Finite Group Databasebeta

← C23×#10269 #19157895

Subgroups — C23×#10269 #19157895

Order 5888.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#486C211yesmaximal, characteristic
2C2×C23×#477C211yesmaximal, characteristic, complemented
3C2×C23×#460C211yesmaximal, characteristic
4C23×#1984C211yesmaximal, characteristic, complemented
5C2×C23×#333C211yesmaximal, characteristic, complemented
6C2×C23×#331C211yesmaximal, characteristic
7C23×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C23—21no—
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×#127C2211yescharacteristic
11C2×C4×Q8×C23C2211yescharacteristic
12C23×#486—22no—
13C23×#486—22no—
14C23×#477—22no—
15C23×#477—22no—
16C23×#460—22no—
17C23×#460—22no—
18C2×C23×#107—21no—
19C2×C23×C4⋊C8C2211yescharacteristic
20C2×C23×C4⋊C8C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C2×C23×#103—21no—
23C2×C23×#102—22no—
24C2×C23×#102—21no—
25C2×C4×C8×C23C2211yescharacteristic
26C23×#336—22no—
27C23×#333—22no—
28C23×#331—22no—
29C22×Q8×C23—21no—
30C22×Q8×C23D411yescharacteristic
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C2×SD16×C23—24no—
35C2×SD16×C23—42no—
36C2×SD16×C23—42no—
37C22×C8×C23—21no—
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C22×C8×C23D412yes—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8—21no—
44C23×C4⋊Q8D412yes—
45C23×#127—21no—
46C23×#127—21no—
47C23×#127—21no—
48C23×#127D412yes—
49C4×Q8×C23—21no—
50C4×Q8×C23—21no—
51C4×Q8×C23—21no—
52C4×Q8×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C42×C23—21no—
58C2×C42×C23C2311yescharacteristic
59C23×#107—24no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8D412yes—
65C23×C4⋊C8D412yes—
66C23×#103—24no—
67C23×#102—24no—
68C23×#102—28no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71C24×C23—41no—
72C2×Q8×C23—21no—
73C2×Q8×C23—21no—
74C2×Q8×C23—21no—
75C2×Q8×C23—22no—
76C2×Q8×C23—21no—
77C2×Q8×C23—21no—
78C2×Q8×C23—41no—
79C2×Q8×C23—82no—
80C2×Q8×C23SD1612yes—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23—22no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—42no—
90C2×D4×C23—41no—
91C2×D4×C23—41no—
92C2×D4×C23—42no—
93C2×D4×C23—82no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—41no—
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23#4111yescharacteristic
100SD16×C23—48no—
101C2×C8×C23—22no—
102C2×C8×C23—22no—
103C2×C8×C23—22no—
104C2×C8×C23—22no—
105C2×C8×C23—22no—
106C2×C8×C23—41no—
107C2×C8×C23—41no—
108C2×C8×C23—41no—
109C2×C8×C23—41no—
110C2×C8×C23—41no—
111C2×C8×C23—42no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—22no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23—41no—
120C42×C23—41no—
121C42×C23—41no—
122C42×C23C2×D411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126#10269C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—42no—
128C23×C23—82no—
129C23×C23—81no—
130C23×C23—81no—
131C23×C23—82no—
132C23×C23#12111yescharacteristic
133Q8×C23—22no—
134Q8×C23—22no—
135Q8×C23—41no—
136Q8×C23—41no—
137Q8×C23—82no—
138Q8×C23—82no—
139D4×C23—42no—
140D4×C23—42no—
141D4×C23—44no—
142D4×C23—82no—
143D4×C23—82no—
144D4×C23—81no—
145D4×C23—81no—
146D4×C23—81no—
147D4×C23—81no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—82no—
156C2×C4×C23#12011yescharacteristic
157C2×C4×C23#12011yescharacteristic
158C2×C4×C23#12011yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C2×C4×C23#12111yescharacteristic
161C2×C4×C23#12711yescharacteristic
162C2×C4×C23C2×SD1611yescharacteristic
163C2×C4×C23#13511yescharacteristic
164C2×C4×C23#13512yes—
165C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C23#13711yescharacteristic
167C2×C4×C23#13712yes—
168C8×C23—42no—
169C8×C23—42no—
170C8×C23—42no—
171C8×C23—81no—
172C2×#486C2×C2311yescharacteristic
173C2×#477C2×C2311yescharacteristic, complemented
174C2×#460C2×C2311yescharacteristic
175#1984C2×C2311yescharacteristic, complemented
176C2×#333C2×C2311yescharacteristic, complemented
177C2×#331C2×C2311yescharacteristic
178#1277C2×C2311yescharacteristic, complemented
179C22×C23—82no—
180C22×C23—82no—
181C22×C23—82no—
182C22×C23—162no—
183C22×C23—162no—
184C22×C23#39211yescharacteristic
185C22×C23#46412yes—
186C22×C23#46511yescharacteristic
187C22×C23#46811yescharacteristic
188C22×C23#47212yes—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—42no—
192C4×C23—81no—
193C4×C23—82no—
194C4×C23#44811yescharacteristic
195C4×C23#44911yescharacteristic
196C4×C23#45011yescharacteristic
197C4×C23#45211yescharacteristic
198C4×C23#45212yes—
199C4×C23#49511yescharacteristic
200C4×C23#49711yescharacteristic
201C22×SD16—21no—
202C2×C4⋊Q8C22×C2311yescharacteristic
203C2×#127C22×C2311yescharacteristic
204C2×C4×Q8C22×C2311yescharacteristic
205#486—22no—
206#486—22no—
207#477—22no—
208#477—22no—
209#460—22no—
210#460—22no—
211C2×#107—21no—
212C2×C4⋊C8C22×C2311yescharacteristic
213C2×C4⋊C8C22×C2311yescharacteristic
214C2×C4⋊C8C22×C2311yescharacteristic
215C2×#103—21no—
216C2×#102—22no—
217C2×#102—21no—
218C2×C4×C8C22×C2311yescharacteristic
219#336—22no—
220#333—22no—
221#331—22no—
222C2×C23—162no—
223C2×C23#167812yes—
224C2×C23#169812yes—
225C2×C23#200011yescharacteristic
226C2×C23#201511yescharacteristic
227C2×C23#201811yescharacteristic
228C22×Q8—21no—
229C22×Q8D4×C2311yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C2×SD16—24no—
234C2×SD16—42no—
235C2×SD16—42no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8D4×C2312yes—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8D4×C2312yes—
244#127—21no—
245#127—21no—
246#127—21no—
247#127D4×C2312yes—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4D4×C2311yescharacteristic
256C2×C42—21no—
257C2×C42C23×C2311yescharacteristic
258#107—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C2312yes—
264C4⋊C8D4×C2312yes—
265#103—24no—
266#102—24no—
267#102—28no—
268C4×C8—21no—
269C4×C8—21no—
270C23#1026911yescharacteristic, direct product factor, Sylow
271C24—41no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8SD16×C2312yes—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4C2×D4×C2311yescharacteristic
298C22×C4C2×D4×C2311yescharacteristic
299C22×C4#41×C2311yescharacteristic
300SD16—48no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C42—41no—
321C42—41no—
322C42C2×D4×C2311yescharacteristic
323C42C2×D4×C2311yescharacteristic
324C42C2×D4×C2311yescharacteristic
325C42C2×D4×C2311yescharacteristic
326C23—42no—
327C23—82no—
328C23—81no—
329C23—81no—
330C23—82no—
331C23C23×#12111yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—82no—
337Q8—82no—
338D4—42no—
339D4—42no—
340D4—44no—
341D4—82no—
342D4—82no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4C23×#12011yescharacteristic
356C2×C4C23×#12011yescharacteristic
357C2×C4C23×#12011yescharacteristic
358C2×C4C23×#12111yescharacteristic
359C2×C4C23×#12111yescharacteristic
360C2×C4C23×#12711yescharacteristic
361C2×C4C2×SD16×C2311yescharacteristic
362C2×C4C23×#13511yescharacteristic
363C2×C4C23×#13512yes—
364C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C23×#13711yescharacteristic
366C2×C4C23×#13712yes—
367C8—42no—
368C8—42no—
369C8—42no—
370C8—81no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—162no—
375C22—162no—
376C22C23×#39211yescharacteristic
377C22C23×#46412yes—
378C22C23×#46511yescharacteristic
379C22C23×#46811yescharacteristic
380C22C23×#47212yes—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4C23×#44811yescharacteristic
387C4C23×#44911yescharacteristic
388C4C23×#45011yescharacteristic
389C4C23×#45211yescharacteristic
390C4C23×#45212yes—
391C4C23×#49511yescharacteristic
392C4C23×#49711yescharacteristic
393C2—162no—
394C2C23×#167812yes—
395C2C23×#169812yes—
396C2C23×#200011yescharacteristic
397C2C23×#201511yescharacteristic
398C2C23×#201811yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×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×#1984C2semidirect
5. C2×C23×#333C2semidirect
7. C23×#1277C2semidirect
126. #10269C23direct (N × H)
173. C2×#477C2×C23semidirect
175. #1984C2×C23semidirect
176. C2×#333C2×C23semidirect
178. #1277C2×C23semidirect
270. C23#10269direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4862
C2×C23×#4772
C2×C23×#4602
C23×#19842
C2×C23×#3332
C2×C23×#3312
C23×#12772
#1026923