Finite Group Databasebeta

← C23×#9211 #19156837

Subgroups — C23×#9211 #19156837

Order 5888.

536 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3106C211yesmaximal, characteristic
2C4×C23×#131C211yesmaximal, characteristic, complemented
3C23×#1525C211yesmaximal, characteristic, complemented
4C23×#1492C211yesmaximal, characteristic, complemented
5C23×#1489C211yesmaximal, characteristic, complemented
6C23×#1299C211yesmaximal, characteristic, complemented
7C23×#1278C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C23×#532C2211yescharacteristic
10C4×#41×C23C2211yescharacteristic
11C23×#482—21no—
12C23×#477C411yescharacteristic
13C23×#476C411yescharacteristic
14C23×#460C411yescharacteristic
15C23×#459C411yescharacteristic
16C23×#449—21no—
17C4×SD16×C23—21no—
18C4×D8×C23—21no—
19C23×#431C2211yescharacteristic
20C23×#423C2211yescharacteristic
21C23×#405—22no—
22C23×#405—21no—
23C4×M16×C23—21no—
24C4×M16×C23C2211yescharacteristic
25C4×M16×C23C2211yescharacteristic
26C2×C4×C8×C23—21no—
27C2×C4×C8×C23C2211yescharacteristic
28C23×#329—22no—
29C23×Q8⋊C8—21no—
30C23×D4⋊C8—21no—
31C23×D4⋊C8—21no—
32C23×D4⋊C8—21no—
33C2×#41×C23—21no—
34C2×#41×C23D411yescharacteristic
35C2×SD16×C23—21no—
36C2×D8×C23—21no—
37C23×#131—24no—
38C23×#131—24no—
39C2×M16×C23—22no—
40C2×M16×C23—21no—
41C2×M16×C23—21no—
42C2×M16×C23—21no—
43C22×C8×C23—22no—
44C22×C8×C23—21no—
45C23×C4⋊Q8C2×C411yescharacteristic
46C23×#127C2×C411yescharacteristic
47C23×#124—21no—
48C23×#121—21no—
49C23×#121—21no—
50C4×Q8×C23C2×C411yescharacteristic
51C4×Q8×C23D411yescharacteristic
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23C2×C411yescharacteristic
57C4×D4×C23D411yescharacteristic
58C23×#117—21no—
59C23×#117D411yescharacteristic
60C2×C42×C23—21no—
61C2×C42×C23C2311yescharacteristic
62C23×#107—21no—
63C23×#106—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8C2×C411yescharacteristic
67C23×C4⋊C8C2×C411yescharacteristic
68C23×#103—21no—
69C23×#102—21no—
70C23×#102—21no—
71C23×#102—21no—
72C23×C22⋊C8—21no—
73C23×#97—22no—
74C23×#97—21no—
75C23×#97D412yes—
76C4×C8×C23—22no—
77C4×C8×C23—21no—
78C4×C8×C23—21no—
79C4×C8×C23—21no—
80C4×C8×C23—21no—
81C4×C8×C23D411yescharacteristic
82C4×C8×C23D411yescharacteristic
83#41×C23—22no—
84#41×C23—22no—
85#41×C23—81no—
86#41×C23—81no—
87C2×Q8×C23—21no—
88C2×Q8×C23C22⋊C411yescharacteristic
89C2×D4×C23—21no—
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—21no—
93C2×D4×C23—41no—
94C2×D4×C23C22⋊C411yescharacteristic
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—41no—
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23#4111yescharacteristic
101C22×C4×C23#4111yescharacteristic
102SD16×C23—42no—
103D8×C23—42no—
104M16×C23—28no—
105M16×C23—22no—
106M16×C23—22no—
107M16×C23—22no—
108M16×C23—41no—
109M16×C23—81no—
110C2×C8×C23—28no—
111C2×C8×C23—22no—
112C2×C8×C23—22no—
113C2×C8×C23—21no—
114C2×C8×C23—22no—
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—21no—
118C2×C8×C23—41no—
119C2×C8×C23—41no—
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—41no—
130C22⋊C4×C23—41no—
131C42×C23—21no—
132C42×C23—21no—
133C42×C23—21no—
134C42×C23C22×C411yescharacteristic
135C42×C23C2×D411yescharacteristic
136C42×C23C2×D411yescharacteristic
137C42×C23C2×D411yescharacteristic
138#9211C2311yesmaximal, characteristic, direct product factor, Sylow
139C23×C23—21no—
140C23×C23—41no—
141C23×C23#12311yescharacteristic
142Q8×C23—21no—
143Q8×C23—81no—
144Q8×C23#10412yes—
145D4×C23—24no—
146D4×C23—21no—
147D4×C23—41no—
148D4×C23—41no—
149D4×C23—41no—
150D4×C23—41no—
151D4×C23—81no—
152D4×C23—82no—
153D4×C23#10412yes—
154C2×C4×C23—24no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—81no—
170C2×C4×C23—81no—
171C2×C4×C23C2×C22⋊C411yescharacteristic
172C2×C4×C23C4×D411yescharacteristic
173C2×C4×C23C4×D411yescharacteristic
174C2×C4×C23#12011yescharacteristic
175C2×C4×C23#12111yescharacteristic
176C2×C4×C23#12111yescharacteristic
177C8×C23—22no—
178C8×C23—24no—
179C8×C23—22no—
180C8×C23—42no—
181C8×C23—41no—
182C8×C23—41no—
183C8×C23—81no—
184#3106C2×C2311yescharacteristic
185C4×#131C2×C2311yescharacteristic, complemented
186#1525C2×C2311yescharacteristic, complemented
187#1492C2×C2311yescharacteristic, complemented
188#1489C2×C2311yescharacteristic, complemented
189#1299C2×C2311yescharacteristic, complemented
190#1278C2×C2311yescharacteristic, complemented
191C22×C23—21no—
192C22×C23—21no—
193C22×C23—22no—
194C22×C23—21no—
195C22×C23—41no—
196C22×C23—41no—
197C22×C23—81no—
198C22×C23—81no—
199C22×C23—81no—
200C22×C23#38611yescharacteristic
201C4×C23—21no—
202C4×C23—21no—
203C4×C23—22no—
204C4×C23—21no—
205C4×C23—41no—
206C4×C23—41no—
207C4×C23—81no—
208C4×C23C2×#10411yescharacteristic
209C4×C23C2×#10411yescharacteristic
210C4×C23#44311yescharacteristic
211C4×C23#44411yescharacteristic
212C4×C23#47111yescharacteristic
213C4×C23#47211yescharacteristic
214C2×#131—21no—
215#532C22×C2311yescharacteristic
216C4×#41C22×C2311yescharacteristic
217#482—21no—
218#477C4×C2311yescharacteristic
219#476C4×C2311yescharacteristic
220#460C4×C2311yescharacteristic
221#459C4×C2311yescharacteristic
222#449—21no—
223C4×SD16—21no—
224C4×D8—21no—
225#431C22×C2311yescharacteristic
226#423C22×C2311yescharacteristic
227#405—21no—
228#405—22no—
229C4×M16—21no—
230C4×M16C22×C2311yescharacteristic
231C4×M16C22×C2311yescharacteristic
232C2×C4×C8—21no—
233C2×C4×C8C22×C2311yescharacteristic
234#329—22no—
235Q8⋊C8—21no—
236D4⋊C8—21no—
237D4⋊C8—21no—
238D4⋊C8—21no—
239C2×C23—41no—
240C2×C23—42no—
241C2×C23—161no—
242C2×C23#187611yescharacteristic
243C2×C23#189611yescharacteristic
244C2×C23#190711yescharacteristic
245C2×#41—21no—
246C2×#41D4×C2311yescharacteristic
247C2×SD16—21no—
248C2×D8—21no—
249#131—24no—
250#131—24no—
251C2×M16—22no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C22×C8—22no—
256C22×C8—21no—
257C4⋊Q8C2×C4×C2311yescharacteristic
258#127C2×C4×C2311yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×Q8C2×C4×C2311yescharacteristic
263C4×Q8D4×C2311yescharacteristic
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4C2×C4×C2311yescharacteristic
269C4×D4D4×C2311yescharacteristic
270#117—21no—
271#117D4×C2311yescharacteristic
272C2×C42—21no—
273C2×C42C23×C2311yescharacteristic
274#107—21no—
275#106—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8C2×C4×C2311yescharacteristic
279C4⋊C8C2×C4×C2311yescharacteristic
280#103—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—21no—
285#97—22no—
286#97—21no—
287#97D4×C2312yes—
288C4×C8—22no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8D4×C2311yescharacteristic
294C4×C8D4×C2311yescharacteristic
295C23#921111yescharacteristic, direct product factor, Sylow
296#41—22no—
297#41—22no—
298#41—81no—
299#41—81no—
300C2×Q8—21no—
301C2×Q8C22⋊C4×C2311yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4C22⋊C4×C2311yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4C2×D4×C2311yescharacteristic
313C22×C4#41×C2311yescharacteristic
314C22×C4#41×C2311yescharacteristic
315SD16—42no—
316D8—42no—
317M16—28no—
318M16—22no—
319M16—22no—
320M16—22no—
321M16—41no—
322M16—81no—
323C2×C8—28no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊C4×C2311yescharacteristic
338C4⋊C4C22⋊C4×C2311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42C22×C4×C2311yescharacteristic
348C42C2×D4×C2311yescharacteristic
349C42C2×D4×C2311yescharacteristic
350C42C2×D4×C2311yescharacteristic
351C23—21no—
352C23—41no—
353C23C23×#12311yescharacteristic
354Q8—21no—
355Q8—81no—
356Q8C23×#10412yes—
357D4—24no—
358D4—21no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—82no—
365D4C23×#10412yes—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4C2×C22⋊C4×C2311yescharacteristic
384C2×C4C4×D4×C2311yescharacteristic
385C2×C4C4×D4×C2311yescharacteristic
386C2×C4C23×#12011yescharacteristic
387C2×C4C23×#12111yescharacteristic
388C2×C4C23×#12111yescharacteristic
389C8—22no—
390C8—24no—
391C8—22no—
392C8—42no—
393C8—41no—
394C8—41no—
395C8—81no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C23×#38611yescharacteristic
406C4—21no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4C2×C23×#10411yescharacteristic
414C4C2×C23×#10411yescharacteristic
415C4C23×#44311yescharacteristic
416C4C23×#44411yescharacteristic
417C4C23×#47111yescharacteristic
418C4C23×#47211yescharacteristic
419C2—41no—
420C2—42no—
421C2—161no—
422C2C23×#187611yescharacteristic
423C2C23×#189611yescharacteristic
424C2C23×#190711yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C23×#131C2semidirect
3. C23×#1525C2semidirect
4. C23×#1492C2semidirect
5. C23×#1489C2semidirect
6. C23×#1299C2semidirect
7. C23×#1278C2semidirect
138. #9211C23direct (N × H)
185. C4×#131C2×C23semidirect
186. #1525C2×C23semidirect
187. #1492C2×C23semidirect
188. #1489C2×C23semidirect
189. #1299C2×C23semidirect
190. #1278C2×C23semidirect
295. C23#9211direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31062
C4×C23×#1312
C23×#15252
C23×#14922
C23×#14892
C23×#12992
C23×#12782
#921123