Finite Group Databasebeta

← C23×#9047 #19156673

Subgroups — C23×#9047 #19156673

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3103C211yesmaximal, characteristic
2C23×#2932C211yesmaximal, characteristic
3C23×#1985C211yesmaximal, characteristic, complemented
4C23×#1540C211yesmaximal, characteristic
5C23×#1487C211yesmaximal, characteristic, complemented
6C23×#1301C211yesmaximal, characteristic, complemented
7C23×#1281C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C23×#532C2211yescharacteristic
10C2×C4×Q8×C23C2211yescharacteristic
11C23×#481C412yes—
12C23×#475—21no—
13C23×#460—21no—
14C23×#448C412yes—
15C4×SD16×C23—22no—
16C23×#435—22no—
17C23×#435—22no—
18C23×#431C2211yescharacteristic
19C23×#431C2211yescharacteristic
20C23×#423C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C23×#408—21no—
23C2×C23×#97C2211yescharacteristic
24C23×#336—21no—
25C23×#336—21no—
26C23×#332—21no—
27C23×#331—21no—
28C23×Q8⋊C8—22no—
29C2×#41×C23—21no—
30C22×Q8×C23C2×C411yescharacteristic
31C2×SD16×C23—22no—
32C23×#131—42no—
33C23×#131—42no—
34C2×M16×C23—22no—
35C2×M16×C23—21no—
36C22×C8×C23—22no—
37C22×C8×C23—21no—
38C22×C8×C23D411yescharacteristic
39C22×C8×C23D411yescharacteristic
40C23×C4⋊Q8D411yescharacteristic
41C23×#127D411yescharacteristic
42C23×#124—21no—
43C23×#121—21no—
44C23×#121C2×C412yes—
45C4×Q8×C23—21no—
46C4×Q8×C23—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23D412yes—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23C2×C411yescharacteristic
53C2×C42×C23—21no—
54C2×C42×C23C2311yescharacteristic
55C23×#106—22no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8D412yes—
62C23×#103—22no—
63C23×#102—22no—
64C23×C22⋊C8—24no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8C2×C412yes—
67C23×#97—21no—
68C23×#97—21no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71#41×C23—41no—
72#41×C23—41no—
73#41×C23—41no—
74#41×C23—41no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23—21no—
78C2×Q8×C23—42no—
79C2×D4×C23—22no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23C22⋊C412yes—
83C22×C4×C23—21no—
84C22×C4×C23—22no—
85C22×C4×C23—21no—
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23#4111yescharacteristic
89SD16×C23—44no—
90M16×C23—44no—
91M16×C23—41no—
92M16×C23—41no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—22no—
96C2×C8×C23—22no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—44no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23SD1612yescomplemented
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23C22⋊C412yes—
112C22⋊C4×C23—22no—
113C22⋊C4×C23—22no—
114C42×C23—41no—
115C42×C23—41no—
116C42×C23C2×D411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23#4111yescharacteristic
120#9047C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—22no—
122C23×C23C4×D411yescharacteristic
123Q8×C23—22no—
124Q8×C23—22no—
125Q8×C23—41no—
126Q8×C23—41no—
127Q8×C23—41no—
128Q8×C23—41no—
129D4×C23—42no—
130D4×C23—42no—
131D4×C23—42no—
132D4×C23—41no—
133D4×C23—41no—
134D4×C23—81no—
135C2×C4×C23—22no—
136C2×C4×C23—22no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—41no—
145C2×C4×C23C2×C22⋊C411yescharacteristic
146C2×C4×C23C4×D411yescharacteristic
147C2×C4×C23#12011yescharacteristic
148C2×C4×C23#12111yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12311yescharacteristic
151C2×C4×C23C2×SD1611yescharacteristic
152C2×C4×C23#13511yescharacteristic
153C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C23#13711yescharacteristic
155C8×C23—22no—
156C8×C23—42no—
157C8×C23—42no—
158C8×C23—41no—
159C8×C23—81no—
160C8×C23—81no—
161#3103C2×C2311yescharacteristic
162#2932C2×C2311yescharacteristic
163#1985C2×C2311yescharacteristic, complemented
164#1540C2×C2311yescharacteristic
165#1487C2×C2311yescharacteristic, complemented
166#1301C2×C2311yescharacteristic, complemented
167#1281C2×C2311yescharacteristic, complemented
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—42no—
171C22×C23—42no—
172C22×C23—42no—
173C22×C23#38611yescharacteristic
174C22×C23C4×SD1611yescharacteristic
175C22×C23#44011yescharacteristic
176C4×C23—21no—
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—21no—
180C4×C23—42no—
181C4×C23—42no—
182C4×C23—81no—
183C4×C23#45011yescharacteristic
184C4×C23#45211yescharacteristic
185C4×C23#46111yescharacteristic
186C4×C23#46311yescharacteristic
187C2×#131—21no—
188#532C22×C2311yescharacteristic
189C2×C4×Q8C22×C2311yescharacteristic
190#481C4×C2312yes—
191#475—21no—
192#460—21no—
193#448C4×C2312yes—
194C4×SD16—22no—
195#435—22no—
196#435—22no—
197#431C22×C2311yescharacteristic
198#431C22×C2311yescharacteristic
199#423C22×C2311yescharacteristic
200C2×C4⋊C8C22×C2311yescharacteristic
201#408—21no—
202C2×#97C22×C2311yescharacteristic
203#336—21no—
204#336—21no—
205#332—21no—
206#331—21no—
207Q8⋊C8—22no—
208C2×C23—21no—
209C2×C23—21no—
210C2×C23—82no—
211C2×C23#187711yescharacteristic
212C2×C23#188111yescharacteristic
213C2×C23#189111yescharacteristic
214C2×#41—21no—
215C22×Q8C2×C4×C2311yescharacteristic
216C2×SD16—22no—
217#131—42no—
218#131—42no—
219C2×M16—22no—
220C2×M16—21no—
221C22×C8—22no—
222C22×C8—21no—
223C22×C8D4×C2311yescharacteristic
224C22×C8D4×C2311yescharacteristic
225C4⋊Q8D4×C2311yescharacteristic
226#127D4×C2311yescharacteristic
227#124—21no—
228#121—21no—
229#121C2×C4×C2312yes—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×D4—21no—
235C4×D4D4×C2312yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C2×C4×C2311yescharacteristic
238C2×C42—21no—
239C2×C42C23×C2311yescharacteristic
240#106—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C2312yes—
247#103—22no—
248#102—22no—
249C22⋊C8—24no—
250C22⋊C8—21no—
251C22⋊C8C2×C4×C2312yes—
252#97—21no—
253#97—21no—
254C4×C8—21no—
255C4×C8—21no—
256C23#904711yescharacteristic, direct product factor, Sylow
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C22⋊C4×C2312yes—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4C22×C4×C2311yescharacteristic
273C22×C4C2×D4×C2311yescharacteristic
274C22×C4#41×C2311yescharacteristic
275SD16—44no—
276M16—44no—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8SD16×C2312yescomplemented
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4C22⋊C4×C2312yes—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C42—41no—
301C42—41no—
302C42C2×D4×C2311yescharacteristic
303C42C2×D4×C2311yescharacteristic
304C42C2×D4×C2311yescharacteristic
305C42#41×C2311yescharacteristic
306C23—22no—
307C23C4×D4×C2311yescharacteristic
308Q8—22no—
309Q8—22no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4C2×C22⋊C4×C2311yescharacteristic
331C2×C4C4×D4×C2311yescharacteristic
332C2×C4C23×#12011yescharacteristic
333C2×C4C23×#12111yescharacteristic
334C2×C4C23×#12111yescharacteristic
335C2×C4C23×#12311yescharacteristic
336C2×C4C2×SD16×C2311yescharacteristic
337C2×C4C23×#13511yescharacteristic
338C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
339C2×C4C23×#13711yescharacteristic
340C8—22no—
341C8—42no—
342C8—42no—
343C8—41no—
344C8—81no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22C23×#38611yescharacteristic
352C22C4×SD16×C2311yescharacteristic
353C22C23×#44011yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4C23×#45011yescharacteristic
362C4C23×#45211yescharacteristic
363C4C23×#46111yescharacteristic
364C4C23×#46311yescharacteristic
365C2—21no—
366C2—21no—
367C2—82no—
368C2C23×#187711yescharacteristic
369C2C23×#188111yescharacteristic
370C2C23×#189111yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C23×#1985C2semidirect
5. C23×#1487C2semidirect
6. C23×#1301C2semidirect
7. C23×#1281C2semidirect
105. C2×C8×C23SD16semidirect
120. #9047C23direct (N × H)
163. #1985C2×C23semidirect
165. #1487C2×C23semidirect
166. #1301C2×C23semidirect
167. #1281C2×C23semidirect
256. C23#9047direct (N × H)
291. C2×C8SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31032
C23×#29322
C23×#19852
C23×#15402
C23×#14872
C23×#13012
C23×#12812
#904723