Finite Group Databasebeta

← C23×#8454 #19156080

Subgroups — C23×#8454 #19156080

Order 5888.

838 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic
2C23×#1821C211yesmaximal, characteristic, complemented
3C23×#1504C212yesmaximal, complemented
4C23×#1504C212yesmaximal, complemented
5C23×#1445C211yesmaximal, characteristic, complemented
6C23×#554—22no—
7C23×#553—22no—
8C23×#552—22no—
9C23×#547—22no—
10C23×#544C411yescharacteristic, complemented
11C23×#539—22no—
12C23×#537C411yescharacteristic, complemented
13C23×#525—22no—
14C23×#525C2212yes—
15C23×#520C412yescomplemented
16C22×C4⋊C4×C23C2211yescharacteristic
17C23×#407C2212yes—
18C2×C23×#95—22no—
19C2×C23×#95C2211yescharacteristic
20C2×C23×#95C2211yescharacteristic
21C23×#343—22no—
22C23×#328—22no—
23C23×#328—22no—
24C23×#327—22no—
25C23×#327—22no—
26C2×#41×C23—22no—
27C2×#41×C23C2×C412yes—
28C23×C4×C23—21no—
29C23×C4×C23—21no—
30C23×C4×C23D412yes—
31C23×C4×C23C2311yescharacteristic
32C2×M16×C23—22no—
33C23×C4⋊Q8—22no—
34C23×#126—24no—
35C23×#125—24no—
36C23×#125—22no—
37C23×#123—24no—
38C23×#123—24no—
39C23×#123—22no—
40C23×C22⋊Q8—24no—
41C23×C22⋊Q8—22no—
42C23×C22⋊Q8—22no—
43C23×C22⋊Q8—22no—
44C23×C22⋊Q8D412yes—
45C23×C22⋊Q8D412yes—
46C23×#121—22no—
47C23×#121—22no—
48C23×#121—22no—
49C23×#121D412yes—
50C23×#121D412yes—
51C23×#120—22no—
52C4×Q8×C23—22no—
53C4×D4×C23—24no—
54C4×D4×C23—22no—
55C4×D4×C23—24no—
56C4×D4×C23—22no—
57C4×D4×C23—22no—
58C23×#117—22no—
59C23×#117—22no—
60C23×#117C2×C412yes—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23—22no—
63C2×C4⋊C4×C23—22no—
64C2×C4⋊C4×C23C2×C411yescharacteristic
65C2×C4⋊C4×C23C2×C411yescharacteristic
66C2×C4⋊C4×C23D412yes—
67C23×C22⋊C8—22no—
68C23×C22⋊C8—22no—
69C23×#95—22no—
70C23×#95—22no—
71C23×#95—44no—
72C24×C23C2×D411yescharacteristic
73#41×C23—44no—
74#41×C23—82no—
75#41×C23—82no—
76C2×Q8×C23—22no—
77C2×Q8×C23C22⋊C412yes—
78C2×D4×C23—22no—
79C2×D4×C23—22no—
80C2×D4×C23—22no—
81C2×D4×C23—22no—
82C2×D4×C23C22⋊C412yes—
83C22×C4×C23—22no—
84C22×C4×C23—24no—
85C22×C4×C23—22no—
86C22×C4×C23—22no—
87C22×C4×C23—22no—
88C22×C4×C23—22no—
89C22×C4×C23—42no—
90C22×C4×C23—42no—
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×C23—42no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23C22⋊C412yes—
101C22×C4×C23C22⋊C412yes—
102C22×C4×C23C22×C411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23C2×D412yes—
105C22×C4×C23C2×D412yes—
106M16×C23—44no—
107C2×C8×C23—42no—
108C4⋊C4×C23—24no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—24no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—22no—
114C4⋊C4×C23—24no—
115C4⋊C4×C23—22no—
116C4⋊C4×C23—22no—
117C4⋊C4×C23C22⋊C412yes—
118C4⋊C4×C23C22⋊C412yes—
119C22⋊C4×C23—24no—
120C22⋊C4×C23—24no—
121C22⋊C4×C23—22no—
122C22⋊C4×C23—22no—
123C22⋊C4×C23—22no—
124C22⋊C4×C23—22no—
125C22⋊C4×C23—22no—
126C42×C23—24no—
127C42×C23—22no—
128#8454C2311yesmaximal, characteristic, direct product factor, Sylow
129C23×C23—22no—
130C23×C23—21no—
131C23×C23—21no—
132C23×C23—21no—
133C23×C23—21no—
134C23×C23—22no—
135C23×C23C2×C22⋊C411yescharacteristic
136C23×C23#12011yescharacteristic
137C23×C23#12011yescharacteristic
138Q8×C23—42no—
139Q8×C23—82no—
140D4×C23—42no—
141D4×C23—44no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23—42no—
145D4×C23—42no—
146D4×C23—82no—
147C2×C4×C23—22no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—22no—
152C2×C4×C23—22no—
153C2×C4×C23—22no—
154C2×C4×C23—42no—
155C2×C4×C23—42no—
156C2×C4×C23—44no—
157C2×C4×C23—42no—
158C2×C4×C23—42no—
159C2×C4×C23—42no—
160C2×C4×C23—42no—
161C2×C4×C23—44no—
162C2×C4×C23—44no—
163C2×C4×C23—42no—
164C2×C4×C23—42no—
165C2×C4×C23—82no—
166C2×C4×C23—84no—
167C2×C4×C23—82no—
168C2×C4×C23—82no—
169C2×C4×C23—81no—
170C2×C4×C23—81no—
171C2×C4×C23—81no—
172C2×C4×C23—81no—
173C2×C4×C23C2×C22⋊C412yes—
174C2×C4×C23#12012yes—
175C8×C23—82no—
176#3492C2×C2311yescharacteristic
177#1821C2×C2311yescharacteristic, complemented
178#1504C2×C2312yescomplemented
179#1504C2×C2312yescomplemented
180#1445C2×C2311yescharacteristic, complemented
181C22×C23—22no—
182C22×C23—22no—
183C22×C23—22no—
184C22×C23—22no—
185C22×C23—21no—
186C22×C23—21no—
187C22×C23—21no—
188C22×C23—21no—
189C22×C23—21no—
190C22×C23—21no—
191C22×C23—42no—
192C22×C23—42no—
193C22×C23—42no—
194C22×C23#37911yescharacteristic
195C22×C23#41012yes—
196C22×C23#45311yescharacteristic
197C22×C23#45411yescharacteristic
198C22×C23#45511yescharacteristic
199C22×C23#45611yescharacteristic
200C4×C23—22no—
201C4×C23—22no—
202C4×C23—42no—
203C4×C23—42no—
204C4×C23—42no—
205C4×C23—42no—
206C4×C23—42no—
207C4×C23—82no—
208C4×C23—82no—
209#554—22no—
210#553—22no—
211#552—22no—
212#547—22no—
213#544C4×C2311yescharacteristic, complemented
214#539—22no—
215#537C4×C2311yescharacteristic, complemented
216#525—22no—
217#525C22×C2312yes—
218#520C4×C2312yescomplemented
219C22×C4⋊C4C22×C2311yescharacteristic
220#407C22×C2312yes—
221C2×#95—22no—
222C2×#95C22×C2311yescharacteristic
223C2×#95C22×C2311yescharacteristic
224#343—22no—
225#328—22no—
226#328—22no—
227#327—22no—
228#327—22no—
229C2×C23—21no—
230C2×C23—21no—
231C2×C23—22no—
232C2×C23—21no—
233C2×C23—21no—
234C2×C23—82no—
235C2×C23#178211yescharacteristic
236C2×C23#179211yescharacteristic
237C2×C23#179311yescharacteristic
238C2×#41—22no—
239C2×#41C2×C4×C2312yes—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4D4×C2312yes—
243C23×C4C23×C2311yescharacteristic
244C2×M16—22no—
245C4⋊Q8—22no—
246#126—24no—
247#125—24no—
248#125—22no—
249#123—24no—
250#123—24no—
251#123—22no—
252C22⋊Q8—22no—
253C22⋊Q8—24no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8D4×C2312yes—
257C22⋊Q8D4×C2312yes—
258#121—22no—
259#121—22no—
260#121—22no—
261#121D4×C2312yes—
262#121D4×C2312yes—
263#120—22no—
264C4×Q8—22no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—22no—
270#117—22no—
271#117—22no—
272#117C2×C4×C2312yes—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C2×C4×C2311yescharacteristic
277C2×C4⋊C4C2×C4×C2311yescharacteristic
278C2×C4⋊C4D4×C2312yes—
279C22⋊C8—22no—
280C22⋊C8—22no—
281#95—22no—
282#95—22no—
283#95—44no—
284C23#845411yescharacteristic, direct product factor, Sylow
285C24C2×D4×C2311yescharacteristic
286#41—44no—
287#41—82no—
288#41—82no—
289C2×Q8—22no—
290C2×Q8C22⋊C4×C2312yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4C22⋊C4×C2312yes—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C22⋊C4×C2312yes—
314C22×C4C22⋊C4×C2312yes—
315C22×C4C22×C4×C2311yescharacteristic
316C22×C4C2×D4×C2311yescharacteristic
317C22×C4C2×D4×C2312yes—
318C22×C4C2×D4×C2312yes—
319M16—44no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4C22⋊C4×C2312yes—
331C4⋊C4C22⋊C4×C2312yes—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C42—24no—
340C42—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23C2×C22⋊C4×C2311yescharacteristic
348C23C23×#12011yescharacteristic
349C23C23×#12011yescharacteristic
350Q8—42no—
351Q8—82no—
352D4—42no—
353D4—44no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4C2×C22⋊C4×C2312yes—
386C2×C4C23×#12012yes—
387C8—82no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22C23×#37911yescharacteristic
402C22C23×#41012yes—
403C22C23×#45311yescharacteristic
404C22C23×#45411yescharacteristic
405C22C23×#45511yescharacteristic
406C22C23×#45611yescharacteristic
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—82no—
415C4—82no—
416C2—21no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—21no—
421C2—82no—
422C2C23×#178211yescharacteristic
423C2C23×#179211yescharacteristic
424C2C23×#179311yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3492C2
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×#1821C2semidirect
3. C23×#1504C2semidirect
4. C23×#1504C2semidirect
5. C23×#1445C2semidirect
10. C23×#544C4semidirect
12. C23×#537C4semidirect
15. C23×#520C4semidirect
128. #8454C23direct (N × H)
177. #1821C2×C23semidirect
178. #1504C2×C23semidirect
179. #1504C2×C23semidirect
180. #1445C2×C23semidirect
213. #544C4×C23semidirect
215. #537C4×C23semidirect
218. #520C4×C23semidirect
284. C23#8454direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34922
C23×#18212
C23×#15042
C23×#14452
#845423