Finite Group Databasebeta

← C31×#2201 #16842134

Subgroups — C31×#2201 #16842134

Order 3968.

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

normal subgroups only

This group has 226 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
1C31×#563C211yesmaximal, characteristic, complemented
2C31×#458C212yesmaximal
3C31×#455C211yesmaximal, characteristic
4C31×#354C211yesmaximal, characteristic, complemented
5C31×#352C212yesmaximal, complemented
6C31×H(5,𝔽2)C2211yescharacteristic
7C31×#137—21no—
8C31×C4⋊Q8—21no—
9C31×#126—22no—
10C31×#125—21no—
11C31×#124C2211yescharacteristic
12C31×#123—22no—
13C31×#123C2212yes—
14C31×C22⋊Q8—22no—
15C31×#104—21no—
16C31×#100C2211yescharacteristic
17C31×#99—22no—
18C31×#99C2212yes—
19#41×C31—21no—
20#41×C31—41no—
21C2×Q8×C31—21no—
22C2×Q8×C31D411yescharacteristic
23C2×D4×C31—22no—
24C2×D4×C31—41no—
25C2×D4×C31C2311yescharacteristic
26C22×C4×C31D412yes—
27Q16×C31—41no—
28SD16×C31—41no—
29M16×C31—21no—
30C4⋊C4×C31—21no—
31C4⋊C4×C31—22no—
32C4⋊C4×C31—22no—
33C4⋊C4×C31—21no—
34C4⋊C4×C31—22no—
35C4⋊C4×C31—42no—
36C22⋊C4×C31—22no—
37C22⋊C4×C31—22no—
38C22⋊C4×C31—22no—
39C22⋊C4×C31—42no—
40C22⋊C4×C31D412yes—
41C42×C31—21no—
42C42×C31D411yescharacteristic
43C23×C31—22no—
44C23×C31C2×D412yes—
45Q8×C31—21no—
46Q8×C31—41no—
47Q8×C31—81no—
48D4×C31—21no—
49D4×C31—41no—
50D4×C31—42no—
51D4×C31—41no—
52C2×C4×C31—22no—
53C2×C4×C31—21no—
54C2×C4×C31—22no—
55C2×C4×C31—21no—
56C2×C4×C31—22no—
57C2×C4×C31—41no—
58C2×C4×C31—42no—
59C2×C4×C31—42no—
60C2×C4×C31C2×D411yescharacteristic
61C8×C31—41no—
62#2201C3111yesmaximal, characteristic, direct product factor, Sylow
63C22×C31—22no—
64C22×C31—42no—
65C22×C31—41no—
66C22×C31—82no—
67C22×C31#12011yescharacteristic
68C4×C31—21no—
69C4×C31—42no—
70C4×C31—41no—
71C4×C31—41no—
72C4×C31—42no—
73C4×C31—41no—
74C4×C31—82no—
75#563C2×C3111yescharacteristic, complemented
76#458C2×C3112yes—
77#455C2×C3111yescharacteristic
78#354C2×C3111yescharacteristic, complemented
79#352C2×C3112yescomplemented
80C2×C31—21no—
81C2×C31—42no—
82C2×C31—81no—
83C2×C31B(4,𝔽2)11yescharacteristic
84H(5,𝔽2)C22×C3111yescharacteristic
85#137—21no—
86C4⋊Q8—21no—
87#126—22no—
88#125—21no—
89#124C22×C3111yescharacteristic
90#123—22no—
91#123C22×C3112yes—
92C22⋊Q8—22no—
93#104—21no—
94#100C22×C3111yescharacteristic
95#99—22no—
96#99C22×C3112yes—
97C31#220111yescharacteristic, direct product factor, Sylow
98#41—21no—
99#41—41no—
100C2×Q8—21no—
101C2×Q8D4×C3111yescharacteristic
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C23×C3111yescharacteristic
105C22×C4D4×C3112yes—
106Q16—41no—
107SD16—41no—
108M16—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—42no—
119C22⋊C4D4×C3112yes—
120C42—21no—
121C42D4×C3111yescharacteristic
122C23—22no—
123C23C2×D4×C3112yes—
124Q8—21no—
125Q8—41no—
126Q8—81no—
127D4—21no—
128D4—41no—
129D4—42no—
130D4—41no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4C2×D4×C3111yescharacteristic
140C8—41no—
141C22—22no—
142C22—42no—
143C22—41no—
144C22—82no—
145C22C31×#12011yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—41no—
150C4—42no—
151C4—41no—
152C4—82no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C31
prime radicalC31×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C31 ◁ C2×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#563C2semidirect
4. C31×#354C2semidirect
5. C31×#352C2semidirect
62. #2201C31direct (N × H)
75. #563C2×C31semidirect
78. #354C2×C31semidirect
79. #352C2×C31semidirect
97. C31#2201direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#5632
C31×#4582
C31×#4552
C31×#3542
C31×#3522
#220131