Finite Group Databasebeta

← C23×#13229 #19160855

Subgroups — C23×#13229 #19160855

Order 5888.

452 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×#3057C211yesmaximal, characteristic
2C23×#2129C211yesmaximal, characteristic, complemented
3C23×#2128C211yesmaximal, characteristic
4C23×#2123C211yesmaximal, characteristic, complemented
5C23×#2122C211yesmaximal, characteristic
6C23×#1812C211yesmaximal, characteristic, complemented
7C23×#1757C211yesmaximal, characteristic, complemented
8C22×SD16×C23C411yescharacteristic, complemented
9C2×C23×#131C2211yescharacteristic
10C23×#536C2211yescharacteristic
11C23×#535C2211yescharacteristic
12C23×#468—21no—
13C23×#468—21no—
14C23×#465—21no—
15C23×#465—21no—
16C23×#427—21no—
17C23×#427C411yescharacteristic, complemented
18C23×#419C411yescharacteristic, complemented
19C23×#418C411yescharacteristic, complemented
20C23×#413C2211yescharacteristic
21C23×#410C2211yescharacteristic
22C23×#410C2211yescharacteristic
23C23×#410C2211yescharacteristic
24C23×#405—21no—
25C23×#354—22no—
26C23×#353—22no—
27C23×#352—22no—
28C23×AΣL(1,𝔽16)—22no—
29C23×#344—21no—
30C23×H(3,ℤ/4)—21no—
31C2×#41×C23C2311yescharacteristic
32C22×Q8×C23C2×C411yescharacteristic
33C22×D4×C23C2×C411yescharacteristic
34C2×SD16×C23—21no—
35C2×SD16×C23—21no—
36C2×SD16×C23—41no—
37C2×SD16×C23—41no—
38C23×#131—41no—
39C23×#131—41no—
40C2×M16×C23—21no—
41C2×M16×C23C2×C411yescharacteristic
42C2×M16×C23D411yescharacteristic
43C2×M16×C23D411yescharacteristic
44C22×C8×C23C2×C411yescharacteristic
45C22×C8×C23D411yescharacteristic
46C22×C8×C23D411yescharacteristic
47C23×C4⋊Q8—21no—
48C23×#127—21no—
49C23×#124—21no—
50C23×#124—21no—
51C23×#123—21no—
52C23×#123—21no—
53C23×C22⋊Q8—21no—
54C23×C22⋊Q8—21no—
55C23×#121—21no—
56C23×#121—21no—
57C23×#120—21no—
58C23×#120—21no—
59C23×#117—21no—
60C23×#117—21no—
61C23×#117—21no—
62C23×#117C2×C411yescharacteristic
63C23×#117C2×C411yescharacteristic
64C23×#107—21no—
65C23×#107—41no—
66C23×#106—21no—
67C23×#106—41no—
68C23×#103—21no—
69C23×#103—21no—
70C23×#102—21no—
71C23×#102—21no—
72C23×#101—21no—
73C23×#100D412yes—
74C23×#99—21no—
75C23×#99—21no—
76C23×#99—21no—
77C23×#99—21no—
78C23×#99—21no—
79C23×#99D412yes—
80C23×#97—41no—
81C4×C8×C23—41no—
82C24×C23—21no—
83#41×C23—41no—
84#41×C23—41no—
85C2×Q8×C23—21no—
86C2×Q8×C23—21no—
87C2×Q8×C23—41no—
88C2×Q8×C23—41no—
89C2×Q8×C23#4111yescharacteristic
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—41no—
93C2×D4×C23—41no—
94C2×D4×C23—41no—
95C2×D4×C23—41no—
96C2×D4×C23C2×D411yescharacteristic
97C2×D4×C23C2×D411yescharacteristic
98C2×D4×C23#4111yescharacteristic
99C22×C4×C23—21no—
100C22×C4×C23C22×C411yescharacteristic
101C22×C4×C23C2×D411yescharacteristic
102C22×C4×C23C2×D411yescharacteristic
103SD16×C23—42no—
104SD16×C23—41no—
105SD16×C23—41no—
106M16×C23—22no—
107M16×C23—41no—
108M16×C23—41no—
109M16×C23—41no—
110M16×C23—41no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23—21no—
114C2×C8×C23—21no—
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—41no—
118C2×C8×C23C22⋊C411yescharacteristic
119C2×C8×C23C22⋊C411yescharacteristic
120C2×C8×C23C22⋊C411yescharacteristic
121C2×C8×C23C22⋊C411yescharacteristic
122C4⋊C4×C23—21no—
123C4⋊C4×C23—21no—
124C4⋊C4×C23—41no—
125C4⋊C4×C23—41no—
126C4⋊C4×C23—41no—
127C4⋊C4×C23—41no—
128C4⋊C4×C23—41no—
129C22⋊C4×C23—21no—
130C22⋊C4×C23—21no—
131C22⋊C4×C23—22no—
132C22⋊C4×C23—41no—
133C22⋊C4×C23—41no—
134C22⋊C4×C23—41no—
135C22⋊C4×C23—41no—
136C22⋊C4×C23—41no—
137C22⋊C4×C23—41no—
138C42×C23—21no—
139C42×C23—21no—
140C42×C23—41no—
141C42×C23—41no—
142C42×C23—41no—
143#13229C2311yesmaximal, characteristic, direct product factor, Sylow
144C23×C23—41no—
145C23×C23—81no—
146C23×C23—161no—
147C23×C23C4×D411yescharacteristic
148C23×C23#12111yescharacteristic
149C23×C23#12711yescharacteristic
150Q8×C23—41no—
151Q8×C23—41no—
152Q8×C23—41no—
153Q8×C23—81no—
154D4×C23—41no—
155D4×C23—41no—
156D4×C23—41no—
157D4×C23—41no—
158D4×C23—41no—
159D4×C23—81no—
160D4×C23—161no—
161D4×C23—161no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—42no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—81no—
173C2×C4×C23—81no—
174C2×C4×C23—81no—
175C2×C4×C23C2×C22⋊C411yescharacteristic
176C2×C4×C23C4×D411yescharacteristic
177C2×C4×C23#12111yescharacteristic
178C2×C4×C23#12411yescharacteristic
179C8×C23—21no—
180C8×C23—21no—
181C8×C23—21no—
182C8×C23—42no—
183C8×C23#9911yescharacteristic
184C8×C23#9911yescharacteristic
185#3057C2×C2311yescharacteristic
186#2129C2×C2311yescharacteristic, complemented
187#2128C2×C2311yescharacteristic
188#2123C2×C2311yescharacteristic, complemented
189#2122C2×C2311yescharacteristic
190#1812C2×C2311yescharacteristic, complemented
191#1757C2×C2311yescharacteristic, complemented
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—21no—
195C22×C23—41no—
196C22×C23—41no—
197C22×C23—41no—
198C22×C23—81no—
199C22×C23—81no—
200C22×C23—81no—
201C22×C23—162no—
202C22×C23#39011yescharacteristic
203C4×C23—21no—
204C4×C23—21no—
205C4×C23—21no—
206C4×C23—81no—
207C4×C23—81no—
208C4×C23—82no—
209C4×C23—81no—
210C4×C23—161no—
211C4×C23—161no—
212C4×C23C2×#9911yescharacteristic
213C4×C23#41011yescharacteristic
214C22×SD16C4×C2311yescharacteristic, complemented
215C2×#131C22×C2311yescharacteristic
216#536C22×C2311yescharacteristic
217#535C22×C2311yescharacteristic
218#468—21no—
219#468—21no—
220#465—21no—
221#465—21no—
222#427—21no—
223#427C4×C2311yescharacteristic, complemented
224#419C4×C2311yescharacteristic, complemented
225#418C4×C2311yescharacteristic, complemented
226#413C22×C2311yescharacteristic
227#410C22×C2311yescharacteristic
228#410C22×C2311yescharacteristic
229#410C22×C2311yescharacteristic
230#405—21no—
231#354—22no—
232#353—22no—
233#352—22no—
234AΣL(1,𝔽16)—22no—
235#344—21no—
236H(3,ℤ/4)—21no—
237C2×C23—21no—
238C2×C23—41no—
239C2×C23—41no—
240C2×C23—41no—
241C2×C23—161no—
242C2×C23#196411yescharacteristic
243C2×#41C23×C2311yescharacteristic
244C22×Q8C2×C4×C2311yescharacteristic
245C22×D4C2×C4×C2311yescharacteristic
246C2×SD16—21no—
247C2×SD16—21no—
248C2×SD16—41no—
249C2×SD16—41no—
250#131—41no—
251#131—41no—
252C2×M16—21no—
253C2×M16C2×C4×C2311yescharacteristic
254C2×M16D4×C2311yescharacteristic
255C2×M16D4×C2311yescharacteristic
256C22×C8C2×C4×C2311yescharacteristic
257C22×C8D4×C2311yescharacteristic
258C22×C8D4×C2311yescharacteristic
259C4⋊Q8—21no—
260#127—21no—
261#124—21no—
262#124—21no—
263#123—21no—
264#123—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267#121—21no—
268#121—21no—
269#120—21no—
270#120—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117C2×C4×C2311yescharacteristic
275#117C2×C4×C2311yescharacteristic
276#107—21no—
277#107—41no—
278#106—21no—
279#106—41no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284#101—21no—
285#100D4×C2312yes—
286#99—21no—
287#99—21no—
288#99—21no—
289#99—21no—
290#99—21no—
291#99D4×C2312yes—
292#97—41no—
293C4×C8—41no—
294C23#1322911yescharacteristic, direct product factor, Sylow
295C24—21no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8#41×C2311yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4C2×D4×C2311yescharacteristic
310C2×D4C2×D4×C2311yescharacteristic
311C2×D4#41×C2311yescharacteristic
312C22×C4—21no—
313C22×C4C22×C4×C2311yescharacteristic
314C22×C4C2×D4×C2311yescharacteristic
315C22×C4C2×D4×C2311yescharacteristic
316SD16—42no—
317SD16—41no—
318SD16—41no—
319M16—22no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8C22⋊C4×C2311yescharacteristic
332C2×C8C22⋊C4×C2311yescharacteristic
333C2×C8C22⋊C4×C2311yescharacteristic
334C2×C8C22⋊C4×C2311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C42—21no—
352C42—21no—
353C42—41no—
354C42—41no—
355C42—41no—
356C23—41no—
357C23—81no—
358C23—161no—
359C23C4×D4×C2311yescharacteristic
360C23C23×#12111yescharacteristic
361C23C23×#12711yescharacteristic
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—81no—
372D4—161no—
373D4—161no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4C2×C22⋊C4×C2311yescharacteristic
388C2×C4C4×D4×C2311yescharacteristic
389C2×C4C23×#12111yescharacteristic
390C2×C4C23×#12411yescharacteristic
391C8—21no—
392C8—21no—
393C8—21no—
394C8—42no—
395C8C23×#9911yescharacteristic
396C8C23×#9911yescharacteristic
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—162no—
407C22C23×#39011yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—82no—
414C4—81no—
415C4—161no—
416C4—161no—
417C4C2×C23×#9911yescharacteristic
418C4C23×#41011yescharacteristic
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—161no—
424C2C23×#196411yescharacteristic
subgroupG/H
center, socleC2×C23#1964
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2129C2semidirect
4. C23×#2123C2semidirect
6. C23×#1812C2semidirect
7. C23×#1757C2semidirect
8. C22×SD16×C23C4semidirect
17. C23×#427C4semidirect
18. C23×#419C4semidirect
19. C23×#418C4semidirect
143. #13229C23direct (N × H)
186. #2129C2×C23semidirect
188. #2123C2×C23semidirect
190. #1812C2×C23semidirect
191. #1757C2×C23semidirect
214. C22×SD16C4×C23semidirect
223. #427C4×C23semidirect
224. #419C4×C23semidirect
225. #418C4×C23semidirect
294. C23#13229direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30572
C23×#21292
C23×#21282
C23×#21232
C23×#21222
C23×#18122
C23×#17572
#1322923