Finite Group Databasebeta

← C23×#1657 #15669276

Subgroups — C23×#1657 #15669276

Order 2944.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C23×#563C211yesmaximal, characteristic, complemented
2C23×#496C211yesmaximal, characteristic
3C23×#490C212yesmaximal, complemented
4C23×#330C212yesmaximal, complemented
5C23×#329C211yesmaximal, characteristic, complemented
6C2×SD16×C23—21no—
7C2×D8×C23—21no—
8C23×C4⋊Q8—21no—
9C23×#127C2211yescharacteristic
10C23×#126—22no—
11C23×#125C2212yes—
12C23×#124C2211yescharacteristic
13C23×#123—21no—
14C23×C22⋊Q8—22no—
15C23×#102—22no—
16C23×#102—22no—
17C23×#97C2212yes—
18C23×#97C2211yescharacteristic
19C2×Q8×C23—21no—
20C2×Q8×C23D411yescharacteristic
21C2×D4×C23—22no—
22C2×D4×C23—21no—
23C2×D4×C23D411yescharacteristic
24C22×C4×C23—21no—
25SD16×C23—42no—
26D8×C23—42no—
27C2×C8×C23—21no—
28C2×C8×C23—22no—
29C4⋊C4×C23—22no—
30C4⋊C4×C23—22no—
31C4⋊C4×C23—22no—
32C4⋊C4×C23D412yes—
33C4⋊C4×C23D412yes—
34C22⋊C4×C23—22no—
35C22⋊C4×C23—21no—
36C22⋊C4×C23—22no—
37C42×C23—21no—
38C42×C23C2311yescharacteristic
39C23×C23—21no—
40C23×C23—41no—
41Q8×C23—41no—
42Q8×C23—81no—
43D4×C23—41no—
44D4×C23—41no—
45D4×C23—41no—
46D4×C23—44no—
47C2×C4×C23—22no—
48C2×C4×C23—21no—
49C2×C4×C23—22no—
50C2×C4×C23—81no—
51C2×C4×C23C2×D412yes—
52C2×C4×C23C2×D411yescharacteristic
53C8×C23—22no—
54C8×C23—42no—
55#1657C2311yesmaximal, characteristic, direct product factor, Sylow
56C22×C23—41no—
57C22×C23—42no—
58C22×C23—81no—
59C22×C23—82no—
60C22×C23#12011yescharacteristic
61C4×C23—21no—
62C4×C23—22no—
63C4×C23—42no—
64C4×C23—41no—
65C4×C23—42no—
66#563C2×C2311yescharacteristic, complemented
67#496C2×C2311yescharacteristic
68#490C2×C2312yescomplemented
69#330C2×C2312yescomplemented
70#329C2×C2311yescharacteristic, complemented
71C2×C23—81no—
72C2×C23—161no—
73C2×C23#45311yescharacteristic
74C2×C23#45412yes—
75C2×SD16—21no—
76C2×D8—21no—
77C4⋊Q8—21no—
78#127C22×C2311yescharacteristic
79#126—22no—
80#125C22×C2312yes—
81#124C22×C2311yescharacteristic
82#123—21no—
83C22⋊Q8—22no—
84#102—22no—
85#102—22no—
86#97C22×C2312yes—
87#97C22×C2311yescharacteristic
88C23#165711yescharacteristic, direct product factor, Sylow
89C2×Q8—21no—
90C2×Q8D4×C2311yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4D4×C2311yescharacteristic
94C22×C4—21no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4D4×C2312yes—
103C4⋊C4D4×C2312yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C23×C2311yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×D4×C2312yes—
122C2×C4C2×D4×C2311yescharacteristic
123C8—22no—
124C8—42no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22C23×#12011yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—161no—
137C2C23×#45311yescharacteristic
138C2C23×#45412yes—
subgroupG/H
center, socleC22×C23#120
commutator, Frattini, cosocleC42C23×C23
prime radicalC23×#496C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#563C2semidirect
3. C23×#490C2semidirect
4. C23×#330C2semidirect
5. C23×#329C2semidirect
55. #1657C23direct (N × H)
66. #563C2×C23semidirect
68. #490C2×C23semidirect
69. #330C2×C23semidirect
70. #329C2×C23semidirect
88. C23#1657direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5632
C23×#4962
C23×#4902
C23×#3302
C23×#3292
#165723