Finite Group Databasebeta

← C23×#10058 #19157684

Subgroups — C23×#10058 #19157684

Order 5888.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#487C211yesmaximal, characteristic, complemented
2C2×C23×#476C211yesmaximal, characteristic
3C23×#2547C211yesmaximal, characteristic
4C23×#1987C211yesmaximal, characteristic, complemented
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C2×C23×D4⋊C8C211yesmaximal, characteristic
7C23×#1279C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×#124—21no—
11C2×C4×D4×C23C2211yescharacteristic
12C23×#487—22no—
13C23×#487—22no—
14C23×#476—22no—
15C23×#476—22no—
16C2×C23×#106—21no—
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×C4⋊C8C2211yescharacteristic
19C2×C23×#103—22no—
20C2×C23×#102—21no—
21C2×C4×C8×C23C2211yescharacteristic
22C23×#397—22no—
23C23×#391—21no—
24C23×#380—22no—
25C4×C4⋊C4×C23C2211yescharacteristic
26C23×#340—22no—
27C23×#332—22no—
28C23×D4⋊C8—22no—
29C22×Q8×C23—21no—
30C22×Q8×C23—21no—
31C22×D4×C23D411yescharacteristic
32C23×C4×C23—21no—
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C22×C8×C23—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8D412yes—
44C23×#124—42no—
45C23×#124—42no—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23D412yes—
54C2×C4⋊C4×C23D411yescharacteristic
55C2×C4⋊C4×C23D412yes—
56C2×C22⋊C4×C23—24no—
57C2×C22⋊C4×C23—21no—
58C2×C42×C23—21no—
59C2×C42×C23C2311yescharacteristic
60C23×#106—24no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8D412yes—
65C23×#103—28no—
66C23×#102—24no—
67C4×C8×C23—21no—
68C4×C8×C23D412yes—
69C23×#95—24no—
70C23×#95—21no—
71C24×C23—21no—
72C2×Q8×C23—22no—
73C2×Q8×C23—44no—
74C2×Q8×C23—41no—
75C2×Q8×C23—41no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×Q8×C23—41no—
80C2×Q8×C23—41no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23—21no—
86C2×D4×C23SD1612yes—
87C22×C4×C23—22no—
88C22×C4×C23—22no—
89C22×C4×C23—21no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23—42no—
94C22×C4×C23—42no—
95C22×C4×C23C2×D411yescharacteristic
96C22×C4×C23C2×D411yescharacteristic
97C22×C4×C23C2×D411yescharacteristic
98C2×C8×C23—22no—
99C2×C8×C23—22no—
100C2×C8×C23—42no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—42no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—22no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C22⋊C4×C23—48no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—41no—
120C42×C23—41no—
121C42×C23—41no—
122C42×C23C2×D411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23#4111yescharacteristic
126#10058C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—22no—
128C23×C23—41no—
129C23×C23—42no—
130C23×C23—41no—
131C23×C23—42no—
132C23×C23#12011yescharacteristic
133Q8×C23—44no—
134Q8×C23—82no—
135Q8×C23—81no—
136Q8×C23—81no—
137Q8×C23—81no—
138Q8×C23—81no—
139D4×C23—22no—
140D4×C23—22no—
141D4×C23—41no—
142D4×C23—41no—
143C2×C4×C23—22no—
144C2×C4×C23—41no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—42no—
150C2×C4×C23—42no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—82no—
154C2×C4×C23—82no—
155C2×C4×C23—81no—
156C2×C4×C23—81no—
157C2×C4×C23#12011yescharacteristic
158C2×C4×C23#12011yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C2×C4×C23#12111yescharacteristic
161C2×C4×C23#12111yescharacteristic
162C2×C4×C23#12711yescharacteristic
163C2×C4×C23C2×SD1611yescharacteristic
164C2×C4×C23#13511yescharacteristic
165C2×C4×C23#13512yes—
166C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C23#13711yescharacteristic
168C2×C4×C23#13712yes—
169C8×C23—42no—
170C8×C23—42no—
171C8×C23—81no—
172C2×#487C2×C2311yescharacteristic, complemented
173C2×#476C2×C2311yescharacteristic
174#2547C2×C2311yescharacteristic
175#1987C2×C2311yescharacteristic, complemented
176C2×#332C2×C2311yescharacteristic, complemented
177C2×D4⋊C8C2×C2311yescharacteristic
178#1279C2×C2311yescharacteristic, complemented
179C22×C23—44no—
180C22×C23—42no—
181C22×C23—82no—
182C22×C23—81no—
183C22×C23—81no—
184C22×C23#39211yescharacteristic
185C22×C23#45011yescharacteristic
186C22×C23#45212yes—
187C22×C23#45211yescharacteristic
188C22×C23#45412yes—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—82no—
192C4×C23—82no—
193C4×C23—82no—
194C4×C23—81no—
195C4×C23#44811yescharacteristic
196C4×C23#44911yescharacteristic
197C4×C23#46412yes—
198C4×C23#46511yescharacteristic
199C4×C23#46811yescharacteristic
200C4×C23#49511yescharacteristic
201C4×C23#49711yescharacteristic
202C2×C4⋊Q8C22×C2311yescharacteristic
203C2×C4⋊Q8C22×C2311yescharacteristic
204C2×#124—21no—
205C2×C4×D4C22×C2311yescharacteristic
206#487—22no—
207#487—22no—
208#476—22no—
209#476—22no—
210C2×#106—21no—
211C2×C4⋊C8C22×C2311yescharacteristic
212C2×C4⋊C8C22×C2311yescharacteristic
213C2×#103—22no—
214C2×#102—21no—
215C2×C4×C8C22×C2311yescharacteristic
216#397—22no—
217#391—21no—
218#380—22no—
219C4×C4⋊C4C22×C2311yescharacteristic
220#340—22no—
221#332—22no—
222D4⋊C8—22no—
223C2×C23—82no—
224C2×C23#163512yes—
225C2×C23#163812yes—
226C2×C23#200011yescharacteristic
227C2×C23#200311yescharacteristic
228C2×C23#201511yescharacteristic
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4D4×C2311yescharacteristic
232C23×C4—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8D4×C2312yes—
244#124—42no—
245#124—42no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4D4×C2311yescharacteristic
254C2×C4⋊C4D4×C2312yes—
255C2×C4⋊C4D4×C2312yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C2311yescharacteristic
260#106—24no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8D4×C2312yes—
265#103—28no—
266#102—24no—
267C4×C8—21no—
268C4×C8D4×C2312yes—
269#95—24no—
270#95—21no—
271C23#1005811yescharacteristic, direct product factor, Sylow
272C24—21no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4SD16×C2312yes—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×D4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298C22×C4C2×D4×C2311yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—48no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42C2×D4×C2311yescharacteristic
324C42C2×D4×C2311yescharacteristic
325C42C2×D4×C2311yescharacteristic
326C42#41×C2311yescharacteristic
327C23—22no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23C23×#12011yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C23×#12011yescharacteristic
358C2×C4C23×#12011yescharacteristic
359C2×C4C23×#12111yescharacteristic
360C2×C4C23×#12111yescharacteristic
361C2×C4C23×#12111yescharacteristic
362C2×C4C23×#12711yescharacteristic
363C2×C4C2×SD16×C2311yescharacteristic
364C2×C4C23×#13511yescharacteristic
365C2×C4C23×#13512yes—
366C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C23×#13711yescharacteristic
368C2×C4C23×#13712yes—
369C8—42no—
370C8—42no—
371C8—81no—
372C22—44no—
373C22—42no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22C23×#39211yescharacteristic
378C22C23×#45011yescharacteristic
379C22C23×#45212yes—
380C22C23×#45211yescharacteristic
381C22C23×#45412yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—81no—
388C4C23×#44811yescharacteristic
389C4C23×#44911yescharacteristic
390C4C23×#46412yes—
391C4C23×#46511yescharacteristic
392C4C23×#46811yescharacteristic
393C4C23×#49511yescharacteristic
394C4C23×#49711yescharacteristic
395C2—82no—
396C2C23×#163512yes—
397C2C23×#163812yes—
398C2C23×#200011yescharacteristic
399C2C23×#200311yescharacteristic
400C2C23×#201511yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
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
1. C2×C23×#487C2semidirect
4. C23×#1987C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1279C2semidirect
126. #10058C23direct (N × H)
172. C2×#487C2×C23semidirect
175. #1987C2×C23semidirect
176. C2×#332C2×C23semidirect
178. #1279C2×C23semidirect
271. C23#10058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4872
C2×C23×#4762
C23×#25472
C23×#19872
C2×C23×#3322
C2×C23×D4⋊C82
C23×#12792
#1005823