Finite Group Databasebeta

← C23×#12121 #19159747

Subgroups — C23×#12121 #19159747

Order 5888.

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

normal subgroups only

This group has 470 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×#1601C212yesmaximal
4C23×#1351C212yesmaximal, complemented
5C23×#1317C211yesmaximal, characteristic, complemented
6C23×#534C2211yescharacteristic
7C2×C23×#123—21no—
8C23×#448—22no—
9C2×C23×#99—21no—
10C23×#399—21no—
11C23×#393C2212yes—
12C23×#391—22no—
13C23×#387—22no—
14C23×#381C2211yescharacteristic
15C4×C22⋊C4×C23—21no—
16C23×#342C2211yescharacteristic
17C23×#327—22no—
18C23×#323C2212yes—
19C22×Q8×C23D412yes—
20C22×D4×C23—21no—
21C23×C4×C23D411yescharacteristic
22C2×SD16×C23—42no—
23C23×#123—42no—
24C23×#123—41no—
25C23×#123—42no—
26C23×#121—22no—
27C23×#120—41no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—41no—
31C2×C22⋊C4×C23—21no—
32C2×C22⋊C4×C23—22no—
33C2×C22⋊C4×C23—41no—
34C2×C22⋊C4×C23D411yescharacteristic
35C2×C22⋊C4×C23C2311yescharacteristic
36C2×C42×C23—21no—
37C2×C42×C23—21no—
38C23×#103—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—42no—
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×C23SD1612yescomplemented
67C22×C4×C23C2×D412yes—
68SD16×C23—84no—
69C2×C8×C23—42no—
70C4⋊C4×C23—22no—
71C4⋊C4×C23—44no—
72C4⋊C4×C23—44no—
73C4⋊C4×C23—81no—
74C22⋊C4×C23—22no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—44no—
77C22⋊C4×C23—42no—
78C22⋊C4×C23—44no—
79C22⋊C4×C23—41no—
80C22⋊C4×C23—41no—
81C42×C23—81no—
82C42×C23—81no—
83#12121C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—21no—
85C23×C23—21no—
86C23×C23—42no—
87C23×C23—41no—
88C23×C23—41no—
89C23×C23—41no—
90C23×C23—82no—
91C23×C23#12011yescharacteristic
92C23×C23C2×SD1611yescharacteristic
93C23×C23#13711yescharacteristic
94Q8×C23—44no—
95Q8×C23—82no—
96D4×C23—82no—
97D4×C23—81no—
98D4×C23—81no—
99D4×C23—81no—
100D4×C23—81no—
101C2×C4×C23—24no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—41no—
116C2×C4×C23—42no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—81no—
120C2×C4×C23—82no—
121C8×C23—82no—
122#2607C2×C2311yescharacteristic, complemented
123#2022C2×C2311yescharacteristic
124#1601C2×C2312yes—
125#1351C2×C2312yescomplemented
126#1317C2×C2311yescharacteristic, complemented
127C22×C23—22no—
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—41no—
131C22×C23—41no—
132C22×C23—42no—
133C22×C23—81no—
134C22×C23—82no—
135C22×C23—81no—
136C22×C23—82no—
137C22×C23—161no—
138C22×C23#44811yescharacteristic
139C22×C23#45311yescharacteristic
140C22×C23B(4,𝔽2)11yescharacteristic
141C4×C23—42no—
142C4×C23—42no—
143C4×C23—44no—
144C4×C23—82no—
145C4×C23—81no—
146C4×C23—82no—
147C4×C23—81no—
148#534C22×C2311yescharacteristic
149C2×#123—21no—
150#448—22no—
151C2×#99—21no—
152#399—21no—
153#393C22×C2312yes—
154#391—22no—
155#387—22no—
156#381C22×C2311yescharacteristic
157C4×C22⋊C4—21no—
158#342C22×C2311yescharacteristic
159#327—22no—
160#323C22×C2312yes—
161C2×C23—21no—
162C2×C23—21no—
163C2×C23—41no—
164C2×C23—41no—
165C2×C23—161no—
166C2×C23#159611yescharacteristic
167C2×C23#219712yes—
168C22×Q8D4×C2312yes—
169C22×D4—21no—
170C23×C4D4×C2311yescharacteristic
171C2×SD16—42no—
172#123—42no—
173#123—41no—
174#123—42no—
175#121—22no—
176#120—41no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4D4×C2311yescharacteristic
184C2×C22⋊C4C23×C2311yescharacteristic
185C2×C42—21no—
186C2×C42—21no—
187#103—42no—
188#99—42no—
189C22⋊C8—22no—
190#95—22no—
191#95—21no—
192#95—22no—
193#95—22no—
194#95D4×C2312yes—
195C23#1212111yescharacteristic, direct product factor, Sylow
196C24—21no—
197C24C2×D4×C2311yescharacteristic
198C2×Q8—44no—
199C2×Q8—42no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4SD16×C2312yescomplemented
217C22×C4C2×D4×C2312yes—
218SD16—84no—
219C2×C8—42no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—81no—
224C22⋊C4—22no—
225C22⋊C4—41no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42—81no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—82no—
240C23C23×#12011yescharacteristic
241C23C2×SD16×C2311yescharacteristic
242C23C23×#13711yescharacteristic
243Q8—44no—
244Q8—82no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—81no—
249D4—81no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—82no—
270C8—82no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—82no—
280C22—81no—
281C22—161no—
282C22C23×#44811yescharacteristic
283C22C23×#45311yescharacteristic
284C22C23×B(4,𝔽2)11yescharacteristic
285C4—42no—
286C4—44no—
287C4—42no—
288C4—82no—
289C4—81no—
290C4—82no—
291C4—81no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—161no—
297C2C23×#159611yescharacteristic
298C2C23×#219712yes—
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×#1351C2semidirect
5. C23×#1317C2semidirect
66. C22×C4×C23SD16semidirect
83. #12121C23direct (N × H)
122. #2607C2×C23semidirect
125. #1351C2×C23semidirect
126. #1317C2×C23semidirect
195. C23#12121direct (N × H)
216. C22×C4SD16×C23semidirect

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×#16012
C23×#13512
C23×#13172
#1212123