Finite Group Databasebeta

← C23×#8368 #19155994

Subgroups — C23×#8368 #19155994

Order 5888.

744 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#556C211yesmaximal, characteristic
2C4×C23×#101C211yesmaximal, characteristic, complemented
3C23×#1527C212yesmaximal
4C23×#1491C212yesmaximal, complemented
5C23×#1467C211yesmaximal, characteristic, complemented
6C23×#556—24no—
7C23×#556—24no—
8C2×C23×C4⋊Q8C411yescharacteristic
9C2×C23×#126—21no—
10C2×C23×#126—21no—
11C2×C23×#124C411yescharacteristic
12C2×C23×#124C2211yescharacteristic
13C2×C23×C22⋊Q8—21no—
14C2×C4×Q8×C23C411yescharacteristic
15C2×C4×Q8×C23C2211yescharacteristic
16C2×C4×Q8×C23C2211yescharacteristic
17C2×C4×D4×C23C411yescharacteristic
18C2×C23×#117—21no—
19C23×#432C2212yes—
20C2×C23×#101—21no—
21C4×M16×C23C2212yes—
22C23×#343—21no—
23C23×#329—24no—
24C23×Q8⋊C8—24no—
25C23×#324—21no—
26C23×#324—21no—
27C22×Q8×C23C2×C411yescharacteristic
28C22×Q8×C23C2×C411yescharacteristic
29C22×Q8×C23D411yescharacteristic
30C22×D4×C23C2×C411yescharacteristic
31C23×C4×C23—21no—
32C2×M16×C23—22no—
33C23×C4⋊Q8—22no—
34C23×C4⋊Q8—22no—
35C23×#126—28no—
36C23×#126—28no—
37C23×#124—22no—
38C23×#124—22no—
39C23×#124—22no—
40C23×#124D414yes—
41C23×C22⋊Q8—28no—
42C4×Q8×C23—22no—
43C4×Q8×C23—22no—
44C4×Q8×C23—22no—
45C4×Q8×C23—22no—
46C4×Q8×C23—22no—
47C4×Q8×C23D414yes—
48C4×D4×C23—22no—
49C4×D4×C23—22no—
50C23×#117—28no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23C2×C411yescharacteristic
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C4⋊C4×C23D411yescharacteristic
58C2×C4⋊C4×C23D411yescharacteristic
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C42×C23—21no—
65C2×C42×C23—21no—
66C2×C42×C23C2×C411yescharacteristic
67C2×C42×C23C2×C411yescharacteristic
68C2×C42×C23C2311yescharacteristic
69C23×C4⋊C8—22no—
70C23×#101—24no—
71C23×C22⋊C8—22no—
72C23×#97—22no—
73C23×#97—22no—
74C4×C8×C23—22no—
75C24×C23—21no—
76C2×Q8×C23—22no—
77C2×Q8×C23—21no—
78C2×Q8×C23—21no—
79C2×Q8×C23—22no—
80C2×Q8×C23—41no—
81C2×Q8×C23—42no—
82C2×Q8×C23—42no—
83C2×Q8×C23C22⋊C412yes—
84C2×Q8×C23C22⋊C412yes—
85C2×D4×C23—21no—
86C2×D4×C23—42no—
87C2×D4×C23C22⋊C412yes—
88C22×C4×C23—24no—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—42no—
93C22×C4×C23—42no—
94C22×C4×C23C22⋊C411yescharacteristic
95C22×C4×C23C22⋊C411yescharacteristic
96C22×C4×C23C22⋊C411yescharacteristic
97C22×C4×C23C22⋊C411yescharacteristic
98C22×C4×C23C22×C411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23C2×D411yescharacteristic
101M16×C23—24no—
102M16×C23—42no—
103C2×C8×C23—42no—
104C2×C8×C23—42no—
105C4⋊C4×C23—24no—
106C4⋊C4×C23—24no—
107C4⋊C4×C23—24no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—24no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23C22⋊C412yes—
114C22⋊C4×C23—24no—
115C22⋊C4×C23—24no—
116C22⋊C4×C23—24no—
117C22⋊C4×C23—24no—
118C22⋊C4×C23—24no—
119C42×C23—24no—
120C42×C23—21no—
121C42×C23—21no—
122C42×C23—21no—
123C42×C23—21no—
124C42×C23—24no—
125C42×C23C2×D412yes—
126C42×C23C2×D412yes—
127#8368C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—22no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23C2×C22⋊C411yescharacteristic
134Q8×C23—24no—
135Q8×C23—44no—
136Q8×C23—42no—
137Q8×C23—42no—
138Q8×C23—42no—
139D4×C23—42no—
140D4×C23—42no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—22no—
146C2×C4×C23—21no—
147C2×C4×C23—22no—
148C2×C4×C23—21no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—41no—
155C2×C4×C23—48no—
156C2×C4×C23—42no—
157C2×C4×C23—42no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23#10412yes—
163C2×C4×C23C2×C22⋊C411yescharacteristic
164C2×C4×C23C2×C22⋊C411yescharacteristic
165C2×C4×C23#12012yes—
166C2×C4×C23#12012yes—
167C8×C23—44no—
168C8×C23—82no—
169C2×#556C2×C2311yescharacteristic
170C4×#101C2×C2311yescharacteristic, complemented
171#1527C2×C2312yes—
172#1491C2×C2312yescomplemented
173#1467C2×C2311yescharacteristic, complemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—42no—
179C22×C23—82no—
180C22×C23—82no—
181C22×C23#37911yescharacteristic
182C22×C23C2×#10411yescharacteristic
183C22×C23#42111yescharacteristic
184C4×C23—22no—
185C4×C23—21no—
186C4×C23—22no—
187C4×C23—22no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—42no—
191C4×C23—42no—
192C4×C23—41no—
193C4×C23—42no—
194C4×C23#41312yes—
195#556—24no—
196#556—24no—
197C2×C4⋊Q8C4×C2311yescharacteristic
198C2×#126—21no—
199C2×#126—21no—
200C2×#124C4×C2311yescharacteristic
201C2×#124C22×C2311yescharacteristic
202C2×C22⋊Q8—21no—
203C2×C4×Q8C4×C2311yescharacteristic
204C2×C4×Q8C22×C2311yescharacteristic
205C2×C4×Q8C22×C2311yescharacteristic
206C2×C4×D4C4×C2311yescharacteristic
207C2×#117—21no—
208#432C22×C2312yes—
209C2×#101—21no—
210C4×M16C22×C2312yes—
211#343—21no—
212#329—24no—
213Q8⋊C8—24no—
214#324—21no—
215#324—21no—
216C2×C23—21no—
217C2×C23—21no—
218C2×C23—82no—
219C2×C23#178511yescharacteristic
220C2×C23#178911yescharacteristic
221C2×C23#179611yescharacteristic
222C22×Q8C2×C4×C2311yescharacteristic
223C22×Q8C2×C4×C2311yescharacteristic
224C22×Q8D4×C2311yescharacteristic
225C22×D4C2×C4×C2311yescharacteristic
226C23×C4—21no—
227C2×M16—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230#126—28no—
231#126—28no—
232#124—22no—
233#124—22no—
234#124—22no—
235#124D4×C2314yes—
236C22⋊Q8—28no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8D4×C2314yes—
243C4×D4—22no—
244C4×D4—22no—
245#117—28no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×C4×C2311yescharacteristic
251C2×C4⋊C4D4×C2311yescharacteristic
252C2×C4⋊C4D4×C2311yescharacteristic
253C2×C4⋊C4D4×C2311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2×C4×C2311yescharacteristic
262C2×C42C2×C4×C2311yescharacteristic
263C2×C42C23×C2311yescharacteristic
264C4⋊C8—22no—
265#101—24no—
266C22⋊C8—22no—
267#97—22no—
268#97—22no—
269C4×C8—22no—
270C23#836811yescharacteristic, direct product factor, Sylow
271C24—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8C22⋊C4×C2312yes—
280C2×Q8C22⋊C4×C2312yes—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4C22⋊C4×C2312yes—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊C4×C2311yescharacteristic
291C22×C4C22⋊C4×C2311yescharacteristic
292C22×C4C22⋊C4×C2311yescharacteristic
293C22×C4C22⋊C4×C2311yescharacteristic
294C22×C4C22×C4×C2311yescharacteristic
295C22×C4C2×D4×C2311yescharacteristic
296C22×C4C2×D4×C2311yescharacteristic
297M16—24no—
298M16—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—21no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4C22⋊C4×C2312yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C42—24no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—24no—
321C42C2×D4×C2312yes—
322C42C2×D4×C2312yes—
323C23—22no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23C2×C22⋊C4×C2311yescharacteristic
329Q8—24no—
330Q8—44no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334D4—42no—
335D4—42no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4C23×#10412yes—
358C2×C4C2×C22⋊C4×C2311yescharacteristic
359C2×C4C2×C22⋊C4×C2311yescharacteristic
360C2×C4C23×#12012yes—
361C2×C4C23×#12012yes—
362C8—44no—
363C8—82no—
364C22—21no—
365C22—21no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—82no—
370C22—82no—
371C22C23×#37911yescharacteristic
372C22C2×C23×#10411yescharacteristic
373C22C23×#42111yescharacteristic
374C4—21no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4C23×#41312yes—
385C2—21no—
386C2—21no—
387C2—82no—
388C2C23×#178511yescharacteristic
389C2C23×#178911yescharacteristic
390C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C23×#101C2semidirect
4. C23×#1491C2semidirect
5. C23×#1467C2semidirect
127. #8368C23direct (N × H)
170. C4×#101C2×C23semidirect
172. #1491C2×C23semidirect
173. #1467C2×C23semidirect
270. C23#8368direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5562
C4×C23×#1012
C23×#15272
C23×#14912
C23×#14672
#836823