Finite Group Databasebeta

← C23×#3125 #15670744

Subgroups — C23×#3125 #15670744

Order 2944.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#542C211yesmaximal, characteristic, complemented
2C23×#520C211yesmaximal, characteristic, complemented
3C23×#483C211yesmaximal, characteristic, complemented
4C23×#481C211yesmaximal, characteristic, complemented
5C23×#476C211yesmaximal, characteristic, complemented
6C23×#475C211yesmaximal, characteristic, complemented
7C23×#464C211yesmaximal, characteristic, complemented
8C23×#463C211yesmaximal, characteristic, complemented
9C23×#452C211yesmaximal, characteristic, complemented
10C23×#449C211yesmaximal, characteristic
11C23×#442C211yesmaximal, characteristic, complemented
12C23×#441C211yesmaximal, characteristic, complemented
13C4×SD16×C23C211yesmaximal, characteristic, complemented
14C4×SD16×C23C211yesmaximal, characteristic, complemented
15C23×#433C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×Q16×C23C2211yescharacteristic
18C2×SD16×C23C2211yescharacteristic
19C2×SD16×C23C2211yescharacteristic
20C2×D8×C23C2211yescharacteristic
21C23×C4⋊Q8C2211yescharacteristic
22C23×#126—21no—
23C23×#124—21no—
24C23×#124C2211yescharacteristic
25C23×#123—21no—
26C23×C22⋊Q8—21no—
27C23×C22⋊Q8C2211yescharacteristic
28C23×#121C2211yescharacteristic
29C4×Q8×C23C2211yescharacteristic
30C4×Q8×C23C2211yescharacteristic
31C4×Q8×C23C2211yescharacteristic
32C4×D4×C23—21no—
33C4×D4×C23—21no—
34C4×D4×C23C2211yescharacteristic
35C4×D4×C23C2211yescharacteristic
36C4×D4×C23C2211yescharacteristic
37C23×#117—21no—
38C23×#117C2211yescharacteristic
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23C2211yescharacteristic
41C23×#106C2211yescharacteristic
42C23×#106C2211yescharacteristic
43C23×#106C2211yescharacteristic
44C23×#106C2211yescharacteristic
45C23×C4⋊C8C2211yescharacteristic
46C23×C4⋊C8C2211yescharacteristic
47C23×#103C2211yescharacteristic
48C23×#103C2211yescharacteristic
49C23×#103C2211yescharacteristic
50C23×#103C2211yescharacteristic
51C23×#102C2211yescharacteristic
52C23×#102C2211yescharacteristic
53C23×#102C2211yescharacteristic
54C23×#102C2211yescharacteristic
55C23×C22⋊C8C2211yescharacteristic
56C23×C22⋊C8C2211yescharacteristic
57C23×#97C2211yescharacteristic
58C4×C8×C23C2211yescharacteristic
59#41×C23—22no—
60#41×C23—22no—
61C2×Q8×C23—21no—
62C2×Q8×C23C2311yescharacteristic
63C2×Q8×C23C2311yescharacteristic
64C2×D4×C23—21no—
65C2×D4×C23—21no—
66C2×D4×C23C2311yescharacteristic
67C2×D4×C23C2311yescharacteristic
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23C2311yescharacteristic
74Q16×C23—22no—
75SD16×C23—22no—
76SD16×C23—22no—
77D8×C23—22no—
78C2×C8×C23C2311yescharacteristic
79C2×C8×C23C2311yescharacteristic
80C2×C8×C23C2311yescharacteristic
81C2×C8×C23C2311yescharacteristic
82C4⋊C4×C23—21no—
83C4⋊C4×C23—21no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23—21no—
86C4⋊C4×C23—21no—
87C4⋊C4×C23—21no—
88C4⋊C4×C23D411yescharacteristic
89C4⋊C4×C23D411yescharacteristic
90C4⋊C4×C23C2311yescharacteristic
91C4⋊C4×C23C2311yescharacteristic
92C4⋊C4×C23C2311yescharacteristic
93C4⋊C4×C23C2311yescharacteristic
94C22⋊C4×C23—21no—
95C22⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C22⋊C4×C23—21no—
99C22⋊C4×C23—21no—
100C22⋊C4×C23—21no—
101C22⋊C4×C23D411yescharacteristic
102C22⋊C4×C23D411yescharacteristic
103C42×C23—21no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23C2311yescharacteristic
107C42×C23C2311yescharacteristic
108C23×C23—21no—
109C23×C23—21no—
110C23×C23C2×D411yescharacteristic
111Q8×C23—21no—
112Q8×C23—21no—
113Q8×C23—21no—
114Q8×C23—81no—
115Q8×C23#4112yes—
116D4×C23—24no—
117D4×C23—21no—
118D4×C23—21no—
119D4×C23—21no—
120D4×C23—81no—
121D4×C23#4112yes—
122C2×C4×C23—24no—
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—81no—
132C2×C4×C23—81no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23C2×D411yescharacteristic
136C2×C4×C23C2×D411yescharacteristic
137C2×C4×C23C2×D411yescharacteristic
138C2×C4×C23C2×D411yescharacteristic
139C2×C4×C23C2×D411yescharacteristic
140C2×C4×C23C2411yescharacteristic
141C8×C23—21no—
142C8×C23—21no—
143C8×C23—21no—
144C8×C23—21no—
145#3125C2311yesmaximal, characteristic, direct product factor, Sylow
146C22×C23—21no—
147C22×C23—22no—
148C22×C23—21no—
149C22×C23—21no—
150C22×C23—41no—
151C22×C23—41no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23—41no—
155C22×C23C22×D411yescharacteristic
156C4×C23—21no—
157C4×C23—22no—
158C4×C23—21no—
159C4×C23—21no—
160C4×C23—21no—
161C4×C23—21no—
162C4×C23—41no—
163C4×C23—41no—
164C4×C23—41no—
165C4×C23—41no—
166C4×C23—41no—
167C4×C23C2×#4111yescharacteristic
168C4×C23C2×#4111yescharacteristic
169#542C2×C2311yescharacteristic, complemented
170#520C2×C2311yescharacteristic, complemented
171#483C2×C2311yescharacteristic, complemented
172#481C2×C2311yescharacteristic, complemented
173#476C2×C2311yescharacteristic, complemented
174#475C2×C2311yescharacteristic, complemented
175#464C2×C2311yescharacteristic, complemented
176#463C2×C2311yescharacteristic, complemented
177#452C2×C2311yescharacteristic, complemented
178#449C2×C2311yescharacteristic
179#442C2×C2311yescharacteristic, complemented
180#441C2×C2311yescharacteristic, complemented
181C4×SD16C2×C2311yescharacteristic, complemented
182C4×SD16C2×C2311yescharacteristic, complemented
183#433C2×C2311yescharacteristic, complemented
184C2×C23—41no—
185C2×C23—42no—
186C2×C23—81no—
187C2×C23#52511yescharacteristic
188C2×C23#57711yescharacteristic
189C2×C23#57711yescharacteristic
190C2×#41C22×C2311yescharacteristic
191C2×Q16C22×C2311yescharacteristic
192C2×SD16C22×C2311yescharacteristic
193C2×SD16C22×C2311yescharacteristic
194C2×D8C22×C2311yescharacteristic
195C4⋊Q8C22×C2311yescharacteristic
196#126—21no—
197#124—21no—
198#124C22×C2311yescharacteristic
199#123—21no—
200C22⋊Q8—21no—
201C22⋊Q8C22×C2311yescharacteristic
202#121C22×C2311yescharacteristic
203C4×Q8C22×C2311yescharacteristic
204C4×Q8C22×C2311yescharacteristic
205C4×Q8C22×C2311yescharacteristic
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C22×C2311yescharacteristic
209C4×D4C22×C2311yescharacteristic
210C4×D4C22×C2311yescharacteristic
211#117—21no—
212#117C22×C2311yescharacteristic
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22×C2311yescharacteristic
215#106C22×C2311yescharacteristic
216#106C22×C2311yescharacteristic
217#106C22×C2311yescharacteristic
218#106C22×C2311yescharacteristic
219C4⋊C8C22×C2311yescharacteristic
220C4⋊C8C22×C2311yescharacteristic
221#103C22×C2311yescharacteristic
222#103C22×C2311yescharacteristic
223#103C22×C2311yescharacteristic
224#103C22×C2311yescharacteristic
225#102C22×C2311yescharacteristic
226#102C22×C2311yescharacteristic
227#102C22×C2311yescharacteristic
228#102C22×C2311yescharacteristic
229C22⋊C8C22×C2311yescharacteristic
230C22⋊C8C22×C2311yescharacteristic
231#97C22×C2311yescharacteristic
232C4×C8C22×C2311yescharacteristic
233C23#312511yescharacteristic, direct product factor, Sylow
234#41—22no—
235#41—22no—
236C2×Q8—21no—
237C2×Q8C23×C2311yescharacteristic
238C2×Q8C23×C2311yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C23×C2311yescharacteristic
242C2×D4C23×C2311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C23×C2311yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—22no—
252D8—22no—
253C2×C8C23×C2311yescharacteristic
254C2×C8C23×C2311yescharacteristic
255C2×C8C23×C2311yescharacteristic
256C2×C8C23×C2311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4D4×C2311yescharacteristic
264C4⋊C4D4×C2311yescharacteristic
265C4⋊C4C23×C2311yescharacteristic
266C4⋊C4C23×C2311yescharacteristic
267C4⋊C4C23×C2311yescharacteristic
268C4⋊C4C23×C2311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4D4×C2311yescharacteristic
277C22⋊C4D4×C2311yescharacteristic
278C42—21no—
279C42—21no—
280C42—21no—
281C42C23×C2311yescharacteristic
282C42C23×C2311yescharacteristic
283C23—21no—
284C23—21no—
285C23C2×D4×C2311yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—81no—
290Q8#41×C2312yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#41×C2312yes—
297C2×C4—24no—
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—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×D4×C2311yescharacteristic
311C2×C4C2×D4×C2311yescharacteristic
312C2×C4C2×D4×C2311yescharacteristic
313C2×C4C2×D4×C2311yescharacteristic
314C2×C4C2×D4×C2311yescharacteristic
315C2×C4C24×C2311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C22×D4×C2311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4C2×#41×C2311yescharacteristic
342C4C2×#41×C2311yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2C23×#52511yescharacteristic
347C2C23×#57711yescharacteristic
348C2C23×#57711yescharacteristic
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#542C2semidirect
2. C23×#520C2semidirect
3. C23×#483C2semidirect
4. C23×#481C2semidirect
5. C23×#476C2semidirect
6. C23×#475C2semidirect
7. C23×#464C2semidirect
8. C23×#463C2semidirect
9. C23×#452C2semidirect
11. C23×#442C2semidirect
12. C23×#441C2semidirect
13. C4×SD16×C23C2semidirect
14. C4×SD16×C23C2semidirect
15. C23×#433C2semidirect
145. #3125C23direct (N × H)
169. #542C2×C23semidirect
170. #520C2×C23semidirect
171. #483C2×C23semidirect
172. #481C2×C23semidirect
173. #476C2×C23semidirect
174. #475C2×C23semidirect
175. #464C2×C23semidirect
176. #463C2×C23semidirect
177. #452C2×C23semidirect
179. #442C2×C23semidirect
180. #441C2×C23semidirect
181. C4×SD16C2×C23semidirect
182. C4×SD16C2×C23semidirect
183. #433C2×C23semidirect
233. C23#3125direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5422
C23×#5202
C23×#4832
C23×#4812
C23×#4762
C23×#4752
C23×#4642
C23×#4632
C23×#4522
C23×#4492
C23×#4422
C23×#4412
C4×SD16×C232
C23×#4332
#312523