Finite Group Databasebeta

← C23×#10625 #19158251

Subgroups — C23×#10625 #19158251

Order 5888.

402 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1954C211yesmaximal, characteristic
2C23×#1834C211yesmaximal, characteristic, complemented
3C23×#1802C211yesmaximal, characteristic
4C23×#1802C211yesmaximal, characteristic
5C23×#1296C211yesmaximal, characteristic, complemented
6C23×#1296C211yesmaximal, characteristic, complemented
7C23×#1277C211yesmaximal, characteristic, complemented
8C2×C4×D4×C23C2211yescharacteristic
9C2×C23×C4⋊C8C2211yescharacteristic
10C2×C23×C4⋊C8C2211yescharacteristic
11C2×C23×C4⋊C8C2211yescharacteristic
12C2×C23×#102—21no—
13C2×C23×#102—21no—
14C2×C23×C22⋊C8—21no—
15C2×C23×C22⋊C8—21no—
16C2×C4×C8×C23C2211yescharacteristic
17C23×#384C2211yescharacteristic
18C23×#384C2211yescharacteristic
19C23×#340—21no—
20C23×#340—21no—
21C23×#340—21no—
22C23×#340—21no—
23C22×D4×C23D411yescharacteristic
24C23×C4×C23—21no—
25C22×C8×C23—21no—
26C22×C8×C23—21no—
27C22×C8×C23—21no—
28C22×C8×C23D411yescharacteristic
29C22×C8×C23D411yescharacteristic
30C4×D4×C23—21no—
31C4×D4×C23—21no—
32C4×D4×C23—21no—
33C4×D4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C42×C23C2311yescharacteristic
43C23×C4⋊C8—21no—
44C23×C4⋊C8—21no—
45C23×C4⋊C8—21no—
46C23×C4⋊C8—21no—
47C23×C4⋊C8—21no—
48C23×C4⋊C8—21no—
49C23×#102—42no—
50C23×#102—42no—
51C23×C22⋊C8—42no—
52C23×C22⋊C8—42no—
53C4×C8×C23—21no—
54C4×C8×C23—21no—
55C24×C23—21no—
56C2×D4×C23—21no—
57C2×D4×C23—21no—
58C2×D4×C23—21no—
59C2×D4×C23—21no—
60C2×D4×C23—21no—
61C2×D4×C23SD1612yes—
62C22×C4×C23—21no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23—41no—
68C22×C4×C23—41no—
69C22×C4×C23—41no—
70C22×C4×C23—41no—
71C22×C4×C23C2×D411yescharacteristic
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23#4111yescharacteristic
74C2×C8×C23—21no—
75C2×C8×C23—21no—
76C2×C8×C23—21no—
77C2×C8×C23—21no—
78C2×C8×C23—21no—
79C2×C8×C23—21no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23—41no—
83C2×C8×C23—41no—
84C2×C8×C23—41no—
85C2×C8×C23—41no—
86C2×C8×C23—41no—
87C2×C8×C23—41no—
88C2×C8×C23—41no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—21no—
91C4⋊C4×C23—41no—
92C4⋊C4×C23—41no—
93C4⋊C4×C23—41no—
94C4⋊C4×C23—41no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—81no—
100C4⋊C4×C23—81no—
101C22⋊C4×C23—41no—
102C22⋊C4×C23—41no—
103C42×C23#4111yescharacteristic
104C42×C23#4111yescharacteristic
105C42×C23#4111yescharacteristic
106C42×C23#4111yescharacteristic
107#10625C2311yesmaximal, characteristic, direct product factor, Sylow
108C23×C23—22no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23#12111yescharacteristic
116D4×C23—22no—
117D4×C23—22no—
118D4×C23—41no—
119D4×C23—41no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
125C2×C4×C23—81no—
126C2×C4×C23—81no—
127C2×C4×C23—81no—
128C2×C4×C23—81no—
129C2×C4×C23—81no—
130C2×C4×C23—81no—
131C2×C4×C23—81no—
132C2×C4×C23—81no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23#12311yescharacteristic
136C2×C4×C23#12311yescharacteristic
137C2×C4×C23#12311yescharacteristic
138C2×C4×C23#12411yescharacteristic
139C2×C4×C23#12611yescharacteristic
140C2×C4×C23#12611yescharacteristic
141C2×C4×C23C2×SD1611yescharacteristic
142C2×C4×C23#13511yescharacteristic
143C2×C4×C23#13511yescharacteristic
144C2×C4×C23#13511yescharacteristic
145C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C23#13711yescharacteristic
149C8×C23—41no—
150C8×C23—41no—
151C8×C23—81no—
152C8×C23—81no—
153C8×C23—81no—
154#1954C2×C2311yescharacteristic
155#1834C2×C2311yescharacteristic, complemented
156#1802C2×C2311yescharacteristic
157#1802C2×C2311yescharacteristic
158#1296C2×C2311yescharacteristic, complemented
159#1296C2×C2311yescharacteristic, complemented
160#1277C2×C2311yescharacteristic, complemented
161C22×C23—42no—
162C22×C23—42no—
163C22×C23—42no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#39711yescharacteristic
169C22×C23#46311yescharacteristic
170C22×C23#46311yescharacteristic
171C22×C23#46511yescharacteristic
172C22×C23#46811yescharacteristic
173C22×C23#47211yescharacteristic
174C22×C23#47311yescharacteristic
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—81no—
178C4×C23—161no—
179C4×C23—161no—
180C4×C23#48111yescharacteristic
181C4×C23#48211yescharacteristic
182C4×C23#48211yescharacteristic
183C4×C23#48211yescharacteristic
184C4×C23#48311yescharacteristic
185C4×C23#48511yescharacteristic
186C4×C23#48811yescharacteristic
187C4×C23#49011yescharacteristic
188C2×C4×D4C22×C2311yescharacteristic
189C2×C4⋊C8C22×C2311yescharacteristic
190C2×C4⋊C8C22×C2311yescharacteristic
191C2×C4⋊C8C22×C2311yescharacteristic
192C2×#102—21no—
193C2×#102—21no—
194C2×C22⋊C8—21no—
195C2×C22⋊C8—21no—
196C2×C4×C8C22×C2311yescharacteristic
197#384C22×C2311yescharacteristic
198#384C22×C2311yescharacteristic
199#340—21no—
200#340—21no—
201#340—21no—
202#340—21no—
203C2×C23—82no—
204C2×C23#167711yescharacteristic
205C2×C23#167911yescharacteristic
206C2×C23#169711yescharacteristic
207C2×C23#169911yescharacteristic
208C2×C23#207311yescharacteristic
209C2×C23#207711yescharacteristic
210C2×C23#207811yescharacteristic
211C22×D4D4×C2311yescharacteristic
212C23×C4—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8D4×C2311yescharacteristic
217C22×C8D4×C2311yescharacteristic
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4D4×C2311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C42C23×C2311yescharacteristic
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237#102—42no—
238#102—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C4×C8—21no—
242C4×C8—21no—
243C23#1062511yescharacteristic, direct product factor, Sylow
244C24—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4SD16×C2312yes—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×D4×C2311yescharacteristic
261C22×C4C2×D4×C2311yescharacteristic
262C22×C4#41×C2311yescharacteristic
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42#41×C2311yescharacteristic
293C42#41×C2311yescharacteristic
294C42#41×C2311yescharacteristic
295C42#41×C2311yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C23×#12111yescharacteristic
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C23×#12311yescharacteristic
324C2×C4C23×#12311yescharacteristic
325C2×C4C23×#12311yescharacteristic
326C2×C4C23×#12411yescharacteristic
327C2×C4C23×#12611yescharacteristic
328C2×C4C23×#12611yescharacteristic
329C2×C4C2×SD16×C2311yescharacteristic
330C2×C4C23×#13511yescharacteristic
331C2×C4C23×#13511yescharacteristic
332C2×C4C23×#13511yescharacteristic
333C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
335C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
336C2×C4C23×#13711yescharacteristic
337C8—41no—
338C8—41no—
339C8—81no—
340C8—81no—
341C8—81no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C23×#39711yescharacteristic
350C22C23×#46311yescharacteristic
351C22C23×#46311yescharacteristic
352C22C23×#46511yescharacteristic
353C22C23×#46811yescharacteristic
354C22C23×#47211yescharacteristic
355C22C23×#47311yescharacteristic
356C4—21no—
357C4—21no—
358C4—81no—
359C4—161no—
360C4—161no—
361C4C23×#48111yescharacteristic
362C4C23×#48211yescharacteristic
363C4C23×#48211yescharacteristic
364C4C23×#48211yescharacteristic
365C4C23×#48311yescharacteristic
366C4C23×#48511yescharacteristic
367C4C23×#48811yescharacteristic
368C4C23×#49011yescharacteristic
369C2—82no—
370C2C23×#167711yescharacteristic
371C2C23×#167911yescharacteristic
372C2C23×#169711yescharacteristic
373C2C23×#169911yescharacteristic
374C2C23×#207311yescharacteristic
375C2C23×#207711yescharacteristic
376C2C23×#207811yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
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. C23×#1834C2semidirect
5. C23×#1296C2semidirect
6. C23×#1296C2semidirect
7. C23×#1277C2semidirect
107. #10625C23direct (N × H)
155. #1834C2×C23semidirect
158. #1296C2×C23semidirect
159. #1296C2×C23semidirect
160. #1277C2×C23semidirect
243. C23#10625direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#19542
C23×#18342
C23×#18022
C23×#12962
C23×#12772
#1062523