Finite Group Databasebeta

← C23×#19346 #19166972

Subgroups — C23×#19346 #19166972

Order 5888.

582 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3237C211yesmaximal, characteristic, complemented
2C23×#3216C211yesmaximal, characteristic
3C23×#3167C211yesmaximal, characteristic
4C23×#2960C211yesmaximal, characteristic, complemented
5C23×#1692C212yesmaximal, complemented
6C23×#1686C212yesmaximal, complemented
7C23×#1680C212yesmaximal, complemented
8C23×#1678C212yesmaximal, complemented
9C23×#1568C211yesmaximal, characteristic, complemented
10C23×#1559C211yesmaximal, characteristic, complemented
11C23×#1557C211yesmaximal, characteristic
12C23×#533C2211yescharacteristic
13C23×#533C2211yescharacteristic
14C23×#532C2211yescharacteristic
15C23×#501—21no—
16C23×#498C2211yescharacteristic
17C23×#497—21no—
18C23×#492C2211yescharacteristic
19C23×#491—21no—
20C23×#489—21no—
21C23×#487C2211yescharacteristic
22C23×#486C2211yescharacteristic
23C23×#485—22no—
24C23×#477C2212yes—
25C23×#475C2212yes—
26C23×#465—21no—
27C23×#465—21no—
28C23×#462C2212yes—
29C23×#460C2212yes—
30C23×#452—22no—
31C23×#446—21no—
32C23×#446C2212yes—
33C8×Q8×C23—21no—
34C8×Q8×C23C2212yes—
35C4×SD16×C23—21no—
36C4×SD16×C23—21no—
37C23×#432C2211yescharacteristic
38C23×#432C2211yescharacteristic
39C23×#424—21no—
40C23×#423C2211yescharacteristic
41C23×#405—21no—
42C2×C4×C8×C23C2211yescharacteristic
43C23×#335C2212yes—
44C23×#334C2212yes—
45C23×#333C2212yes—
46C23×#332C2212yes—
47C23×#332C2212yes—
48C23×#331C2212yes—
49C2×#41×C23—21no—
50C2×Q16×C23—22no—
51C2×SD16×C23—22no—
52C2×SD16×C23—21no—
53C2×SD16×C23—21no—
54C2×M16×C23—21no—
55C22×C8×C23D411yescharacteristic
56C22×C8×C23D411yescharacteristic
57C23×C4⋊Q8D411yescharacteristic
58C23×C4⋊Q8D411yescharacteristic
59C23×C4⋊Q8C2311yescharacteristic
60C23×C4⋊Q8C2311yescharacteristic
61C23×C4⋊Q8C2311yescharacteristic
62C23×#127C2311yescharacteristic
63C23×#125—21no—
64C23×#125—21no—
65C23×#124—21no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8D412yes—
70C23×#121—21no—
71C23×#121—21no—
72C4×Q8×C23—21no—
73C4×Q8×C23—21no—
74C4×Q8×C23—21no—
75C4×Q8×C23C2312yes—
76C4×D4×C23—21no—
77C4×D4×C23—21no—
78C23×#117—21no—
79C23×#117—21no—
80C2×C42×C23C2311yescharacteristic
81C23×#107—22no—
82C23×#106—21no—
83C23×#106—21no—
84C23×#106—22no—
85C23×C4⋊C8—22no—
86C23×C4⋊C8—22no—
87C23×C4⋊C8—21no—
88C23×C4⋊C8C2312yes—
89C23×C4⋊C8C2312yes—
90C23×C4⋊C8C2312yes—
91C23×#103—22no—
92C23×#103—22no—
93C23×#103—21no—
94C23×#103—21no—
95C23×#103—22no—
96C23×#102—21no—
97C23×#102—21no—
98C23×#102—22no—
99C23×C22⋊C8—21no—
100C23×C22⋊C8—21no—
101C23×#97—22no—
102C23×#97—21no—
103C23×#97—21no—
104C4×C8×C23—22no—
105C4×C8×C23D411yescharacteristic
106C4×C8×C23D411yescharacteristic
107C4×C8×C23C2311yescharacteristic
108C4×C8×C23C2311yescharacteristic
109#41×C23—82no—
110C2×Q8×C23—21no—
111C2×Q8×C23—21no—
112C2×Q8×C23—21no—
113C2×Q8×C23—21no—
114C2×Q8×C23C2×D412yes—
115C2×D4×C23—21no—
116C2×D4×C23—21no—
117C2×D4×C23—21no—
118C2×D4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23C2×D411yescharacteristic
121C22×C4×C23C2×D411yescharacteristic
122C22×C4×C23#4111yescharacteristic
123Q16×C23—82no—
124SD16×C23—42no—
125SD16×C23—42no—
126SD16×C23—82no—
127M16×C23—42no—
128C2×C8×C23—22no—
129C2×C8×C23—22no—
130C2×C8×C23—22no—
131C2×C8×C23—22no—
132C2×C8×C23—41no—
133C2×C8×C23C2×D411yescharacteristic
134C2×C8×C23C2×D411yescharacteristic
135C2×C8×C23C2×D411yescharacteristic
136C2×C8×C23C2×D411yescharacteristic
137C4⋊C4×C23—21no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—21no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—21no—
143C4⋊C4×C23—21no—
144C4⋊C4×C23—21no—
145C4⋊C4×C23—21no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—41no—
149C4⋊C4×C23—41no—
150C4⋊C4×C23C2×D412yes—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—41no—
153C22⋊C4×C23—41no—
154C22⋊C4×C23—41no—
155C42×C23—22no—
156C42×C23—41no—
157C42×C23C2×D411yescharacteristic
158C42×C23C2×D411yescharacteristic
159C42×C23#4111yescharacteristic
160C42×C23C2411yescharacteristic
161#19346C2311yesmaximal, characteristic, direct product factor, Sylow
162C23×C23—41no—
163C23×C23#12111yescharacteristic
164Q8×C23—22no—
165Q8×C23—22no—
166Q8×C23—42no—
167Q8×C23—41no—
168Q8×C23—41no—
169Q8×C23—42no—
170Q8×C23—81no—
171D4×C23—41no—
172D4×C23—41no—
173D4×C23—42no—
174D4×C23—81no—
175D4×C23—82no—
176C2×C4×C23—21no—
177C2×C4×C23—21no—
178C2×C4×C23—22no—
179C2×C4×C23—22no—
180C2×C4×C23—22no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—41no—
185C2×C4×C23—82no—
186C2×C4×C23#12111yescharacteristic
187C2×C4×C23#12111yescharacteristic
188C2×C4×C23#12111yescharacteristic
189C2×C4×C23C22×D411yescharacteristic
190C2×C4×C23C22×D411yescharacteristic
191C2×C4×C23C2×#4111yescharacteristic
192C8×C23—21no—
193C8×C23—21no—
194C8×C23—21no—
195C8×C23—42no—
196C8×C23—42no—
197C8×C23—42no—
198C8×C23#13712yes—
199#3237C2×C2311yescharacteristic, complemented
200#3216C2×C2311yescharacteristic
201#3167C2×C2311yescharacteristic
202#2960C2×C2311yescharacteristic, complemented
203#1692C2×C2312yescomplemented
204#1686C2×C2312yescomplemented
205#1680C2×C2312yescomplemented
206#1678C2×C2312yescomplemented
207#1568C2×C2311yescharacteristic, complemented
208#1559C2×C2311yescharacteristic, complemented
209#1557C2×C2311yescharacteristic
210C22×C23—21no—
211C22×C23—21no—
212C22×C23—21no—
213C22×C23—81no—
214C22×C23—81no—
215C22×C23—81no—
216C22×C23C2×#12111yescharacteristic
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—42no—
222C4×C23—42no—
223C4×C23—81no—
224C4×C23—81no—
225C4×C23—81no—
226C4×C23C2×#13711yescharacteristic
227C4×C23#57511yescharacteristic
228C4×C23#57712yes—
229#533C22×C2311yescharacteristic
230#533C22×C2311yescharacteristic
231#532C22×C2311yescharacteristic
232#501—21no—
233#498C22×C2311yescharacteristic
234#497—21no—
235#492C22×C2311yescharacteristic
236#491—21no—
237#489—21no—
238#487C22×C2311yescharacteristic
239#486C22×C2311yescharacteristic
240#485—22no—
241#477C22×C2312yes—
242#475C22×C2312yes—
243#465—21no—
244#465—21no—
245#462C22×C2312yes—
246#460C22×C2312yes—
247#452—22no—
248#446—21no—
249#446C22×C2312yes—
250C8×Q8—21no—
251C8×Q8C22×C2312yes—
252C4×SD16—21no—
253C4×SD16—21no—
254#432C22×C2311yescharacteristic
255#432C22×C2311yescharacteristic
256#424—21no—
257#423C22×C2311yescharacteristic
258#405—21no—
259C2×C4×C8C22×C2311yescharacteristic
260#335C22×C2312yes—
261#334C22×C2312yes—
262#333C22×C2312yes—
263#332C22×C2312yes—
264#332C22×C2312yes—
265#331C22×C2312yes—
266C2×C23—41no—
267C2×C23—161no—
268C2×C23#303711yescharacteristic
269C2×C23#305711yescharacteristic
270C2×C23#306711yescharacteristic
271C2×#41—21no—
272C2×Q16—22no—
273C2×SD16—22no—
274C2×SD16—21no—
275C2×SD16—21no—
276C2×M16—21no—
277C22×C8D4×C2311yescharacteristic
278C22×C8D4×C2311yescharacteristic
279C4⋊Q8D4×C2311yescharacteristic
280C4⋊Q8D4×C2311yescharacteristic
281C4⋊Q8C23×C2311yescharacteristic
282C4⋊Q8C23×C2311yescharacteristic
283C4⋊Q8C23×C2311yescharacteristic
284#127C23×C2311yescharacteristic
285#125—21no—
286#125—21no—
287#124—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8D4×C2312yes—
292#121—21no—
293#121—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8C23×C2312yes—
298C4×D4—21no—
299C4×D4—21no—
300#117—21no—
301#117—21no—
302C2×C42C23×C2311yescharacteristic
303#107—22no—
304#106—21no—
305#106—21no—
306#106—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—21no—
310C4⋊C8C23×C2312yes—
311C4⋊C8C23×C2312yes—
312C4⋊C8C23×C2312yes—
313#103—22no—
314#103—22no—
315#103—21no—
316#103—21no—
317#103—22no—
318#102—21no—
319#102—21no—
320#102—22no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323#97—22no—
324#97—21no—
325#97—21no—
326C4×C8—22no—
327C4×C8D4×C2311yescharacteristic
328C4×C8D4×C2311yescharacteristic
329C4×C8C23×C2311yescharacteristic
330C4×C8C23×C2311yescharacteristic
331C23#1934611yescharacteristic, direct product factor, Sylow
332#41—82no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C2×D4×C2312yes—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—41no—
342C22×C4—41no—
343C22×C4C2×D4×C2311yescharacteristic
344C22×C4C2×D4×C2311yescharacteristic
345C22×C4#41×C2311yescharacteristic
346Q16—82no—
347SD16—42no—
348SD16—42no—
349SD16—82no—
350M16—42no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—41no—
356C2×C8C2×D4×C2311yescharacteristic
357C2×C8C2×D4×C2311yescharacteristic
358C2×C8C2×D4×C2311yescharacteristic
359C2×C8C2×D4×C2311yescharacteristic
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4C2×D4×C2312yes—
374C22⋊C4—22no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C42—22no—
379C42—41no—
380C42C2×D4×C2311yescharacteristic
381C42C2×D4×C2311yescharacteristic
382C42#41×C2311yescharacteristic
383C42C24×C2311yescharacteristic
384C23—41no—
385C23C23×#12111yescharacteristic
386Q8—22no—
387Q8—22no—
388Q8—42no—
389Q8—41no—
390Q8—41no—
391Q8—42no—
392Q8—81no—
393D4—41no—
394D4—41no—
395D4—42no—
396D4—81no—
397D4—82no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4C23×#12111yescharacteristic
409C2×C4C23×#12111yescharacteristic
410C2×C4C23×#12111yescharacteristic
411C2×C4C22×D4×C2311yescharacteristic
412C2×C4C22×D4×C2311yescharacteristic
413C2×C4C2×#41×C2311yescharacteristic
414C8—21no—
415C8—21no—
416C8—21no—
417C8—42no—
418C8—42no—
419C8—42no—
420C8C23×#13712yes—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22C2×C23×#12111yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4C2×C23×#13711yescharacteristic
438C4C23×#57511yescharacteristic
439C4C23×#57712yes—
440C2—41no—
441C2—161no—
442C2C23×#303711yescharacteristic
443C2C23×#305711yescharacteristic
444C2C23×#306711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#121
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3237C2semidirect
4. C23×#2960C2semidirect
5. C23×#1692C2semidirect
6. C23×#1686C2semidirect
7. C23×#1680C2semidirect
8. C23×#1678C2semidirect
9. C23×#1568C2semidirect
10. C23×#1559C2semidirect
161. #19346C23direct (N × H)
199. #3237C2×C23semidirect
202. #2960C2×C23semidirect
203. #1692C2×C23semidirect
204. #1686C2×C23semidirect
205. #1680C2×C23semidirect
206. #1678C2×C23semidirect
207. #1568C2×C23semidirect
208. #1559C2×C23semidirect
331. C23#19346direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32372
C23×#32162
C23×#31672
C23×#29602
C23×#16922
C23×#16862
C23×#16802
C23×#16782
C23×#15682
C23×#15592
C23×#15572
#1934623