Finite Group Databasebeta

← C23×#11875 #19159501

Subgroups — C23×#11875 #19159501

Order 5888.

536 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2535C211yesmaximal, characteristic
2C23×#1799C211yesmaximal, characteristic, complemented
3C23×#1459C211yesmaximal, characteristic, complemented
4C23×#1344C212yesmaximal, complemented
5C23×#1343C212yesmaximal, complemented
6C2×C23×#121—21no—
7C2×C4×D4×C23C411yescharacteristic, complemented
8C2×C4×D4×C23C2211yescharacteristic
9C2×C23×#99—21no—
10C2×C23×C22⋊C8—21no—
11C23×#397C411yescharacteristic, complemented
12C23×#397C2211yescharacteristic
13C23×#392C411yescharacteristic, complemented
14C23×#392C2211yescharacteristic
15C23×#386—21no—
16C23×#386—21no—
17C23×#382C411yescharacteristic, complemented
18C23×#380—21no—
19C23×#380—21no—
20C23×#342C2212yes—
21C23×#324—21no—
22C23×#323C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C22×D4×C23D411yescharacteristic
27C23×C4×C23—21no—
28C23×C4×C23D411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C22×C8×C23—41no—
31C23×#121—42no—
32C23×#121—42no—
33C4×D4×C23—41no—
34C4×D4×C23—41no—
35C4×D4×C23—41no—
36C4×D4×C23—41no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—41no—
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—22no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23C2×C411yescharacteristic
48C2×C22⋊C4×C23C2311yescharacteristic
49C2×C42×C23C2×C411yescharacteristic
50C23×#99—42no—
51C23×C22⋊C8—22no—
52C23×C22⋊C8—41no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—41no—
57C23×#95C2×C411yescharacteristic
58C23×#95C2×C411yescharacteristic
59C24×C23—21no—
60C24×C23C22⋊C411yescharacteristic
61C24×C23C2×D411yescharacteristic
62C2×D4×C23—41no—
63C2×D4×C23—41no—
64C2×D4×C23—41no—
65C2×D4×C23—42no—
66C2×D4×C23—42no—
67C2×D4×C23—42no—
68C2×D4×C23—42no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—81no—
72C2×D4×C23—81no—
73C2×D4×C23D812yes—
74C2×D4×C23SD1612yes—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23—81no—
87C22×C4×C23—81no—
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22⋊C411yescharacteristic
91C22×C4×C23C22×C411yescharacteristic
92C22×C4×C23C2×D411yescharacteristic
93C2×C8×C23—42no—
94C2×C8×C23—82no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—42no—
98C4⋊C4×C23—82no—
99C22⋊C4×C23—41no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—42no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—81no—
109C22⋊C4×C23—81no—
110C42×C23—21no—
111C42×C23—21no—
112#11875C2311yesmaximal, characteristic, direct product factor, Sylow
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—22no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23#10212yes—
129C23×C23C2×C22⋊C411yescharacteristic
130C23×C23C2×D811yescharacteristic
131C23×C23C2×SD1611yescharacteristic
132D4×C23—42no—
133D4×C23—42no—
134D4×C23—42no—
135D4×C23—42no—
136D4×C23—84no—
137D4×C23—82no—
138D4×C23—82no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—81no—
160C2×C4×C23—81no—
161C2×C4×C23—84no—
162C2×C4×C23—82no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165C2×C4×C23#9911yescharacteristic
166C2×C4×C23#9911yescharacteristic
167C2×C4×C23#10212yes—
168C8×C23—82no—
169#2535C2×C2311yescharacteristic
170#1799C2×C2311yescharacteristic, complemented
171#1459C2×C2311yescharacteristic, complemented
172#1344C2×C2312yescomplemented
173#1343C2×C2312yescomplemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—22no—
178C22×C23—21no—
179C22×C23—42no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—42no—
183C22×C23—42no—
184C22×C23—41no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—42no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23C2×#9911yescharacteristic
196C22×C23C2×#10211yescharacteristic
197C22×C23#42111yescharacteristic
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—22no—
201C4×C23—41no—
202C4×C23—41no—
203C4×C23—41no—
204C4×C23—81no—
205C4×C23—81no—
206C4×C23—81no—
207C4×C23—162no—
208C2×#121—21no—
209C2×C4×D4C4×C2311yescharacteristic, complemented
210C2×C4×D4C22×C2311yescharacteristic
211C2×#99—21no—
212C2×C22⋊C8—21no—
213#397C4×C2311yescharacteristic, complemented
214#397C22×C2311yescharacteristic
215#392C4×C2311yescharacteristic, complemented
216#392C22×C2311yescharacteristic
217#386—21no—
218#386—21no—
219#382C4×C2311yescharacteristic, complemented
220#380—21no—
221#380—21no—
222#342C22×C2312yes—
223#324—21no—
224#323C22×C2312yes—
225C2×C23—21no—
226C2×C23—21no—
227C2×C23—41no—
228C2×C23—42no—
229C2×C23—41no—
230C2×C23—82no—
231C2×C23#151911yescharacteristic
232C2×C23#212011yescharacteristic
233C2×C23#212711yescharacteristic
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2×C4×C2311yescharacteristic
237C22×D4D4×C2311yescharacteristic
238C23×C4—21no—
239C23×C4D4×C2311yescharacteristic
240C23×C4D4×C2311yescharacteristic
241C22×C8—41no—
242#121—42no—
243#121—42no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4C2×C4×C2311yescharacteristic
251C2×C4⋊C4D4×C2311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4C2×C4×C2311yescharacteristic
259C2×C22⋊C4C23×C2311yescharacteristic
260C2×C42C2×C4×C2311yescharacteristic
261#99—42no—
262C22⋊C8—22no—
263C22⋊C8—41no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95C2×C4×C2311yescharacteristic
269#95C2×C4×C2311yescharacteristic
270C23#1187511yescharacteristic, direct product factor, Sylow
271C24—21no—
272C24C22⋊C4×C2311yescharacteristic
273C24C2×D4×C2311yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4D8×C2312yes—
286C2×D4SD16×C2312yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊C4×C2311yescharacteristic
301C22×C4C22⋊C4×C2311yescharacteristic
302C22×C4C22⋊C4×C2311yescharacteristic
303C22×C4C22×C4×C2311yescharacteristic
304C22×C4C2×D4×C2311yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—82no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C23×#10212yes—
340C23C2×C22⋊C4×C2311yescharacteristic
341C23C2×D8×C2311yescharacteristic
342C23C2×SD16×C2311yescharacteristic
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—84no—
348D4—82no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4C23×#9911yescharacteristic
377C2×C4C23×#9911yescharacteristic
378C2×C4C23×#10212yes—
379C8—82no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×C23×#9911yescharacteristic
402C22C2×C23×#10211yescharacteristic
403C22C23×#42111yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—162no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—82no—
420C2C23×#151911yescharacteristic
421C2C23×#212011yescharacteristic
422C2C23×#212711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1799C2semidirect
3. C23×#1459C2semidirect
4. C23×#1344C2semidirect
5. C23×#1343C2semidirect
7. C2×C4×D4×C23C4semidirect
11. C23×#397C4semidirect
13. C23×#392C4semidirect
17. C23×#382C4semidirect
112. #11875C23direct (N × H)
170. #1799C2×C23semidirect
171. #1459C2×C23semidirect
172. #1344C2×C23semidirect
173. #1343C2×C23semidirect
209. C2×C4×D4C4×C23semidirect
213. #397C4×C23semidirect
215. #392C4×C23semidirect
219. #382C4×C23semidirect
270. C23#11875direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25352
C23×#17992
C23×#14592
C23×#13442
C23×#13432
#1187523