Finite Group Databasebeta

← C23×#19403 #19167029

Subgroups — C23×#19403 #19167029

Order 5888.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3227C211yesmaximal, characteristic
2C2×C23×#477C211yesmaximal, characteristic, complemented
3C23×#3037C211yesmaximal, characteristic
4C23×#2982C211yesmaximal, characteristic, complemented
5C23×#1678C214yesmaximal, complemented
6C23×#1672C214yesmaximal, complemented
7C23×#1570C211yesmaximal, characteristic, complemented
8C23×#1548C211yesmaximal, characteristic, complemented
9C23×#1540C211yesmaximal, characteristic
10C2×C23×#137—21no—
11C23×#532C2211yescharacteristic
12C2×C23×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8×C23C2211yescharacteristic
14C23×#498C2212yes—
15C23×#486C2212yes—
16C23×#481—22no—
17C23×#477—22no—
18C23×#477C2214yes—
19C23×#476—22no—
20C23×#464—22no—
21C23×#462C2212yes—
22C23×#460C2212yes—
23C8×Q8×C23—22no—
24C8×Q8×C23C2214yes—
25C23×#431—22no—
26C23×#431C2211yescharacteristic
27C23×#431C2211yescharacteristic
28C2×C23×#107—21no—
29C23×#423C2211yescharacteristic
30C2×C23×C4⋊C8C2211yescharacteristic
31C23×#417—21no—
32C2×C23×#103—21no—
33C23×#334C2214yes—
34C23×#333C2214yes—
35C23×#332C2212yes—
36C23×#331C2212yes—
37C2×#41×C23—21no—
38C22×Q8×C23—21no—
39C22×Q8×C23D411yescharacteristic
40C23×#137—44no—
41C2×Q16×C23—22no—
42C2×SD16×C23—22no—
43C2×M16×C23—21no—
44C22×C8×C23—21no—
45C23×C4⋊Q8—21no—
46C23×C4⋊Q8—21no—
47C23×C4⋊Q8—21no—
48C23×C4⋊Q8C2311yescharacteristic
49C23×C4⋊Q8C2312yes—
50C23×#127C2311yescharacteristic
51C23×#124—21no—
52C23×#121—21no—
53C23×#121—21no—
54C4×Q8×C23—21no—
55C4×Q8×C23—21no—
56C4×Q8×C23D412yes—
57C4×Q8×C23C2312yes—
58C4×D4×C23—21no—
59C4×D4×C23—21no—
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23—21no—
62C2×C4⋊C4×C23—21no—
63C2×C4⋊C4×C23D411yescharacteristic
64C2×C42×C23—21no—
65C2×C42×C23C2311yescharacteristic
66C23×#107—24no—
67C23×#106—24no—
68C23×C4⋊C8—24no—
69C23×C4⋊C8—21no—
70C23×C4⋊C8—21no—
71C23×C4⋊C8D412yes—
72C23×C4⋊C8C2312yes—
73C23×C4⋊C8C2312yes—
74C23×C4⋊C8C2312yes—
75C23×#103—24no—
76C23×#103—22no—
77C23×#102—24no—
78C23×#102—22no—
79C23×C22⋊C8—21no—
80C23×C22⋊C8D412yes—
81C4×C8×C23—24no—
82C4×C8×C23—21no—
83C4×C8×C23C2312yes—
84#41×C23—42no—
85#41×C23—42no—
86C2×Q8×C23—21no—
87C2×Q8×C23—22no—
88C2×Q8×C23—21no—
89C2×Q8×C23—22no—
90C2×Q8×C23—41no—
91C2×Q8×C23—82no—
92C2×Q8×C23SD1614yes—
93C2×Q8×C23C2×D412yes—
94C2×D4×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—21no—
97C2×D4×C23—41no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23#4111yescharacteristic
105Q16×C23—44no—
106SD16×C23—44no—
107M16×C23—42no—
108C2×C8×C23—22no—
109C2×C8×C23—22no—
110C2×C8×C23—22no—
111C2×C8×C23—42no—
112C2×C8×C23C2×D414yes—
113C4⋊C4×C23—22no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—22no—
116C4⋊C4×C23—22no—
117C4⋊C4×C23—22no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23C2×D412yes—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C42×C23—22no—
126C42×C23—22no—
127C42×C23C2×D411yescharacteristic
128C42×C23C2×D411yescharacteristic
129C42×C23#4111yescharacteristic
130C42×C23C2411yescharacteristic
131#19403C2311yesmaximal, characteristic, direct product factor, Sylow
132C23×C23—41no—
133C23×C23#12111yescharacteristic
134Q8×C23—22no—
135Q8×C23—24no—
136Q8×C23—41no—
137Q8×C23—41no—
138Q8×C23—82no—
139Q8×C23—82no—
140Q8×C23C2×SD1614yes—
141D4×C23—41no—
142D4×C23—41no—
143D4×C23—42no—
144D4×C23—81no—
145D4×C23—82no—
146C2×C4×C23—22no—
147C2×C4×C23—24no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—42no—
153C2×C4×C23—81no—
154C2×C4×C23—82no—
155C2×C4×C23—82no—
156C2×C4×C23#12111yescharacteristic
157C2×C4×C23#12111yescharacteristic
158C2×C4×C23#12111yescharacteristic
159C2×C4×C23C2×SD1612yes—
160C2×C4×C23#13512yes—
161C2×C4×C23C22×D411yescharacteristic
162C2×C4×C23C22×D411yescharacteristic
163C2×C4×C23C2×#4111yescharacteristic
164C8×C23—24no—
165C8×C23—42no—
166C8×C23—42no—
167C8×C23—42no—
168#3227C2×C2311yescharacteristic
169C2×#477C2×C2311yescharacteristic, complemented
170#3037C2×C2311yescharacteristic
171#2982C2×C2311yescharacteristic, complemented
172#1678C2×C2314yescomplemented
173#1672C2×C2314yescomplemented
174#1570C2×C2311yescharacteristic, complemented
175#1548C2×C2311yescharacteristic, complemented
176#1540C2×C2311yescharacteristic
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—81no—
180C22×C23—81no—
181C22×C23—81no—
182C22×C23#46512yes—
183C22×C23C2×#12111yescharacteristic
184C4×C23—21no—
185C4×C23—24no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—42no—
189C4×C23—81no—
190C4×C23—82no—
191C4×C23#46512yes—
192C4×C23C22×SD1611yescharacteristic
193C4×C23C2×#13511yescharacteristic
194C4×C23#57812yes—
195C2×#137—21no—
196#532C22×C2311yescharacteristic
197C2×C4⋊Q8C22×C2311yescharacteristic
198C2×C4×Q8C22×C2311yescharacteristic
199#498C22×C2312yes—
200#486C22×C2312yes—
201#481—22no—
202#477—22no—
203#477C22×C2314yes—
204#476—22no—
205#464—22no—
206#462C22×C2312yes—
207#460C22×C2312yes—
208C8×Q8—22no—
209C8×Q8C22×C2314yes—
210#431—22no—
211#431C22×C2311yescharacteristic
212#431C22×C2311yescharacteristic
213C2×#107—21no—
214#423C22×C2311yescharacteristic
215C2×C4⋊C8C22×C2311yescharacteristic
216#417—21no—
217C2×#103—21no—
218#334C22×C2314yes—
219#333C22×C2314yes—
220#332C22×C2312yes—
221#331C22×C2312yes—
222C2×C23—22no—
223C2×C23—161no—
224C2×C23#304411yescharacteristic
225C2×C23C2×#46511yescharacteristic
226C2×C23#306811yescharacteristic
227C2×#41—21no—
228C22×Q8—21no—
229C22×Q8D4×C2311yescharacteristic
230#137—44no—
231C2×Q16—22no—
232C2×SD16—22no—
233C2×M16—21no—
234C22×C8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C23×C2312yes—
239C4⋊Q8C23×C2311yescharacteristic
240#127C23×C2311yescharacteristic
241#124—21no—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8D4×C2312yes—
247C4×Q8C23×C2312yes—
248C4×D4—21no—
249C4×D4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4D4×C2311yescharacteristic
254C2×C42—21no—
255C2×C42C23×C2311yescharacteristic
256#107—24no—
257#106—24no—
258C4⋊C8—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8D4×C2312yes—
262C4⋊C8C23×C2312yes—
263C4⋊C8C23×C2312yes—
264C4⋊C8C23×C2312yes—
265#103—24no—
266#103—22no—
267#102—24no—
268#102—22no—
269C22⋊C8—21no—
270C22⋊C8D4×C2312yes—
271C4×C8—24no—
272C4×C8—21no—
273C4×C8C23×C2312yes—
274C23#1940311yescharacteristic, direct product factor, Sylow
275#41—42no—
276#41—42no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8SD16×C2314yes—
284C2×Q8C2×D4×C2312yes—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C2×D4×C2311yescharacteristic
294C22×C4C2×D4×C2311yescharacteristic
295C22×C4#41×C2311yescharacteristic
296Q16—44no—
297SD16—44no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8C2×D4×C2314yes—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4C2×D4×C2312yes—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—22no—
317C42—22no—
318C42C2×D4×C2311yescharacteristic
319C42C2×D4×C2311yescharacteristic
320C42#41×C2311yescharacteristic
321C42C24×C2311yescharacteristic
322C23—41no—
323C23C23×#12111yescharacteristic
324Q8—22no—
325Q8—24no—
326Q8—41no—
327Q8—41no—
328Q8—82no—
329Q8—82no—
330Q8C2×SD16×C2314yes—
331D4—41no—
332D4—41no—
333D4—42no—
334D4—81no—
335D4—82no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4C23×#12111yescharacteristic
347C2×C4C23×#12111yescharacteristic
348C2×C4C23×#12111yescharacteristic
349C2×C4C2×SD16×C2312yes—
350C2×C4C23×#13512yes—
351C2×C4C22×D4×C2311yescharacteristic
352C2×C4C22×D4×C2311yescharacteristic
353C2×C4C2×#41×C2311yescharacteristic
354C8—24no—
355C8—42no—
356C8—42no—
357C8—42no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C23×#46512yes—
364C22C2×C23×#12111yescharacteristic
365C4—21no—
366C4—24no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—82no—
372C4C23×#46512yes—
373C4C22×SD16×C2311yescharacteristic
374C4C2×C23×#13511yescharacteristic
375C4C23×#57812yes—
376C2—22no—
377C2—161no—
378C2C23×#304411yescharacteristic
379C2C2×C23×#46511yescharacteristic
380C2C23×#306811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#121
commutator, Frattini, cosocleC42C24×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. C2×C23×#477C2semidirect
4. C23×#2982C2semidirect
5. C23×#1678C2semidirect
6. C23×#1672C2semidirect
7. C23×#1570C2semidirect
8. C23×#1548C2semidirect
131. #19403C23direct (N × H)
169. C2×#477C2×C23semidirect
171. #2982C2×C23semidirect
172. #1678C2×C23semidirect
173. #1672C2×C23semidirect
174. #1570C2×C23semidirect
175. #1548C2×C23semidirect
274. C23#19403direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32272
C2×C23×#4772
C23×#30372
C23×#29822
C23×#16782
C23×#16722
C23×#15702
C23×#15482
C23×#15402
#1940323