Finite Group Databasebeta

← C23×#10263 #19157889

Subgroups — C23×#10263 #19157889

Order 5888.

626 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#486C211yesmaximal, characteristic
2C2×C23×#462C211yesmaximal, characteristic, complemented
3C2×C23×#460C211yesmaximal, characteristic
4C23×#1984C211yesmaximal, characteristic, complemented
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C2×C23×#331C211yesmaximal, characteristic
7C23×#1277C211yesmaximal, characteristic, complemented
8C22×Q16×C23—21no—
9C22×SD16×C23—21no—
10C2×C23×C4⋊Q8C2211yescharacteristic
11C2×C23×#127C2211yescharacteristic
12C2×C4×Q8×C23C2211yescharacteristic
13C23×#486—22no—
14C23×#486—22no—
15C23×#462—22no—
16C23×#462—22no—
17C23×#460—22no—
18C23×#460—22no—
19C2×C23×C4⋊C8C2211yescharacteristic
20C2×C23×C4⋊C8C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C2×C23×#103—21no—
23C2×C23×#102—22no—
24C2×C23×#102—21no—
25C2×C4×C8×C23C2211yescharacteristic
26C23×#336—22no—
27C23×#332—22no—
28C23×#331—22no—
29C22×Q8×C23—21no—
30C22×Q8×C23D411yescharacteristic
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C2×Q16×C23—24no—
35C2×Q16×C23—42no—
36C2×Q16×C23—42no—
37C2×SD16×C23—24no—
38C2×SD16×C23—42no—
39C2×SD16×C23—42no—
40C22×C8×C23—21no—
41C22×C8×C23—21no—
42C22×C8×C23—21no—
43C22×C8×C23D412yes—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8—21no—
46C23×C4⋊Q8—21no—
47C23×C4⋊Q8D412yes—
48C23×#127—21no—
49C23×#127—21no—
50C23×#127—21no—
51C23×#127D412yes—
52C4×Q8×C23—21no—
53C4×Q8×C23—21no—
54C4×Q8×C23—21no—
55C4×Q8×C23—21no—
56C2×C4⋊C4×C23—21no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23—21no—
59C2×C4⋊C4×C23D411yescharacteristic
60C2×C42×C23—21no—
61C2×C42×C23C2311yescharacteristic
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8D412yes—
67C23×C4⋊C8D412yes—
68C23×#103—24no—
69C23×#102—24no—
70C23×#102—28no—
71C4×C8×C23—21no—
72C4×C8×C23—21no—
73C24×C23—41no—
74C2×Q8×C23—21no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23—22no—
78C2×Q8×C23—21no—
79C2×Q8×C23—21no—
80C2×Q8×C23—41no—
81C2×Q8×C23—41no—
82C2×Q8×C23—41no—
83C2×Q8×C23—41no—
84C2×Q8×C23—41no—
85C2×Q8×C23D812yes—
86C2×D4×C23—22no—
87C2×D4×C23—22no—
88C2×D4×C23—22no—
89C2×D4×C23—41no—
90C2×D4×C23—41no—
91C2×D4×C23—41no—
92C2×D4×C23—41no—
93C2×D4×C23—41no—
94C2×D4×C23—42no—
95C2×D4×C23—41no—
96C2×D4×C23—41no—
97C2×D4×C23—42no—
98C2×D4×C23—82no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—41no—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23#4111yescharacteristic
105Q16×C23—48no—
106SD16×C23—48no—
107C2×C8×C23—22no—
108C2×C8×C23—22no—
109C2×C8×C23—22no—
110C2×C8×C23—22no—
111C2×C8×C23—22no—
112C2×C8×C23—41no—
113C2×C8×C23—41no—
114C2×C8×C23—41no—
115C2×C8×C23—41no—
116C2×C8×C23—41no—
117C2×C8×C23—42no—
118C4⋊C4×C23—22no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—21no—
121C4⋊C4×C23—22no—
122C4⋊C4×C23—41no—
123C4⋊C4×C23—41no—
124C4⋊C4×C23—41no—
125C4⋊C4×C23—41no—
126C42×C23—41no—
127C42×C23—41no—
128C42×C23C2×D411yescharacteristic
129C42×C23C2×D411yescharacteristic
130C42×C23C2×D411yescharacteristic
131C42×C23C2×D411yescharacteristic
132#10263C2311yesmaximal, characteristic, direct product factor, Sylow
133C23×C23—42no—
134C23×C23—82no—
135C23×C23—81no—
136C23×C23—81no—
137C23×C23—82no—
138C23×C23#12111yescharacteristic
139Q8×C23—22no—
140Q8×C23—22no—
141Q8×C23—41no—
142Q8×C23—41no—
143Q8×C23—42no—
144Q8×C23—42no—
145Q8×C23—81no—
146Q8×C23—81no—
147D4×C23—42no—
148D4×C23—42no—
149D4×C23—44no—
150D4×C23—82no—
151D4×C23—82no—
152D4×C23—81no—
153D4×C23—81no—
154D4×C23—81no—
155D4×C23—81no—
156C2×C4×C23—22no—
157C2×C4×C23—22no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—42no—
162C2×C4×C23—42no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23#12011yescharacteristic
166C2×C4×C23#12011yescharacteristic
167C2×C4×C23#12011yescharacteristic
168C2×C4×C23#12111yescharacteristic
169C2×C4×C23#12111yescharacteristic
170C2×C4×C23#12711yescharacteristic
171C2×C4×C23C2×D811yescharacteristic
172C2×C4×C23#13511yescharacteristic
173C2×C4×C23#13512yes—
174C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C23#13712yes—
177C8×C23—42no—
178C8×C23—42no—
179C8×C23—42no—
180C8×C23—81no—
181C2×#486C2×C2311yescharacteristic
182C2×#462C2×C2311yescharacteristic, complemented
183C2×#460C2×C2311yescharacteristic
184#1984C2×C2311yescharacteristic, complemented
185C2×#332C2×C2311yescharacteristic, complemented
186C2×#331C2×C2311yescharacteristic
187#1277C2×C2311yescharacteristic, complemented
188C22×C23—82no—
189C22×C23—82no—
190C22×C23—82no—
191C22×C23—162no—
192C22×C23—162no—
193C22×C23#39211yescharacteristic
194C22×C23#46412yes—
195C22×C23#46611yescharacteristic
196C22×C23#46911yescharacteristic
197C22×C23#47112yes—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—42no—
201C4×C23—82no—
202C4×C23—81no—
203C4×C23#44711yescharacteristic
204C4×C23#44711yescharacteristic
205C4×C23#44911yescharacteristic
206C4×C23#44911yescharacteristic
207C4×C23#45212yes—
208C4×C23#49511yescharacteristic
209C4×C23#49711yescharacteristic
210C22×Q16—21no—
211C22×SD16—21no—
212C2×C4⋊Q8C22×C2311yescharacteristic
213C2×#127C22×C2311yescharacteristic
214C2×C4×Q8C22×C2311yescharacteristic
215#486—22no—
216#486—22no—
217#462—22no—
218#462—22no—
219#460—22no—
220#460—22no—
221C2×C4⋊C8C22×C2311yescharacteristic
222C2×C4⋊C8C22×C2311yescharacteristic
223C2×C4⋊C8C22×C2311yescharacteristic
224C2×#103—21no—
225C2×#102—22no—
226C2×#102—21no—
227C2×C4×C8C22×C2311yescharacteristic
228#336—22no—
229#332—22no—
230#331—22no—
231C2×C23—162no—
232C2×C23#168212yes—
233C2×C23#170212yes—
234C2×C23#199911yescharacteristic
235C2×C23#201511yescharacteristic
236C2×C23#201911yescharacteristic
237C22×Q8—21no—
238C22×Q8D4×C2311yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C2×Q16—24no—
243C2×Q16—42no—
244C2×Q16—42no—
245C2×SD16—24no—
246C2×SD16—42no—
247C2×SD16—42no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8D4×C2312yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8D4×C2312yes—
256#127—21no—
257#127—21no—
258#127—21no—
259#127D4×C2312yes—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4D4×C2311yescharacteristic
268C2×C42—21no—
269C2×C42C23×C2311yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8D4×C2312yes—
275C4⋊C8D4×C2312yes—
276#103—24no—
277#102—24no—
278#102—28no—
279C4×C8—21no—
280C4×C8—21no—
281C23#1026311yescharacteristic, direct product factor, Sylow
282C24—41no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8D8×C2312yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—82no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4C2×D4×C2311yescharacteristic
312C22×C4C2×D4×C2311yescharacteristic
313C22×C4#41×C2311yescharacteristic
314Q16—48no—
315SD16—48no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C42—41no—
336C42—41no—
337C42C2×D4×C2311yescharacteristic
338C42C2×D4×C2311yescharacteristic
339C42C2×D4×C2311yescharacteristic
340C42C2×D4×C2311yescharacteristic
341C23—42no—
342C23—82no—
343C23—81no—
344C23—81no—
345C23—82no—
346C23C23×#12111yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—41no—
350Q8—41no—
351Q8—42no—
352Q8—42no—
353Q8—81no—
354Q8—81no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—82no—
359D4—82no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4C23×#12011yescharacteristic
374C2×C4C23×#12011yescharacteristic
375C2×C4C23×#12011yescharacteristic
376C2×C4C23×#12111yescharacteristic
377C2×C4C23×#12111yescharacteristic
378C2×C4C23×#12711yescharacteristic
379C2×C4C2×D8×C2311yescharacteristic
380C2×C4C23×#13511yescharacteristic
381C2×C4C23×#13512yes—
382C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C23×#13712yes—
385C8—42no—
386C8—42no—
387C8—42no—
388C8—81no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—162no—
393C22—162no—
394C22C23×#39211yescharacteristic
395C22C23×#46412yes—
396C22C23×#46611yescharacteristic
397C22C23×#46911yescharacteristic
398C22C23×#47112yes—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—82no—
403C4—81no—
404C4C23×#44711yescharacteristic
405C4C23×#44711yescharacteristic
406C4C23×#44911yescharacteristic
407C4C23×#44911yescharacteristic
408C4C23×#45212yes—
409C4C23×#49511yescharacteristic
410C4C23×#49711yescharacteristic
411C2—162no—
412C2C23×#168212yes—
413C2C23×#170212yes—
414C2C23×#199911yescharacteristic
415C2C23×#201511yescharacteristic
416C2C23×#201911yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#462C2semidirect
4. C23×#1984C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1277C2semidirect
132. #10263C23direct (N × H)
182. C2×#462C2×C23semidirect
184. #1984C2×C23semidirect
185. C2×#332C2×C23semidirect
187. #1277C2×C23semidirect
281. C23#10263direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4862
C2×C23×#4622
C2×C23×#4602
C23×#19842
C2×C23×#3322
C2×C23×#3312
C23×#12772
#1026323