Finite Group Databasebeta

← C23×#1899 #15669518

Subgroups — C23×#1899 #15669518

Order 2944.

378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×C23×#123C211yesmaximal, characteristic
2C2×C23×#117C211yesmaximal, characteristic, complemented
3C2×C23×#99C212yesmaximal
4C23×#381C211yesmaximal, characteristic, complemented
5C23×#342C212yesmaximal, complemented
6C22×D4×C23C2211yescharacteristic
7C23×C4×C23C2211yescharacteristic
8C23×#123—21no—
9C23×#123—21no—
10C23×#123—22no—
11C23×#117—24no—
12C2×C4⋊C4×C23—21no—
13C2×C4⋊C4×C23C412yescomplemented
14C2×C22⋊C4×C23C412yescomplemented
15C2×C22⋊C4×C23C2212yes—
16C2×C22⋊C4×C23C2211yescharacteristic
17C2×C22⋊C4×C23C2212yes—
18C2×C42×C23—21no—
19C23×#99—24no—
20C23×#95—22no—
21C24×C23C2×C411yescharacteristic
22C24×C23C2311yescharacteristic
23C2×D4×C23—22no—
24C2×D4×C23—21no—
25C2×D4×C23—21no—
26C2×D4×C23—22no—
27C22×C4×C23—22no—
28C22×C4×C23—22no—
29C22×C4×C23—22no—
30C22×C4×C23—22no—
31C22×C4×C23C2×C411yescharacteristic
32C22×C4×C23C2×C412yes—
33C22×C4×C23C2×C412yes—
34C22×C4×C23D411yescharacteristic
35C22×C4×C23D411yescharacteristic
36C22×C4×C23D412yes—
37C4⋊C4×C23—24no—
38C4⋊C4×C23—22no—
39C4⋊C4×C23—22no—
40C22⋊C4×C23—22no—
41C22⋊C4×C23—21no—
42C22⋊C4×C23—21no—
43C22⋊C4×C23—22no—
44C22⋊C4×C23—22no—
45C22⋊C4×C23—22no—
46C22⋊C4×C23—22no—
47C22⋊C4×C23D414yes—
48C42×C23—24no—
49C23×C23—21no—
50C23×C23—22no—
51C23×C23—42no—
52C23×C23—42no—
53C23×C23C22×C411yescharacteristic
54C23×C23C2×D411yescharacteristic
55C23×C23C2×D411yescharacteristic
56C23×C23C2×D412yes—
57C23×C23#4112yes—
58D4×C23—41no—
59D4×C23—42no—
60D4×C23—41no—
61C2×C4×C23—24no—
62C2×C4×C23—21no—
63C2×C4×C23—21no—
64C2×C4×C23—24no—
65C2×C4×C23—21no—
66C2×C4×C23—21no—
67C2×C4×C23—22no—
68C2×C4×C23—21no—
69C2×C4×C23—22no—
70C2×C4×C23—24no—
71C2×C4×C23—44no—
72C2×C4×C23—42no—
73C2×C4×C23—44no—
74C2×C4×C23—42no—
75C2×C4×C23C22⋊C412yes—
76C2×C4×C23C22⋊C412yes—
77#1899C2311yesmaximal, characteristic, direct product factor, Sylow
78C22×C23—22no—
79C22×C23—22no—
80C22×C23—22no—
81C22×C23—42no—
82C22×C23—42no—
83C22×C23—41no—
84C22×C23—42no—
85C22×C23—82no—
86C22×C23C2×C22⋊C411yescharacteristic
87C22×C23C4×D412yes—
88C22×C23#12011yescharacteristic
89C22×C23#12112yes—
90C22×C23#12311yescharacteristic
91C4×C23—22no—
92C4×C23—22no—
93C4×C23—44no—
94C4×C23—41no—
95C4×C23—42no—
96C4×C23—82no—
97C2×#123C2×C2311yescharacteristic
98C2×#117C2×C2311yescharacteristic, complemented
99C2×#99C2×C2312yes—
100#381C2×C2311yescharacteristic, complemented
101#342C2×C2312yescomplemented
102C2×C23—21no—
103C2×C23—22no—
104C2×C23—22no—
105C2×C23—21no—
106C2×C23—81no—
107C2×C23#38611yescharacteristic
108C2×C23#41012yes—
109C22×D4C22×C2311yescharacteristic
110C23×C4C22×C2311yescharacteristic
111#123—22no—
112#123—21no—
113#123—21no—
114#117—24no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C4×C2312yescomplemented
117C2×C22⋊C4C4×C2312yescomplemented
118C2×C22⋊C4C22×C2312yes—
119C2×C22⋊C4C22×C2311yescharacteristic
120C2×C22⋊C4C22×C2312yes—
121C2×C42—21no—
122#99—24no—
123#95—22no—
124C23#189911yescharacteristic, direct product factor, Sylow
125C24C2×C4×C2311yescharacteristic
126C24C23×C2311yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×C4×C2311yescharacteristic
136C22×C4C2×C4×C2312yes—
137C22×C4C2×C4×C2312yes—
138C22×C4D4×C2312yes—
139C22×C4D4×C2311yescharacteristic
140C22×C4D4×C2311yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4D4×C2314yes—
152C42—24no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×C4×C2311yescharacteristic
158C23C2×D4×C2311yescharacteristic
159C23C2×D4×C2311yescharacteristic
160C23C2×D4×C2312yes—
161C23#41×C2312yes—
162D4—41no—
163D4—42no—
164D4—41no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4C22⋊C4×C2312yes—
180C2×C4C22⋊C4×C2312yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—42no—
188C22—82no—
189C22C2×C22⋊C4×C2311yescharacteristic
190C22C4×D4×C2312yes—
191C22C23×#12011yescharacteristic
192C22C23×#12112yes—
193C22C23×#12311yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—41no—
198C4—42no—
199C4—82no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2C23×#38611yescharacteristic
206C2C23×#41012yes—
subgroupG/H
center, socleC22×C23C2×C22⋊C4
commutatorC23C22×C4×C23
Frattini, cosocleC24C23×C23
prime radicalC22×D4×C23C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#117C2semidirect
4. C23×#381C2semidirect
5. C23×#342C2semidirect
13. C2×C4⋊C4×C23C4semidirect
14. C2×C22⋊C4×C23C4semidirect
77. #1899C23direct (N × H)
98. C2×#117C2×C23semidirect
100. #381C2×C23semidirect
101. #342C2×C23semidirect
116. C2×C4⋊C4C4×C23semidirect
117. C2×C22⋊C4C4×C23semidirect
124. C23#1899direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#1232
C2×C23×#1172
C2×C23×#992
C23×#3812
C23×#3422
#189923