Finite Group Databasebeta

← C23×#12112 #19159738

Subgroups — C23×#12112 #19159738

Order 5888.

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3202C211yesmaximal, characteristic
2C23×#2188C212yesmaximal, complemented
3C23×#2020C211yesmaximal, characteristic
4C23×#1350C212yesmaximal, complemented
5C23×#1315C211yesmaximal, characteristic
6C2×C23×#137—21no—
7C22×Q16×C23C2211yescharacteristic
8C23×#536—21no—
9C2×C23×Q32—22no—
10C23×#485—22no—
11C23×#470—21no—
12C23×#467C2212yes—
13C23×#451—22no—
14C2×C23×#101—21no—
15C23×#408C2211yescharacteristic
16C23×#358—22no—
17C23×C22⋊C16C2212yes—
18C23×#341C2211yescharacteristic
19C2×#41×C23D411yescharacteristic
20C22×Q8×C23—21no—
21C22×Q8×C23—21no—
22C23×#137—42no—
23C23×#137—42no—
24C2×Q16×C23—21no—
25C2×Q16×C23—22no—
26C2×Q16×C23—42no—
27C2×Q16×C23—41no—
28C2×Q16×C23D412yes—
29C2×SD16×C23—41no—
30C2×M16×C23—21no—
31C2×M16×C23D411yescharacteristic
32C22×C8×C23C2311yescharacteristic
33C23×C4⋊Q8—41no—
34C23×#124—41no—
35C23×#123—41no—
36C23×#123—41no—
37C23×C22⋊Q8—22no—
38C23×C22⋊Q8—22no—
39C23×#117—21no—
40C23×Q32—44no—
41C2×C16×C23—22no—
42C23×#107D412yes—
43C23×#106—21no—
44C23×#103—22no—
45C23×#103—22no—
46C23×#101—42no—
47C23×C22⋊C8—22no—
48#41×C23—41no—
49#41×C23—41no—
50C2×Q8×C23—22no—
51C2×Q8×C23—22no—
52C2×Q8×C23—41no—
53C2×Q8×C23—41no—
54C2×Q8×C23—41no—
55C2×Q8×C23—41no—
56C2×Q8×C23—41no—
57C2×Q8×C23—41no—
58C2×Q8×C23—81no—
59C2×Q8×C23—81no—
60C2×Q8×C23D811yescharacteristic
61C2×D4×C23—21no—
62C2×D4×C23D811yescharacteristic
63C22×C4×C23—21no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23C2×D411yescharacteristic
67Q16×C23—24no—
68Q16×C23—42no—
69Q16×C23—42no—
70Q16×C23—44no—
71Q16×C23—81no—
72SD16×C23—44no—
73M16×C23—22no—
74M16×C23—41no—
75M16×C23—41no—
76C2×C8×C23—21no—
77C2×C8×C23—41no—
78C2×C8×C23—41no—
79C2×C8×C23C2×D412yes—
80C4⋊C4×C23—22no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—42no—
83C22⋊C4×C23—41no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86C42×C23—41no—
87C16×C23—42no—
88#12112C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—21no—
90C23×C23AGL(1,ℤ/8)11yescharacteristic
91Q8×C23—41no—
92Q8×C23—42no—
93Q8×C23—42no—
94Q8×C23—42no—
95Q8×C23—42no—
96Q8×C23—81no—
97Q8×C23—82no—
98Q8×C23—81no—
99D4×C23—41no—
100D4×C23—81no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—41no—
105C2×C4×C23—41no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—81no—
110C2×C4×C23—82no—
111C2×C4×C23—81no—
112C2×C4×C23#12011yescharacteristic
113C2×C4×C23C2×D811yescharacteristic
114C8×C23—22no—
115C8×C23—42no—
116C8×C23—41no—
117#3202C2×C2311yescharacteristic
118#2188C2×C2312yescomplemented
119#2020C2×C2311yescharacteristic
120#1350C2×C2312yescomplemented
121#1315C2×C2311yescharacteristic
122C22×C23—21no—
123C22×C23—22no—
124C22×C23—41no—
125C22×C23—42no—
126C22×C23#44711yescharacteristic
127C4×C23—21no—
128C4×C23—41no—
129C4×C23—82no—
130C4×C23—82no—
131C4×C23—82no—
132C4×C23#45311yescharacteristic
133C4×C23B(4,𝔽2)11yescharacteristic
134C2×#137—21no—
135C22×Q16C22×C2311yescharacteristic
136#536—21no—
137C2×Q32—22no—
138#485—22no—
139#470—21no—
140#467C22×C2312yes—
141#451—22no—
142C2×#101—21no—
143#408C22×C2311yescharacteristic
144#358—22no—
145C22⋊C16C22×C2312yes—
146#341C22×C2311yescharacteristic
147C2×C23—41no—
148C2×C23—81no—
149C2×C23#159511yescharacteristic
150C2×C23#219512yes—
151C2×#41D4×C2311yescharacteristic
152C22×Q8—21no—
153C22×Q8—21no—
154#137—42no—
155#137—42no—
156C2×Q16—21no—
157C2×Q16—22no—
158C2×Q16—42no—
159C2×Q16—41no—
160C2×Q16D4×C2312yes—
161C2×SD16—41no—
162C2×M16—21no—
163C2×M16D4×C2311yescharacteristic
164C22×C8C23×C2311yescharacteristic
165C4⋊Q8—41no—
166#124—41no—
167#123—41no—
168#123—41no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171#117—21no—
172Q32—44no—
173C2×C16—22no—
174#107D4×C2312yes—
175#106—21no—
176#103—22no—
177#103—22no—
178#101—42no—
179C22⋊C8—22no—
180C23#1211211yescharacteristic, direct product factor, Sylow
181#41—41no—
182#41—41no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8D8×C2311yescharacteristic
194C2×D4—21no—
195C2×D4D8×C2311yescharacteristic
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C2×D4×C2311yescharacteristic
200Q16—24no—
201Q16—42no—
202Q16—42no—
203Q16—44no—
204Q16—81no—
205SD16—44no—
206M16—22no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8C2×D4×C2312yes—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—41no—
220C16—42no—
221C23—21no—
222C23C23×AGL(1,ℤ/8)11yescharacteristic
223Q8—41no—
224Q8—42no—
225Q8—42no—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—82no—
230Q8—81no—
231D4—41no—
232D4—81no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4C23×#12011yescharacteristic
245C2×C4C2×D8×C2311yescharacteristic
246C8—22no—
247C8—42no—
248C8—41no—
249C22—22no—
250C22—21no—
251C22—42no—
252C22—41no—
253C22C23×#44711yescharacteristic
254C4—21no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4C23×#45311yescharacteristic
260C4C23×B(4,𝔽2)11yescharacteristic
261C2—41no—
262C2—81no—
263C2C23×#159511yescharacteristic
264C2C23×#219512yes—
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC2×#41×C23D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C23×#408 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2188C2semidirect
4. C23×#1350C2semidirect
88. #12112C23direct (N × H)
118. #2188C2×C23semidirect
120. #1350C2×C23semidirect
180. C23#12112direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32022
C23×#21882
C23×#20202
C23×#13502
C23×#13152
#1211223