Finite Group Databasebeta

← C23×#12806 #19160432

Subgroups — C23×#12806 #19160432

Order 5888.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2930C211yesmaximal, characteristic
2C23×#2120C212yesmaximal
3C23×#1813C211yesmaximal, characteristic, complemented
4C23×#1812C211yesmaximal, characteristic, complemented
5C23×#1395C212yesmaximal, complemented
6C2×C23×#131C2211yescharacteristic
7C22×M16×C23C411yescharacteristic, complemented
8C23×#525C2211yescharacteristic
9C23×#436—21no—
10C23×#436—21no—
11C23×#435—21no—
12C23×#435—21no—
13C2×C23×#108—21no—
14C23×#427—21no—
15C23×#424C411yescharacteristic, complemented
16C23×#413C2212yes—
17C23×#410C2212yes—
18C23×#408C2211yescharacteristic
19C23×#407C411yescharacteristic, complemented
20C23×#405C411yescharacteristic, complemented
21C23×#352—22no—
22C23×AΣL(1,𝔽16)—22no—
23C23×#338—21no—
24C23×H(3,ℤ/4)—21no—
25C2×#41×C23C2311yescharacteristic
26C23×C4×C23C2×C411yescharacteristic
27C23×#131—41no—
28C23×#131—41no—
29C2×M16×C23—21no—
30C2×M16×C23—21no—
31C2×M16×C23—22no—
32C2×M16×C23—41no—
33C2×M16×C23—41no—
34C2×M16×C23C2×C411yescharacteristic
35C2×M16×C23C2×C411yescharacteristic
36C2×M16×C23D411yescharacteristic
37C2×M16×C23D411yescharacteristic
38C22×C8×C23C2×C411yescharacteristic
39C22×C8×C23C2×C411yescharacteristic
40C22×C8×C23D411yescharacteristic
41C22×C8×C23D411yescharacteristic
42C23×#123D411yescharacteristic
43C23×#123Q811yescharacteristic
44C23×C22⋊Q8—21no—
45C23×#121—21no—
46C23×#120D411yescharacteristic
47C23×#120Q811yescharacteristic
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C23×#117—22no—
51C23×#117C2×C411yescharacteristic
52C23×#108—41no—
53C23×#108—41no—
54C23×#107—41no—
55C23×#106—41no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×#101—22no—
59C23×#100—22no—
60C23×#99—22no—
61C23×#99—22no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×C22⋊C8—21no—
65C23×C22⋊C8—21no—
66C23×#97—21no—
67C4×C8×C23—21no—
68C24×C23C4⋊C411yescharacteristic
69#41×C23—41no—
70#41×C23—41no—
71C2×Q8×C23#4111yescharacteristic
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23C2×D411yescharacteristic
75C2×D4×C23C2×Q811yescharacteristic
76C2×D4×C23#4111yescharacteristic
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—41no—
80C22×C4×C23—41no—
81C22×C4×C23C4⋊C411yescharacteristic
82C22×C4×C23C22×C411yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84C22×C4×C23C2×D411yescharacteristic
85M16×C23—41no—
86M16×C23—41no—
87M16×C23—41no—
88M16×C23—42no—
89M16×C23—41no—
90M16×C23—41no—
91M16×C23—41no—
92M16×C23—42no—
93M16×C23—42no—
94C2×C8×C23—21no—
95C2×C8×C23—21no—
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—21no—
102C2×C8×C23—41no—
103C2×C8×C23—42no—
104C2×C8×C23C22⋊C411yescharacteristic
105C2×C8×C23C22⋊C411yescharacteristic
106C2×C8×C23C22⋊C411yescharacteristic
107C2×C8×C23C22⋊C411yescharacteristic
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—42no—
112C22⋊C4×C23—21no—
113C22⋊C4×C23—21no—
114C22⋊C4×C23—21no—
115C22⋊C4×C23—21no—
116C22⋊C4×C23—42no—
117C22⋊C4×C23C4⋊C411yescharacteristic
118C22⋊C4×C23C4⋊C411yescharacteristic
119C42×C23—21no—
120C42×C23—42no—
121#12806C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—21no—
123C23×C23—41no—
124C23×C23—81no—
125C23×C23C2×C4⋊C411yescharacteristic
126C23×C23#12111yescharacteristic
127C23×C23C22⋊Q811yescharacteristic
128Q8×C23—41no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—81no—
133D4×C23—81no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
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—42no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—82no—
148C2×C4×C23C2×C22⋊C411yescharacteristic
149C2×C4×C23#11711yescharacteristic
150C2×C4×C23#12111yescharacteristic
151C2×C4×C23C22⋊Q811yescharacteristic
152C8×C23—21no—
153C8×C23—21no—
154C8×C23—21no—
155C8×C23—41no—
156C8×C23—41no—
157C8×C23—82no—
158C8×C23#9911yescharacteristic
159C8×C23#9911yescharacteristic
160#2930C2×C2311yescharacteristic
161#2120C2×C2312yes—
162#1813C2×C2311yescharacteristic, complemented
163#1812C2×C2311yescharacteristic, complemented
164#1395C2×C2312yescomplemented
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—41no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23—41no—
174C22×C23—82no—
175C22×C23#38011yescharacteristic
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—162no—
183C4×C23C2×#9911yescharacteristic
184C4×C23#41011yescharacteristic
185C2×#131C22×C2311yescharacteristic
186C22×M16C4×C2311yescharacteristic, complemented
187#525C22×C2311yescharacteristic
188#436—21no—
189#436—21no—
190#435—21no—
191#435—21no—
192C2×#108—21no—
193#427—21no—
194#424C4×C2311yescharacteristic, complemented
195#413C22×C2312yes—
196#410C22×C2312yes—
197#408C22×C2311yescharacteristic
198#407C4×C2311yescharacteristic, complemented
199#405C4×C2311yescharacteristic, complemented
200#352—22no—
201AΣL(1,𝔽16)—22no—
202#338—21no—
203H(3,ℤ/4)—21no—
204C2×C23—21no—
205C2×C23—41no—
206C2×C23—41no—
207C2×C23—41no—
208C2×C23—81no—
209C2×C23#179911yescharacteristic
210C2×#41C23×C2311yescharacteristic
211C23×C4C2×C4×C2311yescharacteristic
212#131—41no—
213#131—41no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—22no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16C2×C4×C2311yescharacteristic
220C2×M16C2×C4×C2311yescharacteristic
221C2×M16D4×C2311yescharacteristic
222C2×M16D4×C2311yescharacteristic
223C22×C8C2×C4×C2311yescharacteristic
224C22×C8C2×C4×C2311yescharacteristic
225C22×C8D4×C2311yescharacteristic
226C22×C8D4×C2311yescharacteristic
227#123D4×C2311yescharacteristic
228#123Q8×C2311yescharacteristic
229C22⋊Q8—21no—
230#121—21no—
231#120D4×C2311yescharacteristic
232#120Q8×C2311yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235#117—22no—
236#117C2×C4×C2311yescharacteristic
237#108—41no—
238#108—41no—
239#107—41no—
240#106—41no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243#101—22no—
244#100—22no—
245#99—22no—
246#99—22no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251#97—21no—
252C4×C8—21no—
253C23#1280611yescharacteristic, direct product factor, Sylow
254C24C4⋊C4×C2311yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8#41×C2311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×D4×C2311yescharacteristic
261C2×D4C2×Q8×C2311yescharacteristic
262C2×D4#41×C2311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C4⋊C4×C2311yescharacteristic
268C22×C4C22×C4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270C22×C4C2×D4×C2311yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—42no—
279M16—42no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8C22⋊C4×C2311yescharacteristic
291C2×C8C22⋊C4×C2311yescharacteristic
292C2×C8C22⋊C4×C2311yescharacteristic
293C2×C8C22⋊C4×C2311yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—42no—
303C22⋊C4C4⋊C4×C2311yescharacteristic
304C22⋊C4C4⋊C4×C2311yescharacteristic
305C42—21no—
306C42—42no—
307C23—21no—
308C23—41no—
309C23—81no—
310C23C2×C4⋊C4×C2311yescharacteristic
311C23C23×#12111yescharacteristic
312C23C23×C22⋊Q811yescharacteristic
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4C2×C22⋊C4×C2311yescharacteristic
334C2×C4C23×#11711yescharacteristic
335C2×C4C23×#12111yescharacteristic
336C2×C4C23×C22⋊Q811yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—41no—
341C8—41no—
342C8—82no—
343C8C23×#9911yescharacteristic
344C8C23×#9911yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22C23×#38011yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—162no—
363C4C2×C23×#9911yescharacteristic
364C4C23×#41011yescharacteristic
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2C23×#179911yescharacteristic
subgroupG/H
centerC4×C23#410
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
socleC2×C23#1799
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C8×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1813C2semidirect
4. C23×#1812C2semidirect
5. C23×#1395C2semidirect
7. C22×M16×C23C4semidirect
15. C23×#424C4semidirect
19. C23×#407C4semidirect
20. C23×#405C4semidirect
121. #12806C23direct (N × H)
162. #1813C2×C23semidirect
163. #1812C2×C23semidirect
164. #1395C2×C23semidirect
186. C22×M16C4×C23semidirect
194. #424C4×C23semidirect
198. #407C4×C23semidirect
199. #405C4×C23semidirect
253. C23#12806direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29302
C23×#21202
C23×#18132
C23×#18122
C23×#13952
#1280623