Finite Group Databasebeta

← C23×#9571 #19157197

Subgroups — C23×#9571 #19157197

Order 5888.

454 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1982C211yesmaximal, characteristic
2C23×#1957C211yesmaximal, characteristic
3C23×#1947C211yesmaximal, characteristic, complemented
4C23×#1770C211yesmaximal, characteristic, complemented
5C2×C23×#331C211yesmaximal, characteristic
6C23×#1296C211yesmaximal, characteristic, complemented
7C23×#1277C211yesmaximal, characteristic, complemented
8C2×C23×#127C2211yescharacteristic
9C2×C23×#107—21no—
10C2×C23×C4⋊C8C2211yescharacteristic
11C2×C23×C4⋊C8C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C2×C23×#102—21no—
14C2×C23×#102—21no—
15C2×C23×#102—21no—
16C2×C23×#102—21no—
17C2×C23×#97—21no—
18C2×C4×C8×C23C2211yescharacteristic
19C23×#384C2211yescharacteristic
20C4×C4⋊C4×C23C2211yescharacteristic
21C23×#340—21no—
22C23×#336—21no—
23C23×#331C414yescomplemented
24C22×D4×C23—21no—
25C22×D4×C23—21no—
26C22×D4×C23—21no—
27C22×C8×C23—21no—
28C22×C8×C23—21no—
29C22×C8×C23—21no—
30C22×C8×C23D411yescharacteristic
31C22×C8×C23D411yescharacteristic
32C23×#127—21no—
33C23×#127—21no—
34C23×#127—21no—
35C23×#127C2×C412yes—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23D411yescharacteristic
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C42×C23—21no—
42C2×C42×C23C2311yescharacteristic
43C23×#107—42no—
44C23×C4⋊C8—21no—
45C23×C4⋊C8—21no—
46C23×C4⋊C8—21no—
47C23×C4⋊C8—21no—
48C23×C4⋊C8C2×C412yes—
49C23×C4⋊C8C2×C412yes—
50C23×#102—42no—
51C23×#102—42no—
52C23×#102—42no—
53C23×#102—42no—
54C23×#97—42no—
55C4×C8×C23—21no—
56C4×C8×C23—21no—
57C23×#95—21no—
58C24×C23—41no—
59C2×D4×C23—22no—
60C2×D4×C23—22no—
61C2×D4×C23—22no—
62C2×D4×C23—41no—
63C2×D4×C23—41no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23—41no—
67C2×D4×C23—41no—
68C2×D4×C23—41no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—81no—
74C2×D4×C23—81no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—41no—
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23#4111yescharacteristic
81C2×C8×C23—21no—
82C2×C8×C23—21no—
83C2×C8×C23—22no—
84C2×C8×C23—22no—
85C2×C8×C23—21no—
86C2×C8×C23—21no—
87C2×C8×C23—21no—
88C2×C8×C23—41no—
89C2×C8×C23—41no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23D812yes—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—81no—
105C42×C23—41no—
106C42×C23—41no—
107C42×C23C22×C411yescharacteristic
108C42×C23#4111yescharacteristic
109C42×C23#4111yescharacteristic
110C42×C23#4111yescharacteristic
111#9571C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—42no—
113C23×C23—81no—
114C23×C23—81no—
115C23×C23—81no—
116C23×C23—81no—
117C23×C23—81no—
118C23×C23—81no—
119C23×C23#12111yescharacteristic
120D4×C23—42no—
121D4×C23—42no—
122D4×C23—42no—
123D4×C23—42no—
124D4×C23—82no—
125D4×C23—82no—
126D4×C23—81no—
127D4×C23—81no—
128D4×C23—81no—
129D4×C23—81no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23#11711yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23C4×D411yescharacteristic
144C2×C4×C23#12311yescharacteristic
145C2×C4×C23#12411yescharacteristic
146C2×C4×C23#12611yescharacteristic
147C2×C4×C23C2×D811yescharacteristic
148C2×C4×C23#13511yescharacteristic
149C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C23#13711yescharacteristic
154C2×C4×C23#13711yescharacteristic
155C8×C23—22no—
156C8×C23—42no—
157C8×C23—42no—
158C8×C23—41no—
159C8×C23—81no—
160#1982C2×C2311yescharacteristic
161#1957C2×C2311yescharacteristic
162#1947C2×C2311yescharacteristic, complemented
163#1770C2×C2311yescharacteristic, complemented
164C2×#331C2×C2311yescharacteristic
165#1296C2×C2311yescharacteristic, complemented
166#1277C2×C2311yescharacteristic, complemented
167C22×C23—82no—
168C22×C23—82no—
169C22×C23—82no—
170C22×C23—161no—
171C22×C23—161no—
172C22×C23—161no—
173C22×C23—161no—
174C22×C23#38811yescharacteristic
175C22×C23#45911yescharacteristic
176C22×C23#46411yescharacteristic
177C22×C23#46811yescharacteristic
178C22×C23#46911yescharacteristic
179C22×C23#47111yescharacteristic
180C22×C23#47211yescharacteristic
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—81no—
184C4×C23—81no—
185C4×C23—161no—
186C4×C23C4×D811yescharacteristic
187C4×C23#44011yescharacteristic
188C4×C23#44011yescharacteristic
189C4×C23#44211yescharacteristic
190C4×C23#48011yescharacteristic
191C4×C23#48211yescharacteristic
192C4×C23#48911yescharacteristic
193C4×C23#49011yescharacteristic
194C2×#127C22×C2311yescharacteristic
195C2×#107—21no—
196C2×C4⋊C8C22×C2311yescharacteristic
197C2×C4⋊C8C22×C2311yescharacteristic
198C2×C4⋊C8C22×C2311yescharacteristic
199C2×#102—21no—
200C2×#102—21no—
201C2×#102—21no—
202C2×#102—21no—
203C2×#97—21no—
204C2×C4×C8C22×C2311yescharacteristic
205#384C22×C2311yescharacteristic
206C4×C4⋊C4C22×C2311yescharacteristic
207#340—21no—
208#336—21no—
209#331C4×C2314yescomplemented
210C2×C23—162no—
211C2×C23#168511yescharacteristic
212C2×C23#168711yescharacteristic
213C2×C23#169811yescharacteristic
214C2×C23#170211yescharacteristic
215C2×C23#192511yescharacteristic
216C2×C23#193111yescharacteristic
217C2×C23#193611yescharacteristic
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8D4×C2311yescharacteristic
225C22×C8D4×C2311yescharacteristic
226#127—21no—
227#127—21no—
228#127—21no—
229#127C2×C4×C2312yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4D4×C2311yescharacteristic
234C2×C4⋊C4D4×C2311yescharacteristic
235C2×C42—21no—
236C2×C42C23×C2311yescharacteristic
237#107—42no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C2×C4×C2312yes—
243C4⋊C8C2×C4×C2312yes—
244#102—42no—
245#102—42no—
246#102—42no—
247#102—42no—
248#97—42no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252C23#957111yescharacteristic, direct product factor, Sylow
253C24—41no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—81no—
269C2×D4—81no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4C2×D4×C2311yescharacteristic
274C22×C4C2×D4×C2311yescharacteristic
275C22×C4#41×C2311yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8D8×C2312yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—81no—
300C42—41no—
301C42—41no—
302C42C22×C4×C2311yescharacteristic
303C42#41×C2311yescharacteristic
304C42#41×C2311yescharacteristic
305C42#41×C2311yescharacteristic
306C23—42no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23C23×#12111yescharacteristic
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—82no—
319D4—82no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4C23×#11711yescharacteristic
336C2×C4C4×D4×C2311yescharacteristic
337C2×C4C4×D4×C2311yescharacteristic
338C2×C4C23×#12311yescharacteristic
339C2×C4C23×#12411yescharacteristic
340C2×C4C23×#12611yescharacteristic
341C2×C4C2×D8×C2311yescharacteristic
342C2×C4C23×#13511yescharacteristic
343C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
345C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
347C2×C4C23×#13711yescharacteristic
348C2×C4C23×#13711yescharacteristic
349C8—22no—
350C8—42no—
351C8—42no—
352C8—41no—
353C8—81no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22C23×#38811yescharacteristic
362C22C23×#45911yescharacteristic
363C22C23×#46411yescharacteristic
364C22C23×#46811yescharacteristic
365C22C23×#46911yescharacteristic
366C22C23×#47111yescharacteristic
367C22C23×#47211yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—161no—
373C4C4×D8×C2311yescharacteristic
374C4C23×#44011yescharacteristic
375C4C23×#44011yescharacteristic
376C4C23×#44211yescharacteristic
377C4C23×#48011yescharacteristic
378C4C23×#48211yescharacteristic
379C4C23×#48911yescharacteristic
380C4C23×#49011yescharacteristic
381C2—162no—
382C2C23×#168511yescharacteristic
383C2C23×#168711yescharacteristic
384C2C23×#169811yescharacteristic
385C2C23×#170211yescharacteristic
386C2C23×#192511yescharacteristic
387C2C23×#193111yescharacteristic
388C2C23×#193611yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1947C2semidirect
4. C23×#1770C2semidirect
6. C23×#1296C2semidirect
7. C23×#1277C2semidirect
23. C23×#331C4semidirect
111. #9571C23direct (N × H)
162. #1947C2×C23semidirect
163. #1770C2×C23semidirect
165. #1296C2×C23semidirect
166. #1277C2×C23semidirect
209. #331C4×C23semidirect
252. C23#9571direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#19822
C23×#19572
C23×#19472
C23×#17702
C2×C23×#3312
C23×#12962
C23×#12772
#957123