Finite Group Databasebeta

← C23×#9097 #19156723

Subgroups — C23×#9097 #19156723

Order 5888.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2965C211yesmaximal, characteristic, complemented
2C4×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic
3C23×#1477C211yesmaximal, characteristic, complemented
4C2×C23×D4⋊C8C211yesmaximal, characteristic, complemented
5C23×#1460C211yesmaximal, characteristic, complemented
6C23×#1283C211yesmaximal, characteristic, complemented
7C23×#1278C211yesmaximal, characteristic, complemented
8C2×C23×AGL(1,ℤ/8)C411yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C2×C4×D4×C23C2211yescharacteristic
11C23×#446—21no—
12C23×#442—21no—
13C23×#440—21no—
14C4×SD16×C23—21no—
15C4×D8×C23—21no—
16C23×#436—21no—
17C23×#435—22no—
18C23×#431—21no—
19C23×#431C2211yescharacteristic
20C23×#431C2211yescharacteristic
21C23×#428C411yescharacteristic
22C2×C23×C4⋊C8C2211yescharacteristic
23C23×#418C411yescharacteristic
24C23×#417C411yescharacteristic
25C4×M16×C23—21no—
26C4×M16×C23C2211yescharacteristic
27C2×C4×C8×C23C2211yescharacteristic
28C23×Q8⋊C8—21no—
29C23×D4⋊C8—21no—
30C23×D4⋊C8—21no—
31C23×D4⋊C8—21no—
32C23×#323—22no—
33C2×#41×C23C2×C411yescharacteristic
34C22×D4×C23C2×C411yescharacteristic
35C23×C4×C23—21no—
36C23×AGL(1,ℤ/8)—22no—
37C23×AGL(1,ℤ/8)—22no—
38C2×SD16×C23—21no—
39C2×D8×C23—21no—
40C2×M16×C23—22no—
41C2×M16×C23C2×C411yescharacteristic
42C2×M16×C23C2×C411yescharacteristic
43C22×C8×C23—21no—
44C22×C8×C23—21no—
45C4×Q8×C23D411yescharacteristic
46C4×Q8×C23D411yescharacteristic
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C4×D4×C23D411yescharacteristic
54C4×D4×C23D411yescharacteristic
55C23×#117—21no—
56C23×#117C2×C411yescharacteristic
57C2×C4⋊C4×C23C2×C411yescharacteristic
58C2×C22⋊C4×C23—21no—
59C2×C42×C23—21no—
60C2×C42×C23C2311yescharacteristic
61C23×#107—21no—
62C23×#106—21no—
63C23×C4⋊C8—22no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×#103—21no—
69C23×#102—21no—
70C23×#102—21no—
71C23×#102—21no—
72C23×C22⋊C8—22no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8D412yes—
75C23×#97—22no—
76C23×#97—21no—
77C4×C8×C23—21no—
78C4×C8×C23—21no—
79C4×C8×C23—21no—
80C4×C8×C23—21no—
81C4×C8×C23D411yescharacteristic
82C4×C8×C23D411yescharacteristic
83C24×C23—21no—
84#41×C23—21no—
85#41×C23—21no—
86#41×C23—21no—
87#41×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—21no—
94C2×D4×C23—21no—
95C2×D4×C23C22⋊C411yescharacteristic
96C2×D4×C23M1612yescomplemented
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—81no—
104C22×C4×C23—81no—
105C22×C4×C23C22×C411yescharacteristic
106C22×C4×C23C2×D411yescharacteristic
107C22×C4×C23#4111yescharacteristic
108SD16×C23—24no—
109D8×C23—24no—
110M16×C23—21no—
111M16×C23—21no—
112M16×C23—21no—
113M16×C23—44no—
114M16×C23M1612yescomplemented
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—21no—
118C2×C8×C23—21no—
119C2×C8×C23—22no—
120C2×C8×C23—42no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C2×C8×C23—81no—
124C4⋊C4×C23—21no—
125C4⋊C4×C23—21no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23C22⋊C411yescharacteristic
129C4⋊C4×C23C22⋊C411yescharacteristic
130C22⋊C4×C23—21no—
131C22⋊C4×C23—21no—
132C22⋊C4×C23—21no—
133C22⋊C4×C23—41no—
134C22⋊C4×C23—41no—
135C42×C23—21no—
136C42×C23—21no—
137C42×C23—21no—
138C42×C23C2×D411yescharacteristic
139C42×C23C2×D411yescharacteristic
140C42×C23C2×D411yescharacteristic
141C42×C23#4111yescharacteristic
142#9097C2311yesmaximal, characteristic, direct product factor, Sylow
143C23×C23—21no—
144C23×C23—22no—
145C23×C23—41no—
146C23×C23—41no—
147C23×C23—81no—
148C23×C23—81no—
149C23×C23C4×D411yescharacteristic
150Q8×C23—21no—
151Q8×C23—21no—
152D4×C23—21no—
153D4×C23—22no—
154D4×C23—21no—
155D4×C23—22no—
156D4×C23—41no—
157D4×C23—41no—
158D4×C23—41no—
159D4×C23—41no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—41no—
166C2×C4×C23—42no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—82no—
173C2×C4×C23—81no—
174C2×C4×C23—81no—
175C2×C4×C23—81no—
176C2×C4×C23C2×C22⋊C411yescharacteristic
177C2×C4×C23C4×D411yescharacteristic
178C2×C4×C23#12011yescharacteristic
179C2×C4×C23#12111yescharacteristic
180C2×C4×C23#12111yescharacteristic
181C2×C4×C23#12311yescharacteristic
182C2×C4×C23C2×M1611yescharacteristic
183C2×C4×C23C2×M1611yescharacteristic
184C2×C4×C23#13111yescharacteristic
185C2×C4×C23#13111yescharacteristic
186C8×C23—22no—
187C8×C23—22no—
188C8×C23—41no—
189C8×C23—42no—
190C8×C23—41no—
191C8×C23—81no—
192C8×C23—81no—
193#2965C2×C2311yescharacteristic, complemented
194C4×AGL(1,ℤ/8)C2×C2311yescharacteristic
195#1477C2×C2311yescharacteristic, complemented
196C2×D4⋊C8C2×C2311yescharacteristic, complemented
197#1460C2×C2311yescharacteristic, complemented
198#1283C2×C2311yescharacteristic, complemented
199#1278C2×C2311yescharacteristic, complemented
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—41no—
203C22×C23—42no—
204C22×C23—41no—
205C22×C23—41no—
206C22×C23—82no—
207C22×C23—82no—
208C22×C23—81no—
209C22×C23—81no—
210C22×C23#38611yescharacteristic
211C22×C23#43511yescharacteristic
212C22×C23#43611yescharacteristic
213C4×C23—21no—
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—41no—
217C4×C23—41no—
218C4×C23—41no—
219C4×C23—81no—
220C4×C23#40711yescharacteristic
221C4×C23#40811yescharacteristic
222C4×C23#43511yescharacteristic
223C4×C23#43611yescharacteristic
224C4×C23#47111yescharacteristic
225C4×C23#47211yescharacteristic
226C2×AGL(1,ℤ/8)C4×C2311yescharacteristic
227C4×#41C22×C2311yescharacteristic
228C2×C4×D4C22×C2311yescharacteristic
229#446—21no—
230#442—21no—
231#440—21no—
232C4×SD16—21no—
233C4×D8—21no—
234#436—21no—
235#435—22no—
236#431—21no—
237#431C22×C2311yescharacteristic
238#431C22×C2311yescharacteristic
239#428C4×C2311yescharacteristic
240C2×C4⋊C8C22×C2311yescharacteristic
241#418C4×C2311yescharacteristic
242#417C4×C2311yescharacteristic
243C4×M16—21no—
244C4×M16C22×C2311yescharacteristic
245C2×C4×C8C22×C2311yescharacteristic
246Q8⋊C8—21no—
247D4⋊C8—21no—
248D4⋊C8—21no—
249D4⋊C8—21no—
250#323—22no—
251C2×C23—21no—
252C2×C23—21no—
253C2×C23—81no—
254C2×C23—82no—
255C2×C23#187011yescharacteristic
256C2×C23#190611yescharacteristic
257C2×C23#190711yescharacteristic
258C2×#41C2×C4×C2311yescharacteristic
259C22×D4C2×C4×C2311yescharacteristic
260C23×C4—21no—
261AGL(1,ℤ/8)—22no—
262AGL(1,ℤ/8)—22no—
263C2×SD16—21no—
264C2×D8—21no—
265C2×M16—22no—
266C2×M16C2×C4×C2311yescharacteristic
267C2×M16C2×C4×C2311yescharacteristic
268C22×C8—21no—
269C22×C8—21no—
270C4×Q8D4×C2311yescharacteristic
271C4×Q8D4×C2311yescharacteristic
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4D4×C2311yescharacteristic
279C4×D4D4×C2311yescharacteristic
280#117—21no—
281#117C2×C4×C2311yescharacteristic
282C2×C4⋊C4C2×C4×C2311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C42—21no—
285C2×C42C23×C2311yescharacteristic
286#107—21no—
287#106—21no—
288C4⋊C8—22no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293#103—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297C22⋊C8—22no—
298C22⋊C8—21no—
299C22⋊C8D4×C2312yes—
300#97—22no—
301#97—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8D4×C2311yescharacteristic
307C4×C8D4×C2311yescharacteristic
308C23#909711yescharacteristic, direct product factor, Sylow
309C24—21no—
310#41—21no—
311#41—21no—
312#41—21no—
313#41—21no—
314C2×Q8C22⋊C4×C2311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C22⋊C4×C2311yescharacteristic
322C2×D4M16×C2312yescomplemented
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C22×C4×C2311yescharacteristic
332C22×C4C2×D4×C2311yescharacteristic
333C22×C4#41×C2311yescharacteristic
334SD16—24no—
335D8—24no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16—44no—
340M16M16×C2312yescomplemented
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—81no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C22⋊C4×C2311yescharacteristic
355C4⋊C4C22⋊C4×C2311yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42C2×D4×C2311yescharacteristic
365C42C2×D4×C2311yescharacteristic
366C42C2×D4×C2311yescharacteristic
367C42#41×C2311yescharacteristic
368C23—21no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—81no—
373C23—81no—
374C23C4×D4×C2311yescharacteristic
375Q8—21no—
376Q8—21no—
377D4—21no—
378D4—22no—
379D4—21no—
380D4—22no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4C2×C22⋊C4×C2311yescharacteristic
402C2×C4C4×D4×C2311yescharacteristic
403C2×C4C23×#12011yescharacteristic
404C2×C4C23×#12111yescharacteristic
405C2×C4C23×#12111yescharacteristic
406C2×C4C23×#12311yescharacteristic
407C2×C4C2×M16×C2311yescharacteristic
408C2×C4C2×M16×C2311yescharacteristic
409C2×C4C23×#13111yescharacteristic
410C2×C4C23×#13111yescharacteristic
411C8—22no—
412C8—22no—
413C8—41no—
414C8—42no—
415C8—41no—
416C8—81no—
417C8—81no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22C23×#38611yescharacteristic
429C22C23×#43511yescharacteristic
430C22C23×#43611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4C23×#40711yescharacteristic
439C4C23×#40811yescharacteristic
440C4C23×#43511yescharacteristic
441C4C23×#43611yescharacteristic
442C4C23×#47111yescharacteristic
443C4C23×#47211yescharacteristic
444C2—21no—
445C2—21no—
446C2—81no—
447C2—82no—
448C2C23×#187011yescharacteristic
449C2C23×#190611yescharacteristic
450C2C23×#190711yescharacteristic
subgroupG/H
centerC2×C4×C23#121
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC2×C23×AGL(1,ℤ/8)C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2965C2semidirect
3. C23×#1477C2semidirect
4. C2×C23×D4⋊C8C2semidirect
5. C23×#1460C2semidirect
6. C23×#1283C2semidirect
7. C23×#1278C2semidirect
96. C2×D4×C23M16semidirect
114. M16×C23M16semidirect
142. #9097C23direct (N × H)
193. #2965C2×C23semidirect
195. #1477C2×C23semidirect
196. C2×D4⋊C8C2×C23semidirect
197. #1460C2×C23semidirect
198. #1283C2×C23semidirect
199. #1278C2×C23semidirect
308. C23#9097direct (N × H)
322. C2×D4M16×C23semidirect
340. M16M16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#29652
C4×C23×AGL(1,ℤ/8)2
C23×#14772
C2×C23×D4⋊C82
C23×#14602
C23×#12832
C23×#12782
#909723