Finite Group Databasebeta

← C23×#8326 #19155952

Subgroups — C23×#8326 #19155952

Order 5888.

812 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 158 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×#540C211yesmaximal, characteristic
2C23×#1478C214yesmaximal, complemented
3C23×#1458C212yesmaximal, complemented
4C23×#540—28no—
5C2×C23×#127C411yescharacteristic
6C2×C23×#125C411yescharacteristic
7C2×C23×#123—22no—
8C2×C23×#121—22no—
9C2×C23×#121C412yes—
10C2×C4×D4×C23C2212yes—
11C2×C23×#117C2211yescharacteristic
12C23×#431C2214yes—
13C23×D4⋊C8—28no—
14C23×#324—22no—
15C23×#323—22no—
16C22×D4×C23—22no—
17C22×D4×C23—22no—
18C22×D4×C23C2×C412yes—
19C23×C4×C23—22no—
20C23×C4×C23C2×C411yescharacteristic
21C23×#127—24no—
22C23×#125—24no—
23C23×#123—216no—
24C23×#121—216no—
25C23×#121—24no—
26C23×#121—24no—
27C4×D4×C23—24no—
28C4×D4×C23D418yes—
29C23×#117—24no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23D412yes—
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23D412yes—
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23C2311yescharacteristic
38C23×C4⋊C8—24no—
39C23×C22⋊C8—24no—
40C4×C8×C23—24no—
41C24×C23—22no—
42C24×C23C22⋊C411yescharacteristic
43C2×D4×C23—28no—
44C2×D4×C23—28no—
45C2×D4×C23—22no—
46C2×D4×C23—28no—
47C2×D4×C23—84no—
48C2×D4×C23—84no—
49C2×D4×C23C22⋊C414yes—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—24no—
53C22×C4×C23—28no—
54C22×C4×C23—84no—
55C22×C4×C23C22⋊C411yescharacteristic
56C22×C4×C23C22⋊C412yes—
57C22×C4×C23C22×C411yescharacteristic
58C22×C4×C23C2×D412yes—
59C2×C8×C23—44no—
60C4⋊C4×C23—28no—
61C4⋊C4×C23—22no—
62C4⋊C4×C23—24no—
63C4⋊C4×C23C22⋊C414yes—
64C22⋊C4×C23—28no—
65C22⋊C4×C23—28no—
66C22⋊C4×C23—24no—
67C42×C23—22no—
68C42×C23C2×D414yes—
69#8326C2311yesmaximal, characteristic, direct product factor, Sylow
70C23×C23—21no—
71C23×C23—24no—
72C23×C23—22no—
73C23×C23—44no—
74C23×C23—44no—
75C23×C23—42no—
76C23×C23—44no—
77C23×C23C2×C22⋊C411yescharacteristic
78D4×C23—28no—
79D4×C23—44no—
80D4×C23—88no—
81D4×C23—88no—
82C2×C4×C23—22no—
83C2×C4×C23—21no—
84C2×C4×C23—24no—
85C2×C4×C23—22no—
86C2×C4×C23—42no—
87C2×C4×C23—44no—
88C2×C4×C23—44no—
89C2×C4×C23—42no—
90C2×C4×C23—88no—
91C2×C4×C23C2×C22⋊C412yes—
92C2×C4×C23#12014yes—
93C2×C4×C23#13114yes—
94C8×C23—84no—
95C2×#540C2×C2311yescharacteristic
96#1478C2×C2314yescomplemented
97#1458C2×C2312yescomplemented
98C22×C23—22no—
99C22×C23—44no—
100C22×C23—42no—
101C22×C23—44no—
102C22×C23—41no—
103C22×C23—44no—
104C22×C23—82no—
105C22×C23—88no—
106C22×C23#37911yescharacteristic
107C22×C23#40812yes—
108C4×C23—24no—
109C4×C23—44no—
110C4×C23—42no—
111C4×C23—41no—
112C4×C23#40814yes—
113#540—28no—
114C2×#127C4×C2311yescharacteristic
115C2×#125C4×C2311yescharacteristic
116C2×#123—22no—
117C2×#121—22no—
118C2×#121C4×C2312yes—
119C2×C4×D4C22×C2312yes—
120C2×#117C22×C2311yescharacteristic
121#431C22×C2314yes—
122D4⋊C8—28no—
123#324—22no—
124#323—22no—
125C2×C23—22no—
126C2×C23—81no—
127C2×C23—84no—
128C2×C23#178011yescharacteristic
129C2×C23#179512yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C2×C4×C2312yes—
133C23×C4—22no—
134C23×C4C2×C4×C2311yescharacteristic
135#127—24no—
136#125—24no—
137#123—216no—
138#121—216no—
139#121—24no—
140#121—24no—
141C4×D4—24no—
142C4×D4D4×C2318yes—
143#117—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×C4×C2312yes—
146C2×C4⋊C4D4×C2312yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4D4×C2312yes—
150C2×C42C2×C4×C2311yescharacteristic
151C2×C42C23×C2311yescharacteristic
152C4⋊C8—24no—
153C22⋊C8—24no—
154C4×C8—24no—
155C23#832611yescharacteristic, direct product factor, Sylow
156C24—22no—
157C24C22⋊C4×C2311yescharacteristic
158C2×D4—28no—
159C2×D4—28no—
160C2×D4—22no—
161C2×D4—28no—
162C2×D4—84no—
163C2×D4—84no—
164C2×D4C22⋊C4×C2314yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—84no—
170C22×C4C22⋊C4×C2311yescharacteristic
171C22×C4C22⋊C4×C2312yes—
172C22×C4C22×C4×C2311yescharacteristic
173C22×C4C2×D4×C2312yes—
174C2×C8—44no—
175C4⋊C4—28no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4C22⋊C4×C2314yes—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—24no—
182C42—22no—
183C42C2×D4×C2314yes—
184C23—21no—
185C23—24no—
186C23—22no—
187C23—44no—
188C23—44no—
189C23—42no—
190C23—44no—
191C23C2×C22⋊C4×C2311yescharacteristic
192D4—28no—
193D4—44no—
194D4—88no—
195D4—88no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4C2×C22⋊C4×C2312yes—
206C2×C4C23×#12014yes—
207C2×C4C23×#13114yes—
208C8—84no—
209C22—22no—
210C22—41no—
211C22—44no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—82no—
216C22—88no—
217C22C23×#37911yescharacteristic
218C22C23×#40812yes—
219C4—24no—
220C4—41no—
221C4—44no—
222C4—42no—
223C4C23×#40814yes—
224C2—22no—
225C2—84no—
226C2—81no—
227C2C23×#178011yescharacteristic
228C2C23×#179512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1478C2semidirect
3. C23×#1458C2semidirect
69. #8326C23direct (N × H)
96. #1478C2×C23semidirect
97. #1458C2×C23semidirect
155. C23#8326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5402
C23×#14782
C23×#14582
#832623