Finite Group Databasebeta

← C23×#9377 #19157003

Subgroups — C23×#9377 #19157003

Order 5888.

434 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3239C211yesmaximal, characteristic
2C23×#2992C211yesmaximal, characteristic
3C23×#1981C211yesmaximal, characteristic, complemented
4C23×#1556C211yesmaximal, characteristic, complemented
5C23×#1509C211yesmaximal, characteristic
6C23×#1290C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C23×#529C2211yescharacteristic
10C23×#501C412yescomplemented
11C23×#489C412yescomplemented
12C23×#484—21no—
13C23×#481—21no—
14C23×#475—21no—
15C23×#474—21no—
16C23×#446—21no—
17C23×#435—21no—
18C23×#435—21no—
19C8×D4×C23—21no—
20C23×#433—21no—
21C23×#432C2211yescharacteristic
22C23×#432C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C23×#424—21no—
25C2×C23×C4⋊C8C2211yescharacteristic
26C2×C23×#103—21no—
27C2×C23×#103—21no—
28C4×C4⋊C4×C23C2211yescharacteristic
29C23×#332—21no—
30C23×#332—21no—
31C23×#328—21no—
32C23×#327—21no—
33C22×Q8×C23—21no—
34C2×M16×C23—21no—
35C22×C8×C23—21no—
36C22×C8×C23D411yescharacteristic
37C22×C8×C23D411yescharacteristic
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8C2×C412yes—
42C23×#126—21no—
43C23×#125C2×C411yescharacteristic
44C23×#124C2×C411yescharacteristic
45C23×#123—21no—
46C23×C22⋊Q8D411yescharacteristic
47C23×#121D411yescharacteristic
48C4×Q8×C23D411yescharacteristic
49C4×D4×C23—21no—
50C4×D4×C23D411yescharacteristic
51C23×#117—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C42×C23—21no—
56C2×C42×C23C2311yescharacteristic
57C23×#107—22no—
58C23×#106—22no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8D411yescharacteristic
66C23×C4⋊C8D411yescharacteristic
67C23×#103—22no—
68C23×#103—42no—
69C23×#103—42no—
70C23×#102—22no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8—21no—
76C23×#97—21no—
77C23×#97—21no—
78C23×#97—21no—
79C23×#97C2×C412yes—
80C4×C8×C23—21no—
81C23×#95—21no—
82C2×Q8×C23—22no—
83C2×Q8×C23—41no—
84C2×Q8×C23—41no—
85C2×Q8×C23—41no—
86C2×Q8×C23—41no—
87C2×Q8×C23—41no—
88C2×Q8×C23C22⋊C411yescharacteristic
89C2×D4×C23—21no—
90C2×D4×C23C22⋊C411yescharacteristic
91C22×C4×C23—21no—
92C22×C4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96C22×C4×C23#4111yescharacteristic
97M16×C23—81no—
98C2×C8×C23—22no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23—41no—
106C2×C8×C23—81no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—22no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23C22⋊C411yescharacteristic
118C4⋊C4×C23C22⋊C411yescharacteristic
119C22⋊C4×C23—21no—
120C22⋊C4×C23—21no—
121C22⋊C4×C23—21no—
122C22⋊C4×C23—21no—
123C22⋊C4×C23—21no—
124C42×C23—21no—
125C42×C23—81no—
126C42×C23C22×C411yescharacteristic
127C42×C23C2×D411yescharacteristic
128C42×C23C2×D411yescharacteristic
129C42×C23#4111yescharacteristic
130#9377C2311yesmaximal, characteristic, direct product factor, Sylow
131C23×C23—21no—
132C23×C23#12111yescharacteristic
133Q8×C23—41no—
134Q8×C23—42no—
135Q8×C23—42no—
136Q8×C23—81no—
137Q8×C23—81no—
138D4×C23—41no—
139D4×C23—41no—
140D4×C23—41no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—81no—
154C2×C4×C23—81no—
155C2×C4×C23—81no—
156C2×C4×C23—81no—
157C2×C4×C23C2×C22⋊C411yescharacteristic
158C2×C4×C23C4×D411yescharacteristic
159C2×C4×C23C4×D411yescharacteristic
160C2×C4×C23#12011yescharacteristic
161C2×C4×C23#12111yescharacteristic
162C2×C4×C23#12311yescharacteristic
163C8×C23—42no—
164C8×C23—41no—
165C8×C23—41no—
166C8×C23—41no—
167C8×C23—41no—
168C8×C23—81no—
169#3239C2×C2311yescharacteristic
170#2992C2×C2311yescharacteristic
171#1981C2×C2311yescharacteristic, complemented
172#1556C2×C2311yescharacteristic, complemented
173#1509C2×C2311yescharacteristic
174#1290C2×C2311yescharacteristic, complemented
175#1285C2×C2311yescharacteristic, complemented
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23#38611yescharacteristic
182C22×C23#47111yescharacteristic
183C22×C23#47311yescharacteristic
184C4×C23—21no—
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—41no—
192C4×C23—82no—
193C4×C23—82no—
194C4×C23#45311yescharacteristic
195C4×C23#45411yescharacteristic
196C2×C4⋊Q8C22×C2311yescharacteristic
197#529C22×C2311yescharacteristic
198#501C4×C2312yescomplemented
199#489C4×C2312yescomplemented
200#484—21no—
201#481—21no—
202#475—21no—
203#474—21no—
204#446—21no—
205#435—21no—
206#435—21no—
207C8×D4—21no—
208#433—21no—
209#432C22×C2311yescharacteristic
210#432C22×C2311yescharacteristic
211#432C22×C2311yescharacteristic
212#424—21no—
213C2×C4⋊C8C22×C2311yescharacteristic
214C2×#103—21no—
215C2×#103—21no—
216C4×C4⋊C4C22×C2311yescharacteristic
217#332—21no—
218#332—21no—
219#328—21no—
220#327—21no—
221C2×C23—21no—
222C2×C23—21no—
223C2×C23—81no—
224C2×C23#188311yescharacteristic
225C2×C23#190811yescharacteristic
226C2×C23#190911yescharacteristic
227C22×Q8—21no—
228C2×M16—21no—
229C22×C8—21no—
230C22×C8D4×C2311yescharacteristic
231C22×C8D4×C2311yescharacteristic
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C2×C4×C2312yes—
236#126—21no—
237#125C2×C4×C2311yescharacteristic
238#124C2×C4×C2311yescharacteristic
239#123—21no—
240C22⋊Q8D4×C2311yescharacteristic
241#121D4×C2311yescharacteristic
242C4×Q8D4×C2311yescharacteristic
243C4×D4—21no—
244C4×D4D4×C2311yescharacteristic
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C2311yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8D4×C2311yescharacteristic
260C4⋊C8D4×C2311yescharacteristic
261#103—22no—
262#103—42no—
263#103—42no—
264#102—22no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97C2×C4×C2312yes—
274C4×C8—21no—
275#95—21no—
276C23#937711yescharacteristic, direct product factor, Sylow
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C22⋊C4×C2311yescharacteristic
284C2×D4—21no—
285C2×D4C22⋊C4×C2311yescharacteristic
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×D4×C2311yescharacteristic
290C22×C4C2×D4×C2311yescharacteristic
291C22×C4#41×C2311yescharacteristic
292M16—81no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—81no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C22⋊C4×C2311yescharacteristic
313C4⋊C4C22⋊C4×C2311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C42—21no—
320C42—81no—
321C42C22×C4×C2311yescharacteristic
322C42C2×D4×C2311yescharacteristic
323C42C2×D4×C2311yescharacteristic
324C42#41×C2311yescharacteristic
325C23—21no—
326C23C23×#12111yescharacteristic
327Q8—41no—
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—81no—
332D4—41no—
333D4—41no—
334D4—41no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C2×C22⋊C4×C2311yescharacteristic
352C2×C4C4×D4×C2311yescharacteristic
353C2×C4C4×D4×C2311yescharacteristic
354C2×C4C23×#12011yescharacteristic
355C2×C4C23×#12111yescharacteristic
356C2×C4C23×#12311yescharacteristic
357C8—42no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—81no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C23×#38611yescharacteristic
369C22C23×#47111yescharacteristic
370C22C23×#47311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—82no—
380C4—82no—
381C4C23×#45311yescharacteristic
382C4C23×#45411yescharacteristic
383C2—21no—
384C2—21no—
385C2—81no—
386C2C23×#188311yescharacteristic
387C2C23×#190811yescharacteristic
388C2C23×#190911yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1981C2semidirect
4. C23×#1556C2semidirect
6. C23×#1290C2semidirect
7. C23×#1285C2semidirect
10. C23×#501C4semidirect
11. C23×#489C4semidirect
130. #9377C23direct (N × H)
171. #1981C2×C23semidirect
172. #1556C2×C23semidirect
174. #1290C2×C23semidirect
175. #1285C2×C23semidirect
198. #501C4×C23semidirect
199. #489C4×C23semidirect
276. C23#9377direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32392
C23×#29922
C23×#19812
C23×#15562
C23×#15092
C23×#12902
C23×#12852
#937723