Finite Group Databasebeta

← C23×#9301 #19156927

Subgroups — C23×#9301 #19156927

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3206C211yesmaximal, characteristic
2C23×#2331C211yesmaximal, characteristic, complemented
3C23×#1963C211yesmaximal, characteristic
4C23×#1548C211yesmaximal, characteristic, complemented
5C23×#1517C211yesmaximal, characteristic
6C23×#1286C211yesmaximal, characteristic, complemented
7C23×#1274C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C23×#492C412yescomplemented
11C23×#487C412yescomplemented
12C23×#461—22no—
13C23×#448—22no—
14C23×#431C2211yescharacteristic
15C23×#431C2211yescharacteristic
16C23×#423C2211yescharacteristic
17C2×C23×#104—21no—
18C2×C23×#104—21no—
19C23×#391—22no—
20C23×#389—21no—
21C23×#387—22no—
22C23×#382C2211yescharacteristic
23C4×C4⋊C4×C23—21no—
24C23×#376C2211yescharacteristic
25C23×#333—22no—
26C23×#327—22no—
27C2×#41×C23—21no—
28C22×Q8×C23D412yes—
29C2×SD16×C23—24no—
30C2×M16×C23—21no—
31C23×C4⋊Q8—21no—
32C23×C4⋊Q8—21no—
33C23×C4⋊Q8C2×C411yescharacteristic
34C23×C4⋊Q8C2×C412yes—
35C23×C4⋊Q8D412yes—
36C23×#127C2×C411yescharacteristic
37C23×#124—21no—
38C23×#121—21no—
39C23×#121—21no—
40C4×D4×C23—21no—
41C4×D4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—22no—
46C2×C42×C23—22no—
47C2×C42×C23D411yescharacteristic
48C2×C42×C23D411yescharacteristic
49C2×C42×C23C2311yescharacteristic
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×#104—42no—
55C23×#104—42no—
56C23×#103—24no—
57C23×C22⋊C8—21no—
58C23×C22⋊C8—21no—
59C4×C8×C23—21no—
60C4×C8×C23C2×C412yes—
61C23×#95—22no—
62C23×#95—22no—
63C23×#95—21no—
64C23×#95—21no—
65C23×#95—21no—
66C23×#95D412yes—
67#41×C23—41no—
68#41×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71C2×Q8×C23—22no—
72C2×Q8×C23—21no—
73C2×Q8×C23—82no—
74C2×Q8×C23C22⋊C414yes—
75C2×D4×C23—21no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—41no—
79C22×C4×C23—21no—
80C22×C4×C23—22no—
81C22×C4×C23—22no—
82C22×C4×C23—21no—
83C22×C4×C23—41no—
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87SD16×C23—84no—
88M16×C23—41no—
89M16×C23—41no—
90C2×C8×C23—22no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—22no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—82no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—41no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23—21no—
107C42×C23—82no—
108C42×C23SD1612yescomplemented
109C42×C23C22×C411yescharacteristic
110C42×C23C2×D411yescharacteristic
111C42×C23#4111yescharacteristic
112C42×C23#4111yescharacteristic
113#9301C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—41no—
115C23×C23#12011yescharacteristic
116Q8×C23—44no—
117Q8×C23—41no—
118Q8×C23—41no—
119Q8×C23—82no—
120D4×C23—42no—
121D4×C23—41no—
122D4×C23—41no—
123D4×C23—81no—
124D4×C23—81no—
125D4×C23—81no—
126C2×C4×C23—24no—
127C2×C4×C23—22no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—42no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—81no—
142C2×C4×C23—81no—
143C2×C4×C23C2×C22⋊C411yescharacteristic
144C2×C4×C23C4×D411yescharacteristic
145C2×C4×C23C4×D411yescharacteristic
146C2×C4×C23#12111yescharacteristic
147C2×C4×C23#12111yescharacteristic
148C2×C4×C23#12311yescharacteristic
149C2×C4×C23C2×SD1611yescharacteristic
150C2×C4×C23#13511yescharacteristic
151C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C23#13711yescharacteristic
153C8×C23—42no—
154C8×C23—81no—
155C8×C23—81no—
156#3206C2×C2311yescharacteristic
157#2331C2×C2311yescharacteristic, complemented
158#1963C2×C2311yescharacteristic
159#1548C2×C2311yescharacteristic, complemented
160#1517C2×C2311yescharacteristic
161#1286C2×C2311yescharacteristic, complemented
162#1274C2×C2311yescharacteristic, complemented
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#38611yescharacteristic
169C22×C23#44811yescharacteristic
170C22×C23#44911yescharacteristic
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—21no—
175C4×C23—42no—
176C4×C23—44no—
177C4×C23—82no—
178C4×C23—81no—
179C4×C23—81no—
180C4×C23C4×SD1611yescharacteristic
181C4×C23#44011yescharacteristic
182C4×C23#46011yescharacteristic
183C4×C23#46411yescharacteristic
184#532C22×C2311yescharacteristic
185C2×C4⋊Q8C22×C2311yescharacteristic
186#492C4×C2312yescomplemented
187#487C4×C2312yescomplemented
188#461—22no—
189#448—22no—
190#431C22×C2311yescharacteristic
191#431C22×C2311yescharacteristic
192#423C22×C2311yescharacteristic
193C2×#104—21no—
194C2×#104—21no—
195#391—22no—
196#389—21no—
197#387—22no—
198#382C22×C2311yescharacteristic
199C4×C4⋊C4—21no—
200#376C22×C2311yescharacteristic
201#333—22no—
202#327—22no—
203C2×C23—21no—
204C2×C23—21no—
205C2×C23—161no—
206C2×C23#188011yescharacteristic
207C2×C23#190511yescharacteristic
208C2×C23#191111yescharacteristic
209C2×#41—21no—
210C22×Q8D4×C2312yes—
211C2×SD16—24no—
212C2×M16—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C2×C4×C2311yescharacteristic
216C4⋊Q8C2×C4×C2312yes—
217C4⋊Q8D4×C2312yes—
218#127C2×C4×C2311yescharacteristic
219#124—21no—
220#121—21no—
221#121—21no—
222C4×D4—21no—
223C4×D4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C42—22no—
229C2×C42D4×C2311yescharacteristic
230C2×C42D4×C2311yescharacteristic
231C2×C42C23×C2311yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#104—42no—
237#104—42no—
238#103—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C4×C8—21no—
242C4×C8C2×C4×C2312yes—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95D4×C2312yes—
249C23#930111yescharacteristic, direct product factor, Sylow
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—82no—
257C2×Q8C22⋊C4×C2314yes—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4C2×D4×C2311yescharacteristic
268C22×C4C2×D4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270SD16—84no—
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—82no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42SD16×C2312yescomplemented
292C42C22×C4×C2311yescharacteristic
293C42C2×D4×C2311yescharacteristic
294C42#41×C2311yescharacteristic
295C42#41×C2311yescharacteristic
296C23—41no—
297C23C23×#12011yescharacteristic
298Q8—44no—
299Q8—41no—
300Q8—41no—
301Q8—82no—
302D4—42no—
303D4—41no—
304D4—41no—
305D4—81no—
306D4—81no—
307D4—81no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4C2×C22⋊C4×C2311yescharacteristic
326C2×C4C4×D4×C2311yescharacteristic
327C2×C4C4×D4×C2311yescharacteristic
328C2×C4C23×#12111yescharacteristic
329C2×C4C23×#12111yescharacteristic
330C2×C4C23×#12311yescharacteristic
331C2×C4C2×SD16×C2311yescharacteristic
332C2×C4C23×#13511yescharacteristic
333C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C23×#13711yescharacteristic
335C8—42no—
336C8—81no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C23×#38611yescharacteristic
344C22C23×#44811yescharacteristic
345C22C23×#44911yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4C4×SD16×C2311yescharacteristic
356C4C23×#44011yescharacteristic
357C4C23×#46011yescharacteristic
358C4C23×#46411yescharacteristic
359C2—21no—
360C2—21no—
361C2—161no—
362C2C23×#188011yescharacteristic
363C2C23×#190511yescharacteristic
364C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
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×#2331C2semidirect
4. C23×#1548C2semidirect
6. C23×#1286C2semidirect
7. C23×#1274C2semidirect
10. C23×#492C4semidirect
11. C23×#487C4semidirect
108. C42×C23SD16semidirect
113. #9301C23direct (N × H)
157. #2331C2×C23semidirect
159. #1548C2×C23semidirect
161. #1286C2×C23semidirect
162. #1274C2×C23semidirect
186. #492C4×C23semidirect
187. #487C4×C23semidirect
249. C23#9301direct (N × H)
291. C42SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32062
C23×#23312
C23×#19632
C23×#15482
C23×#15172
C23×#12862
C23×#12742
#930123