Finite Group Databasebeta

← C23×#9376 #19157002

Subgroups — C23×#9376 #19157002

Order 5888.

430 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3218C211yesmaximal, characteristic
2C23×#2992C211yesmaximal, characteristic
3C23×#1987C211yesmaximal, characteristic, complemented
4C23×#1556C211yesmaximal, characteristic, complemented
5C23×#1509C211yesmaximal, characteristic
6C23×#1290C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C23×#529C2211yescharacteristic
10C23×#497C412yescomplemented
11C23×#491C412yescomplemented
12C23×#462—21no—
13C23×#461—21no—
14C23×#451—21no—
15C23×#448—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
26C4×C4⋊C4×C23C2211yescharacteristic
27C23×#340—21no—
28C23×#340—21no—
29C23×#332—21no—
30C23×#332—21no—
31C23×#328—21no—
32C23×#327—21no—
33C22×Q8×C23—21no—
34C2×Q16×C23—22no—
35C2×SD16×C23—22no—
36C2×M16×C23—21no—
37C22×C8×C23—21no—
38C22×C8×C23D411yescharacteristic
39C22×C8×C23D411yescharacteristic
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8C2×C412yes—
44C23×#126—21no—
45C23×#125C2×C411yescharacteristic
46C23×#124C2×C411yescharacteristic
47C23×#123—21no—
48C23×C22⋊Q8D411yescharacteristic
49C23×#121D411yescharacteristic
50C4×Q8×C23D411yescharacteristic
51C4×D4×C23—21no—
52C4×D4×C23D411yescharacteristic
53C23×#117—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23—21no—
57C2×C42×C23—21no—
58C2×C42×C23C2311yescharacteristic
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—22no—
69C23×C22⋊C8—21no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×#97—21no—
75C23×#97—21no—
76C23×#97—21no—
77C23×#97C2×C412yes—
78C4×C8×C23—21no—
79C23×#95—21no—
80C2×Q8×C23—22no—
81C2×Q8×C23—41no—
82C2×Q8×C23—82no—
83C2×Q8×C23C22⋊C411yescharacteristic
84C2×D4×C23—21no—
85C2×D4×C23C22⋊C411yescharacteristic
86C22×C4×C23—21no—
87C22×C4×C23—41no—
88C22×C4×C23—41no—
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23#4111yescharacteristic
92Q16×C23—82no—
93SD16×C23—82no—
94M16×C23—81no—
95C2×C8×C23—22no—
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—81no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23C22⋊C411yescharacteristic
115C4⋊C4×C23C22⋊C411yescharacteristic
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—21no—
119C22⋊C4×C23—21no—
120C22⋊C4×C23—21no—
121C42×C23—21no—
122C42×C23—81no—
123C42×C23C22×C411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126C42×C23#4111yescharacteristic
127#9376C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—21no—
129C23×C23#12111yescharacteristic
130Q8×C23—41no—
131Q8×C23—42no—
132Q8×C23—42no—
133Q8×C23—82no—
134D4×C23—41no—
135D4×C23—41no—
136D4×C23—41no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—81no—
150C2×C4×C23—82no—
151C2×C4×C23—81no—
152C2×C4×C23C2×C22⋊C411yescharacteristic
153C2×C4×C23C4×D411yescharacteristic
154C2×C4×C23C4×D411yescharacteristic
155C2×C4×C23#12011yescharacteristic
156C2×C4×C23#12111yescharacteristic
157C2×C4×C23#12311yescharacteristic
158C8×C23—42no—
159C8×C23—41no—
160C8×C23—41no—
161C8×C23—41no—
162C8×C23—41no—
163C8×C23—81no—
164#3218C2×C2311yescharacteristic
165#2992C2×C2311yescharacteristic
166#1987C2×C2311yescharacteristic, complemented
167#1556C2×C2311yescharacteristic, complemented
168#1509C2×C2311yescharacteristic
169#1290C2×C2311yescharacteristic, complemented
170#1285C2×C2311yescharacteristic, complemented
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—41no—
174C22×C23—41no—
175C22×C23—41no—
176C22×C23#38611yescharacteristic
177C22×C23#47111yescharacteristic
178C22×C23#47311yescharacteristic
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—41no—
185C4×C23—41no—
186C4×C23—41no—
187C4×C23—82no—
188C4×C23—82no—
189C4×C23#45511yescharacteristic
190C4×C23#45611yescharacteristic
191C2×C4⋊Q8C22×C2311yescharacteristic
192#529C22×C2311yescharacteristic
193#497C4×C2312yescomplemented
194#491C4×C2312yescomplemented
195#462—21no—
196#461—21no—
197#451—21no—
198#448—21no—
199#446—21no—
200#435—21no—
201#435—21no—
202C8×D4—21no—
203#433—21no—
204#432C22×C2311yescharacteristic
205#432C22×C2311yescharacteristic
206#432C22×C2311yescharacteristic
207#424—21no—
208C2×C4⋊C8C22×C2311yescharacteristic
209C4×C4⋊C4C22×C2311yescharacteristic
210#340—21no—
211#340—21no—
212#332—21no—
213#332—21no—
214#328—21no—
215#327—21no—
216C2×C23—21no—
217C2×C23—21no—
218C2×C23—81no—
219C2×C23#188211yescharacteristic
220C2×C23#190811yescharacteristic
221C2×C23#191011yescharacteristic
222C22×Q8—21no—
223C2×Q16—22no—
224C2×SD16—22no—
225C2×M16—21no—
226C22×C8—21no—
227C22×C8D4×C2311yescharacteristic
228C22×C8D4×C2311yescharacteristic
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8C2×C4×C2312yes—
233#126—21no—
234#125C2×C4×C2311yescharacteristic
235#124C2×C4×C2311yescharacteristic
236#123—21no—
237C22⋊Q8D4×C2311yescharacteristic
238#121D4×C2311yescharacteristic
239C4×Q8D4×C2311yescharacteristic
240C4×D4—21no—
241C4×D4D4×C2311yescharacteristic
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C42—21no—
247C2×C42C23×C2311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8D4×C2311yescharacteristic
255C4⋊C8D4×C2311yescharacteristic
256#103—22no—
257#103—22no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#97—21no—
264#97—21no—
265#97—21no—
266#97C2×C4×C2312yes—
267C4×C8—21no—
268#95—21no—
269C23#937611yescharacteristic, direct product factor, Sylow
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—82no—
273C2×Q8C22⋊C4×C2311yescharacteristic
274C2×D4—21no—
275C2×D4C22⋊C4×C2311yescharacteristic
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D4×C2311yescharacteristic
280C22×C4C2×D4×C2311yescharacteristic
281C22×C4#41×C2311yescharacteristic
282Q16—82no—
283SD16—82no—
284M16—81no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4C22⋊C4×C2311yescharacteristic
305C4⋊C4C22⋊C4×C2311yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C42—21no—
312C42—81no—
313C42C22×C4×C2311yescharacteristic
314C42C2×D4×C2311yescharacteristic
315C42C2×D4×C2311yescharacteristic
316C42#41×C2311yescharacteristic
317C23—21no—
318C23C23×#12111yescharacteristic
319Q8—41no—
320Q8—42no—
321Q8—42no—
322Q8—82no—
323D4—41no—
324D4—41no—
325D4—41no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4C2×C22⋊C4×C2311yescharacteristic
342C2×C4C4×D4×C2311yescharacteristic
343C2×C4C4×D4×C2311yescharacteristic
344C2×C4C23×#12011yescharacteristic
345C2×C4C23×#12111yescharacteristic
346C2×C4C23×#12311yescharacteristic
347C8—42no—
348C8—41no—
349C8—41no—
350C8—41no—
351C8—41no—
352C8—81no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C23×#38611yescharacteristic
359C22C23×#47111yescharacteristic
360C22C23×#47311yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4—82no—
371C4C23×#45511yescharacteristic
372C4C23×#45611yescharacteristic
373C2—21no—
374C2—21no—
375C2—81no—
376C2C23×#188211yescharacteristic
377C2C23×#190811yescharacteristic
378C2C23×#191011yescharacteristic
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×#1987C2semidirect
4. C23×#1556C2semidirect
6. C23×#1290C2semidirect
7. C23×#1285C2semidirect
10. C23×#497C4semidirect
11. C23×#491C4semidirect
127. #9376C23direct (N × H)
166. #1987C2×C23semidirect
167. #1556C2×C23semidirect
169. #1290C2×C23semidirect
170. #1285C2×C23semidirect
193. #497C4×C23semidirect
194. #491C4×C23semidirect
269. C23#9376direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32182
C23×#29922
C23×#19872
C23×#15562
C23×#15092
C23×#12902
C23×#12852
#937623