Finite Group Databasebeta

← C23×#9370 #19156996

Subgroups — C23×#9370 #19156996

Order 5888.

424 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3219C211yesmaximal, characteristic
2C23×#2967C211yesmaximal, characteristic, complemented
3C23×#1521C211yesmaximal, characteristic, complemented
4C23×#1514C211yesmaximal, characteristic, complemented
5C23×#1484C211yesmaximal, characteristic, complemented
6C23×#1288C211yesmaximal, characteristic, complemented
7C23×#1287C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C23×#529C2211yescharacteristic
10C23×#497C411yescharacteristic, complemented
11C23×#496C411yescharacteristic, complemented
12C23×#491C411yescharacteristic, complemented
13C23×#490C411yescharacteristic, complemented
14C23×#464—21no—
15C23×#463—21no—
16C23×#452—21no—
17C23×#449—21no—
18C23×#446—21no—
19C23×#436—21no—
20C23×#435—21no—
21C23×#435—21no—
22C23×#433—21no—
23C23×#432C2211yescharacteristic
24C23×#432C2211yescharacteristic
25C23×#431C2211yescharacteristic
26C23×#405—21no—
27C4×M16×C23C2211yescharacteristic
28C23×#383C2211yescharacteristic
29C23×#328—21no—
30C23×#327—21no—
31C23×#327—21no—
32C23×#327—21no—
33C23×D4⋊C8—21no—
34C23×D4⋊C8—21no—
35C2×#41×C23—21no—
36C2×Q16×C23—21no—
37C2×SD16×C23—21no—
38C2×SD16×C23—21no—
39C2×D8×C23—21no—
40C2×M16×C23—21no—
41C2×M16×C23—21no—
42C22×C8×C23—21no—
43C23×C4⋊Q8C2×C411yescharacteristic
44C23×#127C2×C411yescharacteristic
45C23×#126—21no—
46C23×#125C2×C411yescharacteristic
47C23×#124—21no—
48C23×#124C2×C411yescharacteristic
49C23×#123—21no—
50C23×C22⋊Q8D411yescharacteristic
51C23×#121—21no—
52C23×#121—21no—
53C23×#121D411yescharacteristic
54C4×Q8×C23D411yescharacteristic
55C4×D4×C23—21no—
56C4×D4×C23—21no—
57C4×D4×C23—21no—
58C4×D4×C23D411yescharacteristic
59C23×#117—21no—
60C2×C42×C23C2311yescharacteristic
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×#103—21no—
67C23×#103—21no—
68C23×#102—21no—
69C23×#102—21no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8D411yescharacteristic
75C23×C22⋊C8D411yescharacteristic
76C23×#97—21no—
77C23×#97—21no—
78C23×#97—21no—
79C23×#97C2×C411yescharacteristic
80C23×#97C2×C411yescharacteristic
81C4×C8×C23—21no—
82C4×C8×C23—21no—
83C4×C8×C23D411yescharacteristic
84C4×C8×C23D411yescharacteristic
85C23×#95—21no—
86C23×#95—21no—
87C23×#95—21no—
88#41×C23—81no—
89#41×C23—81no—
90C2×Q8×C23—21no—
91C2×Q8×C23C22⋊C411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—41no—
97C2×D4×C23C22⋊C411yescharacteristic
98C22×C4×C23—21no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23C2×D411yescharacteristic
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23#4111yescharacteristic
104Q16×C23—81no—
105SD16×C23—81no—
106SD16×C23—81no—
107D8×C23—81no—
108M16×C23—42no—
109M16×C23—42no—
110C2×C8×C23—21no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23—21no—
114C2×C8×C23—21no—
115C2×C8×C23—21no—
116C2×C8×C23—42no—
117C2×C8×C23—41no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—21no—
121C4⋊C4×C23—21no—
122C4⋊C4×C23—21no—
123C4⋊C4×C23—21no—
124C4⋊C4×C23C22⋊C411yescharacteristic
125C4⋊C4×C23C22⋊C411yescharacteristic
126C22⋊C4×C23—21no—
127C22⋊C4×C23—21no—
128C22⋊C4×C23—21no—
129C22⋊C4×C23—21no—
130C22⋊C4×C23—21no—
131C22⋊C4×C23—41no—
132C22⋊C4×C23—41no—
133C42×C23—21no—
134C42×C23C22×C411yescharacteristic
135C42×C23C2×D411yescharacteristic
136C42×C23C2×D411yescharacteristic
137C42×C23#4111yescharacteristic
138#9370C2311yesmaximal, characteristic, direct product factor, Sylow
139C23×C23—21no—
140C23×C23—41no—
141C23×C23#12111yescharacteristic
142Q8×C23—41no—
143Q8×C23—41no—
144Q8×C23—41no—
145D4×C23—41no—
146D4×C23—41no—
147D4×C23—41no—
148D4×C23—42no—
149D4×C23—81no—
150D4×C23—81no—
151D4×C23—82no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—82no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23C2×C22⋊C411yescharacteristic
166C2×C4×C23C4×D411yescharacteristic
167C2×C4×C23C4×D411yescharacteristic
168C2×C4×C23#12011yescharacteristic
169C2×C4×C23#12111yescharacteristic
170C2×C4×C23#12311yescharacteristic
171C8×C23—22no—
172C8×C23—41no—
173C8×C23—41no—
174C8×C23—41no—
175C8×C23—41no—
176C8×C23—41no—
177C8×C23—81no—
178#3219C2×C2311yescharacteristic
179#2967C2×C2311yescharacteristic, complemented
180#1521C2×C2311yescharacteristic, complemented
181#1514C2×C2311yescharacteristic, complemented
182#1484C2×C2311yescharacteristic, complemented
183#1288C2×C2311yescharacteristic, complemented
184#1287C2×C2311yescharacteristic, complemented
185C22×C23—21no—
186C22×C23—21no—
187C22×C23—21no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—41no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23#38611yescharacteristic
195C4×C23—21no—
196C4×C23—21no—
197C4×C23—21no—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—21no—
201C4×C23—41no—
202C4×C23—41no—
203C4×C23—41no—
204C4×C23—81no—
205C4×C23—161no—
206#532C22×C2311yescharacteristic
207#529C22×C2311yescharacteristic
208#497C4×C2311yescharacteristic, complemented
209#496C4×C2311yescharacteristic, complemented
210#491C4×C2311yescharacteristic, complemented
211#490C4×C2311yescharacteristic, complemented
212#464—21no—
213#463—21no—
214#452—21no—
215#449—21no—
216#446—21no—
217#436—21no—
218#435—21no—
219#435—21no—
220#433—21no—
221#432C22×C2311yescharacteristic
222#432C22×C2311yescharacteristic
223#431C22×C2311yescharacteristic
224#405—21no—
225C4×M16C22×C2311yescharacteristic
226#383C22×C2311yescharacteristic
227#328—21no—
228#327—21no—
229#327—21no—
230#327—21no—
231D4⋊C8—21no—
232D4⋊C8—21no—
233C2×C23—41no—
234C2×C23—81no—
235C2×C23—161no—
236C2×C23#188111yescharacteristic
237C2×C23#190611yescharacteristic
238C2×C23#191011yescharacteristic
239C2×#41—21no—
240C2×Q16—21no—
241C2×SD16—21no—
242C2×SD16—21no—
243C2×D8—21no—
244C2×M16—21no—
245C2×M16—21no—
246C22×C8—21no—
247C4⋊Q8C2×C4×C2311yescharacteristic
248#127C2×C4×C2311yescharacteristic
249#126—21no—
250#125C2×C4×C2311yescharacteristic
251#124—21no—
252#124C2×C4×C2311yescharacteristic
253#123—21no—
254C22⋊Q8D4×C2311yescharacteristic
255#121—21no—
256#121—21no—
257#121D4×C2311yescharacteristic
258C4×Q8D4×C2311yescharacteristic
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4D4×C2311yescharacteristic
263#117—21no—
264C2×C42C23×C2311yescharacteristic
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270#103—21no—
271#103—21no—
272#102—21no—
273#102—21no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8D4×C2311yescharacteristic
279C22⋊C8D4×C2311yescharacteristic
280#97—21no—
281#97—21no—
282#97—21no—
283#97C2×C4×C2311yescharacteristic
284#97C2×C4×C2311yescharacteristic
285C4×C8—21no—
286C4×C8—21no—
287C4×C8D4×C2311yescharacteristic
288C4×C8D4×C2311yescharacteristic
289#95—21no—
290#95—21no—
291#95—21no—
292C23#937011yescharacteristic, direct product factor, Sylow
293#41—81no—
294#41—81no—
295C2×Q8—21no—
296C2×Q8C22⋊C4×C2311yescharacteristic
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4C22⋊C4×C2311yescharacteristic
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C2×D4×C2311yescharacteristic
307C22×C4C2×D4×C2311yescharacteristic
308C22×C4#41×C2311yescharacteristic
309Q16—81no—
310SD16—81no—
311SD16—81no—
312D8—81no—
313M16—42no—
314M16—42no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—42no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊C4×C2311yescharacteristic
330C4⋊C4C22⋊C4×C2311yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C42C22×C4×C2311yescharacteristic
340C42C2×D4×C2311yescharacteristic
341C42C2×D4×C2311yescharacteristic
342C42#41×C2311yescharacteristic
343C23—21no—
344C23—41no—
345C23C23×#12111yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—41no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—42no—
353D4—81no—
354D4—81no—
355D4—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4C2×C22⋊C4×C2311yescharacteristic
370C2×C4C4×D4×C2311yescharacteristic
371C2×C4C4×D4×C2311yescharacteristic
372C2×C4C23×#12011yescharacteristic
373C2×C4C23×#12111yescharacteristic
374C2×C4C23×#12311yescharacteristic
375C8—22no—
376C8—41no—
377C8—41no—
378C8—41no—
379C8—41no—
380C8—41no—
381C8—81no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C23×#38611yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—161no—
403C2—41no—
404C2—81no—
405C2—161no—
406C2C23×#188111yescharacteristic
407C2C23×#190611yescharacteristic
408C2C23×#191011yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3219C2
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×#2967C2semidirect
3. C23×#1521C2semidirect
4. C23×#1514C2semidirect
5. C23×#1484C2semidirect
6. C23×#1288C2semidirect
7. C23×#1287C2semidirect
10. C23×#497C4semidirect
11. C23×#496C4semidirect
12. C23×#491C4semidirect
13. C23×#490C4semidirect
138. #9370C23direct (N × H)
179. #2967C2×C23semidirect
180. #1521C2×C23semidirect
181. #1514C2×C23semidirect
182. #1484C2×C23semidirect
183. #1288C2×C23semidirect
184. #1287C2×C23semidirect
208. #497C4×C23semidirect
209. #496C4×C23semidirect
210. #491C4×C23semidirect
211. #490C4×C23semidirect
292. C23#9370direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32192
C23×#29672
C23×#15212
C23×#15142
C23×#14842
C23×#12882
C23×#12872
#937023