Finite Group Databasebeta

← C2×C23×#1611 #19165743

Subgroups — C2×C23×#1611 #19165743

Order 5888.

1028 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#544C211yesmaximal, characteristic
2C2×C23×#483C212yesmaximal
3C2×C23×#398C211yesmaximal, characteristic
4C23×#1611C218yesmaximal, direct product factor
5C2×C23×#328C212yesmaximal
6C2×C23×#324C211yesmaximal, characteristic
7C23×#544—24no—
8C23×#544—22no—
9C23×#544C2214yes—
10C2×C23×C4⋊Q8—22no—
11C2×C23×#125—22no—
12C2×C23×C22⋊Q8—21no—
13C2×C23×C22⋊Q8C2212yes—
14C2×C23×#117—21no—
15C22×C4⋊C4×C23—21no—
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C4⋊C4×C23C2211yescharacteristic
18C23×#483—24no—
19C23×#483C2218yes—
20C2×C23×#106—22no—
21C2×C23×#103—22no—
22C2×C23×C22⋊C8C2212yes—
23C23×#398—22no—
24C23×#398C2214yes—
25C2×C23×#95—21no—
26C2×C23×#95C2211yescharacteristic
27C23×#328C2218yes—
28C23×#324C2214yes—
29C22×Q8×C23—21no—
30C22×Q8×C23D412yes—
31C23×C4×C23—21no—
32C23×C4×C23—21no—
33C23×C4×C23D411yescharacteristic
34C23×C4×C23D411yescharacteristic
35C23×C4×C23C2311yescharacteristic
36C22×C8×C23—22no—
37C23×C4⋊Q8—28no—
38C23×C4⋊Q8—28no—
39C23×#125—28no—
40C23×#125—28no—
41C23×C22⋊Q8—24no—
42C23×C22⋊Q8—24no—
43C23×C22⋊Q8—22no—
44C23×C22⋊Q8—22no—
45C23×C22⋊Q8D414yes—
46C23×C22⋊Q8C2314yes—
47C23×#117—24no—
48C23×#117—24no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—22no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23—22no—
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—21no—
59C2×C4⋊C4×C23—24no—
60C2×C4⋊C4×C23—44no—
61C2×C4⋊C4×C23D412yes—
62C2×C4⋊C4×C23D412yes—
63C2×C4⋊C4×C23C2312yes—
64C2×C4⋊C4×C23C2312yes—
65C2×C22⋊C4×C23—22no—
66C2×C42×C23—22no—
67C23×#106—28no—
68C23×#103—28no—
69C23×C22⋊C8—22no—
70C23×C22⋊C8C2314yes—
71C23×#95—24no—
72C23×#95—21no—
73C23×#95C2312yes—
74C24×C23C2×D411yescharacteristic
75C2×Q8×C23—24no—
76C2×Q8×C23—22no—
77C2×Q8×C23—44no—
78C2×Q8×C23—82no—
79C2×Q8×C23C2×D414yes—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—22no—
83C22×C4×C23—24no—
84C22×C4×C23—22no—
85C22×C4×C23—22no—
86C22×C4×C23—21no—
87C22×C4×C23—22no—
88C22×C4×C23—21no—
89C22×C4×C23—22no—
90C22×C4×C23—42no—
91C22×C4×C23—42no—
92C22×C4×C23—42no—
93C22×C4×C23—42no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—42no—
97C22×C4×C23—42no—
98C22×C4×C23SD1614yes—
99C22×C4×C23C2×D412yes—
100C22×C4×C23C2×D412yes—
101C22×C4×C23C2×D412yes—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D412yes—
104C22×C4×C23C2411yescharacteristic
105C2×C8×C23—24no—
106C2×C8×C23—44no—
107C4⋊C4×C23—24no—
108C4⋊C4×C23—24no—
109C4⋊C4×C23—24no—
110C4⋊C4×C23—28no—
111C4⋊C4×C23—24no—
112C4⋊C4×C23—24no—
113C4⋊C4×C23—22no—
114C4⋊C4×C23—24no—
115C4⋊C4×C23—24no—
116C4⋊C4×C23—24no—
117C4⋊C4×C23—42no—
118C4⋊C4×C23—48no—
119C4⋊C4×C23C2×D414yes—
120C22⋊C4×C23—24no—
121C22⋊C4×C23—24no—
122C42×C23—28no—
123C2×#1611C2311yesmaximal, characteristic, direct product factor, Sylow
124C23×C23—22no—
125C23×C23—22no—
126C23×C23#12011yescharacteristic
127C23×C23#12011yescharacteristic
128C23×C23C2×SD1612yes—
129C23×C23#13712yes—
130C23×C23C22×D411yescharacteristic
131Q8×C23—44no—
132Q8×C23—82no—
133Q8×C23—84no—
134C2×C4×C23—22no—
135C2×C4×C23—22no—
136C2×C4×C23—24no—
137C2×C4×C23—24no—
138C2×C4×C23—24no—
139C2×C4×C23—24no—
140C2×C4×C23—24no—
141C2×C4×C23—24no—
142C2×C4×C23—22no—
143C2×C4×C23—42no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—48no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—44no—
150C2×C4×C23—42no—
151C2×C4×C23—44no—
152C2×C4×C23—42no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—42no—
156C2×C4×C23—81no—
157C2×C4×C23—82no—
158C2×C4×C23—82no—
159C2×C4×C23#12012yes—
160C2×C4×C23C2×SD1614yes—
161C2×C4×C23C22×D412yes—
162C8×C23—44no—
163C2×#544C2×C2311yescharacteristic
164C2×#483C2×C2312yes—
165C2×#398C2×C2311yescharacteristic
166#1611C2×C2318yesdirect product factor
167C2×#328C2×C2312yes—
168C2×#324C2×C2311yescharacteristic
169C22×C23—24no—
170C22×C23—24no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—21no—
175C22×C23#44812yes—
176C22×C23#44812yes—
177C22×C23#45612yes—
178C22×C23#45812yes—
179C22×C23C2×#12011yescharacteristic
180C22×C23C22×SD1611yescharacteristic
181C22×C23C2×#13711yescharacteristic
182C4×C23—24no—
183C4×C23—22no—
184C4×C23—22no—
185C4×C23—44no—
186C4×C23—44no—
187C4×C23—42no—
188C4×C23—44no—
189C4×C23—82no—
190#544—24no—
191#544—22no—
192#544C22×C2314yes—
193C2×C4⋊Q8—22no—
194C2×#125—22no—
195C2×C22⋊Q8—21no—
196C2×C22⋊Q8C22×C2312yes—
197C2×#117—21no—
198C22×C4⋊C4—21no—
199C22×C4⋊C4C22×C2311yescharacteristic
200C22×C4⋊C4C22×C2311yescharacteristic
201#483—24no—
202#483C22×C2318yes—
203C2×#106—22no—
204C2×#103—22no—
205C2×C22⋊C8C22×C2312yes—
206#398—22no—
207#398C22×C2314yes—
208C2×#95—21no—
209C2×#95C22×C2311yescharacteristic
210#328C22×C2318yes—
211#324C22×C2314yes—
212C2×C23—21no—
213C2×C23—21no—
214C2×C23—22no—
215C2×C23#161114yesdirect product factor
216C2×C23C2×#44811yescharacteristic
217C2×C23C2×#45611yescharacteristic
218C2×C23C2×#45811yescharacteristic
219C22×Q8—21no—
220C22×Q8D4×C2312yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4D4×C2311yescharacteristic
224C23×C4D4×C2311yescharacteristic
225C23×C4C23×C2311yescharacteristic
226C22×C8—22no—
227C4⋊Q8—28no—
228C4⋊Q8—28no—
229#125—28no—
230#125—28no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8D4×C2314yes—
236C22⋊Q8C23×C2314yes—
237#117—24no—
238#117—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4D4×C2312yes—
252C2×C4⋊C4D4×C2312yes—
253C2×C4⋊C4C23×C2312yes—
254C2×C4⋊C4C23×C2312yes—
255C2×C22⋊C4—22no—
256C2×C42—22no—
257#106—28no—
258#103—28no—
259C22⋊C8—22no—
260C22⋊C8C23×C2314yes—
261#95—24no—
262#95—21no—
263#95C23×C2312yes—
264C23C2×#161111yescharacteristic, direct product factor, Sylow
265C24C2×D4×C2311yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8C2×D4×C2314yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4SD16×C2314yes—
290C22×C4C2×D4×C2312yes—
291C22×C4C2×D4×C2312yes—
292C22×C4C2×D4×C2312yes—
293C22×C4C2×D4×C2311yescharacteristic
294C22×C4C2×D4×C2312yes—
295C22×C4C24×C2311yescharacteristic
296C2×C8—24no—
297C2×C8—44no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4C2×D4×C2314yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C42—28no—
314C23—22no—
315C23—22no—
316C23C23×#12011yescharacteristic
317C23C23×#12011yescharacteristic
318C23C2×SD16×C2312yes—
319C23C23×#13712yes—
320C23C22×D4×C2311yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—84no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4C23×#12012yes—
350C2×C4C2×SD16×C2314yes—
351C2×C4C22×D4×C2312yes—
352C8—44no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22C23×#44812yes—
360C22C23×#44812yes—
361C22C23×#45612yes—
362C22C23×#45812yes—
363C22C2×C23×#12011yescharacteristic
364C22C22×SD16×C2311yescharacteristic
365C22C2×C23×#13711yescharacteristic
366C4—24no—
367C4—22no—
368C4—22no—
369C4—44no—
370C4—44no—
371C4—42no—
372C4—44no—
373C4—82no—
374C2—21no—
375C2—21no—
376C2—22no—
377C2C23×#161114yesdirect product factor
378C2C2×C23×#44811yescharacteristic
379C2C2×C23×#45611yescharacteristic
380C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1611C2direct (N × H)
123. C2×#1611C23direct (N × H)
166. #1611C2×C23direct (N × H)
215. C2×C23#1611direct (N × H)
264. C23C2×#1611direct (N × H)
377. C2C23×#1611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5442
C2×C23×#4832
C2×C23×#3982
C23×#16112
C2×C23×#3282
C2×C23×#3242
C2×#161123