Finite Group Databasebeta

← C23×#9501 #19157127

Subgroups — C23×#9501 #19157127

Order 5888.

928 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2405C211yesmaximal, characteristic
2C23×#2309C211yesmaximal, characteristic
3C23×#1819C212yesmaximal, complemented
4C23×#1781C211yesmaximal, characteristic
5C23×#1304C212yesmaximal, complemented
6C23×D4×C23C2211yescharacteristic
7C2×C23×#123C412yescomplemented
8C2×C23×#117—21no—
9C22×C22⋊C4×C23C2211yescharacteristic
10C2×C23×#99—22no—
11C23×#394—22no—
12C23×#390—24no—
13C23×#386C412yescomplemented
14C23×#385—21no—
15C23×#379—21no—
16C23×#379—21no—
17C23×#379C2212yes—
18C23×#379C2211yescharacteristic
19C23×#379C2212yes—
20C23×#342—24no—
21C25×C23D411yescharacteristic
22C25×C23C2311yescharacteristic
23C22×D4×C23—22no—
24C22×D4×C23—22no—
25C22×D4×C23—44no—
26C22×D4×C23C2×C412yes—
27C22×D4×C23D412yes—
28C23×C4×C23C2×C411yescharacteristic
29C23×C4×C23C2×C411yescharacteristic
30C23×C4×C23D411yescharacteristic
31C23×#123—44no—
32C23×#117—44no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23—22no—
44C2×C22⋊C4×C23—22no—
45C2×C22⋊C4×C23—22no—
46C2×C22⋊C4×C23—22no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—22no—
50C2×C22⋊C4×C23C2×C412yes—
51C2×C22⋊C4×C23D414yes—
52C2×C42×C23—22no—
53C23×#99—44no—
54C23×#95—22no—
55C24×C23—21no—
56C24×C23—21no—
57C24×C23—42no—
58C24×C23—42no—
59C24×C23—42no—
60C24×C23—42no—
61C24×C23—81no—
62C24×C23—81no—
63C24×C23C22⋊C412yes—
64C24×C23C22×C411yescharacteristic
65C24×C23C2×D411yescharacteristic
66C24×C23C2×D411yescharacteristic
67C24×C23C2×D412yes—
68C24×C23#4112yes—
69C2×D4×C23—24no—
70C2×D4×C23—24no—
71C2×D4×C23—24no—
72C2×D4×C23—24no—
73C2×D4×C23—42no—
74C2×D4×C23—42no—
75C2×D4×C23—44no—
76C2×D4×C23—44no—
77C2×D4×C23—48no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—24no—
81C22×C4×C23—22no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—24no—
85C22×C4×C23—22no—
86C22×C4×C23—22no—
87C22×C4×C23—42no—
88C22×C4×C23—42no—
89C22×C4×C23—81no—
90C22×C4×C23C22⋊C412yes—
91C4⋊C4×C23—44no—
92C4⋊C4×C23—44no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—44no—
95C22⋊C4×C23—44no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—42no—
98C22⋊C4×C23—41no—
99C22⋊C4×C23—42no—
100C22⋊C4×C23—44no—
101C22⋊C4×C23—44no—
102C22⋊C4×C23—44no—
103C22⋊C4×C23—82no—
104C22⋊C4×C23—81no—
105C22⋊C4×C23—82no—
106C22⋊C4×C23—82no—
107C22⋊C4×C23—82no—
108C22⋊C4×C23—82no—
109C42×C23—44no—
110#9501C2311yesmaximal, characteristic, direct product factor, Sylow
111C23×C23—22no—
112C23×C23—22no—
113C23×C23—22no—
114C23×C23—22no—
115C23×C23—24no—
116C23×C23—22no—
117C23×C23—22no—
118C23×C23—22no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—42no—
124C23×C23—42no—
125C23×C23—42no—
126C23×C23—42no—
127C23×C23—41no—
128C23×C23—42no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—42no—
132C23×C23—42no—
133C23×C23—42no—
134C23×C23—42no—
135C23×C23—81no—
136C23×C23—81no—
137C23×C23—88no—
138C23×C23—88no—
139C23×C23C2×C22⋊C411yescharacteristic
140C23×C23C4×D412yes—
141C23×C23#12011yescharacteristic
142C23×C23#12112yes—
143C23×C23#12311yescharacteristic
144D4×C23—48no—
145D4×C23—48no—
146C2×C4×C23—22no—
147C2×C4×C23—22no—
148C2×C4×C23—24no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—44no—
152C2×C4×C23—42no—
153C2×C4×C23—42no—
154C2×C4×C23—44no—
155C2×C4×C23—42no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—42no—
159C2×C4×C23—42no—
160C2×C4×C23—44no—
161C2×C4×C23—82no—
162C2×C4×C23—82no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165C2×C4×C23—82no—
166C2×C4×C23#9914yes—
167#2405C2×C2311yescharacteristic
168#2309C2×C2311yescharacteristic
169#1819C2×C2312yescomplemented
170#1781C2×C2311yescharacteristic
171#1304C2×C2312yescomplemented
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—22no—
177C22×C23—21no—
178C22×C23—22no—
179C22×C23—21no—
180C22×C23—22no—
181C22×C23—22no—
182C22×C23—22no—
183C22×C23—22no—
184C22×C23—21no—
185C22×C23—44no—
186C22×C23—44no—
187C22×C23—44no—
188C22×C23—42no—
189C22×C23—42no—
190C22×C23—42no—
191C22×C23—84no—
192C22×C23—82no—
193C22×C23—82no—
194C22×C23—84no—
195C22×C23—84no—
196C22×C23—84no—
197C22×C23#38611yescharacteristic
198C22×C23C2×#9912yes—
199C22×C23#45812yes—
200C4×C23—24no—
201C4×C23—84no—
202C4×C23—82no—
203C4×C23—81no—
204C4×C23—81no—
205C4×C23—162no—
206C23×D4C22×C2311yescharacteristic
207C2×#123C4×C2312yescomplemented
208C2×#117—21no—
209C22×C22⋊C4C22×C2311yescharacteristic
210C2×#99—22no—
211#394—22no—
212#390—24no—
213#386C4×C2312yescomplemented
214#385—21no—
215#379—21no—
216#379—21no—
217#379C22×C2312yes—
218#379C22×C2311yescharacteristic
219#379C22×C2312yes—
220#342—24no—
221C2×C23—22no—
222C2×C23—22no—
223C2×C23—42no—
224C2×C23—42no—
225C2×C23—41no—
226C2×C23—82no—
227C2×C23#189011yescharacteristic
228C2×C23#189811yescharacteristic
229C2×C23#191311yescharacteristic
230C25D4×C2311yescharacteristic
231C25C23×C2311yescharacteristic
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—44no—
235C22×D4C2×C4×C2312yes—
236C22×D4D4×C2312yes—
237C23×C4C2×C4×C2311yescharacteristic
238C23×C4C2×C4×C2311yescharacteristic
239C23×C4D4×C2311yescharacteristic
240#123—44no—
241#117—44no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4C2×C4×C2312yes—
260C2×C22⋊C4D4×C2314yes—
261C2×C42—22no—
262#99—44no—
263#95—22no—
264C23#950111yescharacteristic, direct product factor, Sylow
265C24—21no—
266C24—21no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C22⋊C4×C2312yes—
274C24C22×C4×C2311yescharacteristic
275C24C2×D4×C2311yescharacteristic
276C24C2×D4×C2311yescharacteristic
277C24C2×D4×C2312yes—
278C24#41×C2312yes—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—48no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—81no—
300C22×C4C22⋊C4×C2312yes—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C22⋊C4—81no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—44no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—88no—
347C23—88no—
348C23C2×C22⋊C4×C2311yescharacteristic
349C23C4×D4×C2312yes—
350C23C23×#12011yescharacteristic
351C23C23×#12112yes—
352C23C23×#12311yescharacteristic
353D4—48no—
354D4—48no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4C23×#9914yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22—82no—
397C22—82no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22C23×#38611yescharacteristic
402C22C2×C23×#9912yes—
403C22C23×#45812yes—
404C4—24no—
405C4—84no—
406C4—82no—
407C4—81no—
408C4—81no—
409C4—162no—
410C2—22no—
411C2—22no—
412C2—42no—
413C2—41no—
414C2—42no—
415C2—82no—
416C2C23×#189011yescharacteristic
417C2C23×#189811yescharacteristic
418C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1819C2semidirect
5. C23×#1304C2semidirect
7. C2×C23×#123C4semidirect
13. C23×#386C4semidirect
110. #9501C23direct (N × H)
169. #1819C2×C23semidirect
171. #1304C2×C23semidirect
207. C2×#123C4×C23semidirect
213. #386C4×C23semidirect
264. C23#9501direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24052
C23×#23092
C23×#18192
C23×#17812
C23×#13042
#950123