Finite Group Databasebeta

← C23×#12973 #19160599

Subgroups — C23×#12973 #19160599

Order 5888.

460 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2369C211yesmaximal, characteristic
2C2×C23×#356C211yesmaximal, characteristic
3C23×#1886C211yesmaximal, characteristic
4C23×#1857C212yesmaximal
5C23×#1394C212yesmaximal
6C2×C23×C4⋊Q8C2211yescharacteristic
7C23×#519C2211yescharacteristic
8C23×#428—22no—
9C23×#419—21no—
10C2×C23×#101C2211yescharacteristic
11C2×C23×#101C2212yes—
12C23×#391—21no—
13C23×#391C2212yes—
14C23×#389—22no—
15C4×C4⋊C4×C23—21no—
16C4×C4⋊C4×C23—22no—
17C23×#356C414yescomplemented
18C23×#339—21no—
19C23×#339—22no—
20C22×Q8×C23D411yescharacteristic
21C22×Q8×C23C2311yescharacteristic
22C2×M16×C23—22no—
23C2×M16×C23—21no—
24C23×C4⋊Q8—22no—
25C23×C4⋊Q8C2×C412yes—
26C23×C4⋊Q8Q812yes—
27C4×Q8×C23—41no—
28C4×Q8×C23—41no—
29C23×#117—21no—
30C23×#117—22no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23D412yes—
36C2×C42×C23—22no—
37C2×C42×C23—21no—
38C2×C42×C23D411yescharacteristic
39C2×C42×C23D412yes—
40C23×#107—42no—
41C23×#106—42no—
42C23×#103—41no—
43C23×#103—41no—
44C23×#101—22no—
45C23×#101—22no—
46C23×#101C2×C414yes—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—22no—
50C2×Q8×C23—21no—
51C2×Q8×C23—41no—
52C2×Q8×C23—41no—
53C2×Q8×C23—41no—
54C2×Q8×C23—41no—
55C2×Q8×C23C4⋊C412yes—
56C2×Q8×C23C22×C411yescharacteristic
57C2×Q8×C23C2×Q811yescharacteristic
58C2×Q8×C23#4112yes—
59C22×C4×C23—21no—
60C22×C4×C23—22no—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D412yes—
65M16×C23—24no—
66M16×C23—42no—
67M16×C23—41no—
68M16×C23—41no—
69C2×C8×C23—42no—
70C2×C8×C23—41no—
71C4⋊C4×C23—24no—
72C4⋊C4×C23—24no—
73C4⋊C4×C23—24no—
74C4⋊C4×C23—21no—
75C4⋊C4×C23—21no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—42no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—81no—
81C22⋊C4×C23—41no—
82C42×C23—21no—
83C42×C23—42no—
84C42×C23—44no—
85C42×C23—42no—
86C42×C23—41no—
87C42×C23—81no—
88C42×C23C4⋊C412yes—
89#12973C2311yesmaximal, characteristic, direct product factor, Sylow
90C23×C23#12011yescharacteristic
91Q8×C23—41no—
92Q8×C23—41no—
93Q8×C23—42no—
94Q8×C23—42no—
95Q8×C23—81no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—22no—
100C2×C4×C23—21no—
101C2×C4×C23—22no—
102C2×C4×C23—21no—
103C2×C4×C23—24no—
104C2×C4×C23—22no—
105C2×C4×C23—42no—
106C2×C4×C23—41no—
107C2×C4×C23—42no—
108C2×C4×C23—41no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—41no—
112C2×C4×C23—41no—
113C2×C4×C23—42no—
114C2×C4×C23C2×C4⋊C411yescharacteristic
115C2×C4×C23C4×D412yes—
116C2×C4×C23C22⋊Q812yes—
117C2×C4×C23#12311yescharacteristic
118C8×C23—44no—
119C8×C23—81no—
120#2369C2×C2311yescharacteristic
121C2×#356C2×C2311yescharacteristic
122#1886C2×C2311yescharacteristic
123#1857C2×C2312yes—
124#1394C2×C2312yes—
125C22×C23—21no—
126C22×C23—21no—
127C22×C23#38511yescharacteristic
128C22×C23B(4,𝔽2)11yescharacteristic
129C22×C23#45811yescharacteristic
130C4×C23—24no—
131C4×C23—22no—
132C4×C23—21no—
133C4×C23—21no—
134C4×C23—22no—
135C4×C23—44no—
136C4×C23—42no—
137C4×C23—81no—
138C4×C23—82no—
139C4×C23—81no—
140C2×C4⋊Q8C22×C2311yescharacteristic
141#519C22×C2311yescharacteristic
142#428—22no—
143#419—21no—
144C2×#101C22×C2311yescharacteristic
145C2×#101C22×C2312yes—
146#391—21no—
147#391C22×C2312yes—
148#389—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—21no—
151#356C4×C2314yescomplemented
152#339—22no—
153#339—21no—
154C2×C23—21no—
155C2×C23—21no—
156C2×C23#186711yescharacteristic
157C2×C23#220411yescharacteristic
158C2×C23#220511yescharacteristic
159C22×Q8D4×C2311yescharacteristic
160C22×Q8C23×C2311yescharacteristic
161C2×M16—22no—
162C2×M16—21no—
163C4⋊Q8—22no—
164C4⋊Q8C2×C4×C2312yes—
165C4⋊Q8Q8×C2312yes—
166C4×Q8—41no—
167C4×Q8—41no—
168#117—21no—
169#117—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4D4×C2312yes—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42D4×C2311yescharacteristic
178C2×C42D4×C2312yes—
179#107—42no—
180#106—42no—
181#103—41no—
182#103—41no—
183#101—22no—
184#101—22no—
185#101C2×C4×C2314yes—
186#95—22no—
187#95—22no—
188#95—22no—
189C23#1297311yescharacteristic, direct product factor, Sylow
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊C4×C2312yes—
196C2×Q8C22×C4×C2311yescharacteristic
197C2×Q8C2×Q8×C2311yescharacteristic
198C2×Q8#41×C2312yes—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4C2×D4×C2311yescharacteristic
204C22×C4C2×D4×C2312yes—
205M16—24no—
206M16—42no—
207M16—41no—
208M16—41no—
209C2×C8—42no—
210C2×C8—41no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42—44no—
225C42—42no—
226C42—41no—
227C42—81no—
228C42C4⋊C4×C2312yes—
229C23C23×#12011yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4C2×C4⋊C4×C2311yescharacteristic
254C2×C4C4×D4×C2312yes—
255C2×C4C23×C22⋊Q812yes—
256C2×C4C23×#12311yescharacteristic
257C8—44no—
258C8—81no—
259C22—21no—
260C22—21no—
261C22C23×#38511yescharacteristic
262C22C23×B(4,𝔽2)11yescharacteristic
263C22C23×#45811yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C2—21no—
275C2—21no—
276C2C23×#186711yescharacteristic
277C2C23×#220411yescharacteristic
278C2C23×#220511yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×Q8C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC23×C23#120
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
17. C23×#356C4semidirect
89. #12973C23direct (N × H)
151. #356C4×C23semidirect
189. C23#12973direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23692
C2×C23×#3562
C23×#18862
C23×#18572
C23×#13942
#1297323