Finite Group Databasebeta

← C23×#19393 #19167019

Subgroups — C23×#19393 #19167019

Order 5888.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3230C211yesmaximal, characteristic, complemented
2C23×#3077C211yesmaximal, characteristic
3C2×C23×#476C211yesmaximal, characteristic
4C23×#2972C211yesmaximal, characteristic
5C23×#1683C214yesmaximal, complemented
6C23×#1675C214yesmaximal, complemented
7C23×#1571C211yesmaximal, characteristic, complemented
8C23×#1548C211yesmaximal, characteristic, complemented
9C23×#1542C211yesmaximal, characteristic, complemented
10C23×#533C2211yescharacteristic
11C2×C23×C4⋊Q8C2211yescharacteristic
12C2×C4×D4×C23C2211yescharacteristic
13C23×#500C2212yescomplemented
14C23×#487C2212yescomplemented
15C23×#484—22no—
16C23×#478—22no—
17C23×#477—22no—
18C23×#476—22no—
19C23×#476C2212yes—
20C23×#476C2214yes—
21C23×#474C2212yes—
22C8×D4×C23—22no—
23C8×D4×C23C2214yes—
24C23×#431C2211yescharacteristic
25C23×#431C2211yescharacteristic
26C23×#428—21no—
27C2×C23×#106—21no—
28C23×#423C2211yescharacteristic
29C2×C23×C4⋊C8C2211yescharacteristic
30C23×#418—21no—
31C2×C23×#102—21no—
32C2×C23×C22⋊C8—22no—
33C23×#335C2214yescomplemented
34C23×#333C2212yescomplemented
35C23×#333C2214yescomplemented
36C23×#332C2212yescomplemented
37C22×Q8×C23—21no—
38C22×D4×C23D411yescharacteristic
39C23×C4×C23—21no—
40C2×M16×C23—21no—
41C22×C8×C23—21no—
42C22×C8×C23—24no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8—21no—
46C23×C4⋊Q8C2311yescharacteristic
47C23×C4⋊Q8C2312yes—
48C23×C4⋊Q8C2311yescharacteristic
49C23×#125—21no—
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8—21no—
52C4×Q8×C23—21no—
53C4×Q8×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23D412yes—
57C4×D4×C23C2312yes—
58C23×#117—21no—
59C2×C4⋊C4×C23—21no—
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23D411yescharacteristic
62C2×C22⋊C4×C23—21no—
63C2×C42×C23C2311yescharacteristic
64C23×#107—24no—
65C23×#107—22no—
66C23×#106—24no—
67C23×#106—22no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8—21no—
70C23×C4⋊C8D412yes—
71C23×C4⋊C8C2312yes—
72C23×C4⋊C8C2312yes—
73C23×C4⋊C8C2312yes—
74C23×#103—24no—
75C23×#102—24no—
76C23×#102—22no—
77C23×#102—22no—
78C23×C22⋊C8—21no—
79C23×C22⋊C8—24no—
80C23×C22⋊C8D412yes—
81C4×C8×C23—21no—
82C4×C8×C23C2312yes—
83C24×C23—21no—
84C2×Q8×C23—21no—
85C2×Q8×C23—21no—
86C2×Q8×C23—22no—
87C2×Q8×C23—41no—
88C2×Q8×C23—82no—
89C2×D4×C23—21no—
90C2×D4×C23—22no—
91C2×D4×C23Q1614yes—
92C2×D4×C23C2×D412yes—
93C22×C4×C23—21no—
94C22×C4×C23—22no—
95C22×C4×C23—22no—
96C22×C4×C23—41no—
97C22×C4×C23—82no—
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23#4111yescharacteristic
101M16×C23—42no—
102C2×C8×C23—22no—
103C2×C8×C23—22no—
104C2×C8×C23—22no—
105C2×C8×C23—42no—
106C2×C8×C23—84no—
107C2×C8×C23C2×D414yes—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23C2×D412yes—
120C22⋊C4×C23—22no—
121C22⋊C4×C23—22no—
122C22⋊C4×C23—41no—
123C42×C23—41no—
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126C42×C23#4111yescharacteristic
127C42×C23C2411yescharacteristic
128#19393C2311yesmaximal, characteristic, direct product factor, Sylow
129C23×C23—22no—
130C23×C23—24no—
131C23×C23—81no—
132C23×C23—81no—
133C23×C23#12111yescharacteristic
134Q8×C23—42no—
135Q8×C23—81no—
136Q8×C23—82no—
137Q8×C23—82no—
138D4×C23—22no—
139D4×C23—24no—
140D4×C23C2×Q1614yes—
141C2×C4×C23—22no—
142C2×C4×C23—22no—
143C2×C4×C23—41no—
144C2×C4×C23—44no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—42no—
148C2×C4×C23—82no—
149C2×C4×C23—81no—
150C2×C4×C23—82no—
151C2×C4×C23#12111yescharacteristic
152C2×C4×C23#12111yescharacteristic
153C2×C4×C23#12111yescharacteristic
154C2×C4×C23C2×Q1612yes—
155C2×C4×C23#13512yes—
156C2×C4×C23C22×D411yescharacteristic
157C2×C4×C23C22×D411yescharacteristic
158C2×C4×C23C2×#4111yescharacteristic
159C8×C23—24no—
160C8×C23—42no—
161C8×C23—42no—
162C8×C23—42no—
163#3230C2×C2311yescharacteristic, complemented
164#3077C2×C2311yescharacteristic
165C2×#476C2×C2311yescharacteristic
166#2972C2×C2311yescharacteristic
167#1683C2×C2314yescomplemented
168#1675C2×C2314yescomplemented
169#1571C2×C2311yescharacteristic, complemented
170#1548C2×C2311yescharacteristic, complemented
171#1542C2×C2311yescharacteristic, complemented
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—24no—
175C22×C23—42no—
176C22×C23—42no—
177C22×C23—84no—
178C22×C23#46712yes—
179C22×C23C2×#12111yescharacteristic
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—42no—
184C4×C23—81no—
185C4×C23—82no—
186C4×C23—81no—
187C4×C23#46712yes—
188C4×C23C22×Q1611yescharacteristic
189C4×C23C2×#13511yescharacteristic
190C4×C23#57712yes—
191#533C22×C2311yescharacteristic
192C2×C4⋊Q8C22×C2311yescharacteristic
193C2×C4×D4C22×C2311yescharacteristic
194#500C22×C2312yescomplemented
195#487C22×C2312yescomplemented
196#484—22no—
197#478—22no—
198#477—22no—
199#476—22no—
200#476C22×C2312yes—
201#476C22×C2314yes—
202#474C22×C2312yes—
203C8×D4—22no—
204C8×D4C22×C2314yes—
205#431C22×C2311yescharacteristic
206#431C22×C2311yescharacteristic
207#428—21no—
208C2×#106—21no—
209#423C22×C2311yescharacteristic
210C2×C4⋊C8C22×C2311yescharacteristic
211#418—21no—
212C2×#102—21no—
213C2×C22⋊C8—22no—
214#335C22×C2314yescomplemented
215#333C22×C2312yescomplemented
216#333C22×C2314yescomplemented
217#332C22×C2312yescomplemented
218C2×C23—22no—
219C2×C23—44no—
220C2×C23#304211yescharacteristic
221C2×C23C2×#46711yescharacteristic
222C2×C23#306811yescharacteristic
223C22×Q8—21no—
224C22×D4D4×C2311yescharacteristic
225C23×C4—21no—
226C2×M16—21no—
227C22×C8—21no—
228C22×C8—24no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8C23×C2312yes—
233C4⋊Q8C23×C2311yescharacteristic
234C4⋊Q8C23×C2311yescharacteristic
235#125—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4D4×C2312yes—
243C4×D4C23×C2312yes—
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4D4×C2311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42C23×C2311yescharacteristic
250#107—24no—
251#107—22no—
252#106—24no—
253#106—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8D4×C2312yes—
257C4⋊C8C23×C2312yes—
258C4⋊C8C23×C2312yes—
259C4⋊C8C23×C2312yes—
260#103—24no—
261#102—24no—
262#102—22no—
263#102—22no—
264C22⋊C8—21no—
265C22⋊C8—24no—
266C22⋊C8D4×C2312yes—
267C4×C8—21no—
268C4×C8C23×C2312yes—
269C23#1939311yescharacteristic, direct product factor, Sylow
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4Q16×C2314yes—
279C2×D4C2×D4×C2312yes—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4C2×D4×C2311yescharacteristic
286C22×C4C2×D4×C2311yescharacteristic
287C22×C4#41×C2311yescharacteristic
288M16—42no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—84no—
294C2×C8C2×D4×C2314yes—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4C2×D4×C2312yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C42—41no—
311C42C2×D4×C2311yescharacteristic
312C42C2×D4×C2311yescharacteristic
313C42#41×C2311yescharacteristic
314C42C24×C2311yescharacteristic
315C23—22no—
316C23—24no—
317C23—81no—
318C23—81no—
319C23C23×#12111yescharacteristic
320Q8—42no—
321Q8—82no—
322Q8—81no—
323Q8—82no—
324D4—22no—
325D4—24no—
326D4C2×Q16×C2314yes—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4C23×#12111yescharacteristic
338C2×C4C23×#12111yescharacteristic
339C2×C4C23×#12111yescharacteristic
340C2×C4C2×Q16×C2312yes—
341C2×C4C23×#13512yes—
342C2×C4C22×D4×C2311yescharacteristic
343C2×C4C22×D4×C2311yescharacteristic
344C2×C4C2×#41×C2311yescharacteristic
345C8—24no—
346C8—42no—
347C8—42no—
348C8—42no—
349C22—21no—
350C22—21no—
351C22—24no—
352C22—42no—
353C22—42no—
354C22—84no—
355C22C23×#46712yes—
356C22C2×C23×#12111yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—81no—
362C4—82no—
363C4—81no—
364C4C23×#46712yes—
365C4C22×Q16×C2311yescharacteristic
366C4C2×C23×#13511yescharacteristic
367C4C23×#57712yes—
368C2—22no—
369C2—44no—
370C2C23×#304211yescharacteristic
371C2C2×C23×#46711yescharacteristic
372C2C23×#306811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#121
commutator, Frattini, cosocleC42C24×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
1. C23×#3230C2semidirect
5. C23×#1683C2semidirect
6. C23×#1675C2semidirect
7. C23×#1571C2semidirect
8. C23×#1548C2semidirect
9. C23×#1542C2semidirect
13. C23×#500C22semidirect
14. C23×#487C22semidirect
33. C23×#335C22semidirect
34. C23×#333C22semidirect
35. C23×#333C22semidirect
36. C23×#332C22semidirect
128. #19393C23direct (N × H)
163. #3230C2×C23semidirect
167. #1683C2×C23semidirect
168. #1675C2×C23semidirect
169. #1571C2×C23semidirect
170. #1548C2×C23semidirect
171. #1542C2×C23semidirect
194. #500C22×C23semidirect
195. #487C22×C23semidirect
214. #335C22×C23semidirect
215. #333C22×C23semidirect
216. #333C22×C23semidirect
217. #332C22×C23semidirect
269. C23#19393direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32302
C23×#30772
C2×C23×#4762
C23×#29722
C23×#16832
C23×#16752
C23×#15712
C23×#15482
C23×#15422
#1939323