Finite Group Databasebeta

← C23×#8516 #19156142

Subgroups — C23×#8516 #19156142

Order 5888.

1168 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C23×#535C211yesmaximal, characteristic
2C23×#1785C211yesmaximal, characteristic, complemented
3C23×#1782C211yesmaximal, characteristic, complemented
4C23×#1510C214yesmaximal, complemented
5C22×#41×C23C2211yescharacteristic
6C23×D4×C23C411yescharacteristic, complemented
7C23×#535—28no—
8C2×C23×#127—21no—
9C2×C23×#124—21no—
10C2×C23×#121C412yescomplemented
11C2×C23×#121C2212yes—
12C2×C23×#120—21no—
13C2×C23×#120—21no—
14C2×C23×#117C411yescharacteristic, complemented
15C2×C23×#101—21no—
16C2×C23×#100—21no—
17C2×C23×#99—21no—
18C2×C23×#99—21no—
19C23×#407C2212yescomplemented
20C23×#381C2212yescomplemented
21C23×#327—28no—
22C25×C23—21no—
23C2×#41×C23—24no—
24C2×#41×C23—42no—
25C2×#41×C23—42no—
26C22×Q8×C23D411yescharacteristic
27C22×D4×C23—22no—
28C22×D4×C23—22no—
29C22×D4×C23—24no—
30C22×D4×C23—21no—
31C22×D4×C23—42no—
32C22×D4×C23—42no—
33C22×D4×C23C2×C412yes—
34C22×D4×C23D411yescharacteristic
35C22×D4×C23D411yescharacteristic
36C22×D4×C23D411yescharacteristic
37C23×C4×C23C2×C411yescharacteristic
38C23×C4×C23C2×C411yescharacteristic
39C23×C4×C23C2311yescharacteristic
40C2×M16×C23—22no—
41C23×#127—28no—
42C23×#124—28no—
43C23×#121—24no—
44C23×#121—24no—
45C23×#121—24no—
46C23×#121D418yes—
47C23×#120—28no—
48C23×#120—28no—
49C23×#117—24no—
50C2×C4⋊C4×C23C2×C412yes—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23—22no—
55C2×C42×C23—21no—
56C23×#101—42no—
57C23×#100—42no—
58C23×#99—42no—
59C23×#99—42no—
60C23×C22⋊C8—24no—
61C23×#95—24no—
62C24×C23—24no—
63C24×C23—22no—
64C24×C23—42no—
65C24×C23—42no—
66C24×C23—84no—
67C24×C23C22⋊C411yescharacteristic
68C24×C23C22⋊C411yescharacteristic
69C24×C23C2×D411yescharacteristic
70#41×C23—416no—
71C2×Q8×C23—22no—
72C2×Q8×C23—22no—
73C2×Q8×C23—41no—
74C2×D4×C23—24no—
75C2×D4×C23—22no—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—22no—
79C2×D4×C23—28no—
80C2×D4×C23—28no—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23—22no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—42no—
88C2×D4×C23—42no—
89C2×D4×C23—416no—
90C2×D4×C23—42no—
91C2×D4×C23—42no—
92C2×D4×C23—84no—
93C2×D4×C23—84no—
94C2×D4×C23C22⋊C414yes—
95C22×C4×C23—22no—
96C22×C4×C23—22no—
97C22×C4×C23—22no—
98C22×C4×C23—22no—
99C22×C4×C23—22no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—42no—
105C22×C4×C23C22⋊C411yescharacteristic
106C22×C4×C23C22⋊C411yescharacteristic
107C22×C4×C23C22×C411yescharacteristic
108C22×C4×C23C2×D411yescharacteristic
109C22×C4×C23C2×D414yes—
110M16×C23—42no—
111M16×C23—42no—
112C2×C8×C23—42no—
113C4⋊C4×C23—22no—
114C4⋊C4×C23C22⋊C414yes—
115C22⋊C4×C23—24no—
116C22⋊C4×C23—28no—
117C22⋊C4×C23—28no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—42no—
120C42×C23—24no—
121#8516C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—24no—
127C23×C23—21no—
128C23×C23—21no—
129C23×C23—21no—
130C23×C23—42no—
131C23×C23—42no—
132C23×C23—48no—
133C23×C23—44no—
134C23×C23—44no—
135C23×C23—42no—
136C23×C23—42no—
137C23×C23—42no—
138C23×C23—88no—
139C23×C23—88no—
140C23×C23—88no—
141C23×C23#9912yes—
142C23×C23C2×C22⋊C411yescharacteristic
143C23×C23#12012yes—
144Q8×C23—44no—
145D4×C23—44no—
146D4×C23—44no—
147D4×C23—44no—
148D4×C23—44no—
149D4×C23—48no—
150D4×C23—44no—
151D4×C23—88no—
152D4×C23—88no—
153C2×C4×C23—28no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—24no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—44no—
165C2×C4×C23—44no—
166C2×C4×C23—44no—
167C2×C4×C23—82no—
168C2×C4×C23—84no—
169C2×C4×C23#9912yes—
170C2×C4×C23C2×C22⋊C412yes—
171C2×C4×C23#12012yes—
172C8×C23—82no—
173C2×#535C2×C2311yescharacteristic
174#1785C2×C2311yescharacteristic, complemented
175#1782C2×C2311yescharacteristic, complemented
176#1510C2×C2314yescomplemented
177C22×C23—22no—
178C22×C23—24no—
179C22×C23—22no—
180C22×C23—22no—
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—42no—
184C22×C23—42no—
185C22×C23—44no—
186C22×C23—44no—
187C22×C23—41no—
188C22×C23—44no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—84no—
194C22×C23—84no—
195C22×C23—816no—
196C22×C23#37911yescharacteristic
197C22×C23C2×#9911yescharacteristic
198C22×C23C2×#9911yescharacteristic
199C4×C23—22no—
200C4×C23—22no—
201C4×C23—22no—
202C4×C23—44no—
203C4×C23—41no—
204C4×C23—162no—
205C22×#41C22×C2311yescharacteristic
206C23×D4C4×C2311yescharacteristic, complemented
207#535—28no—
208C2×#127—21no—
209C2×#124—21no—
210C2×#121C4×C2312yescomplemented
211C2×#121C22×C2312yes—
212C2×#120—21no—
213C2×#120—21no—
214C2×#117C4×C2311yescharacteristic, complemented
215C2×#101—21no—
216C2×#100—21no—
217C2×#99—21no—
218C2×#99—21no—
219#407C22×C2312yescomplemented
220#381C22×C2312yescomplemented
221#327—28no—
222C2×C23—21no—
223C2×C23—21no—
224C2×C23—42no—
225C2×C23—42no—
226C2×C23—81no—
227C2×C23—84no—
228C2×C23#178111yescharacteristic
229C2×C23#179512yes—
230C25—21no—
231C2×#41—24no—
232C2×#41—42no—
233C2×#41—42no—
234C22×Q8D4×C2311yescharacteristic
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4C2×C4×C2312yes—
242C22×D4D4×C2311yescharacteristic
243C22×D4D4×C2311yescharacteristic
244C22×D4D4×C2311yescharacteristic
245C23×C4C2×C4×C2311yescharacteristic
246C23×C4C2×C4×C2311yescharacteristic
247C23×C4C23×C2311yescharacteristic
248C2×M16—22no—
249#127—28no—
250#124—28no—
251#121—24no—
252#121—24no—
253#121—24no—
254#121D4×C2318yes—
255#120—28no—
256#120—28no—
257#117—24no—
258C2×C4⋊C4C2×C4×C2312yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C42—21no—
264#101—42no—
265#100—42no—
266#99—42no—
267#99—42no—
268C22⋊C8—24no—
269#95—24no—
270C23#851611yescharacteristic, direct product factor, Sylow
271C24—24no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24—84no—
276C24C22⋊C4×C2311yescharacteristic
277C24C22⋊C4×C2311yescharacteristic
278C24C2×D4×C2311yescharacteristic
279#41—416no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—416no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4C22⋊C4×C2314yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4C22⋊C4×C2311yescharacteristic
315C22×C4C22⋊C4×C2311yescharacteristic
316C22×C4C22×C4×C2311yescharacteristic
317C22×C4C2×D4×C2311yescharacteristic
318C22×C4C2×D4×C2314yes—
319M16—42no—
320M16—42no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4C22⋊C4×C2314yes—
324C22⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C42—24no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—24no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—42no—
339C23—42no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—88no—
347C23—88no—
348C23—88no—
349C23C23×#9912yes—
350C23C2×C22⋊C4×C2311yescharacteristic
351C23C23×#12012yes—
352Q8—44no—
353D4—44no—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—48no—
358D4—44no—
359D4—88no—
360D4—88no—
361C2×C4—28no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4C23×#9912yes—
378C2×C4C2×C22⋊C4×C2312yes—
379C2×C4C23×#12012yes—
380C8—82no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—44no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—84no—
398C22—84no—
399C22—816no—
400C22C23×#37911yescharacteristic
401C22C2×C23×#9911yescharacteristic
402C22C2×C23×#9911yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—44no—
408C4—162no—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—42no—
413C2—84no—
414C2—81no—
415C2C23×#178111yescharacteristic
416C2C23×#179512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#535C2
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×#1785C2semidirect
3. C23×#1782C2semidirect
4. C23×#1510C2semidirect
6. C23×D4×C23C4semidirect
10. C2×C23×#121C4semidirect
14. C2×C23×#117C4semidirect
19. C23×#407C22semidirect
20. C23×#381C22semidirect
121. #8516C23direct (N × H)
174. #1785C2×C23semidirect
175. #1782C2×C23semidirect
176. #1510C2×C23semidirect
206. C23×D4C4×C23semidirect
210. C2×#121C4×C23semidirect
214. C2×#117C4×C23semidirect
219. #407C22×C23semidirect
220. #381C22×C23semidirect
270. C23#8516direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5352
C23×#17852
C23×#17822
C23×#15102
#851623