Finite Group Databasebeta

← C23×#33468 #19181094

Subgroups — C23×#33468 #19181094

Order 5888.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3414C211yesmaximal, characteristic
2C2×C23×#444C211yesmaximal, characteristic, complemented
3C23×#2943C211yesmaximal, characteristic, complemented
4C23×AGL(1,ℤ/16)C214yesmaximal, complemented
5C23×#2180C214yesmaximal, complemented
6C23×#2155C211yesmaximal, characteristic, complemented
7C23×#2148C211yesmaximal, characteristic, complemented
8C23×#2145C211yesmaximal, characteristic, complemented
9C2×C23×#347C211yesmaximal, characteristic, complemented
10C2×C23×#135C2211yescharacteristic
11C2×C23×#135C2211yescharacteristic
12C2×C23×#131—21no—
13C4×#41×C23—21no—
14C23×#510C412yescomplemented
15C23×#510C2212yes—
16C23×#509C412yescomplemented
17C23×#509C2212yes—
18C23×#508C414yescomplemented
19C2×C23×M32C2211yescharacteristic
20C2×C23×M32C2211yescharacteristic
21C23×#444—22no—
22C23×#444C2214yes—
23C23×#442C2212yescomplemented
24C23×#441C2212yescomplemented
25C23×#440—22no—
26C2×C23×#108C2211yescharacteristic
27C23×#427C2211yescharacteristic
28C2×C23×#104—21no—
29C23×#416—21no—
30C2×C23×#97C2211yescharacteristic
31C23×#365C2214yescomplemented
32C23×#362C2212yescomplemented
33C23×#361C2212yescomplemented
34C23×#360C2214yes—
35C23×#347C2214yescomplemented
36C2×#41×C23—21no—
37C2×#41×C23—21no—
38C23×#135—21no—
39C23×#135—21no—
40C23×#135—21no—
41C23×#135—21no—
42C23×#135C2×C412yes—
43C23×#135C2×C412yes—
44C23×#135C2×C412yescomplemented
45C23×#135C2312yes—
46C2×Q16×C23C2×C411yescharacteristic, complemented
47C2×Q16×C23C2311yescharacteristic
48C2×SD16×C23—21no—
49C2×SD16×C23—21no—
50C2×D8×C23C2×C411yescharacteristic, complemented
51C2×D8×C23C2311yescharacteristic
52C23×#131—24no—
53C23×#131—42no—
54C2×M16×C23—21no—
55C2×M16×C23—41no—
56C22×C8×C23—41no—
57C22×C8×C23D411yescharacteristic
58C22×C8×C23C2311yescharacteristic
59C4×Q8×C23—22no—
60C4×D4×C23—22no—
61C4×D4×C23—42no—
62C23×#117—21no—
63C23×#117—41no—
64C2×C42×C23—41no—
65C2×C42×C23D411yescharacteristic
66C23×Q32C2×C414yescomplemented
67C23×SD32C2×C414yescomplemented
68C23×SD32C2×C414yescomplemented
69C23×D16C2×C414yescomplemented
70C23×M32—21no—
71C23×M32C2×C412yescomplemented
72C23×M32C2312yes—
73C23×M32C2312yes—
74C2×C16×C23—21no—
75C2×C16×C23C2×C412yescomplemented
76C23×#108—21no—
77C23×#108C2312yes—
78C23×#107—21no—
79C23×#106C2312yes—
80C23×#104—24no—
81C23×#103—22no—
82C23×#102—22no—
83C23×#97D412yes—
84C23×#97C2312yes—
85#41×C23—22no—
86#41×C23—22no—
87#41×C23—22no—
88#41×C23—22no—
89#41×C23—41no—
90#41×C23—41no—
91#41×C23—41no—
92#41×C23—41no—
93C2×Q8×C23—21no—
94C2×Q8×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—21no—
97C2×D4×C23—41no—
98C2×D4×C23—41no—
99C22×C4×C23—21no—
100C22×C4×C23—42no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23C2×D411yescharacteristic
104Q16×C23—21no—
105Q16×C23—21no—
106Q16×C23C22×C412yes—
107Q16×C23C22×C412yes—
108SD16×C23—22no—
109SD16×C23—22no—
110SD16×C23—22no—
111SD16×C23—22no—
112D8×C23—21no—
113D8×C23—21no—
114D8×C23C22×C412yes—
115D8×C23C22×C412yes—
116M16×C23—22no—
117M16×C23—44no—
118M16×C23—41no—
119C2×C8×C23—21no—
120C2×C8×C23—21no—
121C2×C8×C23—44no—
122C2×C8×C23—41no—
123C2×C8×C23C22×C411yescharacteristic
124C2×C8×C23C22×C411yescharacteristic
125C2×C8×C23C2×D412yes—
126C2×C8×C23C2×D411yescharacteristic
127C2×C8×C23#4111yescharacteristic
128C2×C8×C23#4111yescharacteristic
129C2×C8×C23C2411yescharacteristic
130C4⋊C4×C23—22no—
131C4⋊C4×C23—42no—
132C22⋊C4×C23—42no—
133C22⋊C4×C23—41no—
134C42×C23—21no—
135C42×C23—42no—
136C42×C23—41no—
137C42×C23C2×D412yes—
138C16×C23—22no—
139C16×C23C22×C414yes—
140#33468C2311yesmaximal, characteristic, direct product factor, Sylow
141C23×C23—41no—
142C23×C23—41no—
143C23×C23AGL(1,ℤ/8)11yescharacteristic
144Q8×C23—22no—
145Q8×C23—22no—
146Q8×C23—41no—
147Q8×C23—41no—
148D4×C23—22no—
149D4×C23—22no—
150D4×C23—42no—
151D4×C23—42no—
152D4×C23—42no—
153D4×C23—42no—
154D4×C23—41no—
155D4×C23—41no—
156C2×C4×C23—22no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—42no—
160C2×C4×C23—42no—
161C2×C4×C23—42no—
162C2×C4×C23—42no—
163C2×C4×C23—42no—
164C2×C4×C23—81no—
165C2×C4×C23—84no—
166C2×C4×C23C4×D411yescharacteristic
167C2×C4×C23C4×D411yescharacteristic
168C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C23C22×D411yescharacteristic
172C8×C23—22no—
173C8×C23—42no—
174C8×C23C4×D411yescharacteristic
175C8×C23C4×D411yescharacteristic
176C8×C23C23×C411yescharacteristic
177C8×C23C2×#4111yescharacteristic
178#3414C2×C2311yescharacteristic
179C2×#444C2×C2311yescharacteristic, complemented
180#2943C2×C2311yescharacteristic, complemented
181AGL(1,ℤ/16)C2×C2314yescomplemented
182#2180C2×C2314yescomplemented
183#2155C2×C2311yescharacteristic, complemented
184#2148C2×C2311yescharacteristic, complemented
185#2145C2×C2311yescharacteristic, complemented
186C2×#347C2×C2311yescharacteristic, complemented
187C22×C23—41no—
188C22×C23—42no—
189C22×C23—42no—
190C22×C23—82no—
191C22×C23—82no—
192C22×C23#44211yescharacteristic
193C22×C23#44211yescharacteristic
194C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
195C4×C23—42no—
196C4×C23—42no—
197C4×C23—42no—
198C4×C23—82no—
199C4×C23#44211yescharacteristic
200C4×C23#44211yescharacteristic
201C4×C23C2×C4×D411yescharacteristic
202C4×C23C2×AGL(1,ℤ/8)11yescharacteristic
203C2×#135C22×C2311yescharacteristic
204C2×#135C22×C2311yescharacteristic
205C2×#131—21no—
206C4×#41—21no—
207#510C4×C2312yescomplemented
208#510C22×C2312yes—
209#509C4×C2312yescomplemented
210#509C22×C2312yes—
211#508C4×C2314yescomplemented
212C2×M32C22×C2311yescharacteristic
213C2×M32C22×C2311yescharacteristic
214#444—22no—
215#444C22×C2314yes—
216#442C22×C2312yescomplemented
217#441C22×C2312yescomplemented
218#440—22no—
219C2×#108C22×C2311yescharacteristic
220#427C22×C2311yescharacteristic
221C2×#104—21no—
222#416—21no—
223C2×#97C22×C2311yescharacteristic
224#365C22×C2314yescomplemented
225#362C22×C2312yescomplemented
226#361C22×C2312yescomplemented
227#360C22×C2314yes—
228#347C22×C2314yescomplemented
229C2×C23—21no—
230C2×C23—21no—
231C2×C23—21no—
232C2×C23—82no—
233C2×C23—82no—
234C2×C23C2×#44211yescharacteristic
235C2×#41—21no—
236C2×#41—21no—
237#135—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135C2×C4×C2312yes—
242#135C2×C4×C2312yes—
243#135C2×C4×C2312yescomplemented
244#135C23×C2312yes—
245C2×Q16C2×C4×C2311yescharacteristic, complemented
246C2×Q16C23×C2311yescharacteristic
247C2×SD16—21no—
248C2×SD16—21no—
249C2×D8C2×C4×C2311yescharacteristic, complemented
250C2×D8C23×C2311yescharacteristic
251#131—24no—
252#131—42no—
253C2×M16—21no—
254C2×M16—41no—
255C22×C8—41no—
256C22×C8D4×C2311yescharacteristic
257C22×C8C23×C2311yescharacteristic
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—42no—
261#117—21no—
262#117—41no—
263C2×C42—41no—
264C2×C42D4×C2311yescharacteristic
265Q32C2×C4×C2314yescomplemented
266SD32C2×C4×C2314yescomplemented
267SD32C2×C4×C2314yescomplemented
268D16C2×C4×C2314yescomplemented
269M32—21no—
270M32C2×C4×C2312yescomplemented
271M32C23×C2312yes—
272M32C23×C2312yes—
273C2×C16—21no—
274C2×C16C2×C4×C2312yescomplemented
275#108—21no—
276#108C23×C2312yes—
277#107—21no—
278#106C23×C2312yes—
279#104—24no—
280#103—22no—
281#102—22no—
282#97D4×C2312yes—
283#97C23×C2312yes—
284C23#3346811yescharacteristic, direct product factor, Sylow
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4C2×D4×C2311yescharacteristic
304Q16—21no—
305Q16—21no—
306Q16C22×C4×C2312yes—
307Q16C22×C4×C2312yes—
308SD16—22no—
309SD16—22no—
310SD16—22no—
311SD16—22no—
312D8—21no—
313D8—21no—
314D8C22×C4×C2312yes—
315D8C22×C4×C2312yes—
316M16—22no—
317M16—44no—
318M16—41no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—44no—
322C2×C8—41no—
323C2×C8C22×C4×C2311yescharacteristic
324C2×C8C22×C4×C2311yescharacteristic
325C2×C8C2×D4×C2312yes—
326C2×C8C2×D4×C2311yescharacteristic
327C2×C8#41×C2311yescharacteristic
328C2×C8#41×C2311yescharacteristic
329C2×C8C24×C2311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C42—21no—
335C42—42no—
336C42—41no—
337C42C2×D4×C2312yes—
338C16—22no—
339C16C22×C4×C2314yes—
340C23—41no—
341C23—41no—
342C23C23×AGL(1,ℤ/8)11yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—41no—
346Q8—41no—
347D4—22no—
348D4—22no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—41no—
354D4—41no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—84no—
365C2×C4C4×D4×C2311yescharacteristic
366C2×C4C4×D4×C2311yescharacteristic
367C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
370C2×C4C22×D4×C2311yescharacteristic
371C8—22no—
372C8—42no—
373C8C4×D4×C2311yescharacteristic
374C8C4×D4×C2311yescharacteristic
375C8C23×C4×C2311yescharacteristic
376C8C2×#41×C2311yescharacteristic
377C22—41no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22C23×#44211yescharacteristic
383C22C23×#44211yescharacteristic
384C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
385C4—42no—
386C4—42no—
387C4—42no—
388C4—82no—
389C4C23×#44211yescharacteristic
390C4C23×#44211yescharacteristic
391C4C2×C4×D4×C2311yescharacteristic
392C4C2×C23×AGL(1,ℤ/8)11yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×C23×#44211yescharacteristic
subgroupG/H
centerC4×C23#442
commutatorC8C23×C4×C23
Frattini, cosocleC2×C8C24×C23
socleC2×C23C2×#442
prime radicalC23×#3414C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C22×C4×C23 ◁ C2×C23×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#444C2semidirect
3. C23×#2943C2semidirect
4. C23×AGL(1,ℤ/16)C2semidirect
5. C23×#2180C2semidirect
6. C23×#2155C2semidirect
7. C23×#2148C2semidirect
8. C23×#2145C2semidirect
9. C2×C23×#347C2semidirect
14. C23×#510C4semidirect
16. C23×#509C4semidirect
18. C23×#508C4semidirect
23. C23×#442C22semidirect
24. C23×#441C22semidirect
31. C23×#365C22semidirect
32. C23×#362C22semidirect
33. C23×#361C22semidirect
35. C23×#347C22semidirect
44. C23×#135C2×C4semidirect
46. C2×Q16×C23C2×C4semidirect
50. C2×D8×C23C2×C4semidirect
66. C23×Q32C2×C4semidirect
67. C23×SD32C2×C4semidirect
68. C23×SD32C2×C4semidirect
69. C23×D16C2×C4semidirect
71. C23×M32C2×C4semidirect
75. C2×C16×C23C2×C4semidirect
140. #33468C23direct (N × H)
179. C2×#444C2×C23semidirect
180. #2943C2×C23semidirect
181. AGL(1,ℤ/16)C2×C23semidirect
182. #2180C2×C23semidirect
183. #2155C2×C23semidirect
184. #2148C2×C23semidirect
185. #2145C2×C23semidirect
186. C2×#347C2×C23semidirect
207. #510C4×C23semidirect
209. #509C4×C23semidirect
211. #508C4×C23semidirect
216. #442C22×C23semidirect
217. #441C22×C23semidirect
224. #365C22×C23semidirect
225. #362C22×C23semidirect
226. #361C22×C23semidirect
228. #347C22×C23semidirect
243. #135C2×C4×C23semidirect
245. C2×Q16C2×C4×C23semidirect
249. C2×D8C2×C4×C23semidirect
265. Q32C2×C4×C23semidirect
266. SD32C2×C4×C23semidirect
267. SD32C2×C4×C23semidirect
268. D16C2×C4×C23semidirect
270. M32C2×C4×C23semidirect
274. C2×C16C2×C4×C23semidirect
284. C23#33468direct (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×#4442
C23×#29432
C23×AGL(1,ℤ/16)2
C23×#21802
C23×#21552
C23×#21482
C23×#21452
C2×C23×#3472
#3346823