Finite Group Databasebeta

← C23×#10388 #19158014

Subgroups — C23×#10388 #19158014

Order 5888.

594 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2608C211yesmaximal, characteristic, complemented
2C23×#1789C212yesmaximal
3C23×#1782C211yesmaximal, characteristic
4C23×#1306C211yesmaximal, characteristic, complemented
5C23×#1284C212yesmaximal, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×#104—22no—
8C2×C23×#104—22no—
9C2×C23×#99—22no—
10C23×#407C2212yes—
11C23×#396—22no—
12C23×#389—22no—
13C23×#387—22no—
14C23×#381C2211yescharacteristic
15C23×#381C2211yescharacteristic
16C4×C22⋊C4×C23C2212yes—
17C2×#41×C23—22no—
18C2×#41×C23—22no—
19C2×#41×C23—44no—
20C22×Q8×C23Q811yescharacteristic
21C22×D4×C23—21no—
22C22×D4×C23Q811yescharacteristic
23C23×C4×C23—21no—
24C23×C4×C23C2311yescharacteristic
25C2×M16×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—22no—
29C2×C22⋊C4×C23—22no—
30C2×C22⋊C4×C23—21no—
31C2×C22⋊C4×C23D412yes—
32C2×C42×C23D412yes—
33C2×C42×C23D412yes—
34C23×#104—44no—
35C23×#104—44no—
36C23×#99—44no—
37C23×C22⋊C8—22no—
38C23×C22⋊C8—22no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95—22no—
43C24×C23—21no—
44C24×C23C2×D411yescharacteristic
45#41×C23—24no—
46#41×C23—24no—
47#41×C23—42no—
48#41×C23—42no—
49#41×C23—44no—
50#41×C23—44no—
51C2×Q8×C23—22no—
52C2×Q8×C23—42no—
53C2×D4×C23—24no—
54C2×D4×C23—22no—
55C2×D4×C23—42no—
56C2×D4×C23—42no—
57C2×D4×C23—41no—
58C2×D4×C23—41no—
59C22×C4×C23—22no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—22no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23—42no—
67C22×C4×C23—41no—
68C22×C4×C23C2×D412yes—
69C22×C4×C23C2×Q811yescharacteristic
70C22×C4×C23#4112yes—
71C22×C4×C23#4111yescharacteristic
72M16×C23—42no—
73M16×C23—42no—
74C2×C8×C23—42no—
75C4⋊C4×C23—44no—
76C4⋊C4×C23—44no—
77C4⋊C4×C23—82no—
78C22⋊C4×C23—44no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—42no—
81C42×C23—42no—
82C42×C23—42no—
83#10388C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—22no—
85C23×C23—22no—
86C23×C23—42no—
87C23×C23—41no—
88C23×C23—41no—
89C23×C23—82no—
90C23×C23#12011yescharacteristic
91C23×C23C22⋊Q811yescharacteristic
92C23×C23#12411yescharacteristic
93Q8×C23—24no—
94Q8×C23—42no—
95D4×C23—24no—
96D4×C23—42no—
97D4×C23—44no—
98D4×C23—44no—
99C2×C4×C23—22no—
100C2×C4×C23—22no—
101C2×C4×C23—24no—
102C2×C4×C23—22no—
103C2×C4×C23—41no—
104C2×C4×C23—41no—
105C2×C4×C23—42no—
106C2×C4×C23—44no—
107C2×C4×C23—44no—
108C2×C4×C23—42no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—81no—
119C2×C4×C23—82no—
120C2×C4×C23C22⋊Q812yes—
121C2×C4×C23#12412yes—
122C8×C23—82no—
123#2608C2×C2311yescharacteristic, complemented
124#1789C2×C2312yes—
125#1782C2×C2311yescharacteristic
126#1306C2×C2311yescharacteristic, complemented
127#1284C2×C2312yescomplemented
128C22×C23—22no—
129C22×C23—21no—
130C22×C23—21no—
131C22×C23—22no—
132C22×C23—22no—
133C22×C23—42no—
134C22×C23—82no—
135C22×C23—82no—
136C22×C23—82no—
137C22×C23—82no—
138C22×C23#39311yescharacteristic
139C22×C23B(4,𝔽2)12yes—
140C4×C23—22no—
141C4×C23—22no—
142C4×C23—44no—
143C4×C23—42no—
144C4×C23—82no—
145C4×C23—82no—
146C4×C23—161no—
147C22×#41C22×C2311yescharacteristic
148C2×#104—22no—
149C2×#104—22no—
150C2×#99—22no—
151#407C22×C2312yes—
152#396—22no—
153#389—22no—
154#387—22no—
155#381C22×C2311yescharacteristic
156#381C22×C2311yescharacteristic
157C4×C22⋊C4C22×C2312yes—
158C2×C23—22no—
159C2×C23—41no—
160C2×C23—41no—
161C2×C23—82no—
162C2×C23#202911yescharacteristic
163C2×C23#203112yes—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—44no—
167C22×Q8Q8×C2311yescharacteristic
168C22×D4—21no—
169C22×D4Q8×C2311yescharacteristic
170C23×C4—21no—
171C23×C4C23×C2311yescharacteristic
172C2×M16—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4D4×C2312yes—
179C2×C42D4×C2312yes—
180C2×C42D4×C2312yes—
181#104—44no—
182#104—44no—
183#99—44no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190C23#1038811yescharacteristic, direct product factor, Sylow
191C24—21no—
192C24C2×D4×C2311yescharacteristic
193#41—24no—
194#41—24no—
195#41—42no—
196#41—42no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4C2×D4×C2312yes—
217C22×C4C2×Q8×C2311yescharacteristic
218C22×C4#41×C2312yes—
219C22×C4#41×C2311yescharacteristic
220M16—42no—
221M16—42no—
222C2×C8—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—82no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42—42no—
230C42—42no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23C23×#12011yescharacteristic
238C23C23×C22⋊Q811yescharacteristic
239C23C23×#12411yescharacteristic
240Q8—24no—
241Q8—42no—
242D4—24no—
243D4—42no—
244D4—44no—
245D4—44no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4C23×C22⋊Q812yes—
268C2×C4C23×#12412yes—
269C8—82no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22C23×#39311yescharacteristic
281C22C23×B(4,𝔽2)12yes—
282C4—22no—
283C4—22no—
284C4—44no—
285C4—42no—
286C4—82no—
287C4—82no—
288C4—161no—
289C2—22no—
290C2—41no—
291C2—41no—
292C2—82no—
293C2C23×#202911yescharacteristic
294C2C23×#203112yes—
subgroupG/H
center, socleC22×C23#393
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC22×#41×C23C22
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
1. C23×#2608C2semidirect
4. C23×#1306C2semidirect
5. C23×#1284C2semidirect
83. #10388C23direct (N × H)
123. #2608C2×C23semidirect
126. #1306C2×C23semidirect
127. #1284C2×C23semidirect
190. C23#10388direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26082
C23×#17892
C23×#17822
C23×#13062
C23×#12842
#1038823