#20942
C31×#12702
#782231