Finite Group Databasebeta

← C23×#9052 #19156678

Subgroups — C23×#9052 #19156678

Order 5888.

488 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3105C211yesmaximal, characteristic
2C23×#2992C211yesmaximal, characteristic, complemented
3C23×#1955C211yesmaximal, characteristic, complemented
4C23×#1524C211yesmaximal, characteristic, complemented
5C23×#1490C211yesmaximal, characteristic, complemented
6C23×#1302C211yesmaximal, characteristic, complemented
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#529C2211yescharacteristic
9C2×C4×D4×C23C2211yescharacteristic
10C23×#484C411yescharacteristic
11C23×#481C411yescharacteristic
12C23×#478—21no—
13C23×#463—21no—
14C23×#450C411yescharacteristic
15C23×#447C411yescharacteristic
16C23×#446—21no—
17C4×SD16×C23—21no—
18C4×D8×C23—21no—
19C23×#435—21no—
20C23×#435—21no—
21C8×D4×C23—21no—
22C23×#433—21no—
23C23×#432C2211yescharacteristic
24C23×#431C2211yescharacteristic
25C23×#424—21no—
26C2×C23×C4⋊C8C2211yescharacteristic
27C2×C23×C22⋊C8—21no—
28C2×C23×C22⋊C8—21no—
29C4×M16×C23C2211yescharacteristic
30C2×C23×#97C2211yescharacteristic
31C23×#330—21no—
32C23×#329—21no—
33C23×D4⋊C8—21no—
34C23×D4⋊C8—21no—
35C22×D4×C23C2×C411yescharacteristic
36C23×C4×C23—21no—
37C2×SD16×C23—21no—
38C2×D8×C23—21no—
39C2×M16×C23—21no—
40C2×M16×C23—21no—
41C22×C8×C23—21no—
42C22×C8×C23—21no—
43C22×C8×C23D411yescharacteristic
44C22×C8×C23D411yescharacteristic
45C23×#126—21no—
46C23×#125D411yescharacteristic
47C23×#124D411yescharacteristic
48C23×#123—21no—
49C23×C22⋊Q8C2×C411yescharacteristic
50C23×#121C2×C411yescharacteristic
51C4×Q8×C23D411yescharacteristic
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23—21no—
57C4×D4×C23D411yescharacteristic
58C23×#117—21no—
59C2×C4⋊C4×C23C2×C411yescharacteristic
60C2×C22⋊C4×C23—21no—
61C2×C42×C23C2311yescharacteristic
62C23×#107—21no—
63C23×#106—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8D411yescharacteristic
70C23×C4⋊C8D411yescharacteristic
71C23×#103—21no—
72C23×#102—21no—
73C23×#102—21no—
74C23×#102—21no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8—42no—
79C23×C22⋊C8—42no—
80C23×C22⋊C8C2×C411yescharacteristic
81C23×C22⋊C8C2×C411yescharacteristic
82C23×#97—21no—
83C23×#97—21no—
84C23×#97—21no—
85C23×#97—21no—
86C23×#97—21no—
87C4×C8×C23—21no—
88C4×C8×C23—21no—
89C4×C8×C23—21no—
90C24×C23—21no—
91C2×Q8×C23C22⋊C411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—42no—
96C2×D4×C23C22⋊C411yescharacteristic
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23C22×C411yescharacteristic
106C22×C4×C23C2×D411yescharacteristic
107C22×C4×C23#4111yescharacteristic
108SD16×C23—42no—
109D8×C23—42no—
110M16×C23—22no—
111M16×C23—41no—
112M16×C23—81no—
113C2×C8×C23—22no—
114C2×C8×C23—21no—
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—21no—
118C2×C8×C23—21no—
119C2×C8×C23—21no—
120C2×C8×C23—41no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C2×C8×C23—41no—
124C2×C8×C23—41no—
125C2×C8×C23—81no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—21no—
131C4⋊C4×C23—21no—
132C4⋊C4×C23C22⋊C411yescharacteristic
133C4⋊C4×C23C22⋊C411yescharacteristic
134C22⋊C4×C23—21no—
135C22⋊C4×C23—21no—
136C22⋊C4×C23—21no—
137C22⋊C4×C23—21no—
138C22⋊C4×C23—21no—
139C22⋊C4×C23—41no—
140C22⋊C4×C23—41no—
141C42×C23—21no—
142C42×C23C2×D411yescharacteristic
143C42×C23C2×D411yescharacteristic
144C42×C23C2×D411yescharacteristic
145C42×C23#4111yescharacteristic
146#9052C2311yesmaximal, characteristic, direct product factor, Sylow
147C23×C23—22no—
148C23×C23—21no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23—41no—
154C23×C23—41no—
155C23×C23C4×D411yescharacteristic
156Q8×C23—41no—
157D4×C23—22no—
158D4×C23—22no—
159D4×C23—41no—
160D4×C23—41no—
161D4×C23—41no—
162D4×C23—41no—
163D4×C23—41no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—41no—
177C2×C4×C23—81no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23—81no—
181C2×C4×C23C2×C22⋊C411yescharacteristic
182C2×C4×C23C4×D411yescharacteristic
183C2×C4×C23#12011yescharacteristic
184C2×C4×C23#12111yescharacteristic
185C2×C4×C23#12111yescharacteristic
186C2×C4×C23#12311yescharacteristic
187C8×C23—42no—
188C8×C23—41no—
189C8×C23—41no—
190C8×C23—42no—
191C8×C23—41no—
192C8×C23—41no—
193C8×C23—41no—
194C8×C23—41no—
195#3105C2×C2311yescharacteristic
196#2992C2×C2311yescharacteristic, complemented
197#1955C2×C2311yescharacteristic, complemented
198#1524C2×C2311yescharacteristic, complemented
199#1490C2×C2311yescharacteristic, complemented
200#1302C2×C2311yescharacteristic, complemented
201#1283C2×C2311yescharacteristic, complemented
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—42no—
205C22×C23—42no—
206C22×C23—41no—
207C22×C23—41no—
208C22×C23—42no—
209C22×C23—41no—
210C22×C23—81no—
211C22×C23—81no—
212C22×C23—81no—
213C22×C23—81no—
214C22×C23#38611yescharacteristic
215C22×C23#44311yescharacteristic
216C22×C23#44411yescharacteristic
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—21no—
222C4×C23—41no—
223C4×C23—41no—
224C4×C23—41no—
225C4×C23—81no—
226C4×C23#45311yescharacteristic
227C4×C23#45411yescharacteristic
228#529C22×C2311yescharacteristic
229C2×C4×D4C22×C2311yescharacteristic
230#484C4×C2311yescharacteristic
231#481C4×C2311yescharacteristic
232#478—21no—
233#463—21no—
234#450C4×C2311yescharacteristic
235#447C4×C2311yescharacteristic
236#446—21no—
237C4×SD16—21no—
238C4×D8—21no—
239#435—21no—
240#435—21no—
241C8×D4—21no—
242#433—21no—
243#432C22×C2311yescharacteristic
244#431C22×C2311yescharacteristic
245#424—21no—
246C2×C4⋊C8C22×C2311yescharacteristic
247C2×C22⋊C8—21no—
248C2×C22⋊C8—21no—
249C4×M16C22×C2311yescharacteristic
250C2×#97C22×C2311yescharacteristic
251#330—21no—
252#329—21no—
253D4⋊C8—21no—
254D4⋊C8—21no—
255C2×C23—21no—
256C2×C23—21no—
257C2×C23—81no—
258C2×C23—82no—
259C2×C23#187611yescharacteristic
260C2×C23#188311yescharacteristic
261C2×C23#189111yescharacteristic
262C22×D4C2×C4×C2311yescharacteristic
263C23×C4—21no—
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16—21no—
267C2×M16—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8D4×C2311yescharacteristic
271C22×C8D4×C2311yescharacteristic
272#126—21no—
273#125D4×C2311yescharacteristic
274#124D4×C2311yescharacteristic
275#123—21no—
276C22⋊Q8C2×C4×C2311yescharacteristic
277#121C2×C4×C2311yescharacteristic
278C4×Q8D4×C2311yescharacteristic
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4D4×C2311yescharacteristic
285#117—21no—
286C2×C4⋊C4C2×C4×C2311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C23×C2311yescharacteristic
289#107—21no—
290#106—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8D4×C2311yescharacteristic
297C4⋊C8D4×C2311yescharacteristic
298#103—21no—
299#102—21no—
300#102—21no—
301#102—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8C2×C4×C2311yescharacteristic
308C22⋊C8C2×C4×C2311yescharacteristic
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C23#905211yescharacteristic, direct product factor, Sylow
318C24—21no—
319C2×Q8C22⋊C4×C2311yescharacteristic
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4C22⋊C4×C2311yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4C22×C4×C2311yescharacteristic
334C22×C4C2×D4×C2311yescharacteristic
335C22×C4#41×C2311yescharacteristic
336SD16—42no—
337D8—42no—
338M16—22no—
339M16—41no—
340M16—81no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—81no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4C22⋊C4×C2311yescharacteristic
361C4⋊C4C22⋊C4×C2311yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C42—21no—
370C42C2×D4×C2311yescharacteristic
371C42C2×D4×C2311yescharacteristic
372C42C2×D4×C2311yescharacteristic
373C42#41×C2311yescharacteristic
374C23—22no—
375C23—21no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C4×D4×C2311yescharacteristic
383Q8—41no—
384D4—22no—
385D4—22no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4C2×C22⋊C4×C2311yescharacteristic
409C2×C4C4×D4×C2311yescharacteristic
410C2×C4C23×#12011yescharacteristic
411C2×C4C23×#12111yescharacteristic
412C2×C4C23×#12111yescharacteristic
413C2×C4C23×#12311yescharacteristic
414C8—42no—
415C8—41no—
416C8—41no—
417C8—42no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—41no—
422C22—21no—
423C22—21no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22C23×#38611yescharacteristic
435C22C23×#44311yescharacteristic
436C22C23×#44411yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—81no—
446C4C23×#45311yescharacteristic
447C4C23×#45411yescharacteristic
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2C23×#187611yescharacteristic
453C2C23×#188311yescharacteristic
454C2C23×#189111yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2992C2semidirect
3. C23×#1955C2semidirect
4. C23×#1524C2semidirect
5. C23×#1490C2semidirect
6. C23×#1302C2semidirect
7. C23×#1283C2semidirect
146. #9052C23direct (N × H)
196. #2992C2×C23semidirect
197. #1955C2×C23semidirect
198. #1524C2×C23semidirect
199. #1490C2×C23semidirect
200. #1302C2×C23semidirect
201. #1283C2×C23semidirect
317. C23#9052direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31052
C23×#29922
C23×#19552
C23×#15242
C23×#14902
C23×#13022
C23×#12832
#905223