Finite Group Databasebeta

← C23×#9991 #19157617

Subgroups — C23×#9991 #19157617

Order 5888.

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3195C211yesmaximal, characteristic
2C2×C23×#451C212yesmaximal
3C23×#1784C211yesmaximal, characteristic
4C23×#1472C212yesmaximal
5C23×#1298C211yesmaximal, characteristic
6C23×Q8×C23C2211yescharacteristic
7C22×Q16×C23—22no—
8C2×C23×C22⋊Q8C2212yes—
9C23×#483—22no—
10C23×#470—22no—
11C23×#451—24no—
12C23×#451—22no—
13C23×#451—24no—
14C2×C23×#103—22no—
15C2×C23×#101—22no—
16C23×#407C2211yescharacteristic
17C23×#407C2211yescharacteristic
18C2×C23×C22⋊C8C2212yes—
19C23×#324—24no—
20C22×Q8×C23—21no—
21C22×Q8×C23—22no—
22C22×Q8×C23—22no—
23C22×Q8×C23—44no—
24C22×Q8×C23D412yes—
25C22×Q8×C23D412yes—
26C23×C4×C23—21no—
27C23×C4×C23C2311yescharacteristic
28C2×Q16×C23—28no—
29C2×Q16×C23—22no—
30C2×Q16×C23—48no—
31C2×M16×C23—21no—
32C2×M16×C23D412yes—
33C22×C8×C23—22no—
34C23×C22⋊Q8—22no—
35C23×C22⋊Q8—22no—
36C23×C22⋊Q8—22no—
37C23×C22⋊Q8—22no—
38C2×C4⋊C4×C23—22no—
39C2×C4⋊C4×C23D412yes—
40C2×C22⋊C4×C23—22no—
41C23×#106—22no—
42C23×#103—28no—
43C23×#103—24no—
44C23×#101—44no—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8D414yes—
49C24×C23#4111yescharacteristic
50C2×Q8×C23—24no—
51C2×Q8×C23—28no—
52C2×Q8×C23—24no—
53C2×Q8×C23—22no—
54C2×Q8×C23—22no—
55C2×Q8×C23—44no—
56C2×Q8×C23—44no—
57C2×Q8×C23—42no—
58C2×Q8×C23—44no—
59C2×Q8×C23—48no—
60C2×Q8×C23—41no—
61C2×Q8×C23—41no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—42no—
66C22×C4×C23—44no—
67C22×C4×C23—42no—
68C22×C4×C23C2×D412yes—
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23C2×D411yescharacteristic
71C22×C4×C23C2×D412yes—
72Q16×C23—416no—
73Q16×C23—82no—
74M16×C23—42no—
75M16×C23—42no—
76C2×C8×C23—22no—
77C2×C8×C23—24no—
78C2×C8×C23—44no—
79C2×C8×C23—41no—
80C4⋊C4×C23—22no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—42no—
83C4⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86#9991C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—24no—
88C23×C23—41no—
89C23×C23#12111yescharacteristic
90C23×C23#12112yes—
91Q8×C23—28no—
92Q8×C23—44no—
93Q8×C23—44no—
94Q8×C23—44no—
95Q8×C23—44no—
96Q8×C23—44no—
97C2×C4×C23—24no—
98C2×C4×C23—22no—
99C2×C4×C23—24no—
100C2×C4×C23—41no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—48no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—44no—
108C2×C4×C23—82no—
109C2×C4×C23#12012yes—
110C2×C4×C23#12011yescharacteristic
111C2×C4×C23#12711yescharacteristic
112C8×C23—44no—
113C8×C23—42no—
114C8×C23—81no—
115#3195C2×C2311yescharacteristic
116C2×#451C2×C2312yes—
117#1784C2×C2311yescharacteristic
118#1472C2×C2312yes—
119#1298C2×C2311yescharacteristic
120C22×C23—22no—
121C22×C23—22no—
122C22×C23—22no—
123C22×C23—21no—
124C22×C23—44no—
125C22×C23#39211yescharacteristic
126C22×C23#47314yes—
127C4×C23—22no—
128C4×C23—21no—
129C4×C23—44no—
130C4×C23—82no—
131C4×C23—82no—
132C4×C23B(4,𝔽2)12yes—
133C23×Q8C22×C2311yescharacteristic
134C22×Q16—22no—
135C2×C22⋊Q8C22×C2312yes—
136#483—22no—
137#470—22no—
138#451—22no—
139#451—24no—
140#451—24no—
141C2×#103—22no—
142C2×#101—22no—
143#407C22×C2311yescharacteristic
144#407C22×C2311yescharacteristic
145C2×C22⋊C8C22×C2312yes—
146#324—24no—
147C2×C23—24no—
148C2×C23—41no—
149C2×C23#202012yes—
150C2×C23#202111yescharacteristic
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—44no—
155C22×Q8D4×C2312yes—
156C22×Q8D4×C2312yes—
157C23×C4—21no—
158C23×C4C23×C2311yescharacteristic
159C2×Q16—28no—
160C2×Q16—22no—
161C2×Q16—48no—
162C2×M16—21no—
163C2×M16D4×C2312yes—
164C22×C8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4D4×C2312yes—
171C2×C22⋊C4—22no—
172#106—22no—
173#103—28no—
174#103—24no—
175#101—44no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8D4×C2314yes—
180C23#999111yescharacteristic, direct product factor, Sylow
181C24#41×C2311yescharacteristic
182C2×Q8—24no—
183C2×Q8—28no—
184C2×Q8—24no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—48no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4C2×D4×C2312yes—
201C22×C4C2×D4×C2311yescharacteristic
202C22×C4C2×D4×C2311yescharacteristic
203C22×C4C2×D4×C2312yes—
204Q16—416no—
205Q16—82no—
206M16—42no—
207M16—42no—
208C2×C8—22no—
209C2×C8—24no—
210C2×C8—44no—
211C2×C8—41no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C23—24no—
219C23—41no—
220C23C23×#12111yescharacteristic
221C23C23×#12112yes—
222Q8—28no—
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—44no—
227Q8—44no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4C23×#12012yes—
241C2×C4C23×#12011yescharacteristic
242C2×C4C23×#12711yescharacteristic
243C8—44no—
244C8—42no—
245C8—81no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—44no—
251C22C23×#39211yescharacteristic
252C22C23×#47314yes—
253C4—22no—
254C4—21no—
255C4—44no—
256C4—82no—
257C4—82no—
258C4C23×B(4,𝔽2)12yes—
259C2—24no—
260C2—41no—
261C2C23×#202012yes—
262C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
86. #9991C23direct (N × H)
180. C23#9991direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31952
C2×C23×#4512
C23×#17842
C23×#14722
C23×#12982
#999123