Finite Group Databasebeta

← C23×#9529 #19157155

Subgroups — C23×#9529 #19157155

Order 5888.

468 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3109C211yesmaximal, characteristic
2C23×#2981C211yesmaximal, characteristic, complemented
3C23×#1557C211yesmaximal, characteristic, complemented
4C23×#1486C211yesmaximal, characteristic, complemented
5C23×#1466C211yesmaximal, characteristic, complemented
6C23×#1307C211yesmaximal, characteristic, complemented
7C23×#1287C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C4×D4×C23C2211yescharacteristic
10C23×#481C412yes—
11C23×#476—21no—
12C23×#459—21no—
13C23×#447C412yes—
14C23×#442—22no—
15C23×#436—21no—
16C23×#435—22no—
17C8×D4×C23—21no—
18C23×#432C2211yescharacteristic
19C23×#432C2211yescharacteristic
20C23×#432C2211yescharacteristic
21C23×#431C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C23×#408—22no—
24C23×#332—21no—
25C23×#331—21no—
26C23×D4⋊C8—21no—
27C23×D4⋊C8—21no—
28C23×#323—22no—
29C2×#41×C23—21no—
30C22×D4×C23C2×C411yescharacteristic
31C23×C4×C23—21no—
32C2×D8×C23—22no—
33C2×M16×C23—22no—
34C2×M16×C23—21no—
35C22×C8×C23—22no—
36C23×C4⋊Q8D411yescharacteristic
37C23×#127D411yescharacteristic
38C23×#124—21no—
39C23×#121—21no—
40C23×#121C2×C412yes—
41C4×D4×C23—21no—
42C4×D4×C23—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23—21no—
46C4×D4×C23D412yes—
47C2×C4⋊C4×C23C2×C411yescharacteristic
48C2×C22⋊C4×C23—21no—
49C2×C42×C23C2311yescharacteristic
50C23×#106—22no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×#102—22no—
58C23×#102—22no—
59C23×C22⋊C8—22no—
60C23×C22⋊C8—22no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8C2×C412yes—
64C23×C22⋊C8D412yes—
65C23×#97—21no—
66C23×#97—21no—
67C23×#97—21no—
68C4×C8×C23D411yescharacteristic
69C4×C8×C23D411yescharacteristic
70C24×C23—21no—
71#41×C23—82no—
72C2×Q8×C23—21no—
73C2×D4×C23—22no—
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—42no—
79C2×D4×C23C22⋊C412yes—
80C22×C4×C23—22no—
81C22×C4×C23—21no—
82C22×C4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23—81no—
85C22×C4×C23—81no—
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23#4111yescharacteristic
89D8×C23—44no—
90M16×C23—82no—
91M16×C23—81no—
92C2×C8×C23—22no—
93C2×C8×C23—21no—
94C2×C8×C23—22no—
95C2×C8×C23—21no—
96C2×C8×C23—41no—
97C2×C8×C23—41no—
98C2×C8×C23—82no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23C22⋊C412yes—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—22no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C42×C23C2×D411yescharacteristic
108C42×C23C2×D411yescharacteristic
109C42×C23C2×D411yescharacteristic
110C42×C23#4111yescharacteristic
111#9529C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—22no—
113C23×C23—22no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—81no—
117C23×C23—81no—
118C23×C23C4×D411yescharacteristic
119Q8×C23—81no—
120D4×C23—22no—
121D4×C23—22no—
122D4×C23—42no—
123D4×C23—41no—
124D4×C23—41no—
125D4×C23—41no—
126D4×C23—41no—
127D4×C23—81no—
128D4×C23—82no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—22no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—41no—
137C2×C4×C23—82no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23C2×C22⋊C411yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23#12011yescharacteristic
144C2×C4×C23#12111yescharacteristic
145C2×C4×C23#12111yescharacteristic
146C2×C4×C23#12311yescharacteristic
147C8×C23—41no—
148C8×C23—42no—
149C8×C23—41no—
150C8×C23—42no—
151C8×C23—81no—
152C8×C23—81no—
153#3109C2×C2311yescharacteristic
154#2981C2×C2311yescharacteristic, complemented
155#1557C2×C2311yescharacteristic, complemented
156#1486C2×C2311yescharacteristic, complemented
157#1466C2×C2311yescharacteristic, complemented
158#1307C2×C2311yescharacteristic, complemented
159#1287C2×C2311yescharacteristic, complemented
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—42no—
166C22×C23—42no—
167C22×C23—82no—
168C22×C23—82no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23#38611yescharacteristic
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—42no—
178C4×C23—81no—
179C4×C23#47111yescharacteristic
180C4×C23#47211yescharacteristic
181#532C22×C2311yescharacteristic
182C2×C4×D4C22×C2311yescharacteristic
183#481C4×C2312yes—
184#476—21no—
185#459—21no—
186#447C4×C2312yes—
187#442—22no—
188#436—21no—
189#435—22no—
190C8×D4—21no—
191#432C22×C2311yescharacteristic
192#432C22×C2311yescharacteristic
193#432C22×C2311yescharacteristic
194#431C22×C2311yescharacteristic
195#423C22×C2311yescharacteristic
196#408—22no—
197#332—21no—
198#331—21no—
199D4⋊C8—21no—
200D4⋊C8—21no—
201#323—22no—
202C2×C23—41no—
203C2×C23—82no—
204C2×C23—82no—
205C2×C23#189111yescharacteristic
206C2×C23#190411yescharacteristic
207C2×C23#190711yescharacteristic
208C2×#41—21no—
209C22×D4C2×C4×C2311yescharacteristic
210C23×C4—21no—
211C2×D8—22no—
212C2×M16—22no—
213C2×M16—21no—
214C22×C8—22no—
215C4⋊Q8D4×C2311yescharacteristic
216#127D4×C2311yescharacteristic
217#124—21no—
218#121—21no—
219#121C2×C4×C2312yes—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4D4×C2312yes—
226C2×C4⋊C4C2×C4×C2311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C42C23×C2311yescharacteristic
229#106—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#102—22no—
237#102—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8C2×C4×C2312yes—
243C22⋊C8D4×C2312yes—
244#97—21no—
245#97—21no—
246#97—21no—
247C4×C8D4×C2311yescharacteristic
248C4×C8D4×C2311yescharacteristic
249C23#952911yescharacteristic, direct product factor, Sylow
250C24—21no—
251#41—82no—
252C2×Q8—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—42no—
259C2×D4C22⋊C4×C2312yes—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C22×C4×C2311yescharacteristic
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4#41×C2311yescharacteristic
269D8—44no—
270M16—82no—
271M16—81no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—82no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4C22⋊C4×C2312yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42C2×D4×C2311yescharacteristic
288C42C2×D4×C2311yescharacteristic
289C42C2×D4×C2311yescharacteristic
290C42#41×C2311yescharacteristic
291C23—22no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—81no—
296C23—81no—
297C23C4×D4×C2311yescharacteristic
298Q8—81no—
299D4—22no—
300D4—22no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—81no—
307D4—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C2×C22⋊C4×C2311yescharacteristic
321C2×C4C4×D4×C2311yescharacteristic
322C2×C4C23×#12011yescharacteristic
323C2×C4C23×#12111yescharacteristic
324C2×C4C23×#12111yescharacteristic
325C2×C4C23×#12311yescharacteristic
326C8—41no—
327C8—42no—
328C8—41no—
329C8—42no—
330C8—81no—
331C8—81no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22—81no—
342C22—81no—
343C22C23×#38611yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4C23×#47111yescharacteristic
352C4C23×#47211yescharacteristic
353C2—41no—
354C2—82no—
355C2—82no—
356C2C23×#189111yescharacteristic
357C2C23×#190411yescharacteristic
358C2C23×#190711yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3109C2
LCSG ▷ C22×C4 ▷ 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×#2981C2semidirect
3. C23×#1557C2semidirect
4. C23×#1486C2semidirect
5. C23×#1466C2semidirect
6. C23×#1307C2semidirect
7. C23×#1287C2semidirect
111. #9529C23direct (N × H)
154. #2981C2×C23semidirect
155. #1557C2×C23semidirect
156. #1486C2×C23semidirect
157. #1466C2×C23semidirect
158. #1307C2×C23semidirect
159. #1287C2×C23semidirect
249. C23#9529direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31092
C23×#29812
C23×#15572
C23×#14862
C23×#14662
C23×#13072
C23×#12872
#952923