Finite Group Databasebeta

← C23×#11527 #19159153

Subgroups — C23×#11527 #19159153

Order 5888.

530 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2321C211yesmaximal, characteristic
2C23×#1466C211yesmaximal, characteristic, complemented
3C23×#1465C211yesmaximal, characteristic, complemented
4C23×#1326C212yesmaximal, complemented
5C23×#1316C212yesmaximal, complemented
6C2×C23×#120C411yescharacteristic
7C2×C4×D4×C23C2211yescharacteristic
8C22×C22⋊C4×C23C411yescharacteristic
9C23×#432—21no—
10C23×#407—21no—
11C23×#390—21no—
12C23×#388—21no—
13C23×#388—21no—
14C23×#386—21no—
15C23×#386C411yescharacteristic
16C23×#385C2211yescharacteristic
17C23×#379C2211yescharacteristic
18C4×C22⋊C4×C23C411yescharacteristic
19C23×#324—21no—
20C23×#323—21no—
21C23×#323C2212yes—
22C23×#323C2212yes—
23C25×C23C2×C411yescharacteristic
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23C2×C411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C2×M16×C23—41no—
30C23×#120—22no—
31C23×#120—22no—
32C4×D4×C23—41no—
33C4×D4×C23—41no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23D411yescharacteristic
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—41no—
43C2×C22⋊C4×C23—41no—
44C2×C22⋊C4×C23C2×C411yescharacteristic
45C2×C22⋊C4×C23C2×C411yescharacteristic
46C2×C22⋊C4×C23D411yescharacteristic
47C2×C22⋊C4×C23C2311yescharacteristic
48C2×C42×C23—21no—
49C2×C42×C23D411yescharacteristic
50C23×C4⋊C8—41no—
51C23×C22⋊C8—22no—
52C23×C22⋊C8—22no—
53C23×C22⋊C8—41no—
54C23×#97—41no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C24×C23—21no—
59C24×C23—21no—
60C24×C23—41no—
61C24×C23—41no—
62C24×C23—81no—
63C24×C23—81no—
64C24×C23C22⋊C411yescharacteristic
65C24×C23M1612yescomplemented
66C24×C23C22×C411yescharacteristic
67C2×D4×C23—21no—
68C2×D4×C23—21no—
69C2×D4×C23—24no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—81no—
73C2×D4×C23—81no—
74C22×C4×C23—21no—
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—41no—
82C22×C4×C23—81no—
83C22×C4×C23—81no—
84C22×C4×C23—81no—
85C22×C4×C23C22⋊C411yescharacteristic
86C22×C4×C23C22⋊C411yescharacteristic
87C22×C4×C23C22⋊C411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90M16×C23—82no—
91C2×C8×C23—42no—
92C2×C8×C23—42no—
93C4⋊C4×C23—41no—
94C4⋊C4×C23—81no—
95C22⋊C4×C23—24no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—41no—
99C22⋊C4×C23—42no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—41no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—81no—
104C22⋊C4×C23—81no—
105C22⋊C4×C23—81no—
106C22⋊C4×C23M1612yescomplemented
107C42×C23—21no—
108C42×C23—21no—
109C42×C23—81no—
110#11527C2311yesmaximal, characteristic, direct product factor, Sylow
111C23×C23—21no—
112C23×C23—21no—
113C23×C23—22no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—42no—
126C23×C23—42no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—82no—
130C23×C23—82no—
131C23×C23—82no—
132C23×C23—82no—
133C23×C23—84no—
134C23×C23C2×C22⋊C411yescharacteristic
135C23×C23C2×M1611yescharacteristic
136C23×C23C2×M1611yescharacteristic
137D4×C23—42no—
138D4×C23—42no—
139D4×C23—84no—
140C2×C4×C23—21no—
141C2×C4×C23—22no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—81no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23—82no—
164C2×C4×C23#9911yescharacteristic
165C2×C4×C23#9911yescharacteristic
166C8×C23—82no—
167C8×C23—82no—
168#2321C2×C2311yescharacteristic
169#1466C2×C2311yescharacteristic, complemented
170#1465C2×C2311yescharacteristic, complemented
171#1326C2×C2312yescomplemented
172#1316C2×C2312yescomplemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—42no—
180C22×C23—42no—
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—41no—
184C22×C23—41no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—42no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—84no—
193C22×C23—84no—
194C22×C23—81no—
195C22×C23—81no—
196C22×C23#40711yescharacteristic
197C22×C23C2×#9911yescharacteristic
198C22×C23#41311yescharacteristic
199C4×C23—21no—
200C4×C23—21no—
201C4×C23—42no—
202C4×C23—41no—
203C4×C23—41no—
204C4×C23—41no—
205C4×C23—41no—
206C4×C23—81no—
207C4×C23—81no—
208C2×#120C4×C2311yescharacteristic
209C2×C4×D4C22×C2311yescharacteristic
210C22×C22⋊C4C4×C2311yescharacteristic
211#432—21no—
212#407—21no—
213#390—21no—
214#388—21no—
215#388—21no—
216#386—21no—
217#386C4×C2311yescharacteristic
218#385C22×C2311yescharacteristic
219#379C22×C2311yescharacteristic
220C4×C22⋊C4C4×C2311yescharacteristic
221#324—21no—
222#323—21no—
223#323C22×C2312yes—
224#323C22×C2312yes—
225C2×C23—21no—
226C2×C23—21no—
227C2×C23—41no—
228C2×C23—41no—
229C2×C23—81no—
230C2×C23—82no—
231C2×C23#146511yescharacteristic
232C2×C23#212111yescharacteristic
233C2×C23#212211yescharacteristic
234C25C2×C4×C2311yescharacteristic
235C22×D4—21no—
236C22×D4C2×C4×C2311yescharacteristic
237C23×C4C2×C4×C2311yescharacteristic
238C23×C4C2×C4×C2311yescharacteristic
239C23×C4D4×C2311yescharacteristic
240C2×M16—41no—
241#120—22no—
242#120—22no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C2311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4C2×C4×C2311yescharacteristic
256C2×C22⋊C4C2×C4×C2311yescharacteristic
257C2×C22⋊C4D4×C2311yescharacteristic
258C2×C22⋊C4C23×C2311yescharacteristic
259C2×C42—21no—
260C2×C42D4×C2311yescharacteristic
261C4⋊C8—41no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265#97—41no—
266#95—21no—
267#95—21no—
268#95—21no—
269C23#1152711yescharacteristic, direct product factor, Sylow
270C24—21no—
271C24—21no—
272C24—41no—
273C24—41no—
274C24—81no—
275C24—81no—
276C24C22⋊C4×C2311yescharacteristic
277C24M16×C2312yescomplemented
278C24C22×C4×C2311yescharacteristic
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—81no—
285C2×D4—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C22⋊C4×C2311yescharacteristic
298C22×C4C22⋊C4×C2311yescharacteristic
299C22×C4C22⋊C4×C2311yescharacteristic
300C22×C4C2×D4×C2311yescharacteristic
301C22×C4C2×D4×C2311yescharacteristic
302M16—82no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C22⋊C4—24no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4M16×C2312yescomplemented
319C42—21no—
320C42—21no—
321C42—81no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—84no—
345C23C2×C22⋊C4×C2311yescharacteristic
346C23C2×M16×C2311yescharacteristic
347C23C2×M16×C2311yescharacteristic
348D4—42no—
349D4—42no—
350D4—84no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4C23×#9911yescharacteristic
376C2×C4C23×#9911yescharacteristic
377C8—82no—
378C8—82no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22C23×#40711yescharacteristic
403C22C2×C23×#9911yescharacteristic
404C22C23×#41311yescharacteristic
405C4—21no—
406C4—21no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—81no—
419C2—82no—
420C2C23×#146511yescharacteristic
421C2C23×#212111yescharacteristic
422C2C23×#212211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC2×C23×#120C4
LCSG ▷ C24 ▷ C22 ▷ 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×#1466C2semidirect
3. C23×#1465C2semidirect
4. C23×#1326C2semidirect
5. C23×#1316C2semidirect
65. C24×C23M16semidirect
106. C22⋊C4×C23M16semidirect
110. #11527C23direct (N × H)
169. #1466C2×C23semidirect
170. #1465C2×C23semidirect
171. #1326C2×C23semidirect
172. #1316C2×C23semidirect
269. C23#11527direct (N × H)
277. C24M16×C23semidirect
318. C22⋊C4M16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#23212
C23×#14662
C23×#14652
C23×#13262
C23×#13162
#1152723