Finite Group Databasebeta

← C23×#12122 #19159748

Subgroups — C23×#12122 #19159748

Order 5888.

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

normal subgroups only

This group has 458 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×#2607C211yesmaximal, characteristic, complemented
2C23×#2022C211yesmaximal, characteristic
3C23×#1613C212yesmaximal
4C23×#1352C212yesmaximal, complemented
5C23×#1317C211yesmaximal, characteristic, complemented
6C23×#534C2211yescharacteristic
7C2×C23×#123—21no—
8C23×#481—22no—
9C2×C23×#99—21no—
10C23×#399C2212yes—
11C23×#393—21no—
12C23×#391—22no—
13C23×#387—22no—
14C23×#381C2211yescharacteristic
15C4×C22⋊C4×C23—21no—
16C23×#342C2211yescharacteristic
17C23×#327—22no—
18C23×#323C2212yes—
19C22×Q8×C23—21no—
20C22×D4×C23—21no—
21C23×C4×C23D411yescharacteristic
22C23×#123—42no—
23C23×#123—41no—
24C23×#123—42no—
25C23×#121—22no—
26C23×#120—41no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—41no—
29C2×C4⋊C4×C23D412yes—
30C2×C22⋊C4×C23—21no—
31C2×C22⋊C4×C23—22no—
32C2×C22⋊C4×C23—41no—
33C2×C22⋊C4×C23D411yescharacteristic
34C2×C22⋊C4×C23C2311yescharacteristic
35C2×C42×C23—21no—
36C2×C42×C23—21no—
37C23×#106—42no—
38C23×#102—42no—
39C23×#99—42no—
40C23×C22⋊C8—22no—
41C23×#95—22no—
42C23×#95—21no—
43C23×#95—22no—
44C23×#95—22no—
45C23×#95D412yes—
46C24×C23—21no—
47C24×C23C2×D411yescharacteristic
48C2×Q8×C23—44no—
49C2×Q8×C23—81no—
50C2×D4×C23—22no—
51C2×D4×C23—41no—
52C2×D4×C23—41no—
53C2×D4×C23—41no—
54C2×D4×C23—41no—
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—21no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23SD1612yes—
67C22×C4×C23C2×D412yes—
68C2×C8×C23—42no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—42no—
71C4⋊C4×C23—44no—
72C4⋊C4×C23—44no—
73C22⋊C4×C23—22no—
74C22⋊C4×C23—41no—
75C22⋊C4×C23—44no—
76C22⋊C4×C23—42no—
77C22⋊C4×C23—44no—
78C22⋊C4×C23—41no—
79C22⋊C4×C23—41no—
80C42×C23—81no—
81C42×C23—81no—
82#12122C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—21no—
84C23×C23—21no—
85C23×C23—42no—
86C23×C23—41no—
87C23×C23—41no—
88C23×C23—41no—
89C23×C23—82no—
90C23×C23#12011yescharacteristic
91C23×C23C2×SD1611yescharacteristic
92C23×C23#13711yescharacteristic
93Q8×C23—84no—
94D4×C23—82no—
95D4×C23—81no—
96D4×C23—81no—
97D4×C23—81no—
98D4×C23—81no—
99C2×C4×C23—24no—
100C2×C4×C23—22no—
101C2×C4×C23—22no—
102C2×C4×C23—42no—
103C2×C4×C23—41no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—41no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—42no—
114C2×C4×C23—41no—
115C2×C4×C23—42no—
116C2×C4×C23—41no—
117C2×C4×C23—81no—
118C2×C4×C23—82no—
119C8×C23—82no—
120#2607C2×C2311yescharacteristic, complemented
121#2022C2×C2311yescharacteristic
122#1613C2×C2312yes—
123#1352C2×C2312yescomplemented
124#1317C2×C2311yescharacteristic, complemented
125C22×C23—22no—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—41no—
129C22×C23—41no—
130C22×C23—42no—
131C22×C23—81no—
132C22×C23—82no—
133C22×C23—81no—
134C22×C23—82no—
135C22×C23—161no—
136C22×C23#44811yescharacteristic
137C22×C23#45311yescharacteristic
138C22×C23B(4,𝔽2)11yescharacteristic
139C4×C23—44no—
140C4×C23—42no—
141C4×C23—42no—
142C4×C23—81no—
143C4×C23—82no—
144C4×C23—81no—
145C4×C23—82no—
146#534C22×C2311yescharacteristic
147C2×#123—21no—
148#481—22no—
149C2×#99—21no—
150#399C22×C2312yes—
151#393—21no—
152#391—22no—
153#387—22no—
154#381C22×C2311yescharacteristic
155C4×C22⋊C4—21no—
156#342C22×C2311yescharacteristic
157#327—22no—
158#323C22×C2312yes—
159C2×C23—21no—
160C2×C23—21no—
161C2×C23—41no—
162C2×C23—41no—
163C2×C23—161no—
164C2×C23#159611yescharacteristic
165C2×C23#219812yes—
166C22×Q8—21no—
167C22×D4—21no—
168C23×C4D4×C2311yescharacteristic
169#123—42no—
170#123—41no—
171#123—42no—
172#121—22no—
173#120—41no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4D4×C2312yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4D4×C2311yescharacteristic
181C2×C22⋊C4C23×C2311yescharacteristic
182C2×C42—21no—
183C2×C42—21no—
184#106—42no—
185#102—42no—
186#99—42no—
187C22⋊C8—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95D4×C2312yes—
193C23#1212211yescharacteristic, direct product factor, Sylow
194C24—21no—
195C24C2×D4×C2311yescharacteristic
196C2×Q8—44no—
197C2×Q8—81no—
198C2×D4—22no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4SD16×C2312yes—
215C22×C4C2×D4×C2312yes—
216C2×C8—42no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—22no—
222C22⋊C4—41no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—81no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23C23×#12011yescharacteristic
238C23C2×SD16×C2311yescharacteristic
239C23C23×#13711yescharacteristic
240Q8—84no—
241D4—82no—
242D4—81no—
243D4—81no—
244D4—81no—
245D4—81no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—82no—
266C8—82no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—161no—
278C22C23×#44811yescharacteristic
279C22C23×#45311yescharacteristic
280C22C23×B(4,𝔽2)11yescharacteristic
281C4—44no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—82no—
286C4—82no—
287C4—81no—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2—161no—
293C2C23×#159611yescharacteristic
294C2C23×#219812yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2607C2semidirect
4. C23×#1352C2semidirect
5. C23×#1317C2semidirect
82. #12122C23direct (N × H)
120. #2607C2×C23semidirect
123. #1352C2×C23semidirect
124. #1317C2×C23semidirect
193. C23#12122direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26072
C23×#20222
C23×#16132
C23×#13522
C23×#13172
#1212223