Finite Group Databasebeta

← C23×#9761 #19157387

Subgroups — C23×#9761 #19157387

Order 5888.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3167C211yesmaximal, characteristic
2C23×#1726C211yesmaximal, characteristic, complemented
3C23×#1568C211yesmaximal, characteristic, complemented
4C23×#1531C211yesmaximal, characteristic, complemented
5C23×#1530C211yesmaximal, characteristic
6C23×#1506C211yesmaximal, characteristic, complemented
7C23×#1505C211yesmaximal, characteristic
8C23×#533C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C23×#498C411yescharacteristic, complemented
11C23×#492C411yescharacteristic, complemented
12C23×#487C411yescharacteristic, complemented
13C23×#486C411yescharacteristic, complemented
14C23×#465—21no—
15C23×#465—21no—
16C4×SD16×C23—21no—
17C4×SD16×C23—21no—
18C23×#432C2211yescharacteristic
19C23×#432C2211yescharacteristic
20C2×C23×#97—21no—
21C2×C23×#97C2211yescharacteristic
22C2×C4×C8×C23—21no—
23C2×C4×C8×C23C2211yescharacteristic
24C23×#376C2211yescharacteristic
25C23×#335—21no—
26C23×#334—21no—
27C23×#330—22no—
28C23×#329—22no—
29C23×#328—22no—
30C23×#327—22no—
31C2×#41×C23—21no—
32C2×SD16×C23—21no—
33C2×SD16×C23—21no—
34C22×C8×C23—21no—
35C22×C8×C23D411yescharacteristic
36C22×C8×C23D411yescharacteristic
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×#127C2×C411yescharacteristic
41C23×#125—21no—
42C23×#124—21no—
43C23×C22⋊Q8—21no—
44C23×C22⋊Q8—21no—
45C23×#121—21no—
46C23×#121—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C23×#117—21no—
52C2×C42×C23—21no—
53C2×C42×C23C2311yescharacteristic
54C23×#106—21no—
55C23×#106—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×#103—21no—
59C23×#103—21no—
60C23×#102—21no—
61C23×#102—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×#97—24no—
65C23×#97—21no—
66C23×#97—21no—
67C23×#97—21no—
68C23×#97D412yes—
69C4×C8×C23—24no—
70C4×C8×C23C2×C411yescharacteristic
71C4×C8×C23C2×C411yescharacteristic
72C4×C8×C23D411yescharacteristic
73C4×C8×C23D411yescharacteristic
74C23×#95—21no—
75C23×#95D412yes—
76#41×C23—82no—
77C2×Q8×C23—21no—
78C2×Q8×C23—21no—
79C2×Q8×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—41no—
84C22×C4×C23—22no—
85C22×C4×C23—41no—
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23#4111yescharacteristic
89SD16×C23—42no—
90SD16×C23—42no—
91C2×C8×C23—22no—
92C2×C8×C23—22no—
93C2×C8×C23—22no—
94C2×C8×C23—22no—
95C2×C8×C23—22no—
96C2×C8×C23—41no—
97C2×C8×C23—41no—
98C2×C8×C23C22⋊C411yescharacteristic
99C2×C8×C23C22⋊C411yescharacteristic
100C2×C8×C23C22⋊C411yescharacteristic
101C2×C8×C23C22⋊C411yescharacteristic
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—41no—
109C4⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C42×C23—22no—
114C42×C23—22no—
115C42×C23—41no—
116C42×C23C22×C411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23#4111yescharacteristic
120#9761C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—41no—
122C23×C23#12111yescharacteristic
123Q8×C23—41no—
124Q8×C23—41no—
125Q8×C23—81no—
126Q8×C23—81no—
127D4×C23—41no—
128D4×C23—41no—
129D4×C23—81no—
130D4×C23—81no—
131D4×C23—82no—
132C2×C4×C23—22no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—24no—
136C2×C4×C23—41no—
137C2×C4×C23—42no—
138C2×C4×C23—42no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—82no—
143C2×C4×C23C2×C22⋊C411yescharacteristic
144C2×C4×C23C4×D411yescharacteristic
145C2×C4×C23C4×D411yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23#12411yescharacteristic
148C2×C4×C23#12711yescharacteristic
149C8×C23—24no—
150C8×C23—21no—
151C8×C23—21no—
152C8×C23—81no—
153C8×C23—81no—
154C8×C23#10412yes—
155C8×C23#10412yes—
156#3167C2×C2311yescharacteristic
157#1726C2×C2311yescharacteristic, complemented
158#1568C2×C2311yescharacteristic, complemented
159#1531C2×C2311yescharacteristic, complemented
160#1530C2×C2311yescharacteristic
161#1506C2×C2311yescharacteristic, complemented
162#1505C2×C2311yescharacteristic
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#39011yescharacteristic
169C22×C23#47112yes—
170C4×C23—21no—
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—44no—
176C4×C23—81no—
177C4×C23—81no—
178C4×C23—81no—
179C4×C23C2×#10411yescharacteristic
180C4×C23C2×#10411yescharacteristic
181#533C22×C2311yescharacteristic
182#532C22×C2311yescharacteristic
183#498C4×C2311yescharacteristic, complemented
184#492C4×C2311yescharacteristic, complemented
185#487C4×C2311yescharacteristic, complemented
186#486C4×C2311yescharacteristic, complemented
187#465—21no—
188#465—21no—
189C4×SD16—21no—
190C4×SD16—21no—
191#432C22×C2311yescharacteristic
192#432C22×C2311yescharacteristic
193C2×#97—21no—
194C2×#97C22×C2311yescharacteristic
195C2×C4×C8—21no—
196C2×C4×C8C22×C2311yescharacteristic
197#376C22×C2311yescharacteristic
198#335—21no—
199#334—21no—
200#330—22no—
201#329—22no—
202#328—22no—
203#327—22no—
204C2×C23—22no—
205C2×C23—161no—
206C2×C23#196211yescharacteristic
207C2×C23#197611yescharacteristic
208C2×C23#197711yescharacteristic
209C2×#41—21no—
210C2×SD16—21no—
211C2×SD16—21no—
212C22×C8—21no—
213C22×C8D4×C2311yescharacteristic
214C22×C8D4×C2311yescharacteristic
215C4⋊Q8C2×C4×C2311yescharacteristic
216C4⋊Q8C2×C4×C2311yescharacteristic
217C4⋊Q8C2×C4×C2311yescharacteristic
218#127C2×C4×C2311yescharacteristic
219#125—21no—
220#124—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223#121—21no—
224#121—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229#117—21no—
230C2×C42—21no—
231C2×C42C23×C2311yescharacteristic
232#106—21no—
233#106—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#103—21no—
237#103—21no—
238#102—21no—
239#102—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242#97—24no—
243#97—21no—
244#97—21no—
245#97—21no—
246#97D4×C2312yes—
247C4×C8—24no—
248C4×C8C2×C4×C2311yescharacteristic
249C4×C8C2×C4×C2311yescharacteristic
250C4×C8D4×C2311yescharacteristic
251C4×C8D4×C2311yescharacteristic
252#95—21no—
253#95D4×C2312yes—
254C23#976111yescharacteristic, direct product factor, Sylow
255#41—82no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4C2×D4×C2311yescharacteristic
267C22×C4#41×C2311yescharacteristic
268SD16—42no—
269SD16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22⋊C4×C2311yescharacteristic
278C2×C8C22⋊C4×C2311yescharacteristic
279C2×C8C22⋊C4×C2311yescharacteristic
280C2×C8C22⋊C4×C2311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42—22no—
293C42—22no—
294C42—41no—
295C42C22×C4×C2311yescharacteristic
296C42C2×D4×C2311yescharacteristic
297C42C2×D4×C2311yescharacteristic
298C42#41×C2311yescharacteristic
299C23—41no—
300C23C23×#12111yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—82no—
321C2×C4C2×C22⋊C4×C2311yescharacteristic
322C2×C4C4×D4×C2311yescharacteristic
323C2×C4C4×D4×C2311yescharacteristic
324C2×C4C23×#12111yescharacteristic
325C2×C4C23×#12411yescharacteristic
326C2×C4C23×#12711yescharacteristic
327C8—24no—
328C8—21no—
329C8—21no—
330C8—81no—
331C8—81no—
332C8C23×#10412yes—
333C8C23×#10412yes—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22C23×#39011yescharacteristic
340C22C23×#47112yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4C2×C23×#10411yescharacteristic
351C4C2×C23×#10411yescharacteristic
352C2—22no—
353C2—161no—
354C2C23×#196211yescharacteristic
355C2C23×#197611yescharacteristic
356C2C23×#197711yescharacteristic
subgroupG/H
center, socleC22×C23#390
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
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
2. C23×#1726C2semidirect
3. C23×#1568C2semidirect
4. C23×#1531C2semidirect
6. C23×#1506C2semidirect
10. C23×#498C4semidirect
11. C23×#492C4semidirect
12. C23×#487C4semidirect
13. C23×#486C4semidirect
120. #9761C23direct (N × H)
157. #1726C2×C23semidirect
158. #1568C2×C23semidirect
159. #1531C2×C23semidirect
161. #1506C2×C23semidirect
183. #498C4×C23semidirect
184. #492C4×C23semidirect
185. #487C4×C23semidirect
186. #486C4×C23semidirect
254. C23#9761direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31672
C23×#17262
C23×#15682
C23×#15312
C23×#15302
C23×#15062
C23×#15052
#976123