Finite Group Databasebeta

← C23×#9327 #19156953

Subgroups — C23×#9327 #19156953

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3110C211yesmaximal, characteristic
2C23×#2319C211yesmaximal, characteristic
3C23×#1986C211yesmaximal, characteristic, complemented
4C23×#1506C211yesmaximal, characteristic, complemented
5C23×#1490C211yesmaximal, characteristic
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C4×D4×C23C2211yescharacteristic
10C23×#484—21no—
11C23×#474C412yescomplemented
12C23×#461C412yescomplemented
13C23×#450—21no—
14C23×#440—22no—
15C23×#432C2211yescharacteristic
16C23×#432C2211yescharacteristic
17C4×M16×C23C2211yescharacteristic
18C23×#390—21no—
19C23×#389—21no—
20C23×#388—22no—
21C23×#381—21no—
22C23×#380—21no—
23C4×C4⋊C4×C23C2211yescharacteristic
24C23×#376C2211yescharacteristic
25C23×#339—22no—
26C23×#328—21no—
27C23×#328—21no—
28C23×D4⋊C8—22no—
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C2×SD16×C23—22no—
32C2×M16×C23—21no—
33C23×C4⋊Q8C2×C411yescharacteristic
34C23×C4⋊Q8C2×C411yescharacteristic
35C23×#125—21no—
36C23×C22⋊Q8—21no—
37C23×C22⋊Q8—21no—
38C4×Q8×C23—21no—
39C4×Q8×C23—21no—
40C4×D4×C23—21no—
41C4×D4×C23—21no—
42C4×D4×C23C2×C412yes—
43C4×D4×C23D412yes—
44C23×#117—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—22no—
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—22no—
50C2×C22⋊C4×C23—22no—
51C2×C42×C23—21no—
52C2×C42×C23D412yes—
53C2×C42×C23C2311yescharacteristic
54C23×#107—22no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8C2×C412yes—
57C23×#103—22no—
58C23×#102—22no—
59C23×C22⋊C8—21no—
60C23×C22⋊C8—21no—
61C23×#97—21no—
62C23×#97—21no—
63C23×#97—21no—
64C4×C8×C23—21no—
65C23×#95—22no—
66C23×#95—21no—
67C23×#95—21no—
68C23×#95D411yescharacteristic
69C23×#95D411yescharacteristic
70C24×C23—21no—
71C2×Q8×C23—21no—
72C2×Q8×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—42no—
75C2×D4×C23C22⋊C412yes—
76C22×C4×C23—21no—
77C22×C4×C23—22no—
78C22×C4×C23—22no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—22no—
82C22×C4×C23—82no—
83C22×C4×C23C2×D411yescharacteristic
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23C2×D411yescharacteristic
86SD16×C23—44no—
87M16×C23—22no—
88M16×C23—41no—
89C2×C8×C23—22no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—42no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—82no—
101C4⋊C4×C23C22⋊C412yes—
102C22⋊C4×C23—22no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—82no—
106C22⋊C4×C23—82no—
107C42×C23—42no—
108C42×C23—42no—
109C42×C23—41no—
110C42×C23C22×C411yescharacteristic
111C42×C23C2×D411yescharacteristic
112C42×C23#4111yescharacteristic
113C42×C23#4111yescharacteristic
114#9327C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—22no—
116C23×C23—42no—
117C23×C23—42no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23#12011yescharacteristic
121Q8×C23—42no—
122Q8×C23—81no—
123D4×C23—22no—
124D4×C23—22no—
125D4×C23—42no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—22no—
129C2×C4×C23—22no—
130C2×C4×C23—41no—
131C2×C4×C23—42no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—42no—
136C2×C4×C23—42no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—82no—
144C2×C4×C23—82no—
145C2×C4×C23C2×C22⋊C411yescharacteristic
146C2×C4×C23C4×D411yescharacteristic
147C2×C4×C23C4×D411yescharacteristic
148C2×C4×C23#12111yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12311yescharacteristic
151C8×C23—42no—
152C8×C23—42no—
153C8×C23—81no—
154#3110C2×C2311yescharacteristic
155#2319C2×C2311yescharacteristic
156#1986C2×C2311yescharacteristic, complemented
157#1506C2×C2311yescharacteristic, complemented
158#1490C2×C2311yescharacteristic
159#1300C2×C2311yescharacteristic, complemented
160#1285C2×C2311yescharacteristic, complemented
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—42no—
166C22×C23—82no—
167C22×C23—82no—
168C22×C23#38611yescharacteristic
169C22×C23#45511yescharacteristic
170C22×C23#45611yescharacteristic
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—42no—
177C4×C23—81no—
178C4×C23—82no—
179C4×C23—81no—
180C4×C23—81no—
181C4×C23—81no—
182C4×C23#42112yes—
183#533C22×C2311yescharacteristic
184C2×C4×D4C22×C2311yescharacteristic
185#484—21no—
186#474C4×C2312yescomplemented
187#461C4×C2312yescomplemented
188#450—21no—
189#440—22no—
190#432C22×C2311yescharacteristic
191#432C22×C2311yescharacteristic
192C4×M16C22×C2311yescharacteristic
193#390—21no—
194#389—21no—
195#388—22no—
196#381—21no—
197#380—21no—
198C4×C4⋊C4C22×C2311yescharacteristic
199#376C22×C2311yescharacteristic
200#339—22no—
201#328—21no—
202#328—21no—
203D4⋊C8—22no—
204C2×C23—21no—
205C2×C23—21no—
206C2×C23—82no—
207C2×C23#188211yescharacteristic
208C2×C23#189711yescharacteristic
209C2×C23#190411yescharacteristic
210C22×D4D4×C2311yescharacteristic
211C23×C4—21no—
212C2×SD16—22no—
213C2×M16—21no—
214C4⋊Q8C2×C4×C2311yescharacteristic
215C4⋊Q8C2×C4×C2311yescharacteristic
216#125—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4C2×C4×C2312yes—
224C4×D4D4×C2312yes—
225#117—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4D4×C2311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42D4×C2312yes—
234C2×C42C23×C2311yescharacteristic
235#107—22no—
236C4⋊C8—21no—
237C4⋊C8C2×C4×C2312yes—
238#103—22no—
239#102—22no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242#97—21no—
243#97—21no—
244#97—21no—
245C4×C8—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95D4×C2311yescharacteristic
250#95D4×C2311yescharacteristic
251C23#932711yescharacteristic, direct product factor, Sylow
252C24—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—42no—
257C2×D4C22⋊C4×C2312yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—82no—
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4C2×D4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268SD16—44no—
269M16—22no—
270M16—41no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—82no—
283C4⋊C4C22⋊C4×C2312yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C42—42no—
290C42—42no—
291C42—41no—
292C42C22×C4×C2311yescharacteristic
293C42C2×D4×C2311yescharacteristic
294C42#41×C2311yescharacteristic
295C42#41×C2311yescharacteristic
296C23—22no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23C23×#12011yescharacteristic
302Q8—42no—
303Q8—81no—
304D4—22no—
305D4—22no—
306D4—42no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C2×C22⋊C4×C2311yescharacteristic
327C2×C4C4×D4×C2311yescharacteristic
328C2×C4C4×D4×C2311yescharacteristic
329C2×C4C23×#12111yescharacteristic
330C2×C4C23×#12111yescharacteristic
331C2×C4C23×#12311yescharacteristic
332C8—42no—
333C8—42no—
334C8—81no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—82no—
342C22C23×#38611yescharacteristic
343C22C23×#45511yescharacteristic
344C22C23×#45611yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—81no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4C23×#42112yes—
357C2—21no—
358C2—21no—
359C2—82no—
360C2C23×#188211yescharacteristic
361C2C23×#189711yescharacteristic
362C2C23×#190411yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
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×#1986C2semidirect
4. C23×#1506C2semidirect
6. C23×#1300C2semidirect
7. C23×#1285C2semidirect
11. C23×#474C4semidirect
12. C23×#461C4semidirect
114. #9327C23direct (N × H)
156. #1986C2×C23semidirect
157. #1506C2×C23semidirect
159. #1300C2×C23semidirect
160. #1285C2×C23semidirect
186. #474C4×C23semidirect
187. #461C4×C23semidirect
251. C23#9327direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31102
C23×#23192
C23×#19862
C23×#15062
C23×#14902
C23×#13002
C23×#12852
#932723