Finite Group Databasebeta

← C23×#33596 #19181222

Subgroups — C23×#33596 #19181222

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3414C211yesmaximal, characteristic
2C2×C23×#510C211yesmaximal, characteristic, complemented
3C23×#3156C211yesmaximal, characteristic, complemented
4C23×#2957C211yesmaximal, characteristic, complemented
5C23×#2214C214yesmaximal, complemented
6C23×#2212C214yesmaximal, complemented
7C23×#2153C211yesmaximal, characteristic, complemented
8C23×#2148C211yesmaximal, characteristic, complemented
9C2×C23×#362C211yesmaximal, characteristic, complemented
10C2×C23×#137—21no—
11C2×C23×#135C2211yescharacteristic
12C2×C23×#135C2211yescharacteristic
13C22×Q16×C23C2211yescharacteristic
14C23×#568—21no—
15C23×#536—21no—
16C23×#510—22no—
17C23×#510—21no—
18C23×#510C2214yes—
19C23×#510C2212yes—
20C23×#509—21no—
21C23×#509C2212yes—
22C23×#508—22no—
23C2×C23×Q32—21no—
24C2×C23×SD32—21no—
25C2×C23×M32C2211yescharacteristic
26C2×C23×M32C2211yescharacteristic
27C23×#497—22no—
28C23×#495C2212yescomplemented
29C23×#494C2212yescomplemented
30C23×#444—22no—
31C23×#443C2214yes—
32C2×C23×#108C2211yescharacteristic
33C23×#421—21no—
34C23×#405C2211yescharacteristic
35C23×#364C2214yescomplemented
36C23×#362C2214yescomplemented
37C23×#362C2212yescomplemented
38C23×#361C2212yescomplemented
39C2×#41×C23—21no—
40C2×#41×C23—21no—
41C22×Q8×C23—21no—
42C23×#137—42no—
43C23×#137—42no—
44C23×#135—21no—
45C23×#135—21no—
46C23×#135—21no—
47C23×#135—21no—
48C23×#135—21no—
49C23×#135—21no—
50C23×#135D412yes—
51C23×#135C2312yes—
52C2×Q16×C23—21no—
53C2×Q16×C23—21no—
54C2×Q16×C23—21no—
55C2×Q16×C23—22no—
56C2×Q16×C23—21no—
57C2×Q16×C23—21no—
58C2×Q16×C23D411yescharacteristic
59C2×Q16×C23C2312yes—
60C2×Q16×C23C2311yescharacteristic
61C2×SD16×C23—21no—
62C2×SD16×C23—22no—
63C2×SD16×C23—21no—
64C2×D8×C23D411yescharacteristic
65C2×D8×C23C2311yescharacteristic
66C23×#131—24no—
67C23×#131—42no—
68C2×M16×C23—21no—
69C2×M16×C23—42no—
70C2×M16×C23D411yescharacteristic
71C22×C8×C23C2311yescharacteristic
72C23×C4⋊Q8—22no—
73C23×#124—22no—
74C23×#123—42no—
75C23×C22⋊Q8—42no—
76C23×#117D411yescharacteristic
77C23×Q32—24no—
78C23×Q32—22no—
79C23×SD32—22no—
80C23×SD32—24no—
81C23×SD32—22no—
82C23×D16—22no—
83C23×M32—21no—
84C23×M32—21no—
85C23×M32C2312yes—
86C23×M32C2312yes—
87C2×C16×C23—21no—
88C2×C16×C23—21no—
89C23×#108—21no—
90C23×#108C2312yes—
91C23×#104—24no—
92C23×#97D412yes—
93C4×C8×C23C2312yes—
94#41×C23—22no—
95#41×C23—22no—
96#41×C23—41no—
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—41no—
100#41×C23—41no—
101#41×C23—41no—
102C2×Q8×C23—21no—
103C2×Q8×C23—22no—
104C2×Q8×C23—21no—
105C2×Q8×C23—41no—
106C2×Q8×C23—41no—
107C2×Q8×C23—41no—
108C2×Q8×C23—41no—
109C2×Q8×C23—41no—
110C2×D4×C23—21no—
111C2×D4×C23—21no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C22×C4×C23—41no—
115C22×C4×C23—41no—
116C22×C4×C23—41no—
117C22×C4×C23C2×D411yescharacteristic
118Q16×C23—21no—
119Q16×C23—22no—
120Q16×C23—21no—
121Q16×C23—22no—
122Q16×C23—21no—
123Q16×C23—41no—
124Q16×C23—41no—
125Q16×C23—44no—
126Q16×C23C2×D412yes—
127SD16×C23—22no—
128SD16×C23—22no—
129SD16×C23—41no—
130SD16×C23—41no—
131SD16×C23—44no—
132D8×C23—21no—
133D8×C23—21no—
134D8×C23—21no—
135D8×C23C2×D412yes—
136M16×C23—22no—
137M16×C23—41no—
138M16×C23—44no—
139M16×C23D812yes—
140M16×C23D812yes—
141C2×C8×C23—41no—
142C2×C8×C23—44no—
143C2×C8×C23C2×D411yescharacteristic
144C2×C8×C23C2×D411yescharacteristic
145C2×C8×C23C2×D411yescharacteristic
146C2×C8×C23C2×D412yes—
147C2×C8×C23#4111yescharacteristic
148C2×C8×C23#4111yescharacteristic
149C2×C8×C23C2411yescharacteristic
150C4⋊C4×C23—21no—
151C4⋊C4×C23—44no—
152C22⋊C4×C23—21no—
153C22⋊C4×C23—44no—
154C42×C23C2×D412yes—
155C16×C23—22no—
156C16×C23—22no—
157#33596C2311yesmaximal, characteristic, direct product factor, Sylow
158C23×C23—41no—
159C23×C23—41no—
160C23×C23AGL(1,ℤ/8)11yescharacteristic
161Q8×C23—22no—
162Q8×C23—41no—
163Q8×C23—41no—
164Q8×C23—42no—
165Q8×C23—42no—
166Q8×C23—41no—
167Q8×C23—81no—
168Q8×C23—81no—
169D4×C23—22no—
170D4×C23—42no—
171D4×C23—42no—
172D4×C23—41no—
173D4×C23—41no—
174D4×C23—41no—
175D4×C23—81no—
176D4×C23—81no—
177C2×C4×C23—22no—
178C2×C4×C23—41no—
179C2×C4×C23—42no—
180C2×C4×C23—42no—
181C2×C4×C23—42no—
182C2×C4×C23—41no—
183C2×C4×C23—81no—
184C2×C4×C23—81no—
185C2×C4×C23—81no—
186C2×C4×C23—81no—
187C2×C4×C23#12111yescharacteristic
188C2×C4×C23#12111yescharacteristic
189C2×C4×C23C2×D811yescharacteristic
190C2×C4×C23C2×D811yescharacteristic
191C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C23C22×D411yescharacteristic
193C8×C23—42no—
194C8×C23#12111yescharacteristic
195C8×C23#12111yescharacteristic
196C8×C23C2×D814yes—
197C8×C23C22×D411yescharacteristic
198C8×C23C2×#4111yescharacteristic
199#3414C2×C2311yescharacteristic
200C2×#510C2×C2311yescharacteristic, complemented
201#3156C2×C2311yescharacteristic, complemented
202#2957C2×C2311yescharacteristic, complemented
203#2214C2×C2314yescomplemented
204#2212C2×C2314yescomplemented
205#2153C2×C2311yescharacteristic, complemented
206#2148C2×C2311yescharacteristic, complemented
207C2×#362C2×C2311yescharacteristic, complemented
208C22×C23—41no—
209C22×C23—42no—
210C22×C23—82no—
211C22×C23—81no—
212C22×C23—161no—
213C22×C23#45911yescharacteristic
214C22×C23#45911yescharacteristic
215C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
216C4×C23—42no—
217C4×C23—42no—
218C4×C23—81no—
219C4×C23—82no—
220C4×C23#45911yescharacteristic
221C4×C23#45911yescharacteristic
222C4×C23C2×#12111yescharacteristic
223C4×C23C22×D811yescharacteristic
224C2×#137—21no—
225C2×#135C22×C2311yescharacteristic
226C2×#135C22×C2311yescharacteristic
227C22×Q16C22×C2311yescharacteristic
228#568—21no—
229#536—21no—
230#510—22no—
231#510—21no—
232#510C22×C2314yes—
233#510C22×C2312yes—
234#509—21no—
235#509C22×C2312yes—
236#508—22no—
237C2×Q32—21no—
238C2×SD32—21no—
239C2×M32C22×C2311yescharacteristic
240C2×M32C22×C2311yescharacteristic
241#497—22no—
242#495C22×C2312yescomplemented
243#494C22×C2312yescomplemented
244#444—22no—
245#443C22×C2314yes—
246C2×#108C22×C2311yescharacteristic
247#421—21no—
248#405C22×C2311yescharacteristic
249#364C22×C2314yescomplemented
250#362C22×C2314yescomplemented
251#362C22×C2312yescomplemented
252#361C22×C2312yescomplemented
253C2×C23—21no—
254C2×C23—21no—
255C2×C23—21no—
256C2×C23—82no—
257C2×C23—161no—
258C2×C23C2×#45911yescharacteristic
259C2×#41—21no—
260C2×#41—21no—
261C22×Q8—21no—
262#137—42no—
263#137—42no—
264#135—21no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—21no—
269#135—21no—
270#135D4×C2312yes—
271#135C23×C2312yes—
272C2×Q16—21no—
273C2×Q16—21no—
274C2×Q16—21no—
275C2×Q16—22no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×Q16D4×C2311yescharacteristic
279C2×Q16C23×C2312yes—
280C2×Q16C23×C2311yescharacteristic
281C2×SD16—21no—
282C2×SD16—22no—
283C2×SD16—21no—
284C2×D8D4×C2311yescharacteristic
285C2×D8C23×C2311yescharacteristic
286#131—24no—
287#131—42no—
288C2×M16—21no—
289C2×M16—42no—
290C2×M16D4×C2311yescharacteristic
291C22×C8C23×C2311yescharacteristic
292C4⋊Q8—22no—
293#124—22no—
294#123—42no—
295C22⋊Q8—42no—
296#117D4×C2311yescharacteristic
297Q32—24no—
298Q32—22no—
299SD32—22no—
300SD32—24no—
301SD32—22no—
302D16—22no—
303M32—21no—
304M32—21no—
305M32C23×C2312yes—
306M32C23×C2312yes—
307C2×C16—21no—
308C2×C16—21no—
309#108—21no—
310#108C23×C2312yes—
311#104—24no—
312#97D4×C2312yes—
313C4×C8C23×C2312yes—
314C23#3359611yescharacteristic, direct product factor, Sylow
315#41—22no—
316#41—22no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—21no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4C2×D4×C2311yescharacteristic
339Q16—21no—
340Q16—22no—
341Q16—21no—
342Q16—22no—
343Q16—21no—
344Q16—41no—
345Q16—41no—
346Q16—44no—
347Q16C2×D4×C2312yes—
348SD16—22no—
349SD16—22no—
350SD16—41no—
351SD16—41no—
352SD16—44no—
353D8—21no—
354D8—21no—
355D8—21no—
356D8C2×D4×C2312yes—
357M16—22no—
358M16—41no—
359M16—44no—
360M16D8×C2312yes—
361M16D8×C2312yes—
362C2×C8—41no—
363C2×C8—44no—
364C2×C8C2×D4×C2311yescharacteristic
365C2×C8C2×D4×C2311yescharacteristic
366C2×C8C2×D4×C2311yescharacteristic
367C2×C8C2×D4×C2312yes—
368C2×C8#41×C2311yescharacteristic
369C2×C8#41×C2311yescharacteristic
370C2×C8C24×C2311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—44no—
373C22⋊C4—21no—
374C22⋊C4—44no—
375C42C2×D4×C2312yes—
376C16—22no—
377C16—22no—
378C23—41no—
379C23—41no—
380C23C23×AGL(1,ℤ/8)11yescharacteristic
381Q8—22no—
382Q8—41no—
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—42no—
387Q8—81no—
388Q8—81no—
389D4—22no—
390D4—42no—
391D4—42no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—81no—
396D4—81no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C23×#12111yescharacteristic
408C2×C4C23×#12111yescharacteristic
409C2×C4C2×D8×C2311yescharacteristic
410C2×C4C2×D8×C2311yescharacteristic
411C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C22×D4×C2311yescharacteristic
413C8—42no—
414C8C23×#12111yescharacteristic
415C8C23×#12111yescharacteristic
416C8C2×D8×C2314yes—
417C8C22×D4×C2311yescharacteristic
418C8C2×#41×C2311yescharacteristic
419C22—41no—
420C22—42no—
421C22—82no—
422C22—81no—
423C22—161no—
424C22C23×#45911yescharacteristic
425C22C23×#45911yescharacteristic
426C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
427C4—42no—
428C4—42no—
429C4—81no—
430C4—82no—
431C4C23×#45911yescharacteristic
432C4C23×#45911yescharacteristic
433C4C2×C23×#12111yescharacteristic
434C4C22×D8×C2311yescharacteristic
435C2—21no—
436C2—21no—
437C2—21no—
438C2—82no—
439C2—161no—
440C2C2×C23×#45911yescharacteristic
subgroupG/H
center, socleC2×C23C2×#459
commutator, Frattini, cosocleC2×C8C24×C23
prime radicalC23×#3414C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ C23×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#510C2semidirect
3. C23×#3156C2semidirect
4. C23×#2957C2semidirect
5. C23×#2214C2semidirect
6. C23×#2212C2semidirect
7. C23×#2153C2semidirect
8. C23×#2148C2semidirect
9. C2×C23×#362C2semidirect
28. C23×#495C22semidirect
29. C23×#494C22semidirect
35. C23×#364C22semidirect
36. C23×#362C22semidirect
37. C23×#362C22semidirect
38. C23×#361C22semidirect
157. #33596C23direct (N × H)
200. C2×#510C2×C23semidirect
201. #3156C2×C23semidirect
202. #2957C2×C23semidirect
203. #2214C2×C23semidirect
204. #2212C2×C23semidirect
205. #2153C2×C23semidirect
206. #2148C2×C23semidirect
207. C2×#362C2×C23semidirect
242. #495C22×C23semidirect
243. #494C22×C23semidirect
249. #364C22×C23semidirect
250. #362C22×C23semidirect
251. #362C22×C23semidirect
252. #361C22×C23semidirect
314. C23#33596direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34142
C2×C23×#5102
C23×#31562
C23×#29572
C23×#22142
C23×#22122
C23×#21532
C23×#21482
C2×C23×#3622
#3359623