Finite Group Databasebeta

← C23×#9367 #19156993

Subgroups — C23×#9367 #19156993

Order 5888.

542 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3094C211yesmaximal, characteristic
2C23×#2346C211yesmaximal, characteristic, complemented
3C23×#1964C211yesmaximal, characteristic
4C23×#1826C211yesmaximal, characteristic, complemented
5C23×#1507C211yesmaximal, characteristic
6C23×#1305C211yesmaximal, characteristic, complemented
7C23×#1297C211yesmaximal, characteristic, complemented
8C2×C23×#121C2211yescharacteristic
9C2×C23×#117C2211yescharacteristic
10C23×#482—22no—
11C23×#481—21no—
12C23×#480—21no—
13C23×#428C412yescomplemented
14C23×#418C412yescomplemented
15C2×C23×#99—22no—
16C23×#407C2211yescharacteristic
17C23×#407C2211yescharacteristic
18C23×#389—21no—
19C23×#388—22no—
20C23×#382—21no—
21C23×#380C2211yescharacteristic
22C4×C4⋊C4×C23—21no—
23C4×C22⋊C4×C23C2211yescharacteristic
24C4×C22⋊C4×C23C2211yescharacteristic
25C23×#376—21no—
26C23×#339—22no—
27C23×#327—21no—
28C23×#327—21no—
29C22×D4×C23—21no—
30C22×D4×C23C2×C411yescharacteristic
31C23×C4×C23C2311yescharacteristic
32C2×M16×C23—21no—
33C2×M16×C23C2×C412yes—
34C23×#121—21no—
35C23×#121—21no—
36C23×#121—21no—
37C23×#121—21no—
38C23×#117—21no—
39C23×#117—21no—
40C23×#117C2×C412yes—
41C23×#117D412yes—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—22no—
44C2×C4⋊C4×C23C2×C411yescharacteristic
45C2×C4⋊C4×C23D411yescharacteristic
46C2×C4⋊C4×C23D411yescharacteristic
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23D412yes—
52C2×C42×C23—21no—
53C2×C42×C23—22no—
54C2×C42×C23—21no—
55C2×C42×C23—21no—
56C23×#107—22no—
57C23×#106—22no—
58C23×#102—22no—
59C23×#102—22no—
60C23×#99—44no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×C22⋊C8—21no—
65C23×#95—22no—
66C23×#95—22no—
67C23×#95—21no—
68C23×#95—21no—
69C23×#95D411yescharacteristic
70C23×#95D411yescharacteristic
71C24×C23—21no—
72C24×C23C2×D411yescharacteristic
73C2×D4×C23—21no—
74C2×D4×C23—21no—
75C2×D4×C23—42no—
76C2×D4×C23—42no—
77C2×D4×C23—42no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—22no—
86C22×C4×C23—42no—
87C22×C4×C23—41no—
88C22×C4×C23C22×C411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23#4111yescharacteristic
93C22×C4×C23#4111yescharacteristic
94M16×C23—42no—
95M16×C23—42no—
96C2×C8×C23—22no—
97C2×C8×C23—41no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—42no—
103C4⋊C4×C23—82no—
104C4⋊C4×C23C22⋊C412yes—
105C4⋊C4×C23C22⋊C412yes—
106C22⋊C4×C23—22no—
107C22⋊C4×C23—22no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—42no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—42no—
114C42×C23—22no—
115C42×C23—22no—
116C42×C23—42no—
117C42×C23—44no—
118C42×C23—42no—
119#9367C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—21no—
121C23×C23—21no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—42no—
125C23×C23—41no—
126C23×C23—81no—
127C23×C23—81no—
128C23×C23C4×D411yescharacteristic
129C23×C23#12011yescharacteristic
130C23×C23#12111yescharacteristic
131D4×C23—42no—
132D4×C23—81no—
133D4×C23—82no—
134D4×C23—81no—
135D4×C23—81no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—22no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—42no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—42no—
155C2×C4×C23—42no—
156C2×C4×C23—41no—
157C2×C4×C23—42no—
158C2×C4×C23—41no—
159C2×C4×C23—81no—
160C2×C4×C23—81no—
161C2×C4×C23C2×C22⋊C411yescharacteristic
162C2×C4×C23C4×D411yescharacteristic
163C2×C4×C23#12111yescharacteristic
164C2×C4×C23#12311yescharacteristic
165C8×C23—42no—
166C8×C23—81no—
167#3094C2×C2311yescharacteristic
168#2346C2×C2311yescharacteristic, complemented
169#1964C2×C2311yescharacteristic
170#1826C2×C2311yescharacteristic, complemented
171#1507C2×C2311yescharacteristic
172#1305C2×C2311yescharacteristic, complemented
173#1297C2×C2311yescharacteristic, complemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—21no—
181C22×C23—21no—
182C22×C23—81no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—162no—
188C22×C23#38611yescharacteristic
189C22×C23#45311yescharacteristic
190C22×C23#45411yescharacteristic
191C4×C23—21no—
192C4×C23—21no—
193C4×C23—21no—
194C4×C23—42no—
195C4×C23—42no—
196C4×C23—42no—
197C4×C23—81no—
198C4×C23—82no—
199C4×C23—82no—
200C4×C23—81no—
201C4×C23#41012yes—
202C2×#121C22×C2311yescharacteristic
203C2×#117C22×C2311yescharacteristic
204#482—22no—
205#481—21no—
206#480—21no—
207#428C4×C2312yescomplemented
208#418C4×C2312yescomplemented
209C2×#99—22no—
210#407C22×C2311yescharacteristic
211#407C22×C2311yescharacteristic
212#389—21no—
213#388—22no—
214#382—21no—
215#380C22×C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C22⋊C4C22×C2311yescharacteristic
218C4×C22⋊C4C22×C2311yescharacteristic
219#376—21no—
220#339—22no—
221#327—21no—
222#327—21no—
223C2×C23—21no—
224C2×C23—21no—
225C2×C23—41no—
226C2×C23—41no—
227C2×C23—161no—
228C2×C23#188311yescharacteristic
229C2×C23#189911yescharacteristic
230C2×C23#190611yescharacteristic
231C22×D4—21no—
232C22×D4C2×C4×C2311yescharacteristic
233C23×C4C23×C2311yescharacteristic
234C2×M16—21no—
235C2×M16C2×C4×C2312yes—
236#121—21no—
237#121—21no—
238#121—21no—
239#121—21no—
240#117—21no—
241#117—21no—
242#117C2×C4×C2312yes—
243#117D4×C2312yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C2×C4×C2311yescharacteristic
247C2×C4⋊C4D4×C2311yescharacteristic
248C2×C4⋊C4D4×C2311yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4D4×C2312yes—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258#107—22no—
259#106—22no—
260#102—22no—
261#102—22no—
262#99—44no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95D4×C2311yescharacteristic
272#95D4×C2311yescharacteristic
273C23#936711yescharacteristic, direct product factor, Sylow
274C24—21no—
275C24C2×D4×C2311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4C22×C4×C2311yescharacteristic
292C22×C4C2×D4×C2311yescharacteristic
293C22×C4C2×D4×C2311yescharacteristic
294C22×C4C2×D4×C2311yescharacteristic
295C22×C4#41×C2311yescharacteristic
296C22×C4#41×C2311yescharacteristic
297M16—42no—
298M16—42no—
299C2×C8—22no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C4⋊C4C22⋊C4×C2312yes—
308C4⋊C4C22⋊C4×C2312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—44no—
321C42—42no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—81no—
329C23—81no—
330C23C4×D4×C2311yescharacteristic
331C23C23×#12011yescharacteristic
332C23C23×#12111yescharacteristic
333D4—42no—
334D4—81no—
335D4—82no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C2×C22⋊C4×C2311yescharacteristic
364C2×C4C4×D4×C2311yescharacteristic
365C2×C4C23×#12111yescharacteristic
366C2×C4C23×#12311yescharacteristic
367C8—42no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—162no—
383C22C23×#38611yescharacteristic
384C22C23×#45311yescharacteristic
385C22C23×#45411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—82no—
394C4—82no—
395C4—81no—
396C4C23×#41012yes—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2C23×#188311yescharacteristic
403C2C23×#189911yescharacteristic
404C2C23×#190611yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2346C2semidirect
4. C23×#1826C2semidirect
6. C23×#1305C2semidirect
7. C23×#1297C2semidirect
13. C23×#428C4semidirect
14. C23×#418C4semidirect
119. #9367C23direct (N × H)
168. #2346C2×C23semidirect
170. #1826C2×C23semidirect
172. #1305C2×C23semidirect
173. #1297C2×C23semidirect
207. #428C4×C23semidirect
208. #418C4×C23semidirect
273. C23#9367direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30942
C23×#23462
C23×#19642
C23×#18262
C23×#15072
C23×#13052
C23×#12972
#936723