Finite Group Databasebeta

← C23×#1610 #15669229

Subgroups — C23×#1610 #15669229

Order 2944.

254 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 68 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×#537C211yesmaximal, characteristic
2C23×#480C212yesmaximal, complemented
3C23×#398C211yesmaximal, characteristic, complemented
4C23×#327C212yesmaximal, complemented
5C23×#324C211yesmaximal, characteristic, complemented
6C2×#41×C23—21no—
7C23×C22⋊Q8—22no—
8C23×#121—22no—
9C23×#121—21no—
10C23×#121C2212yes—
11C2×C4⋊C4×C23—21no—
12C2×C4⋊C4×C23C2211yescharacteristic
13C2×C4⋊C4×C23C2211yescharacteristic
14C23×#107—22no—
15C23×#102—22no—
16C23×C22⋊C8C2212yes—
17C23×#95—21no—
18C23×#95C2211yescharacteristic
19#41×C23—42no—
20#41×C23—42no—
21C2×Q8×C23—21no—
22C2×D4×C23—21no—
23C2×D4×C23—22no—
24C2×D4×C23—21no—
25C2×D4×C23D412yes—
26C22×C4×C23—22no—
27C22×C4×C23—21no—
28C22×C4×C23—21no—
29C22×C4×C23D411yescharacteristic
30C22×C4×C23D411yescharacteristic
31C22×C4×C23C2311yescharacteristic
32C2×C8×C23—22no—
33C4⋊C4×C23—21no—
34C4⋊C4×C23—22no—
35C4⋊C4×C23—22no—
36C4⋊C4×C23—42no—
37C4⋊C4×C23D412yes—
38C22⋊C4×C23—22no—
39C22⋊C4×C23—22no—
40C23×C23—22no—
41C23×C23C2×D411yescharacteristic
42Q8×C23—42no—
43D4×C23—42no—
44D4×C23—42no—
45D4×C23—42no—
46D4×C23—42no—
47D4×C23—81no—
48C2×C4×C23—21no—
49C2×C4×C23—21no—
50C2×C4×C23—22no—
51C2×C4×C23—21no—
52C2×C4×C23—22no—
53C2×C4×C23—41no—
54C2×C4×C23—44no—
55C2×C4×C23—41no—
56C2×C4×C23—41no—
57C2×C4×C23—41no—
58C2×C4×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23—41no—
61C2×C4×C23D812yes—
62C2×C4×C23C2×D412yes—
63C8×C23—42no—
64#1610C2311yesmaximal, characteristic, direct product factor, Sylow
65C22×C23—21no—
66C22×C23—21no—
67C22×C23—42no—
68C22×C23—42no—
69C22×C23—42no—
70C22×C23#12011yescharacteristic
71C22×C23C2×D811yescharacteristic
72C22×C23AGL(1,ℤ/8)11yescharacteristic
73C4×C23—22no—
74C4×C23—22no—
75C4×C23—42no—
76C4×C23—42no—
77C4×C23—41no—
78C4×C23—81no—
79#537C2×C2311yescharacteristic
80#480C2×C2312yescomplemented
81#398C2×C2311yescharacteristic, complemented
82#327C2×C2312yescomplemented
83#324C2×C2311yescharacteristic, complemented
84C2×C23—21no—
85C2×C23—21no—
86C2×C23—82no—
87C2×C23#44711yescharacteristic
88C2×C23#45611yescharacteristic
89C2×C23#45811yescharacteristic
90C2×#41—21no—
91C22⋊Q8—22no—
92#121—22no—
93#121—21no—
94#121C22×C2312yes—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C22×C2311yescharacteristic
97C2×C4⋊C4C22×C2311yescharacteristic
98#107—22no—
99#102—22no—
100C22⋊C8C22×C2312yes—
101#95—21no—
102#95C22×C2311yescharacteristic
103C23#161011yescharacteristic, direct product factor, Sylow
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4D4×C2312yes—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4D4×C2311yescharacteristic
115C22×C4D4×C2311yescharacteristic
116C22×C4C23×C2311yescharacteristic
117C2×C8—22no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—42no—
122C4⋊C4D4×C2312yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×D4×C2311yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4D8×C2312yes—
147C2×C4C2×D4×C2312yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22C23×#12011yescharacteristic
155C22C2×D8×C2311yescharacteristic
156C22C23×AGL(1,ℤ/8)11yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—42no—
161C4—41no—
162C4—81no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2C23×#44711yescharacteristic
167C2C23×#45611yescharacteristic
168C2C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23#120
commutator, Frattini, cosocleC22×C4C23×C23
prime radicalC23×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#480C2semidirect
3. C23×#398C2semidirect
4. C23×#327C2semidirect
5. C23×#324C2semidirect
64. #1610C23direct (N × H)
80. #480C2×C23semidirect
81. #398C2×C23semidirect
82. #327C2×C23semidirect
83. #324C2×C23semidirect
103. C23#1610direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5372
C23×#4802
C23×#3982
C23×#3272
C23×#3242
#161023