Finite Group Databasebeta

← C23×#2022 #15669641

Subgroups — C23×#2022 #15669641

Order 2944.

430 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 92 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×#534C211yesmaximal, characteristic
2C2×C23×#123C212yesmaximal, complemented
3C2×C23×#99C212yesmaximal, complemented
4C23×#381C211yesmaximal, characteristic, complemented
5C23×#342C211yesmaximal, characteristic, complemented
6C22×D4×C23C2212yes—
7C23×C4×C23C2211yescharacteristic
8C23×#123—22no—
9C23×#123—24no—
10C23×#123—24no—
11C23×#121—22no—
12C23×#120—22no—
13C2×C4⋊C4×C23—22no—
14C2×C22⋊C4×C23—22no—
15C2×C22⋊C4×C23C2211yescharacteristic
16C2×C22⋊C4×C23C2211yescharacteristic
17C2×C22⋊C4×C23C2212yes—
18C23×#99—24no—
19C23×#95—22no—
20C24×C23D412yes—
21C24×C23C2311yescharacteristic
22C2×D4×C23—22no—
23C2×D4×C23—22no—
24C2×D4×C23—22no—
25C2×D4×C23—22no—
26C2×D4×C23—22no—
27C2×D4×C23—22no—
28C2×D4×C23—22no—
29C22×C4×C23—22no—
30C22×C4×C23—22no—
31C22×C4×C23—22no—
32C22×C4×C23—21no—
33C22×C4×C23—21no—
34C22×C4×C23—21no—
35C22×C4×C23D412yes—
36C22×C4×C23D412yes—
37C22×C4×C23D412yescomplemented
38C4⋊C4×C23—28no—
39C4⋊C4×C23—22no—
40C22⋊C4×C23—28no—
41C22⋊C4×C23—24no—
42C22⋊C4×C23—22no—
43C22⋊C4×C23—22no—
44C22⋊C4×C23—22no—
45C22⋊C4×C23D414yescomplemented
46C23×C23—22no—
47C23×C23—22no—
48C23×C23—22no—
49C23×C23—42no—
50C23×C23—44no—
51C23×C23C2×D412yes—
52C23×C23C2×D411yescharacteristic
53C23×C23C2×D411yescharacteristic
54C23×C23C2×D412yes—
55C23×C23#4111yescharacteristic
56D4×C23—42no—
57D4×C23—42no—
58D4×C23—42no—
59D4×C23—42no—
60D4×C23—82no—
61C2×C4×C23—24no—
62C2×C4×C23—24no—
63C2×C4×C23—22no—
64C2×C4×C23—22no—
65C2×C4×C23—22no—
66C2×C4×C23—22no—
67C2×C4×C23—22no—
68C2×C4×C23—44no—
69C2×C4×C23—41no—
70C2×C4×C23—42no—
71C2×C4×C23—41no—
72C2×C4×C23—42no—
73C2×C4×C23—42no—
74C2×C4×C23—41no—
75#2022C2311yesmaximal, characteristic, direct product factor, Sylow
76C22×C23—22no—
77C22×C23—24no—
78C22×C23—41no—
79C22×C23—42no—
80C22×C23—41no—
81C22×C23—44no—
82C22×C23—42no—
83C22×C23—44no—
84C22×C23—82no—
85C22×C23#12012yes—
86C22×C23#12011yescharacteristic
87C22×C23#12111yescharacteristic
88C22×C23#12112yes—
89C22×C23#12711yescharacteristic
90C4×C23—44no—
91C4×C23—42no—
92C4×C23—42no—
93C4×C23—42no—
94C4×C23—81no—
95#534C2×C2311yescharacteristic
96C2×#123C2×C2312yescomplemented
97C2×#99C2×C2312yescomplemented
98#381C2×C2311yescharacteristic, complemented
99#342C2×C2311yescharacteristic, complemented
100C2×C23—22no—
101C2×C23—21no—
102C2×C23—22no—
103C2×C23—21no—
104C2×C23—82no—
105C2×C23#39211yescharacteristic
106C2×C23#45812yes—
107C22×D4C22×C2312yes—
108C23×C4C22×C2311yescharacteristic
109#123—22no—
110#123—24no—
111#123—24no—
112#121—22no—
113#120—22no—
114C2×C4⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4C22×C2311yescharacteristic
117C2×C22⋊C4C22×C2311yescharacteristic
118C2×C22⋊C4C22×C2312yes—
119#99—24no—
120#95—22no—
121C23#202211yescharacteristic, direct product factor, Sylow
122C24D4×C2312yes—
123C24C23×C2311yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4D4×C2312yes—
138C22×C4D4×C2312yes—
139C22×C4D4×C2312yescomplemented
140C4⋊C4—28no—
141C4⋊C4—22no—
142C22⋊C4—28no—
143C22⋊C4—24no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4D4×C2314yescomplemented
148C23—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—44no—
153C23C2×D4×C2312yes—
154C23C2×D4×C2311yescharacteristic
155C23C2×D4×C2311yescharacteristic
156C23C2×D4×C2312yes—
157C23#41×C2311yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—82no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—44no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—44no—
183C22—42no—
184C22—44no—
185C22—82no—
186C22C23×#12012yes—
187C22C23×#12011yescharacteristic
188C22C23×#12111yescharacteristic
189C22C23×#12112yes—
190C22C23×#12711yescharacteristic
191C4—44no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—81no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2C23×#39211yescharacteristic
202C2C23×#45812yes—
subgroupG/H
center, socleC22×C23#120
commutator, Frattini, cosocleC24C23×C23
prime radicalC23×#534C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#123C2semidirect
3. C2×C23×#99C2semidirect
4. C23×#381C2semidirect
5. C23×#342C2semidirect
37. C22×C4×C23D4semidirect
45. C22⋊C4×C23D4semidirect
75. #2022C23direct (N × H)
96. C2×#123C2×C23semidirect
97. C2×#99C2×C23semidirect
98. #381C2×C23semidirect
99. #342C2×C23semidirect
121. C23#2022direct (N × H)
139. C22×C4D4×C23semidirect
147. C22⋊C4D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5342
C2×C23×#1232
C2×C23×#992
C23×#3812
C23×#3422
#202223