Finite Group Databasebeta

← C23×#12120 #19159746

Subgroups — C23×#12120 #19159746

Order 5888.

664 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2437C211yesmaximal, characteristic, complemented
2C23×#2021C211yesmaximal, characteristic
3C23×#1613C211yesmaximal, characteristic, complemented
4C23×#1601C211yesmaximal, characteristic, complemented
5C23×#1352C211yesmaximal, characteristic, complemented
6C23×#1351C211yesmaximal, characteristic, complemented
7C23×#1316C211yesmaximal, characteristic, complemented
8C23×#534C2211yescharacteristic
9C2×C23×C22⋊Q8—21no—
10C2×C23×#120—21no—
11C22×C22⋊C4×C23—21no—
12C23×#481—21no—
13C23×#448—21no—
14C2×C23×#99—21no—
15C23×#399C2211yescharacteristic
16C23×#396—21no—
17C23×#393C2211yescharacteristic
18C23×#385—21no—
19C23×#385—21no—
20C23×#380—21no—
21C23×#379C2211yescharacteristic
22C23×#342C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C25×C23D411yescharacteristic
28C22×Q8×C23D411yescharacteristic, complemented
29C22×D4×C23—21no—
30C22×D4×C23—41no—
31C23×C4×C23—21no—
32C23×C4×C23—21no—
33C2×SD16×C23—41no—
34C23×#123—41no—
35C23×C22⋊Q8—44no—
36C23×#121—21no—
37C23×#121—21no—
38C23×#120—42no—
39C23×#120—41no—
40C23×#120—42no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23D411yescharacteristic, complemented
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—44no—
54C2×C22⋊C4×C23—41no—
55C2×C22⋊C4×C23D411yescharacteristic
56C2×C22⋊C4×C23C2311yescharacteristic
57C23×#106—41no—
58C23×#103—41no—
59C23×#102—41no—
60C23×#99—42no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×#95—21no—
64C23×#95—21no—
65C23×#95D411yescharacteristic, complemented
66C23×#95D411yescharacteristic, complemented
67C24×C23—21no—
68C24×C23—21no—
69C24×C23—21no—
70C24×C23—42no—
71C24×C23—41no—
72C24×C23—42no—
73C24×C23—41no—
74C24×C23SD1612yescomplemented
75C24×C23C2×D411yescharacteristic
76C2×Q8×C23—42no—
77C2×Q8×C23—41no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—44no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—82no—
88C2×D4×C23—82no—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—42no—
96C22×C4×C23—42no—
97C22×C4×C23—41no—
98C22×C4×C23—42no—
99C22×C4×C23—42no—
100C22×C4×C23—81no—
101C22×C4×C23—81no—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104SD16×C23—82no—
105C2×C8×C23—41no—
106C2×C8×C23—41no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—42no—
111C4⋊C4×C23—42no—
112C4⋊C4×C23—42no—
113C4⋊C4×C23—81no—
114C22⋊C4×C23—24no—
115C22⋊C4×C23—22no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—42no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—41no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—42no—
122C22⋊C4×C23—42no—
123C22⋊C4×C23—42no—
124C22⋊C4×C23—44no—
125C22⋊C4×C23—41no—
126C22⋊C4×C23—41no—
127#12120C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—22no—
129C23×C23—22no—
130C23×C23—22no—
131C23×C23—21no—
132C23×C23—21no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—42no—
137C23×C23—44no—
138C23×C23—41no—
139C23×C23—44no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23—41no—
144C23×C23—42no—
145C23×C23—42no—
146C23×C23—41no—
147C23×C23—81no—
148C23×C23—84no—
149C23×C23—81no—
150C23×C23—81no—
151C23×C23—81no—
152C23×C23—81no—
153C23×C23#12011yescharacteristic
154C23×C23C2×SD1611yescharacteristic
155C23×C23#13711yescharacteristic
156Q8×C23—42no—
157Q8×C23—81no—
158D4×C23—81no—
159D4×C23—81no—
160D4×C23—81no—
161D4×C23—81no—
162D4×C23—88no—
163D4×C23—81no—
164D4×C23—81no—
165C2×C4×C23—22no—
166C2×C4×C23—22no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—42no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—41no—
179C2×C4×C23—41no—
180C2×C4×C23—41no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—41no—
185C2×C4×C23—41no—
186C2×C4×C23—41no—
187C2×C4×C23—41no—
188C2×C4×C23—84no—
189C2×C4×C23—84no—
190C2×C4×C23—81no—
191C2×C4×C23—81no—
192C8×C23—81no—
193C8×C23—81no—
194#2437C2×C2311yescharacteristic, complemented
195#2021C2×C2311yescharacteristic
196#1613C2×C2311yescharacteristic, complemented
197#1601C2×C2311yescharacteristic, complemented
198#1352C2×C2311yescharacteristic, complemented
199#1351C2×C2311yescharacteristic, complemented
200#1316C2×C2311yescharacteristic, complemented
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—24no—
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—41no—
207C22×C23—41no—
208C22×C23—42no—
209C22×C23—42no—
210C22×C23—42no—
211C22×C23—44no—
212C22×C23—42no—
213C22×C23—42no—
214C22×C23—41no—
215C22×C23—41no—
216C22×C23—42no—
217C22×C23—81no—
218C22×C23—82no—
219C22×C23—82no—
220C22×C23—82no—
221C22×C23—82no—
222C22×C23—81no—
223C22×C23—81no—
224C22×C23—81no—
225C22×C23—161no—
226C22×C23#44811yescharacteristic
227C22×C23#45311yescharacteristic
228C22×C23B(4,𝔽2)11yescharacteristic
229C4×C23—42no—
230C4×C23—41no—
231C4×C23—42no—
232C4×C23—41no—
233C4×C23—81no—
234C4×C23—81no—
235C4×C23—82no—
236C4×C23—81no—
237C4×C23—81no—
238#534C22×C2311yescharacteristic
239C2×C22⋊Q8—21no—
240C2×#120—21no—
241C22×C22⋊C4—21no—
242#481—21no—
243#448—21no—
244C2×#99—21no—
245#399C22×C2311yescharacteristic
246#396—21no—
247#393C22×C2311yescharacteristic
248#385—21no—
249#385—21no—
250#380—21no—
251#379C22×C2311yescharacteristic
252#342C22×C2311yescharacteristic
253#327—21no—
254#327—21no—
255#323C22×C2311yescharacteristic
256#323C22×C2311yescharacteristic
257C2×C23—21no—
258C2×C23—21no—
259C2×C23—41no—
260C2×C23—44no—
261C2×C23—41no—
262C2×C23—161no—
263C2×C23#159611yescharacteristic
264C2×C23#219611yescharacteristic
265C2×C23#219711yescharacteristic
266C25D4×C2311yescharacteristic
267C22×Q8D4×C2311yescharacteristic, complemented
268C22×D4—21no—
269C22×D4—41no—
270C23×C4—21no—
271C23×C4—21no—
272C2×SD16—41no—
273#123—41no—
274C22⋊Q8—44no—
275#121—21no—
276#121—21no—
277#120—42no—
278#120—41no—
279#120—42no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4D4×C2311yescharacteristic, complemented
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4D4×C2311yescharacteristic
295C2×C22⋊C4C23×C2311yescharacteristic
296#106—41no—
297#103—41no—
298#102—41no—
299#99—42no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302#95—21no—
303#95—21no—
304#95D4×C2311yescharacteristic, complemented
305#95D4×C2311yescharacteristic, complemented
306C23#1212011yescharacteristic, direct product factor, Sylow
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—41no—
312C24—42no—
313C24—41no—
314C24SD16×C2312yescomplemented
315C24C2×D4×C2311yescharacteristic
316C2×Q8—42no—
317C2×Q8—41no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—44no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—82no—
328C2×D4—82no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4C2×D4×C2311yescharacteristic
343C22×C4C2×D4×C2311yescharacteristic
344SD16—82no—
345C2×C8—41no—
346C2×C8—41no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—44no—
377C23—41no—
378C23—44no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—81no—
387C23—84no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23C23×#12011yescharacteristic
393C23C2×SD16×C2311yescharacteristic
394C23C23×#13711yescharacteristic
395Q8—42no—
396Q8—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—88no—
402D4—81no—
403D4—81no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—81no—
430C2×C4—81no—
431C8—81no—
432C8—81no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—161no—
458C22C23×#44811yescharacteristic
459C22C23×#45311yescharacteristic
460C22C23×B(4,𝔽2)11yescharacteristic
461C4—42no—
462C4—41no—
463C4—42no—
464C4—41no—
465C4—81no—
466C4—81no—
467C4—82no—
468C4—81no—
469C4—81no—
470C2—21no—
471C2—21no—
472C2—41no—
473C2—44no—
474C2—41no—
475C2—161no—
476C2C23×#159611yescharacteristic
477C2C23×#219611yescharacteristic
478C2C23×#219711yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2437C2semidirect
3. C23×#1613C2semidirect
4. C23×#1601C2semidirect
5. C23×#1352C2semidirect
6. C23×#1351C2semidirect
7. C23×#1316C2semidirect
28. C22×Q8×C23D4semidirect
45. C2×C4⋊C4×C23D4semidirect
65. C23×#95D4semidirect
66. C23×#95D4semidirect
74. C24×C23SD16semidirect
127. #12120C23direct (N × H)
194. #2437C2×C23semidirect
196. #1613C2×C23semidirect
197. #1601C2×C23semidirect
198. #1352C2×C23semidirect
199. #1351C2×C23semidirect
200. #1316C2×C23semidirect
267. C22×Q8D4×C23semidirect
284. C2×C4⋊C4D4×C23semidirect
304. #95D4×C23semidirect
305. #95D4×C23semidirect
306. C23#12120direct (N × H)
314. C24SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#24372
C23×#20212
C23×#16132
C23×#16012
C23×#13522
C23×#13512
C23×#13162
#1212023