Finite Group Databasebeta

← C23×#10150 #19157776

Subgroups — C23×#10150 #19157776

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3226C211yesmaximal, characteristic, complemented
2C23×#3151C211yesmaximal, characteristic
3C2×C23×#486C211yesmaximal, characteristic, complemented
4C23×#1966C211yesmaximal, characteristic, complemented
5C23×#1545C211yesmaximal, characteristic, complemented
6C23×#1492C211yesmaximal, characteristic, complemented
7C23×#1278C211yesmaximal, characteristic, complemented
8C2×C23×AGL(1,ℤ/8)—22no—
9C23×#532C2211yescharacteristic
10C2×C23×C4⋊Q8C2211yescharacteristic
11C2×C23×#127C2211yescharacteristic
12C23×#500—21no—
13C23×#496—22no—
14C23×#493—21no—
15C23×#492—21no—
16C23×#486—22no—
17C23×#486—21no—
18C23×#486—22no—
19C23×#480—22no—
20C23×#474—22no—
21C23×#431C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C2×C23×#102—22no—
24C2×C23×#101—22no—
25C4×M16×C23C2211yescharacteristic
26C2×C4×C8×C23C2211yescharacteristic
27C23×#331—22no—
28C23×D4⋊C8—22no—
29C2×#41×C23—21no—
30C22×Q8×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C23×AGL(1,ℤ/8)—44no—
35C23×AGL(1,ℤ/8)—44no—
36C2×SD16×C23—24no—
37C2×D8×C23—24no—
38C2×M16×C23—21no—
39C2×M16×C23D412yes—
40C22×C8×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8D411yescharacteristic
46C23×#127—21no—
47C23×#127—21no—
48C23×#127—21no—
49C23×#127D411yescharacteristic
50C23×#127D412yes—
51C23×#124—21no—
52C23×#121—21no—
53C23×#121—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C2×C4⋊C4×C23—21no—
57C2×C4⋊C4×C23D412yes—
58C2×C42×C23C2311yescharacteristic
59C23×#107—24no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8D412yes—
63C23×#102—28no—
64C23×#102—24no—
65C23×#101—44no—
66C23×C22⋊C8—21no—
67C23×#97—21no—
68C4×C8×C23—21no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71C4×C8×C23D412yes—
72C24×C23—41no—
73#41×C23—42no—
74#41×C23—42no—
75C2×Q8×C23—21no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—42no—
79C2×Q8×C23—42no—
80C2×D4×C23—21no—
81C2×D4×C23—22no—
82C2×D4×C23—21no—
83C2×D4×C23—22no—
84C2×D4×C23—21no—
85C2×D4×C23—22no—
86C2×D4×C23—44no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—41no—
91C2×D4×C23—42no—
92C2×D4×C23—42no—
93C2×D4×C23—82no—
94C22×C4×C23—22no—
95C22×C4×C23—41no—
96C22×C4×C23C2×D411yescharacteristic
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23#4111yescharacteristic
99SD16×C23—48no—
100D8×C23—48no—
101M16×C23—22no—
102M16×C23—42no—
103M16×C23D814yes—
104C2×C8×C23—22no—
105C2×C8×C23—22no—
106C2×C8×C23—22no—
107C2×C8×C23—42no—
108C2×C8×C23—41no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C42×C23C2×D411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23C2×D411yescharacteristic
120C42×C23C2×D411yescharacteristic
121#10150C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—41no—
123C23×C23—42no—
124C23×C23—82no—
125C23×C23—82no—
126C23×C23—82no—
127C23×C23#12111yescharacteristic
128Q8×C23—42no—
129Q8×C23—82no—
130Q8×C23—81no—
131Q8×C23—81no—
132D4×C23—44no—
133D4×C23—42no—
134D4×C23—44no—
135D4×C23—42no—
136D4×C23—42no—
137D4×C23—82no—
138D4×C23—82no—
139D4×C23—82no—
140D4×C23—82no—
141D4×C23—81no—
142D4×C23—81no—
143C2×C4×C23—22no—
144C2×C4×C23—22no—
145C2×C4×C23—41no—
146C2×C4×C23—42no—
147C2×C4×C23—44no—
148C2×C4×C23—81no—
149C2×C4×C23—81no—
150C2×C4×C23#12011yescharacteristic
151C2×C4×C23#12011yescharacteristic
152C2×C4×C23#12011yescharacteristic
153C2×C4×C23#12111yescharacteristic
154C2×C4×C23#12111yescharacteristic
155C2×C4×C23#12711yescharacteristic
156C2×C4×C23C2×D812yes—
157C2×C4×C23AGL(1,ℤ/8)12yes—
158C8×C23—24no—
159C8×C23—42no—
160C8×C23—42no—
161C8×C23—81no—
162#3226C2×C2311yescharacteristic, complemented
163#3151C2×C2311yescharacteristic
164C2×#486C2×C2311yescharacteristic, complemented
165#1966C2×C2311yescharacteristic, complemented
166#1545C2×C2311yescharacteristic, complemented
167#1492C2×C2311yescharacteristic, complemented
168#1278C2×C2311yescharacteristic, complemented
169C22×C23—21no—
170C22×C23—21no—
171C22×C23—82no—
172C22×C23—81no—
173C22×C23—81no—
174C22×C23—82no—
175C22×C23—81no—
176C22×C23—82no—
177C22×C23—164no—
178C22×C23#39211yescharacteristic
179C22×C23#45912yes—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—82no—
183C4×C23—81no—
184C4×C23#44712yes—
185C4×C23#45412yes—
186C4×C23#47112yes—
187C4×C23#49311yescharacteristic
188C4×C23#49611yescharacteristic
189C2×AGL(1,ℤ/8)—22no—
190#532C22×C2311yescharacteristic
191C2×C4⋊Q8C22×C2311yescharacteristic
192C2×#127C22×C2311yescharacteristic
193#500—21no—
194#496—22no—
195#493—21no—
196#492—21no—
197#486—22no—
198#486—21no—
199#486—22no—
200#480—22no—
201#474—22no—
202#431C22×C2311yescharacteristic
203#423C22×C2311yescharacteristic
204C2×#102—22no—
205C2×#101—22no—
206C4×M16C22×C2311yescharacteristic
207C2×C4×C8C22×C2311yescharacteristic
208#331—22no—
209D4⋊C8—22no—
210C2×C23—22no—
211C2×C23—161no—
212C2×C23—162no—
213C2×C23#200311yescharacteristic
214C2×C23#201111yescharacteristic
215C2×C23#201911yescharacteristic
216C2×#41—21no—
217C22×Q8—21no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—21no—
221AGL(1,ℤ/8)—44no—
222AGL(1,ℤ/8)—44no—
223C2×SD16—24no—
224C2×D8—24no—
225C2×M16—21no—
226C2×M16D4×C2312yes—
227C22×C8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8D4×C2311yescharacteristic
233#127—21no—
234#127—21no—
235#127—21no—
236#127D4×C2312yes—
237#127D4×C2311yescharacteristic
238#124—21no—
239#121—21no—
240#121—21no—
241C4×D4—21no—
242C4×D4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4D4×C2312yes—
245C2×C42C23×C2311yescharacteristic
246#107—24no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8D4×C2312yes—
250#102—28no—
251#102—24no—
252#101—44no—
253C22⋊C8—21no—
254#97—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8D4×C2312yes—
259C23#1015011yescharacteristic, direct product factor, Sylow
260C24—41no—
261#41—42no—
262#41—42no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—21no—
269C2×D4—22no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—82no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4C2×D4×C2311yescharacteristic
285C22×C4C2×D4×C2311yescharacteristic
286C22×C4#41×C2311yescharacteristic
287SD16—48no—
288D8—48no—
289M16—22no—
290M16—42no—
291M16D8×C2314yes—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—41no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42C2×D4×C2311yescharacteristic
306C42C2×D4×C2311yescharacteristic
307C42C2×D4×C2311yescharacteristic
308C42C2×D4×C2311yescharacteristic
309C23—41no—
310C23—42no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23C23×#12111yescharacteristic
315Q8—42no—
316Q8—82no—
317Q8—81no—
318Q8—81no—
319D4—44no—
320D4—42no—
321D4—44no—
322D4—42no—
323D4—42no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—81no—
329D4—81no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4C23×#12011yescharacteristic
338C2×C4C23×#12011yescharacteristic
339C2×C4C23×#12011yescharacteristic
340C2×C4C23×#12111yescharacteristic
341C2×C4C23×#12111yescharacteristic
342C2×C4C23×#12711yescharacteristic
343C2×C4C2×D8×C2312yes—
344C2×C4C23×AGL(1,ℤ/8)12yes—
345C8—24no—
346C8—42no—
347C8—42no—
348C8—81no—
349C22—21no—
350C22—21no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—82no—
355C22—81no—
356C22—82no—
357C22—164no—
358C22C23×#39211yescharacteristic
359C22C23×#45912yes—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—81no—
364C4C23×#44712yes—
365C4C23×#45412yes—
366C4C23×#47112yes—
367C4C23×#49311yescharacteristic
368C4C23×#49611yescharacteristic
369C2—22no—
370C2—162no—
371C2—161no—
372C2C23×#200311yescharacteristic
373C2C23×#201111yescharacteristic
374C2C23×#201911yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3151C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3226C2semidirect
3. C2×C23×#486C2semidirect
4. C23×#1966C2semidirect
5. C23×#1545C2semidirect
6. C23×#1492C2semidirect
7. C23×#1278C2semidirect
121. #10150C23direct (N × H)
162. #3226C2×C23semidirect
164. C2×#486C2×C23semidirect
165. #1966C2×C23semidirect
166. #1545C2×C23semidirect
167. #1492C2×C23semidirect
168. #1278C2×C23semidirect
259. C23#10150direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32262
C23×#31512
C2×C23×#4862
C23×#19662
C23×#15452
C23×#14922
C23×#12782
#1015023