Finite Group Databasebeta

← C23×#23895 #19171521

Subgroups — C23×#23895 #19171521

Order 5888.

2002 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2002 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3484C212yesmaximal, complemented
2C23×#3026C214yesmaximal, complemented
3C23×#2021C214yesmaximal, complemented
4C23×#1913C214yesmaximal, complemented
5C23×#1781C211yesmaximal, characteristic, complemented
6C2×C23×H(5,𝔽2)C2212yescomplemented
7C23×D4×C23C2211yescharacteristic
8C23×#546—28no—
9D42×C23—28no—
10C23×#535—22no—
11C23×#534—22no—
12C23×#534C2214yes—
13C23×#534C2214yescomplemented
14C23×#525—22no—
15C2×C23×#120—22no—
16C2×C23×#120C2214yescomplemented
17C23×#518C2212yescomplemented
18C23×#458—28no—
19C23×B(4,𝔽2)—28no—
20C2×C23×#99C2218yescomplemented
21C2×C23×#99C2214yescomplemented
22C23×#379C2212yescomplemented
23C23×#342C2214yescomplemented
24C25×C23D411yescharacteristic, complemented
25C25×C23C2311yescharacteristic
26C23×H(5,𝔽2)—28no—
27C23×H(5,𝔽2)—44no—
28C2×#41×C23—22no—
29C2×#41×C23—24no—
30C22×D4×C23—22no—
31C22×D4×C23—24no—
32C22×D4×C23—24no—
33C22×D4×C23—24no—
34C22×D4×C23—28no—
35C22×D4×C23—42no—
36C22×D4×C23D412yescomplemented
37C22×D4×C23C2312yes—
38C22×D4×C23C2314yes—
39C23×C4×C23D411yescharacteristic, complemented
40C23×#127—24no—
41C23×#124—24no—
42C23×#123—24no—
43C23×#123—24no—
44C23×#123—28no—
45C23×#123D418yescomplemented
46C23×C22⋊Q8—24no—
47C23×#121—24no—
48C23×#121—28no—
49C23×#121—28no—
50C23×#121—28no—
51C23×#121—44no—
52C23×#120—28no—
53C23×#120—24no—
54C23×#120—24no—
55C23×#120—24no—
56C23×#120—24no—
57C23×#120—28no—
58C23×#120—28no—
59C23×#120D418yescomplemented
60C4×D4×C23—28no—
61C4×D4×C23—28no—
62C23×#117—22no—
63C2×C22⋊C4×C23—24no—
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23—24no—
66C2×C22⋊C4×C23—42no—
67C2×C22⋊C4×C23C2314yes—
68C2×C22⋊C4×C23C2314yes—
69C23×#99—28no—
70C23×#99—216no—
71C24×C23—22no—
72C24×C23—24no—
73C24×C23—24no—
74C24×C23—22no—
75C24×C23—41no—
76C24×C23—82no—
77C24×C23—82no—
78C24×C23C2×D411yescharacteristic
79C24×C23C2×D411yescharacteristic
80C24×C23C2×D412yes—
81C24×C23C2×D414yescomplemented
82C24×C23C2411yescharacteristic
83#41×C23—416no—
84#41×C23—44no—
85#41×C23—44no—
86C2×Q8×C23—22no—
87C2×D4×C23—24no—
88C2×D4×C23—24no—
89C2×D4×C23—28no—
90C2×D4×C23—24no—
91C2×D4×C23—22no—
92C2×D4×C23—24no—
93C2×D4×C23—28no—
94C2×D4×C23—24no—
95C2×D4×C23—24no—
96C2×D4×C23—28no—
97C2×D4×C23—28no—
98C2×D4×C23—28no—
99C2×D4×C23—24no—
100C2×D4×C23—42no—
101C2×D4×C23—416no—
102C2×D4×C23—416no—
103C2×D4×C23—44no—
104C2×D4×C23—44no—
105C2×D4×C23—44no—
106C2×D4×C23—88no—
107C2×D4×C23—88no—
108C2×D4×C23C2×D418yes—
109C22×C4×C23—22no—
110C22×C4×C23—24no—
111C22×C4×C23—24no—
112C22×C4×C23—24no—
113C22×C4×C23—24no—
114C22×C4×C23—24no—
115C22×C4×C23—41no—
116C22×C4×C23C2×D412yes—
117C22×C4×C23C2×D414yescomplemented
118C4⋊C4×C23—24no—
119C4⋊C4×C23—28no—
120C4⋊C4×C23—44no—
121C22⋊C4×C23—28no—
122C22⋊C4×C23—24no—
123C22⋊C4×C23—28no—
124C22⋊C4×C23—28no—
125C22⋊C4×C23—24no—
126C22⋊C4×C23—28no—
127C22⋊C4×C23—28no—
128C22⋊C4×C23—48no—
129C22⋊C4×C23—84no—
130C22⋊C4×C23C2×D418yescomplemented
131C42×C23—24no—
132#23895C2311yesmaximal, characteristic, direct product factor, Sylow
133C23×C23—22no—
134C23×C23—22no—
135C23×C23—21no—
136C23×C23—22no—
137C23×C23—22no—
138C23×C23—24no—
139C23×C23—21no—
140C23×C23—28no—
141C23×C23—42no—
142C23×C23—44no—
143C23×C23—44no—
144C23×C23—48no—
145C23×C23—48no—
146C23×C23—44no—
147C23×C23—44no—
148C23×C23—82no—
149C23×C23—81no—
150C23×C23—81no—
151C23×C23—82no—
152C23×C23—84no—
153C23×C23—82no—
154C23×C23—84no—
155C23×C23—84no—
156C23×C23—84no—
157C23×C23—82no—
158C23×C23#12014yes—
159C23×C23C22×D411yescharacteristic
160C23×C23C22×D414yes—
161C23×C23H(5,𝔽2)12yes—
162Q8×C23—44no—
163D4×C23—216no—
164D4×C23—48no—
165D4×C23—44no—
166D4×C23—48no—
167D4×C23—416no—
168D4×C23—44no—
169D4×C23—44no—
170D4×C23—48no—
171D4×C23—88no—
172D4×C23—88no—
173D4×C23—164no—
174C2×C4×C23—22no—
175C2×C4×C23—22no—
176C2×C4×C23—28no—
177C2×C4×C23—24no—
178C2×C4×C23—28no—
179C2×C4×C23—44no—
180C2×C4×C23—48no—
181C2×C4×C23—44no—
182C2×C4×C23—44no—
183C2×C4×C23—84no—
184C2×C4×C23—84no—
185C2×C4×C23—84no—
186C2×C4×C23#12014yes—
187#3484C2×C2312yescomplemented
188#3026C2×C2314yescomplemented
189#2021C2×C2314yescomplemented
190#1913C2×C2314yescomplemented
191#1781C2×C2311yescharacteristic, complemented
192C22×C23—24no—
193C22×C23—22no—
194C22×C23—24no—
195C22×C23—42no—
196C22×C23—44no—
197C22×C23—41no—
198C22×C23—44no—
199C22×C23—44no—
200C22×C23—42no—
201C22×C23—41no—
202C22×C23—48no—
203C22×C23—44no—
204C22×C23—42no—
205C22×C23—41no—
206C22×C23—44no—
207C22×C23—41no—
208C22×C23—82no—
209C22×C23—88no—
210C22×C23—88no—
211C22×C23—88no—
212C22×C23—164no—
213C22×C23C2×#12012yes—
214C22×C23#53411yescharacteristic
215C22×C23D4214yes—
216C4×C23—24no—
217C4×C23—48no—
218C4×C23—44no—
219C4×C23—84no—
220C2×H(5,𝔽2)C22×C2312yescomplemented
221C23×D4C22×C2311yescharacteristic
222#546—28no—
223D42—28no—
224#535—22no—
225#534—22no—
226#534C22×C2314yes—
227#534C22×C2314yescomplemented
228#525—22no—
229C2×#120—22no—
230C2×#120C22×C2314yescomplemented
231#518C22×C2312yescomplemented
232#458—28no—
233B(4,𝔽2)—28no—
234C2×#99C22×C2318yescomplemented
235C2×#99C22×C2314yescomplemented
236#379C22×C2312yescomplemented
237#342C22×C2314yescomplemented
238C2×C23—22no—
239C2×C23—24no—
240C2×C23—21no—
241C2×C23—42no—
242C2×C23—44no—
243C2×C23—81no—
244C2×C23—84no—
245C2×C23#240311yescharacteristic
246C25D4×C2311yescharacteristic, complemented
247C25C23×C2311yescharacteristic
248H(5,𝔽2)—28no—
249H(5,𝔽2)—44no—
250C2×#41—22no—
251C2×#41—24no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4—24no—
255C22×D4—24no—
256C22×D4—28no—
257C22×D4—42no—
258C22×D4D4×C2312yescomplemented
259C22×D4C23×C2312yes—
260C22×D4C23×C2314yes—
261C23×C4D4×C2311yescharacteristic, complemented
262#127—24no—
263#124—24no—
264#123—24no—
265#123—24no—
266#123—28no—
267#123D4×C2318yescomplemented
268C22⋊Q8—24no—
269#121—24no—
270#121—28no—
271#121—28no—
272#121—28no—
273#121—44no—
274#120—28no—
275#120—24no—
276#120—24no—
277#120—24no—
278#120—24no—
279#120—28no—
280#120—28no—
281#120D4×C2318yescomplemented
282C4×D4—28no—
283C4×D4—28no—
284#117—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4C23×C2314yes—
290C2×C22⋊C4C23×C2314yes—
291#99—28no—
292#99—216no—
293C23#2389511yescharacteristic, direct product factor, Sylow
294C24—22no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—41no—
299C24—82no—
300C24—82no—
301C24C2×D4×C2311yescharacteristic
302C24C2×D4×C2311yescharacteristic
303C24C2×D4×C2312yes—
304C24C2×D4×C2314yescomplemented
305C24C24×C2311yescharacteristic
306#41—416no—
307#41—44no—
308#41—44no—
309C2×Q8—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—28no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4C2×D4×C2318yes—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—41no—
339C22×C4C2×D4×C2312yes—
340C22×C4C2×D4×C2314yescomplemented
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—48no—
352C22⋊C4—84no—
353C22⋊C4C2×D4×C2318yescomplemented
354C42—24no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—21no—
362C23—28no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—48no—
367C23—48no—
368C23—44no—
369C23—44no—
370C23—82no—
371C23—81no—
372C23—81no—
373C23—82no—
374C23—84no—
375C23—82no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23—82no—
380C23C23×#12014yes—
381C23C22×D4×C2311yescharacteristic
382C23C22×D4×C2314yes—
383C23C23×H(5,𝔽2)12yes—
384Q8—44no—
385D4—216no—
386D4—48no—
387D4—44no—
388D4—48no—
389D4—416no—
390D4—44no—
391D4—44no—
392D4—48no—
393D4—88no—
394D4—88no—
395D4—164no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4C23×#12014yes—
409C22—24no—
410C22—22no—
411C22—24no—
412C22—42no—
413C22—44no—
414C22—41no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—41no—
419C22—48no—
420C22—44no—
421C22—42no—
422C22—41no—
423C22—44no—
424C22—41no—
425C22—82no—
426C22—88no—
427C22—88no—
428C22—88no—
429C22—164no—
430C22C2×C23×#12012yes—
431C22C23×#53411yescharacteristic
432C22D42×C2314yes—
433C4—24no—
434C4—48no—
435C4—44no—
436C4—84no—
437C2—22no—
438C2—24no—
439C2—21no—
440C2—42no—
441C2—44no—
442C2—81no—
443C2—84no—
444C2C23×#240311yescharacteristic
subgroupG/H
center, socleC2×C23#2403
commutator, Frattini, cosocleC24C24×C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3484C2semidirect
2. C23×#3026C2semidirect
3. C23×#2021C2semidirect
4. C23×#1913C2semidirect
5. C23×#1781C2semidirect
6. C2×C23×H(5,𝔽2)C22semidirect
13. C23×#534C22semidirect
16. C2×C23×#120C22semidirect
17. C23×#518C22semidirect
20. C2×C23×#99C22semidirect
21. C2×C23×#99C22semidirect
22. C23×#379C22semidirect
23. C23×#342C22semidirect
24. C25×C23D4semidirect
36. C22×D4×C23D4semidirect
39. C23×C4×C23D4semidirect
45. C23×#123D4semidirect
59. C23×#120D4semidirect
81. C24×C23C2×D4semidirect
117. C22×C4×C23C2×D4semidirect
130. C22⋊C4×C23C2×D4semidirect
132. #23895C23direct (N × H)
187. #3484C2×C23semidirect
188. #3026C2×C23semidirect
189. #2021C2×C23semidirect
190. #1913C2×C23semidirect
191. #1781C2×C23semidirect
220. C2×H(5,𝔽2)C22×C23semidirect
227. #534C22×C23semidirect
230. C2×#120C22×C23semidirect
231. #518C22×C23semidirect
234. C2×#99C22×C23semidirect
235. C2×#99C22×C23semidirect
236. #379C22×C23semidirect
237. #342C22×C23semidirect
246. C25D4×C23semidirect
258. C22×D4D4×C23semidirect
261. C23×C4D4×C23semidirect
267. #123D4×C23semidirect
281. #120D4×C23semidirect
293. C23#23895direct (N × H)
304. C24C2×D4×C23semidirect
340. C22×C4C2×D4×C23semidirect
353. C22⋊C4C2×D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34842
C23×#30262
C23×#20212
C23×#19132
C23×#17812
#2389523