Finite Group Databasebeta

← C23×#13436 #19161062

Subgroups — C23×#13436 #19161062

Order 5888.

438 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3199C211yesmaximal, characteristic
2C23×#2122C212yesmaximal, complemented
3C23×#1910C212yesmaximal, complemented
4C23×#1858C211yesmaximal, characteristic, complemented
5C23×#1391C211yesmaximal, characteristic, complemented
6C23×#568C2211yescharacteristic
7C23×#535C2212yes—
8C23×#482—22no—
9C23×#469—21no—
10C23×#466—21no—
11C23×#447—22no—
12C23×#430—21no—
13C23×#418—22no—
14C23×#413C2211yescharacteristic
15C23×#410C2212yes—
16C23×#408C2211yescharacteristic
17C23×AΣL(1,𝔽16)—24no—
18C23×#341—21no—
19C23×H(3,ℤ/4)—22no—
20C2×#41×C23C2311yescharacteristic
21C22×D4×C23D412yes—
22C2×D8×C23—21no—
23C2×D8×C23—21no—
24C23×#131—42no—
25C2×M16×C23—22no—
26C2×M16×C23—21no—
27C2×M16×C23—21no—
28C2×M16×C23D411yescharacteristic
29C22×C8×C23D411yescharacteristic
30C23×#127—22no—
31C23×#124—22no—
32C23×#121—22no—
33C23×#121—22no—
34C23×#120—22no—
35C23×#120—22no—
36C23×#117—22no—
37C23×#117D412yes—
38C23×#108—42no—
39C23×#107—21no—
40C23×#106—21no—
41C23×#102—22no—
42C23×#102—22no—
43C23×#102—44no—
44C23×#101—21no—
45C23×#100D412yes—
46C23×#99—22no—
47C23×#99—22no—
48C23×C22⋊C8—22no—
49C24×C23—22no—
50#41×C23—41no—
51#41×C23—41no—
52C2×Q8×C23#4111yescharacteristic
53C2×D4×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—42no—
56C2×D4×C23—42no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23C2×D411yescharacteristic
60C2×D4×C23#4112yes—
61C22×C4×C23C2×D411yescharacteristic
62C22×C4×C23C2×D412yes—
63D8×C23—81no—
64D8×C23—81no—
65M16×C23—22no—
66M16×C23—22no—
67M16×C23—41no—
68M16×C23—42no—
69M16×C23—41no—
70M16×C23—41no—
71C2×C8×C23—24no—
72C2×C8×C23—21no—
73C2×C8×C23—21no—
74C2×C8×C23—41no—
75C2×C8×C23—41no—
76C2×C8×C23—41no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—42no—
79C22⋊C4×C23—22no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—42no—
82C22⋊C4×C23—42no—
83C42×C23—22no—
84C42×C23—42no—
85#13436C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—42no—
87C23×C23—82no—
88C23×C23—162no—
89C23×C23#12112yes—
90C23×C23#12411yescharacteristic
91Q8×C23—41no—
92D4×C23—41no—
93D4×C23—42no—
94D4×C23—44no—
95D4×C23—42no—
96D4×C23—42no—
97D4×C23—162no—
98D4×C23—162no—
99C2×C4×C23—22no—
100C2×C4×C23—21no—
101C2×C4×C23—22no—
102C2×C4×C23—21no—
103C2×C4×C23—42no—
104C2×C4×C23—82no—
105C2×C4×C23#12011yescharacteristic
106C2×C4×C23#12111yescharacteristic
107C2×C4×C23#12312yes—
108C8×C23—42no—
109C8×C23—42no—
110C8×C23—41no—
111C8×C23—41no—
112#3199C2×C2311yescharacteristic
113#2122C2×C2312yescomplemented
114#1910C2×C2312yescomplemented
115#1858C2×C2311yescharacteristic, complemented
116#1391C2×C2311yescharacteristic, complemented
117C22×C23—22no—
118C22×C23—21no—
119C22×C23—41no—
120C22×C23—42no—
121C22×C23—82no—
122C22×C23—82no—
123C22×C23—82no—
124C22×C23—164no—
125C22×C23#39411yescharacteristic
126C4×C23—21no—
127C4×C23—22no—
128C4×C23—82no—
129C4×C23—162no—
130C4×C23B(4,𝔽2)11yescharacteristic
131C4×C23#45811yescharacteristic
132#568C22×C2311yescharacteristic
133#535C22×C2312yes—
134#482—22no—
135#469—21no—
136#466—21no—
137#447—22no—
138#430—21no—
139#418—22no—
140#413C22×C2311yescharacteristic
141#410C22×C2312yes—
142#408C22×C2311yescharacteristic
143AΣL(1,𝔽16)—24no—
144#341—21no—
145H(3,ℤ/4)—22no—
146C2×C23—21no—
147C2×C23—42no—
148C2×C23—41no—
149C2×C23—162no—
150C2×C23#204411yescharacteristic
151C2×#41C23×C2311yescharacteristic
152C22×D4D4×C2312yes—
153C2×D8—21no—
154C2×D8—21no—
155#131—42no—
156C2×M16—22no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16D4×C2311yescharacteristic
160C22×C8D4×C2311yescharacteristic
161#127—22no—
162#124—22no—
163#121—22no—
164#121—22no—
165#120—22no—
166#120—22no—
167#117—22no—
168#117D4×C2312yes—
169#108—42no—
170#107—21no—
171#106—21no—
172#102—22no—
173#102—22no—
174#102—44no—
175#101—21no—
176#100D4×C2312yes—
177#99—22no—
178#99—22no—
179C22⋊C8—22no—
180C23#1343611yescharacteristic, direct product factor, Sylow
181C24—22no—
182#41—41no—
183#41—41no—
184C2×Q8#41×C2311yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4C2×D4×C2311yescharacteristic
192C2×D4#41×C2312yes—
193C22×C4C2×D4×C2311yescharacteristic
194C22×C4C2×D4×C2312yes—
195D8—81no—
196D8—81no—
197M16—22no—
198M16—22no—
199M16—42no—
200M16—41no—
201M16—41no—
202M16—41no—
203C2×C8—24no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—22no—
216C42—42no—
217C23—42no—
218C23—82no—
219C23—162no—
220C23C23×#12112yes—
221C23C23×#12411yescharacteristic
222Q8—41no—
223D4—41no—
224D4—42no—
225D4—44no—
226D4—42no—
227D4—42no—
228D4—162no—
229D4—162no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4C23×#12011yescharacteristic
237C2×C4C23×#12111yescharacteristic
238C2×C4C23×#12312yes—
239C8—42no—
240C8—42no—
241C8—41no—
242C8—41no—
243C22—22no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—164no—
251C22C23×#39411yescharacteristic
252C4—21no—
253C4—22no—
254C4—82no—
255C4—162no—
256C4C23×B(4,𝔽2)11yescharacteristic
257C4C23×#45811yescharacteristic
258C2—21no—
259C2—42no—
260C2—41no—
261C2—162no—
262C2C23×#204411yescharacteristic
subgroupG/H
center, socleC2×C23#2044
commutator, Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3199C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2122C2semidirect
3. C23×#1910C2semidirect
4. C23×#1858C2semidirect
5. C23×#1391C2semidirect
85. #13436C23direct (N × H)
113. #2122C2×C23semidirect
114. #1910C2×C23semidirect
115. #1858C2×C23semidirect
116. #1391C2×C23semidirect
180. C23#13436direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31992
C23×#21222
C23×#19102
C23×#18582
C23×#13912
#1343623