Finite Group Databasebeta

← C23×#12990 #19160616

Subgroups — C23×#12990 #19160616

Order 5888.

518 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3050C211yesmaximal, characteristic
2C22×C23×#108C211yesmaximal, characteristic, complemented
3C2×C23×#359C212yesmaximal, complemented
4C2×C23×C22⋊C16C211yesmaximal, characteristic, complemented
5C23×#1382C212yesmaximal, complemented
6C2×C23×#135C2211yescharacteristic
7C22×M16×C23—21no—
8C23×C8×C23C2211yescharacteristic
9C23×#525—21no—
10C22×C16×C23C2212yes—
11C23×#467C411yescharacteristic
12C23×#467C411yescharacteristic
13C23×#466C411yescharacteristic
14C23×#466C411yescharacteristic
15C23×#465—21no—
16C23×#465—21no—
17C2×C23×#108—21no—
18C2×C23×#108—24no—
19C2×C23×#108C2212yes—
20C23×#427C2211yescharacteristic
21C23×#416—21no—
22C23×#359—24no—
23C23×C22⋊C16—21no—
24C23×C22⋊C16—21no—
25C2×#41×C23—21no—
26C23×C4×C23D411yescharacteristic
27C23×#135—22no—
28C23×#135—22no—
29C2×Q16×C23C2×C411yescharacteristic
30C2×SD16×C23—21no—
31C2×D8×C23C2×C411yescharacteristic
32C2×M16×C23—24no—
33C2×M16×C23—22no—
34C2×M16×C23—42no—
35C2×M16×C23—41no—
36C22×C8×C23—22no—
37C22×C8×C23—22no—
38C22×C8×C23—41no—
39C22×C8×C23C2×C411yescharacteristic
40C22×C8×C23C2×C411yescharacteristic
41C22×C8×C23D411yescharacteristic
42C22×C8×C23D411yescharacteristic
43C22×C8×C23D411yescharacteristic
44C22×C8×C23C2311yescharacteristic
45C23×#123—41no—
46C23×C22⋊Q8—21no—
47C23×C22⋊Q8—21no—
48C23×#121—21no—
49C23×#121—21no—
50C23×#120—41no—
51C4×D4×C23—41no—
52C4×D4×C23—41no—
53C23×#117—21no—
54C2×C16×C23—22no—
55C2×C16×C23—22no—
56C2×C16×C23—22no—
57C23×#108—24no—
58C23×#108—22no—
59C23×#108D414yescomplemented
60C23×#107C2×C411yescharacteristic
61C23×#107C2×C411yescharacteristic
62C23×#106—21no—
63C23×#103—21no—
64C23×#103—21no—
65C23×#102—21no—
66C23×#102—21no—
67C24×C23SD1611yescharacteristic
68#41×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72C2×Q8×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—41no—
82C22×C4×C23C22⋊C411yescharacteristic
83C22×C4×C23C22⋊C411yescharacteristic
84C22×C4×C23D811yescharacteristic
85C22×C4×C23D811yescharacteristic
86C22×C4×C23SD1611yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88Q16×C23—22no—
89SD16×C23—42no—
90D8×C23—22no—
91M16×C23—24no—
92M16×C23—44no—
93M16×C23—42no—
94C2×C8×C23—21no—
95C2×C8×C23—21no—
96C2×C8×C23—22no—
97C2×C8×C23—22no—
98C2×C8×C23—22no—
99C2×C8×C23—22no—
100C2×C8×C23—44no—
101C2×C8×C23—42no—
102C2×C8×C23C22⋊C411yescharacteristic
103C2×C8×C23C22⋊C411yescharacteristic
104C2×C8×C23C22×C411yescharacteristic
105C2×C8×C23C2×D412yes—
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23#4112yes—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C42×C23—41no—
118C16×C23—42no—
119#12990C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—22no—
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23#10211yescharacteristic
126C23×C23#10211yescharacteristic
127C23×C23C2×SD1611yescharacteristic
128Q8×C23—41no—
129Q8×C23—41no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—82no—
133D4×C23—82no—
134D4×C23—81no—
135D4×C23—81no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—81no—
150C2×C4×C23—81no—
151C2×C4×C23#10211yescharacteristic
152C2×C4×C23#10211yescharacteristic
153C2×C4×C23C2×C22⋊C411yescharacteristic
154C2×C4×C23#12011yescharacteristic
155C2×C4×C23#12311yescharacteristic
156C2×C4×C23C2×D811yescharacteristic
157C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C23#13711yescharacteristic
159C8×C23—21no—
160C8×C23—21no—
161C8×C23—44no—
162C8×C23C4×D412yes—
163C8×C23#12112yes—
164#3050C2×C2311yescharacteristic
165C22×#108C2×C2311yescharacteristic, complemented
166C2×#359C2×C2312yescomplemented
167C2×C22⋊C16C2×C2311yescharacteristic, complemented
168#1382C2×C2312yescomplemented
169C22×C23—21no—
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—81no—
179C22×C23—81no—
180C22×C23—81no—
181C22×C23#35912yescomplemented
182C22×C23C2×#10211yescharacteristic
183C22×C23#44811yescharacteristic
184C22×C23#48111yescharacteristic
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—81no—
188C4×C23—81no—
189C4×C23—81no—
190C4×C23#38611yescharacteristic
191C4×C23#41711yescharacteristic
192C4×C23#44711yescharacteristic
193C4×C23#48011yescharacteristic
194C2×#135C22×C2311yescharacteristic
195C22×M16—21no—
196C23×C8C22×C2311yescharacteristic
197#525—21no—
198C22×C16C22×C2312yes—
199#467C4×C2311yescharacteristic
200#467C4×C2311yescharacteristic
201#466C4×C2311yescharacteristic
202#466C4×C2311yescharacteristic
203#465—21no—
204#465—21no—
205C2×#108—21no—
206C2×#108—24no—
207C2×#108C22×C2312yes—
208#427C22×C2311yescharacteristic
209#416—21no—
210#359—24no—
211C22⋊C16—21no—
212C22⋊C16—21no—
213C2×C23—22no—
214C2×C23—21no—
215C2×C23—21no—
216C2×C23—41no—
217C2×C23—161no—
218C2×C23#187411yescharacteristic
219C2×C23C2×#35911yescharacteristic
220C2×C23#214311yescharacteristic
221C2×#41—21no—
222C23×C4D4×C2311yescharacteristic
223#135—22no—
224#135—22no—
225C2×Q16C2×C4×C2311yescharacteristic
226C2×SD16—21no—
227C2×D8C2×C4×C2311yescharacteristic
228C2×M16—24no—
229C2×M16—22no—
230C2×M16—42no—
231C2×M16—41no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—41no—
235C22×C8C2×C4×C2311yescharacteristic
236C22×C8C2×C4×C2311yescharacteristic
237C22×C8D4×C2311yescharacteristic
238C22×C8D4×C2311yescharacteristic
239C22×C8D4×C2311yescharacteristic
240C22×C8C23×C2311yescharacteristic
241#123—41no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244#121—21no—
245#121—21no—
246#120—41no—
247C4×D4—41no—
248C4×D4—41no—
249#117—21no—
250C2×C16—22no—
251C2×C16—22no—
252C2×C16—22no—
253#108—24no—
254#108—22no—
255#108D4×C2314yescomplemented
256#107C2×C4×C2311yescharacteristic
257#107C2×C4×C2311yescharacteristic
258#106—21no—
259#103—21no—
260#103—21no—
261#102—21no—
262#102—21no—
263C23#1299011yescharacteristic, direct product factor, Sylow
264C24SD16×C2311yescharacteristic
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4C22⋊C4×C2311yescharacteristic
280C22×C4C22⋊C4×C2311yescharacteristic
281C22×C4D8×C2311yescharacteristic
282C22×C4D8×C2311yescharacteristic
283C22×C4SD16×C2311yescharacteristic
284C22×C4C2×D4×C2311yescharacteristic
285Q16—22no—
286SD16—42no—
287D8—22no—
288M16—24no—
289M16—44no—
290M16—42no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8C22⋊C4×C2311yescharacteristic
300C2×C8C22⋊C4×C2311yescharacteristic
301C2×C8C22×C4×C2311yescharacteristic
302C2×C8C2×D4×C2312yes—
303C2×C8C2×D4×C2311yescharacteristic
304C2×C8#41×C2312yes—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C42—41no—
315C16—42no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23C23×#10211yescharacteristic
322C23C23×#10211yescharacteristic
323C23C2×SD16×C2311yescharacteristic
324Q8—41no—
325Q8—41no—
326D4—41no—
327D4—41no—
328D4—82no—
329D4—82no—
330D4—81no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4C23×#10211yescharacteristic
348C2×C4C23×#10211yescharacteristic
349C2×C4C2×C22⋊C4×C2311yescharacteristic
350C2×C4C23×#12011yescharacteristic
351C2×C4C23×#12311yescharacteristic
352C2×C4C2×D8×C2311yescharacteristic
353C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C23×#13711yescharacteristic
355C8—21no—
356C8—21no—
357C8—44no—
358C8C4×D4×C2312yes—
359C8C23×#12112yes—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—42no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C23×#35912yescomplemented
373C22C2×C23×#10211yescharacteristic
374C22C23×#44811yescharacteristic
375C22C23×#48111yescharacteristic
376C4—21no—
377C4—21no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4C23×#38611yescharacteristic
382C4C23×#41711yescharacteristic
383C4C23×#44711yescharacteristic
384C4C23×#48011yescharacteristic
385C2—22no—
386C2—21no—
387C2—21no—
388C2—41no—
389C2—161no—
390C2C23×#187411yescharacteristic
391C2C2×C23×#35911yescharacteristic
392C2C23×#214311yescharacteristic
subgroupG/H
centerC2×C4×C23C2×D8
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
socleC22×C23C2×#102
prime radicalC23×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C23×C4×C23 ◁ C23×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C23×#108C2semidirect
3. C2×C23×#359C2semidirect
4. C2×C23×C22⋊C16C2semidirect
5. C23×#1382C2semidirect
59. C23×#108D4semidirect
119. #12990C23direct (N × H)
165. C22×#108C2×C23semidirect
166. C2×#359C2×C23semidirect
167. C2×C22⋊C16C2×C23semidirect
168. #1382C2×C23semidirect
181. C22×C23#359semidirect
255. #108D4×C23semidirect
263. C23#12990direct (N × H)
372. C22C23×#359semidirect

Maximal subgroups

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

subgroupindex
C23×#30502
C22×C23×#1082
C2×C23×#3592
C2×C23×C22⋊C162
C23×#13822
#1299023