Finite Group Databasebeta

← C23×#8382 #19156008

Subgroups — C23×#8382 #19156008

Order 5888.

780 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#552C211yesmaximal, characteristic
2C23×#1823C212yesmaximal, complemented
3C23×#1799C211yesmaximal, characteristic
4C23×#1520C212yesmaximal, complemented
5C23×#1459C211yesmaximal, characteristic
6C23×#552—24no—
7C23×#552—24no—
8C2×C23×#126—21no—
9C2×C23×#125—21no—
10C2×C23×#125C411yescharacteristic, complemented
11C2×C23×#123—21no—
12C2×C23×C22⋊Q8C411yescharacteristic, complemented
13C2×C23×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8×C23C411yescharacteristic, complemented
15C2×C4×D4×C23C2211yescharacteristic
16C2×C23×#117—21no—
17C2×C23×#117C411yescharacteristic, complemented
18C22×C4⋊C4×C23C2211yescharacteristic
19C2×C23×#99—21no—
20C2×C23×C22⋊C8C2212yes—
21C23×#380C2212yes—
22C23×#342—21no—
23C23×#340—24no—
24C23×#328—24no—
25C23×#324—21no—
26C23×#323—21no—
27C22×Q8×C23C2×C411yescharacteristic
28C22×D4×C23D411yescharacteristic
29C23×C4×C23—21no—
30C23×C4×C23C2×C411yescharacteristic
31C23×C4×C23C2311yescharacteristic
32C22×C8×C23—22no—
33C23×#126—28no—
34C23×#125—28no—
35C23×#125—22no—
36C23×#125—22no—
37C23×#123—28no—
38C23×C22⋊Q8—22no—
39C23×C22⋊Q8—22no—
40C23×C22⋊Q8—22no—
41C23×C22⋊Q8D414yes—
42C4×Q8×C23—22no—
43C4×Q8×C23—22no—
44C4×D4×C23—22no—
45C4×D4×C23—22no—
46C23×#117—28no—
47C23×#117—22no—
48C23×#117—22no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23C2×C411yescharacteristic
58C2×C4⋊C4×C23C2×C411yescharacteristic
59C2×C4⋊C4×C23C2×C411yescharacteristic
60C2×C4⋊C4×C23D411yescharacteristic
61C2×C4⋊C4×C23D414yes—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—21no—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23D411yescharacteristic
67C2×C22⋊C4×C23D411yescharacteristic
68C2×C42×C23—21no—
69C2×C42×C23—21no—
70C2×C42×C23C2×C411yescharacteristic
71C23×#99—42no—
72C23×C22⋊C8—22no—
73C23×C22⋊C8—22no—
74C23×#95—22no—
75C24×C23C22⋊C411yescharacteristic
76C24×C23C2×D411yescharacteristic
77C2×Q8×C23—21no—
78C2×Q8×C23—42no—
79C2×Q8×C23C22⋊C412yes—
80C2×D4×C23—22no—
81C2×D4×C23—22no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—24no—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—42no—
92C22×C4×C23—42no—
93C22×C4×C23—42no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—42no—
97C22×C4×C23C22⋊C411yescharacteristic
98C22×C4×C23C22⋊C411yescharacteristic
99C22×C4×C23C22⋊C411yescharacteristic
100C22×C4×C23C22×C411yescharacteristic
101C22×C4×C23C2×D412yes—
102C22×C4×C23C2×D412yes—
103C22×C4×C23C2×D411yescharacteristic
104C2×C8×C23—44no—
105C2×C8×C23—42no—
106C4⋊C4×C23—24no—
107C4⋊C4×C23—24no—
108C4⋊C4×C23—24no—
109C4⋊C4×C23—24no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—24no—
114C4⋊C4×C23—24no—
115C4⋊C4×C23—22no—
116C4⋊C4×C23—44no—
117C4⋊C4×C23C22⋊C412yes—
118C4⋊C4×C23C22⋊C412yes—
119C4⋊C4×C23C22⋊C412yes—
120C22⋊C4×C23—24no—
121C22⋊C4×C23—24no—
122C22⋊C4×C23—24no—
123C22⋊C4×C23—22no—
124C22⋊C4×C23—22no—
125C22⋊C4×C23—42no—
126C22⋊C4×C23—42no—
127C42×C23—24no—
128C42×C23—24no—
129C42×C23—21no—
130C42×C23—21no—
131#8382C2311yesmaximal, characteristic, direct product factor, Sylow
132C23×C23—21no—
133C23×C23—21no—
134C23×C23—21no—
135C23×C23—41no—
136C23×C23—41no—
137C23×C23—41no—
138C23×C23—41no—
139C23×C23C2×C22⋊C411yescharacteristic
140C23×C23#12012yes—
141C23×C23#13512yes—
142C23×C23AGL(1,ℤ/8)11yescharacteristic
143C23×C23#13711yescharacteristic
144Q8×C23—42no—
145Q8×C23—42no—
146D4×C23—42no—
147D4×C23—42no—
148C2×C4×C23—24no—
149C2×C4×C23—22no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—22no—
153C2×C4×C23—22no—
154C2×C4×C23—22no—
155C2×C4×C23—21no—
156C2×C4×C23—22no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—41no—
160C2×C4×C23—42no—
161C2×C4×C23—42no—
162C2×C4×C23—42no—
163C2×C4×C23—42no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—44no—
169C2×C4×C23—44no—
170C2×C4×C23—42no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—82no—
174C2×C4×C23—82no—
175C2×C4×C23—82no—
176C2×C4×C23C2×C22⋊C411yescharacteristic
177C2×C4×C23C2×C22⋊C411yescharacteristic
178C2×C4×C23#12012yes—
179C8×C23—82no—
180C2×#552C2×C2311yescharacteristic
181#1823C2×C2312yescomplemented
182#1799C2×C2311yescharacteristic
183#1520C2×C2312yescomplemented
184#1459C2×C2311yescharacteristic
185C22×C23—21no—
186C22×C23—22no—
187C22×C23—21no—
188C22×C23—22no—
189C22×C23—42no—
190C22×C23—41no—
191C22×C23—41no—
192C22×C23—41no—
193C22×C23—41no—
194C22×C23—41no—
195C22×C23—81no—
196C22×C23—81no—
197C22×C23#37911yescharacteristic
198C22×C23#41611yescharacteristic
199C22×C23#41711yescharacteristic
200C22×C23#44912yes—
201C22×C23#45212yes—
202C4×C23—22no—
203C4×C23—21no—
204C4×C23—41no—
205C4×C23—42no—
206C4×C23—41no—
207C4×C23—42no—
208C4×C23—41no—
209C4×C23—42no—
210C4×C23—42no—
211C4×C23—162no—
212C4×C23#41012yes—
213#552—24no—
214#552—24no—
215C2×#126—21no—
216C2×#125—21no—
217C2×#125C4×C2311yescharacteristic, complemented
218C2×#123—21no—
219C2×C22⋊Q8C4×C2311yescharacteristic, complemented
220C2×C22⋊Q8C22×C2311yescharacteristic
221C2×C4×Q8C4×C2311yescharacteristic, complemented
222C2×C4×D4C22×C2311yescharacteristic
223C2×#117—21no—
224C2×#117C4×C2311yescharacteristic, complemented
225C22×C4⋊C4C22×C2311yescharacteristic
226C2×#99—21no—
227C2×C22⋊C8C22×C2312yes—
228#380C22×C2312yes—
229#342—21no—
230#340—24no—
231#328—24no—
232#324—21no—
233#323—21no—
234C2×C23—22no—
235C2×C23—21no—
236C2×C23—21no—
237C2×C23—22no—
238C2×C23—81no—
239C2×C23#178211yescharacteristic
240C2×C23#178811yescharacteristic
241C2×C23#179611yescharacteristic
242C22×Q8C2×C4×C2311yescharacteristic
243C22×D4D4×C2311yescharacteristic
244C23×C4—21no—
245C23×C4C2×C4×C2311yescharacteristic
246C23×C4C23×C2311yescharacteristic
247C22×C8—22no—
248#126—28no—
249#125—28no—
250#125—22no—
251#125—22no—
252#123—28no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8D4×C2314yes—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—22no—
261#117—28no—
262#117—22no—
263#117—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C2×C4×C2311yescharacteristic
273C2×C4⋊C4C2×C4×C2311yescharacteristic
274C2×C4⋊C4C2×C4×C2311yescharacteristic
275C2×C4⋊C4D4×C2311yescharacteristic
276C2×C4⋊C4D4×C2314yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4D4×C2311yescharacteristic
282C2×C22⋊C4D4×C2311yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C2×C4×C2311yescharacteristic
286#99—42no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289#95—22no—
290C23#838211yescharacteristic, direct product factor, Sylow
291C24C22⋊C4×C2311yescharacteristic
292C24C2×D4×C2311yescharacteristic
293C2×Q8—21no—
294C2×Q8—42no—
295C2×Q8C22⋊C4×C2312yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4C22⋊C4×C2311yescharacteristic
314C22×C4C22⋊C4×C2311yescharacteristic
315C22×C4C22⋊C4×C2311yescharacteristic
316C22×C4C22×C4×C2311yescharacteristic
317C22×C4C2×D4×C2312yes—
318C22×C4C2×D4×C2312yes—
319C22×C4C2×D4×C2311yescharacteristic
320C2×C8—44no—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4C22⋊C4×C2312yes—
334C4⋊C4C22⋊C4×C2312yes—
335C4⋊C4C22⋊C4×C2312yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C42—24no—
344C42—24no—
345C42—21no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C2×C22⋊C4×C2311yescharacteristic
355C23C23×#12012yes—
356C23C23×#13512yes—
357C23C23×AGL(1,ℤ/8)11yescharacteristic
358C23C23×#13711yescharacteristic
359Q8—42no—
360Q8—42no—
361D4—42no—
362D4—42no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4C2×C22⋊C4×C2311yescharacteristic
392C2×C4C2×C22⋊C4×C2311yescharacteristic
393C2×C4C23×#12012yes—
394C8—82no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22C23×#37911yescharacteristic
408C22C23×#41611yescharacteristic
409C22C23×#41711yescharacteristic
410C22C23×#44912yes—
411C22C23×#45212yes—
412C4—22no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—162no—
422C4C23×#41012yes—
423C2—22no—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—81no—
428C2C23×#178211yescharacteristic
429C2C23×#178811yescharacteristic
430C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1823C2semidirect
4. C23×#1520C2semidirect
10. C2×C23×#125C4semidirect
12. C2×C23×C22⋊Q8C4semidirect
14. C2×C4×Q8×C23C4semidirect
17. C2×C23×#117C4semidirect
131. #8382C23direct (N × H)
181. #1823C2×C23semidirect
183. #1520C2×C23semidirect
217. C2×#125C4×C23semidirect
219. C2×C22⋊Q8C4×C23semidirect
221. C2×C4×Q8C4×C23semidirect
224. C2×#117C4×C23semidirect
290. C23#8382direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5522
C23×#18232
C23×#17992
C23×#15202
C23×#14592
#838223