Finite Group Databasebeta

← C23×#9162 #19156788

Subgroups — C23×#9162 #19156788

Order 5888.

472 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3115C211yesmaximal, characteristic
2C23×#2971C211yesmaximal, characteristic, complemented
3C23×#1759C211yesmaximal, characteristic, complemented
4C23×#1525C211yesmaximal, characteristic, complemented
5C23×#1505C211yesmaximal, characteristic, complemented
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1291C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C23×#482—21no—
11C23×#475C411yescharacteristic, complemented
12C23×#474C411yescharacteristic, complemented
13C23×#460C411yescharacteristic, complemented
14C23×#459C411yescharacteristic, complemented
15C23×#449—21no—
16C8×Q8×C23—21no—
17C23×#442—21no—
18C23×#440—21no—
19C23×#436—22no—
20C23×#436—21no—
21C23×#432—21no—
22C23×#432C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C2×C23×#104—21no—
25C4×M16×C23—21no—
26C4×M16×C23C2211yescharacteristic
27C4×M16×C23C2211yescharacteristic
28C23×#376C2211yescharacteristic
29C23×#339—21no—
30C23×#329—21no—
31C23×#329—21no—
32C23×#327—22no—
33C2×#41×C23—21no—
34C2×#41×C23D411yescharacteristic
35C2×SD16×C23—21no—
36C2×D8×C23—21no—
37C2×M16×C23—22no—
38C2×M16×C23—21no—
39C2×M16×C23—21no—
40C23×C4⋊Q8C2×C411yescharacteristic
41C23×#127C2×C411yescharacteristic
42C23×#124—21no—
43C23×#121—21no—
44C23×#121—21no—
45C4×Q8×C23C2×C411yescharacteristic
46C4×Q8×C23D411yescharacteristic
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23C2×C411yescharacteristic
52C4×D4×C23D411yescharacteristic
53C23×#117—21no—
54C23×#117D411yescharacteristic
55C2×C42×C23—21no—
56C2×C42×C23—21no—
57C2×C42×C23C2311yescharacteristic
58C23×#107—21no—
59C23×#106—21no—
60C23×C4⋊C8—22no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8C2×C411yescharacteristic
63C23×C4⋊C8C2×C411yescharacteristic
64C23×#104—24no—
65C23×#103—21no—
66C23×#102—21no—
67C23×#102—21no—
68C23×#102—21no—
69C23×C22⋊C8—22no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8D412yes—
72C23×#97—21no—
73C23×#97—21no—
74C23×#97—21no—
75C23×#97D411yescharacteristic
76C23×#97D411yescharacteristic
77C4×C8×C23—22no—
78C4×C8×C23—21no—
79C4×C8×C23—21no—
80C23×#95—21no—
81C23×#95—21no—
82#41×C23—22no—
83#41×C23—41no—
84#41×C23—82no—
85C2×Q8×C23—21no—
86C2×Q8×C23C22⋊C411yescharacteristic
87C2×D4×C23—21no—
88C2×D4×C23—21no—
89C2×D4×C23—21no—
90C2×D4×C23—21no—
91C2×D4×C23—41no—
92C2×D4×C23C22⋊C411yescharacteristic
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—41no—
97C22×C4×C23—41no—
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23#4111yescharacteristic
101SD16×C23—42no—
102D8×C23—42no—
103M16×C23—22no—
104M16×C23—42no—
105M16×C23—44no—
106M16×C23—41no—
107C2×C8×C23—21no—
108C2×C8×C23—21no—
109C2×C8×C23—21no—
110C2×C8×C23—22no—
111C2×C8×C23—21no—
112C2×C8×C23—41no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23C22⋊C411yescharacteristic
118C4⋊C4×C23C22⋊C411yescharacteristic
119C22⋊C4×C23—21no—
120C22⋊C4×C23—21no—
121C22⋊C4×C23—21no—
122C22⋊C4×C23—41no—
123C22⋊C4×C23—41no—
124C42×C23—21no—
125C42×C23—21no—
126C42×C23—21no—
127C42×C23—22no—
128C42×C23—41no—
129C42×C23C22×C411yescharacteristic
130C42×C23C2×D411yescharacteristic
131C42×C23C2×D411yescharacteristic
132C42×C23#4111yescharacteristic
133#9162C2311yesmaximal, characteristic, direct product factor, Sylow
134C23×C23—21no—
135C23×C23—41no—
136C23×C23#12111yescharacteristic
137Q8×C23—21no—
138Q8×C23—81no—
139Q8×C23#10412yes—
140D4×C23—21no—
141D4×C23—21no—
142D4×C23—42no—
143D4×C23—41no—
144D4×C23—41no—
145D4×C23—81no—
146D4×C23—81no—
147D4×C23—82no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—42no—
156C2×C4×C23—42no—
157C2×C4×C23—42no—
158C2×C4×C23—41no—
159C2×C4×C23—81no—
160C2×C4×C23—82no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23C2×C22⋊C411yescharacteristic
164C2×C4×C23C4×D411yescharacteristic
165C2×C4×C23C4×D411yescharacteristic
166C2×C4×C23#12011yescharacteristic
167C2×C4×C23#12111yescharacteristic
168C2×C4×C23#12311yescharacteristic
169C8×C23—22no—
170C8×C23—42no—
171C8×C23—41no—
172C8×C23—42no—
173C8×C23—41no—
174C8×C23—41no—
175#3115C2×C2311yescharacteristic
176#2971C2×C2311yescharacteristic, complemented
177#1759C2×C2311yescharacteristic, complemented
178#1525C2×C2311yescharacteristic, complemented
179#1505C2×C2311yescharacteristic, complemented
180#1300C2×C2311yescharacteristic, complemented
181#1291C2×C2311yescharacteristic, complemented
182C22×C23—21no—
183C22×C23—21no—
184C22×C23—21no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23#38611yescharacteristic
192C4×C23—21no—
193C4×C23—22no—
194C4×C23—21no—
195C4×C23—21no—
196C4×C23—21no—
197C4×C23—21no—
198C4×C23—41no—
199C4×C23—41no—
200C4×C23—82no—
201C4×C23—81no—
202C4×C23C2×#10411yescharacteristic
203C4×C23#42111yescharacteristic
204#532C22×C2311yescharacteristic
205C4×#41C22×C2311yescharacteristic
206#482—21no—
207#475C4×C2311yescharacteristic, complemented
208#474C4×C2311yescharacteristic, complemented
209#460C4×C2311yescharacteristic, complemented
210#459C4×C2311yescharacteristic, complemented
211#449—21no—
212C8×Q8—21no—
213#442—21no—
214#440—21no—
215#436—21no—
216#436—22no—
217#432—21no—
218#432C22×C2311yescharacteristic
219#432C22×C2311yescharacteristic
220C2×#104—21no—
221C4×M16—21no—
222C4×M16C22×C2311yescharacteristic
223C4×M16C22×C2311yescharacteristic
224#376C22×C2311yescharacteristic
225#339—21no—
226#329—21no—
227#329—21no—
228#327—22no—
229C2×C23—41no—
230C2×C23—81no—
231C2×C23—161no—
232C2×C23#187111yescharacteristic
233C2×C23#190311yescharacteristic
234C2×C23#190611yescharacteristic
235C2×#41—21no—
236C2×#41D4×C2311yescharacteristic
237C2×SD16—21no—
238C2×D8—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—21no—
242C4⋊Q8C2×C4×C2311yescharacteristic
243#127C2×C4×C2311yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8C2×C4×C2311yescharacteristic
248C4×Q8D4×C2311yescharacteristic
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C2×C4×C2311yescharacteristic
254C4×D4D4×C2311yescharacteristic
255#117—21no—
256#117D4×C2311yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C23×C2311yescharacteristic
260#107—21no—
261#106—21no—
262C4⋊C8—22no—
263C4⋊C8—21no—
264C4⋊C8C2×C4×C2311yescharacteristic
265C4⋊C8C2×C4×C2311yescharacteristic
266#104—24no—
267#103—21no—
268#102—21no—
269#102—21no—
270#102—21no—
271C22⋊C8—22no—
272C22⋊C8—21no—
273C22⋊C8D4×C2312yes—
274#97—21no—
275#97—21no—
276#97—21no—
277#97D4×C2311yescharacteristic
278#97D4×C2311yescharacteristic
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282#95—21no—
283#95—21no—
284C23#916211yescharacteristic, direct product factor, Sylow
285#41—22no—
286#41—41no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8C22⋊C4×C2311yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4C22⋊C4×C2311yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C2×D4×C2311yescharacteristic
302C22×C4C2×D4×C2311yescharacteristic
303C22×C4#41×C2311yescharacteristic
304SD16—42no—
305D8—42no—
306M16—22no—
307M16—42no—
308M16—44no—
309M16—41no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊C4×C2311yescharacteristic
321C4⋊C4C22⋊C4×C2311yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—41no—
332C42C22×C4×C2311yescharacteristic
333C42C2×D4×C2311yescharacteristic
334C42C2×D4×C2311yescharacteristic
335C42#41×C2311yescharacteristic
336C23—21no—
337C23—41no—
338C23C23×#12111yescharacteristic
339Q8—21no—
340Q8—81no—
341Q8C23×#10412yes—
342D4—21no—
343D4—21no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4C2×C22⋊C4×C2311yescharacteristic
366C2×C4C4×D4×C2311yescharacteristic
367C2×C4C4×D4×C2311yescharacteristic
368C2×C4C23×#12011yescharacteristic
369C2×C4C23×#12111yescharacteristic
370C2×C4C23×#12311yescharacteristic
371C8—22no—
372C8—42no—
373C8—41no—
374C8—42no—
375C8—41no—
376C8—41no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C23×#38611yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—81no—
397C4C2×C23×#10411yescharacteristic
398C4C23×#42111yescharacteristic
399C2—41no—
400C2—81no—
401C2—161no—
402C2C23×#187111yescharacteristic
403C2C23×#190311yescharacteristic
404C2C23×#190611yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3115C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2971C2semidirect
3. C23×#1759C2semidirect
4. C23×#1525C2semidirect
5. C23×#1505C2semidirect
6. C23×#1300C2semidirect
7. C23×#1291C2semidirect
11. C23×#475C4semidirect
12. C23×#474C4semidirect
13. C23×#460C4semidirect
14. C23×#459C4semidirect
133. #9162C23direct (N × H)
176. #2971C2×C23semidirect
177. #1759C2×C23semidirect
178. #1525C2×C23semidirect
179. #1505C2×C23semidirect
180. #1300C2×C23semidirect
181. #1291C2×C23semidirect
207. #475C4×C23semidirect
208. #474C4×C23semidirect
209. #460C4×C23semidirect
210. #459C4×C23semidirect
284. C23#9162direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31152
C23×#29712
C23×#17592
C23×#15252
C23×#15052
C23×#13002
C23×#12912
#916223