Finite Group Databasebeta

← C23×#11821 #19159447

Subgroups — C23×#11821 #19159447

Order 5888.

582 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2397C211yesmaximal, characteristic
2C23×#1799C211yesmaximal, characteristic, complemented
3C23×#1465C211yesmaximal, characteristic, complemented
4C23×#1352C212yesmaximal, complemented
5C23×#1343C212yesmaximal, complemented
6C2×C4×D4×C23—21no—
7C2×C4×D4×C23—21no—
8C2×C4×D4×C23C411yescharacteristic, complemented
9C2×C4×D4×C23C2211yescharacteristic
10C2×C23×#99—21no—
11C23×#407—21no—
12C23×#399—21no—
13C23×#399C411yescharacteristic, complemented
14C23×#399C2211yescharacteristic
15C23×#392—21no—
16C23×#392C411yescharacteristic, complemented
17C23×#392C2211yescharacteristic
18C23×#380—21no—
19C23×#376C411yescharacteristic, complemented
20C23×#342C2212yes—
21C23×#324—21no—
22C23×#323C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C22×D4×C23D411yescharacteristic
27C23×C4×C23—21no—
28C23×C4×C23D411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C2×M16×C23—41no—
31C4×D4×C23—41no—
32C4×D4×C23—41no—
33C4×D4×C23—44no—
34C4×D4×C23—44no—
35C4×D4×C23—41no—
36C4×D4×C23—41no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—41no—
40C2×C4⋊C4×C23C2×C411yescharacteristic
41C2×C4⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23C2×C411yescharacteristic
49C2×C22⋊C4×C23C2311yescharacteristic
50C2×C42×C23—21no—
51C2×C42×C23C2×C411yescharacteristic
52C23×#99—42no—
53C23×C22⋊C8—22no—
54C23×C22⋊C8—41no—
55C23×#95—21no—
56C23×#95—41no—
57C23×#95C2×C411yescharacteristic
58C23×#95C2×C411yescharacteristic
59C24×C23—21no—
60C24×C23C22⋊C411yescharacteristic
61C24×C23C2×D411yescharacteristic
62C2×D4×C23—21no—
63C2×D4×C23—21no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23—41no—
67C2×D4×C23—42no—
68C2×D4×C23—42no—
69C2×D4×C23—42no—
70C2×D4×C23—42no—
71C2×D4×C23—42no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C2×D4×C23—42no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23—42no—
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22⋊C411yescharacteristic
91C22×C4×C23C22⋊C411yescharacteristic
92C22×C4×C23C22×C411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94M16×C23—82no—
95C2×C8×C23—42no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—42no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23—82no—
101C22⋊C4×C23—41no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—42no—
106C22⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—42no—
109C22⋊C4×C23—42no—
110C42×C23—24no—
111C42×C23—21no—
112C42×C23—21no—
113#11821C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—22no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—21no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23C2×C22⋊C411yescharacteristic
131C23×C23AGL(1,ℤ/8)11yescharacteristic
132C23×C23#13711yescharacteristic
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—42no—
136D4×C23—44no—
137D4×C23—44no—
138D4×C23—48no—
139D4×C23—42no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—22no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165C2×C4×C23—82no—
166C2×C4×C23—82no—
167C2×C4×C23—82no—
168C2×C4×C23#9911yescharacteristic
169C2×C4×C23#9911yescharacteristic
170C8×C23—82no—
171#2397C2×C2311yescharacteristic
172#1799C2×C2311yescharacteristic, complemented
173#1465C2×C2311yescharacteristic, complemented
174#1352C2×C2312yescomplemented
175#1343C2×C2312yescomplemented
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—42no—
183C22×C23—42no—
184C22×C23—42no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—42no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—41no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23—81no—
196C22×C23—81no—
197C22×C23C2×#9911yescharacteristic
198C22×C23#41711yescharacteristic
199C22×C23#42111yescharacteristic
200C4×C23—21no—
201C4×C23—42no—
202C4×C23—42no—
203C4×C23—41no—
204C4×C23—41no—
205C4×C23—41no—
206C4×C23—41no—
207C4×C23—81no—
208C4×C23—162no—
209C4×C23#35312yes—
210C2×C4×D4—21no—
211C2×C4×D4—21no—
212C2×C4×D4C4×C2311yescharacteristic, complemented
213C2×C4×D4C22×C2311yescharacteristic
214C2×#99—21no—
215#407—21no—
216#399—21no—
217#399C4×C2311yescharacteristic, complemented
218#399C22×C2311yescharacteristic
219#392—21no—
220#392C4×C2311yescharacteristic, complemented
221#392C22×C2311yescharacteristic
222#380—21no—
223#376C4×C2311yescharacteristic, complemented
224#342C22×C2312yes—
225#324—21no—
226#323C22×C2312yes—
227C2×C23—21no—
228C2×C23—21no—
229C2×C23—41no—
230C2×C23—41no—
231C2×C23—81no—
232C2×C23—82no—
233C2×C23#151311yescharacteristic
234C2×C23#212011yescharacteristic
235C2×C23C2×#35311yescharacteristic
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×C4×C2311yescharacteristic
239C22×D4D4×C2311yescharacteristic
240C23×C4—21no—
241C23×C4D4×C2311yescharacteristic
242C23×C4D4×C2311yescharacteristic
243C2×M16—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4C2×C4×C2311yescharacteristic
254C2×C4⋊C4D4×C2311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C2×C4×C2311yescharacteristic
262C2×C22⋊C4C23×C2311yescharacteristic
263C2×C42—21no—
264C2×C42C2×C4×C2311yescharacteristic
265#99—42no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—21no—
269#95—41no—
270#95C2×C4×C2311yescharacteristic
271#95C2×C4×C2311yescharacteristic
272C23#1182111yescharacteristic, direct product factor, Sylow
273C24—21no—
274C24C22⋊C4×C2311yescharacteristic
275C24C2×D4×C2311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4C22⋊C4×C2311yescharacteristic
304C22×C4C22⋊C4×C2311yescharacteristic
305C22×C4C22⋊C4×C2311yescharacteristic
306C22×C4C22×C4×C2311yescharacteristic
307C22×C4C2×D4×C2311yescharacteristic
308M16—82no—
309C2×C8—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C42—24no—
325C42—21no—
326C42—21no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23C2×C22⋊C4×C2311yescharacteristic
344C23C23×AGL(1,ℤ/8)11yescharacteristic
345C23C23×#13711yescharacteristic
346D4—42no—
347D4—42no—
348D4—42no—
349D4—44no—
350D4—44no—
351D4—48no—
352D4—42no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4C23×#9911yescharacteristic
382C2×C4C23×#9911yescharacteristic
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C2×C23×#9911yescharacteristic
406C22C23×#41711yescharacteristic
407C22C23×#42111yescharacteristic
408C4—21no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—81no—
416C4—162no—
417C4C23×#35312yes—
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—81no—
423C2—82no—
424C2C23×#151311yescharacteristic
425C2C23×#212011yescharacteristic
426C2C2×C23×#35311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#2397C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1799C2semidirect
3. C23×#1465C2semidirect
4. C23×#1352C2semidirect
5. C23×#1343C2semidirect
8. C2×C4×D4×C23C4semidirect
13. C23×#399C4semidirect
16. C23×#392C4semidirect
19. C23×#376C4semidirect
113. #11821C23direct (N × H)
172. #1799C2×C23semidirect
173. #1465C2×C23semidirect
174. #1352C2×C23semidirect
175. #1343C2×C23semidirect
212. C2×C4×D4C4×C23semidirect
217. #399C4×C23semidirect
220. #392C4×C23semidirect
223. #376C4×C23semidirect
272. C23#11821direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23972
C23×#17992
C23×#14652
C23×#13522
C23×#13432
#1182123