Finite Group Databasebeta

← C23×#8877 #19156503

Subgroups — C23×#8877 #19156503

Order 5888.

360 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2992C211yesmaximal, characteristic
2C23×#1832C211yesmaximal, characteristic, complemented
3C23×#1573C211yesmaximal, characteristic, complemented
4C23×#1500C211yesmaximal, characteristic
5C23×#1483C211yesmaximal, characteristic
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#529C2211yescharacteristic
9C23×#446—21no—
10C23×#435—21no—
11C23×#435—21no—
12C8×D4×C23—21no—
13C23×#433—21no—
14C23×#432C2211yescharacteristic
15C23×#431C2211yescharacteristic
16C23×#424—21no—
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×C4⋊C8C2211yescharacteristic
19C4×M16×C23C2211yescharacteristic
20C23×#376C2211yescharacteristic
21C23×#336—21no—
22C23×#336—21no—
23C23×#335—21no—
24C23×#334—21no—
25C23×#328C411yescharacteristic
26C23×#328C411yescharacteristic
27C23×#327C411yescharacteristic
28C23×#327C411yescharacteristic
29C23×Q8⋊C8—21no—
30C23×D4⋊C8—21no—
31C2×M16×C23—21no—
32C2×M16×C23—21no—
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C22×C8×C23D411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C23×#126—21no—
38C23×#125Q811yescharacteristic
39C23×#124Q811yescharacteristic
40C23×#123—21no—
41C23×C22⋊Q8C2×C411yescharacteristic
42C23×#121C2×C411yescharacteristic
43C4×Q8×C23D411yescharacteristic
44C4×D4×C23—21no—
45C4×D4×C23D411yescharacteristic
46C23×#117—21no—
47C2×C42×C23—21no—
48C2×C42×C23C2311yescharacteristic
49C23×C4⋊C8—21no—
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8D411yescharacteristic
57C23×C4⋊C8D411yescharacteristic
58C23×C22⋊C8—21no—
59C23×C22⋊C8—21no—
60C23×C22⋊C8—21no—
61C23×C22⋊C8C2×C411yescharacteristic
62C23×C22⋊C8C2×C411yescharacteristic
63C23×#97—21no—
64C23×#97—21no—
65C23×#97—21no—
66C4×C8×C23—21no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C23×#95—21no—
70C23×#95C2×C411yescharacteristic
71C23×#95C2×C411yescharacteristic
72C2×Q8×C23C4⋊C411yescharacteristic
73C2×D4×C23—21no—
74C2×D4×C23C4⋊C411yescharacteristic
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23C22×C411yescharacteristic
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23#4111yescharacteristic
81M16×C23—22no—
82M16×C23—41no—
83M16×C23—81no—
84C2×C8×C23—21no—
85C2×C8×C23—21no—
86C2×C8×C23—21no—
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—42no—
94C2×C8×C23—41no—
95C2×C8×C23—81no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23C4⋊C411yescharacteristic
101C4⋊C4×C23C4⋊C411yescharacteristic
102C22⋊C4×C23—21no—
103C22⋊C4×C23—21no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23—21no—
107C42×C23—21no—
108C42×C23—41no—
109C42×C23—41no—
110C42×C23C2×D411yescharacteristic
111C42×C23C2×D411yescharacteristic
112C42×C23C2×Q811yescharacteristic
113C42×C23#4111yescharacteristic
114#8877C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—21no—
116C23×C23C4×D411yescharacteristic
117Q8×C23—41no—
118D4×C23—41no—
119D4×C23—41no—
120D4×C23—41no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23C2×C4⋊C411yescharacteristic
138C2×C4×C23C4×D411yescharacteristic
139C2×C4×C23#12011yescharacteristic
140C2×C4×C23C22⋊Q811yescharacteristic
141C2×C4×C23C22⋊Q811yescharacteristic
142C2×C4×C23#12311yescharacteristic
143C8×C23—41no—
144C8×C23—41no—
145C8×C23—42no—
146C8×C23—41no—
147C8×C23—41no—
148C8×C23—41no—
149C8×C23—41no—
150C8×C23—81no—
151#2992C2×C2311yescharacteristic
152#1832C2×C2311yescharacteristic, complemented
153#1573C2×C2311yescharacteristic, complemented
154#1500C2×C2311yescharacteristic
155#1483C2×C2311yescharacteristic
156#1300C2×C2311yescharacteristic, complemented
157#1283C2×C2311yescharacteristic, complemented
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23#38511yescharacteristic
164C22×C23#44311yescharacteristic
165C22×C23#44411yescharacteristic
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23#45411yescharacteristic
177C4×C23#45511yescharacteristic
178#529C22×C2311yescharacteristic
179#446—21no—
180#435—21no—
181#435—21no—
182C8×D4—21no—
183#433—21no—
184#432C22×C2311yescharacteristic
185#431C22×C2311yescharacteristic
186#424—21no—
187C2×C4⋊C8C22×C2311yescharacteristic
188C2×C4⋊C8C22×C2311yescharacteristic
189C4×M16C22×C2311yescharacteristic
190#376C22×C2311yescharacteristic
191#336—21no—
192#336—21no—
193#335—21no—
194#334—21no—
195#328C4×C2311yescharacteristic
196#328C4×C2311yescharacteristic
197#327C4×C2311yescharacteristic
198#327C4×C2311yescharacteristic
199Q8⋊C8—21no—
200D4⋊C8—21no—
201C2×C23—21no—
202C2×C23—21no—
203C2×C23—81no—
204C2×C23#186011yescharacteristic
205C2×C23#186611yescharacteristic
206C2×C23#186811yescharacteristic
207C2×M16—21no—
208C2×M16—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8D4×C2311yescharacteristic
212C22×C8D4×C2311yescharacteristic
213#126—21no—
214#125Q8×C2311yescharacteristic
215#124Q8×C2311yescharacteristic
216#123—21no—
217C22⋊Q8C2×C4×C2311yescharacteristic
218#121C2×C4×C2311yescharacteristic
219C4×Q8D4×C2311yescharacteristic
220C4×D4—21no—
221C4×D4D4×C2311yescharacteristic
222#117—21no—
223C2×C42—21no—
224C2×C42C23×C2311yescharacteristic
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8D4×C2311yescharacteristic
233C4⋊C8D4×C2311yescharacteristic
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8C2×C4×C2311yescharacteristic
238C22⋊C8C2×C4×C2311yescharacteristic
239#97—21no—
240#97—21no—
241#97—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—21no—
246#95C2×C4×C2311yescharacteristic
247#95C2×C4×C2311yescharacteristic
248C23#887711yescharacteristic, direct product factor, Sylow
249C2×Q8C4⋊C4×C2311yescharacteristic
250C2×D4—21no—
251C2×D4C4⋊C4×C2311yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4C22×C4×C2311yescharacteristic
256C22×C4C2×D4×C2311yescharacteristic
257C22×C4#41×C2311yescharacteristic
258M16—22no—
259M16—41no—
260M16—81no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—81no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4C4⋊C4×C2311yescharacteristic
278C4⋊C4C4⋊C4×C2311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C42—21no—
285C42—41no—
286C42—41no—
287C42C2×D4×C2311yescharacteristic
288C42C2×D4×C2311yescharacteristic
289C42C2×Q8×C2311yescharacteristic
290C42#41×C2311yescharacteristic
291C23—21no—
292C23C4×D4×C2311yescharacteristic
293Q8—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4C2×C4⋊C4×C2311yescharacteristic
314C2×C4C4×D4×C2311yescharacteristic
315C2×C4C23×#12011yescharacteristic
316C2×C4C23×C22⋊Q811yescharacteristic
317C2×C4C23×C22⋊Q811yescharacteristic
318C2×C4C23×#12311yescharacteristic
319C8—41no—
320C8—41no—
321C8—42no—
322C8—41no—
323C8—41no—
324C8—41no—
325C8—41no—
326C8—81no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C23×#38511yescharacteristic
333C22C23×#44311yescharacteristic
334C22C23×#44411yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4C23×#45411yescharacteristic
346C4C23×#45511yescharacteristic
347C2—21no—
348C2—21no—
349C2—81no—
350C2C23×#186011yescharacteristic
351C2C23×#186611yescharacteristic
352C2C23×#186811yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#385
prime radicalC23×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1832C2semidirect
3. C23×#1573C2semidirect
6. C23×#1300C2semidirect
7. C23×#1283C2semidirect
114. #8877C23direct (N × H)
152. #1832C2×C23semidirect
153. #1573C2×C23semidirect
156. #1300C2×C23semidirect
157. #1283C2×C23semidirect
248. C23#8877direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29922
C23×#18322
C23×#15732
C23×#15002
C23×#14832
C23×#13002
C23×#12832
#887723