Finite Group Databasebeta

← C23×#2759 #15670378

Subgroups — C23×#2759 #15670378

Order 2944.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#401C211yesmaximal, characteristic, complemented
2C23×#397C211yesmaximal, characteristic
3C23×#397C211yesmaximal, characteristic
4C23×#395C211yesmaximal, characteristic, complemented
5C23×#393C211yesmaximal, characteristic
6C23×#393C211yesmaximal, characteristic
7C23×#391C211yesmaximal, characteristic, complemented
8C23×#391C211yesmaximal, characteristic, complemented
9C23×#391C211yesmaximal, characteristic, complemented
10C23×#388C211yesmaximal, characteristic
11C23×#388C211yesmaximal, characteristic
12C23×#388C211yesmaximal, characteristic
13C23×#387C211yesmaximal, characteristic, complemented
14C23×#387C211yesmaximal, characteristic, complemented
15C23×#383C211yesmaximal, characteristic, complemented
16C22×Q8×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic
18C2×C4⋊C4×C23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C4⋊C4×C23C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C22⋊C4×C23C2211yescharacteristic
24C2×C22⋊C4×C23C2211yescharacteristic
25C2×C22⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C22⋊C4×C23C2211yescharacteristic
29C2×C22⋊C4×C23C2211yescharacteristic
30C2×C42×C23C2211yescharacteristic
31C2×C42×C23C2211yescharacteristic
32C2×C42×C23C2211yescharacteristic
33C23×#95C2211yescharacteristic
34C23×#95C2211yescharacteristic
35C23×#95C2211yescharacteristic
36C23×#95C2211yescharacteristic
37C23×#95C2211yescharacteristic
38C23×#95C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23×#95C2211yescharacteristic
43C23×#95C2211yescharacteristic
44C23×#95C2211yescharacteristic
45C23×#95C2211yescharacteristic
46C23×#95C2211yescharacteristic
47C23×#95C2211yescharacteristic
48C23×#95C2211yescharacteristic
49C23×#95C2211yescharacteristic
50C23×#95C2211yescharacteristic
51C24×C23C2311yescharacteristic
52C2×Q8×C23—22no—
53C2×Q8×C23—22no—
54C2×Q8×C23—22no—
55C2×Q8×C23—41no—
56C2×Q8×C23—41no—
57C2×Q8×C23—41no—
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C22×C4×C23C2311yescharacteristic
62C22×C4×C23C2311yescharacteristic
63C22×C4×C23C2311yescharacteristic
64C22×C4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C22×C4×C23C2311yescharacteristic
70C22×C4×C23C2311yescharacteristic
71C22×C4×C23C2311yescharacteristic
72C4⋊C4×C23—22no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C22⋊C4×C23—22no—
78C22⋊C4×C23—22no—
79C22⋊C4×C23—22no—
80C22⋊C4×C23—41no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—41no—
84C42×C23—22no—
85C42×C23—41no—
86C42×C23—41no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—21no—
92C23×C23—21no—
93C23×C23—21no—
94C23×C23C2411yescharacteristic
95Q8×C23—44no—
96Q8×C23—44no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
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—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23#4112yes—
137C2×C4×C23#4112yes—
138C2×C4×C23#4112yes—
139#2759C2311yesmaximal, characteristic, direct product factor, Sylow
140C22×C23—41no—
141C22×C23—41no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23C2×#4111yescharacteristic
148C22×C23C2×#4111yescharacteristic
149C22×C23C2×#4111yescharacteristic
150C22×C23H(5,𝔽2)11yescharacteristic
151C22×C23H(5,𝔽2)11yescharacteristic
152C22×C232−1+411yescharacteristic
153C22×C232−1+411yescharacteristic
154C4×C23—22no—
155C4×C23—22no—
156C4×C23—22no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—41no—
163C4×C23—41no—
164C4×C23—41no—
165C4×C23—41no—
166C4×C23—41no—
167C4×C23—41no—
168#401C2×C2311yescharacteristic, complemented
169#397C2×C2311yescharacteristic
170#397C2×C2311yescharacteristic
171#395C2×C2311yescharacteristic, complemented
172#393C2×C2311yescharacteristic
173#393C2×C2311yescharacteristic
174#391C2×C2311yescharacteristic, complemented
175#391C2×C2311yescharacteristic, complemented
176#391C2×C2311yescharacteristic, complemented
177#388C2×C2311yescharacteristic
178#388C2×C2311yescharacteristic
179#388C2×C2311yescharacteristic
180#387C2×C2311yescharacteristic, complemented
181#387C2×C2311yescharacteristic, complemented
182#383C2×C2311yescharacteristic, complemented
183C2×C23—81no—
184C2×C23#53811yescharacteristic
185C2×C23#54211yescharacteristic
186C2×C23#54211yescharacteristic
187C2×C23#55111yescharacteristic
188C2×C23#55611yescharacteristic
189C2×C23#55611yescharacteristic
190C2×C23#56311yescharacteristic
191C22×Q8C22×C2311yescharacteristic
192C22×Q8C22×C2311yescharacteristic
193C2×C4⋊C4C22×C2311yescharacteristic
194C2×C4⋊C4C22×C2311yescharacteristic
195C2×C4⋊C4C22×C2311yescharacteristic
196C2×C4⋊C4C22×C2311yescharacteristic
197C2×C4⋊C4C22×C2311yescharacteristic
198C2×C22⋊C4C22×C2311yescharacteristic
199C2×C22⋊C4C22×C2311yescharacteristic
200C2×C22⋊C4C22×C2311yescharacteristic
201C2×C22⋊C4C22×C2311yescharacteristic
202C2×C22⋊C4C22×C2311yescharacteristic
203C2×C22⋊C4C22×C2311yescharacteristic
204C2×C22⋊C4C22×C2311yescharacteristic
205C2×C42C22×C2311yescharacteristic
206C2×C42C22×C2311yescharacteristic
207C2×C42C22×C2311yescharacteristic
208#95C22×C2311yescharacteristic
209#95C22×C2311yescharacteristic
210#95C22×C2311yescharacteristic
211#95C22×C2311yescharacteristic
212#95C22×C2311yescharacteristic
213#95C22×C2311yescharacteristic
214#95C22×C2311yescharacteristic
215#95C22×C2311yescharacteristic
216#95C22×C2311yescharacteristic
217#95C22×C2311yescharacteristic
218#95C22×C2311yescharacteristic
219#95C22×C2311yescharacteristic
220#95C22×C2311yescharacteristic
221#95C22×C2311yescharacteristic
222#95C22×C2311yescharacteristic
223#95C22×C2311yescharacteristic
224#95C22×C2311yescharacteristic
225#95C22×C2311yescharacteristic
226C23#275911yescharacteristic, direct product factor, Sylow
227C24C23×C2311yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4C23×C2311yescharacteristic
235C22×C4C23×C2311yescharacteristic
236C22×C4C23×C2311yescharacteristic
237C22×C4C23×C2311yescharacteristic
238C22×C4C23×C2311yescharacteristic
239C22×C4C23×C2311yescharacteristic
240C22×C4C23×C2311yescharacteristic
241C22×C4C23×C2311yescharacteristic
242C22×C4C23×C2311yescharacteristic
243C22×C4C23×C2311yescharacteristic
244C22×C4C23×C2311yescharacteristic
245C22×C4C23×C2311yescharacteristic
246C22×C4C23×C2311yescharacteristic
247C22×C4C23×C2311yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—22no—
261C42—41no—
262C42—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23C24×C2311yescharacteristic
271Q8—44no—
272Q8—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4#41×C2312yes—
313C2×C4#41×C2312yes—
314C2×C4#41×C2312yes—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22C2×#41×C2311yescharacteristic
323C22C2×#41×C2311yescharacteristic
324C22C2×#41×C2311yescharacteristic
325C22C23×H(5,𝔽2)11yescharacteristic
326C22C23×H(5,𝔽2)11yescharacteristic
327C22C23×2−1+411yescharacteristic
328C22C23×2−1+411yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C2—81no—
344C2C23×#53811yescharacteristic
345C2C23×#54211yescharacteristic
346C2C23×#54211yescharacteristic
347C2C23×#55111yescharacteristic
348C2C23×#55611yescharacteristic
349C2C23×#55611yescharacteristic
350C2C23×#56311yescharacteristic
subgroupG/H
center, socleC23×C23C24
commutator, Frattini, cosocleC23C24×C23
prime radicalC24×C23C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#401C2semidirect
4. C23×#395C2semidirect
7. C23×#391C2semidirect
8. C23×#391C2semidirect
9. C23×#391C2semidirect
13. C23×#387C2semidirect
14. C23×#387C2semidirect
15. C23×#383C2semidirect
139. #2759C23direct (N × H)
168. #401C2×C23semidirect
171. #395C2×C23semidirect
174. #391C2×C23semidirect
175. #391C2×C23semidirect
176. #391C2×C23semidirect
180. #387C2×C23semidirect
181. #387C2×C23semidirect
182. #383C2×C23semidirect
226. C23#2759direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#4012
C23×#3972
C23×#3952
C23×#3932
C23×#3912
C23×#3882
C23×#3872
C23×#3832
#275923