Finite Group Databasebeta

← C23×#9980 #19157606

Subgroups — C23×#9980 #19157606

Order 5888.

822 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3006C212yesmaximal, complemented
2C23×#2815C211yesmaximal, characteristic, complemented
3C23×#1782C211yesmaximal, characteristic, complemented
4C23×#1510C212yesmaximal, complemented
5C23×#1306C211yesmaximal, characteristic, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×AGL(1,ℤ/8)—22no—
8C2×C23×#121C2212yes—
9C23×#449—24no—
10C23×#448—22no—
11C23×#447—22no—
12C23×#417—22no—
13C2×C23×#99—22no—
14C23×#407C2212yes—
15C23×#397—22no—
16C23×#392—22no—
17C23×#392—22no—
18C23×#381C2211yescharacteristic
19C23×#381C2211yescharacteristic
20C23×#327—24no—
21C2×#41×C23—22no—
22C2×#41×C23—22no—
23C2×#41×C23—44no—
24C22×Q8×C23D411yescharacteristic, complemented
25C22×D4×C23—21no—
26C22×D4×C23—22no—
27C22×D4×C23—22no—
28C22×D4×C23D412yes—
29C22×D4×C23D411yescharacteristic, complemented
30C23×C4×C23—21no—
31C23×C4×C23C2311yescharacteristic
32C23×AGL(1,ℤ/8)—48no—
33C2×SD16×C23—24no—
34C2×D8×C23—24no—
35C2×M16×C23—22no—
36C23×#121—22no—
37C23×#121—22no—
38C23×#121—22no—
39C23×#121—22no—
40C2×C4⋊C4×C23D412yes—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—24no—
44C2×C22⋊C4×C23D412yes—
45C23×#103—24no—
46C23×#102—24no—
47C23×#99—44no—
48C23×C22⋊C8—22no—
49C23×C22⋊C8D414yescomplemented
50C23×#95—22no—
51C23×#95—22no—
52C23×#95—22no—
53C24×C23—21no—
54C24×C23—22no—
55C24×C23C2×D411yescharacteristic
56#41×C23—28no—
57#41×C23—44no—
58#41×C23—44no—
59#41×C23—44no—
60C2×Q8×C23—22no—
61C2×Q8×C23—42no—
62C2×D4×C23—24no—
63C2×D4×C23—22no—
64C2×D4×C23—22no—
65C2×D4×C23—22no—
66C2×D4×C23—44no—
67C2×D4×C23—42no—
68C2×D4×C23—42no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—42no—
72C2×D4×C23—42no—
73C2×D4×C23—84no—
74C2×D4×C23—84no—
75C2×D4×C23—82no—
76C22×C4×C23—21no—
77C22×C4×C23—22no—
78C22×C4×C23—22no—
79C22×C4×C23—42no—
80C22×C4×C23—42no—
81C22×C4×C23—42no—
82C22×C4×C23—42no—
83C22×C4×C23—41no—
84C22×C4×C23C2×D412yes—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D412yes—
87C22×C4×C23#4111yescharacteristic
88SD16×C23—48no—
89D8×C23—48no—
90M16×C23—44no—
91C2×C8×C23—24no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—22no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—42no—
98C22⋊C4×C23—84no—
99#9980C2311yesmaximal, characteristic, direct product factor, Sylow
100C23×C23—22no—
101C23×C23—22no—
102C23×C23—42no—
103C23×C23—41no—
104C23×C23—42no—
105C23×C23—42no—
106C23×C23—44no—
107C23×C23—41no—
108C23×C23—42no—
109C23×C23—44no—
110C23×C23—82no—
111C23×C23#12011yescharacteristic
112C23×C23#12111yescharacteristic
113C23×C23#12711yescharacteristic
114Q8×C23—24no—
115Q8×C23—42no—
116D4×C23—24no—
117D4×C23—42no—
118D4×C23—44no—
119D4×C23—44no—
120D4×C23—44no—
121D4×C23—82no—
122D4×C23—84no—
123D4×C23—84no—
124D4×C23—88no—
125C2×C4×C23—24no—
126C2×C4×C23—22no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—42no—
130C2×C4×C23—48no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—44no—
134C2×C4×C23—44no—
135C2×C4×C23—42no—
136C2×C4×C23—42no—
137C2×C4×C23—81no—
138C2×C4×C23—82no—
139C2×C4×C23#12012yes—
140C2×C4×C23#12112yes—
141C8×C23—44no—
142#3006C2×C2312yescomplemented
143#2815C2×C2311yescharacteristic, complemented
144#1782C2×C2311yescharacteristic, complemented
145#1510C2×C2312yescomplemented
146#1306C2×C2311yescharacteristic, complemented
147C22×C23—22no—
148C22×C23—22no—
149C22×C23—22no—
150C22×C23—21no—
151C22×C23—21no—
152C22×C23—42no—
153C22×C23—82no—
154C22×C23—84no—
155C22×C23—84no—
156C22×C23—82no—
157C22×C23—82no—
158C22×C23—82no—
159C22×C23—82no—
160C22×C23—82no—
161C22×C23—82no—
162C22×C23#39211yescharacteristic
163C22×C23B(4,𝔽2)12yes—
164C4×C23—22no—
165C4×C23—22no—
166C4×C23—42no—
167C4×C23—82no—
168C4×C23—82no—
169C4×C23—161no—
170C22×#41C22×C2311yescharacteristic
171C2×AGL(1,ℤ/8)—22no—
172C2×#121C22×C2312yes—
173#449—24no—
174#448—22no—
175#447—22no—
176#417—22no—
177C2×#99—22no—
178#407C22×C2312yes—
179#397—22no—
180#392—22no—
181#392—22no—
182#381C22×C2311yescharacteristic
183#381C22×C2311yescharacteristic
184#327—24no—
185C2×C23—22no—
186C2×C23—41no—
187C2×C23—41no—
188C2×C23—82no—
189C2×C23—162no—
190C2×C23#200812yes—
191C2×C23#202111yescharacteristic
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—44no—
195C22×Q8D4×C2311yescharacteristic, complemented
196C22×D4—21no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4D4×C2312yes—
200C22×D4D4×C2311yescharacteristic, complemented
201C23×C4—21no—
202C23×C4C23×C2311yescharacteristic
203AGL(1,ℤ/8)—48no—
204C2×SD16—24no—
205C2×D8—24no—
206C2×M16—22no—
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211C2×C4⋊C4D4×C2312yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4D4×C2312yes—
216#103—24no—
217#102—24no—
218#99—44no—
219C22⋊C8—22no—
220C22⋊C8D4×C2314yescomplemented
221#95—22no—
222#95—22no—
223#95—22no—
224C23#998011yescharacteristic, direct product factor, Sylow
225C24—21no—
226C24—22no—
227C24C2×D4×C2311yescharacteristic
228#41—28no—
229#41—44no—
230#41—44no—
231#41—44no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—82no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4C2×D4×C2312yes—
257C22×C4C2×D4×C2311yescharacteristic
258C22×C4C2×D4×C2312yes—
259C22×C4#41×C2311yescharacteristic
260SD16—48no—
261D8—48no—
262M16—44no—
263C2×C8—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—84no—
271C23—22no—
272C23—22no—
273C23—42no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—41no—
279C23—42no—
280C23—44no—
281C23—82no—
282C23C23×#12011yescharacteristic
283C23C23×#12111yescharacteristic
284C23C23×#12711yescharacteristic
285Q8—24no—
286Q8—42no—
287D4—24no—
288D4—42no—
289D4—44no—
290D4—44no—
291D4—44no—
292D4—82no—
293D4—84no—
294D4—84no—
295D4—88no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4C23×#12012yes—
311C2×C4C23×#12112yes—
312C8—44no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—82no—
320C22—84no—
321C22—84no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22C23×#39211yescharacteristic
329C22C23×B(4,𝔽2)12yes—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—82no—
334C4—82no—
335C4—161no—
336C2—22no—
337C2—41no—
338C2—41no—
339C2—82no—
340C2—162no—
341C2C23×#200812yes—
342C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
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×#3006C2semidirect
2. C23×#2815C2semidirect
3. C23×#1782C2semidirect
4. C23×#1510C2semidirect
5. C23×#1306C2semidirect
24. C22×Q8×C23D4semidirect
29. C22×D4×C23D4semidirect
49. C23×C22⋊C8D4semidirect
99. #9980C23direct (N × H)
142. #3006C2×C23semidirect
143. #2815C2×C23semidirect
144. #1782C2×C23semidirect
145. #1510C2×C23semidirect
146. #1306C2×C23semidirect
195. C22×Q8D4×C23semidirect
200. C22×D4D4×C23semidirect
220. C22⋊C8D4×C23semidirect
224. C23#9980direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30062
C23×#28152
C23×#17822
C23×#15102
C23×#13062
#998023