Finite Group Databasebeta

← C23×#7647 #19155273

Subgroups — C23×#7647 #19155273

Order 5888.

584 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C23×#121C211yesmaximal, characteristic
2C23×#1817C211yesmaximal, characteristic, complemented
3C23×#1797C211yesmaximal, characteristic, complemented
4C23×#1271C212yesmaximal, complemented
5C23×#1270C212yesmaximal, complemented
6C2×C23×#121C411yescharacteristic
7C2×C4×D4×C23—21no—
8C2×C4×D4×C23C411yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C2×C23×C4⋊C8—21no—
11C2×C23×C22⋊C8—21no—
12C2×C23×C22⋊C8—21no—
13C23×#390C2211yescharacteristic
14C23×#389C2211yescharacteristic
15C23×#388—21no—
16C23×#386—21no—
17C23×#380C411yescharacteristic
18C4×C4⋊C4×C23C411yescharacteristic
19C4×C22⋊C4×C23—21no—
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23—21no—
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C22×C8×C23—22no—
30C22×C8×C23—22no—
31C23×#121—22no—
32C23×#121—22no—
33C4×D4×C23—42no—
34C4×D4×C23—42no—
35C4×D4×C23—42no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23C2×C411yescharacteristic
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C42×C23—21no—
46C2×C42×C23—42no—
47C2×C42×C23C2×C411yescharacteristic
48C2×C42×C23D411yescharacteristic
49C2×C42×C23D411yescharacteristic
50C2×C42×C23C2311yescharacteristic
51C23×C4⋊C8—42no—
52C23×C22⋊C8—42no—
53C23×C22⋊C8—42no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C24×C23—21no—
58C24×C23C22⋊C411yescharacteristic
59C2×D4×C23—24no—
60C2×D4×C23—21no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23—42no—
64C2×D4×C23—42no—
65C2×D4×C23M1612yes—
66C22×C4×C23—21no—
67C22×C4×C23—22no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—22no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—41no—
74C22×C4×C23—41no—
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23—41no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23—42no—
81C22×C4×C23C22⋊C411yescharacteristic
82C22×C4×C23C22⋊C411yescharacteristic
83C22×C4×C23C22⋊C411yescharacteristic
84C22×C4×C23D812yes—
85C22×C4×C23SD1612yes—
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89C2×C8×C23—42no—
90C2×C8×C23—42no—
91C2×C8×C23—42no—
92C2×C8×C23—42no—
93C2×C8×C23—42no—
94C2×C8×C23—42no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23M1612yes—
101C22⋊C4×C23—24no—
102C22⋊C4×C23—42no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—41no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—42no—
107C42×C23—22no—
108C42×C23—22no—
109C42×C23—41no—
110C42×C23—41no—
111C42×C23—42no—
112#7647C2311yesmaximal, characteristic, direct product factor, Sylow
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—22no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23#10212yes—
127C23×C23C2×C22⋊C411yescharacteristic
128D4×C23—42no—
129D4×C23—42no—
130D4×C23—48no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—22no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—44no—
148C2×C4×C23—42no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—84no—
159C2×C4×C23#9912yes—
160C2×C4×C23#10212yes—
161C2×C4×C23C2×M1611yescharacteristic
162C2×C4×C23C2×M1611yescharacteristic
163C2×C4×C23C2×D811yescharacteristic
164C2×C4×C23C2×SD1611yescharacteristic
165C2×C4×C23#13512yes—
166C8×C23—82no—
167C8×C23—82no—
168C4×#121C2×C2311yescharacteristic
169#1817C2×C2311yescharacteristic, complemented
170#1797C2×C2311yescharacteristic, complemented
171#1271C2×C2312yescomplemented
172#1270C2×C2312yescomplemented
173C22×C23—22no—
174C22×C23—22no—
175C22×C23—22no—
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—42no—
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23#40711yescharacteristic
188C22×C23C2×#9911yescharacteristic
189C22×C23#41311yescharacteristic
190C22×C23C2×#10211yescharacteristic
191C22×C23#41611yescharacteristic
192C22×C23#42111yescharacteristic
193C22×C23#42111yescharacteristic
194C4×C23—21no—
195C4×C23—21no—
196C4×C23—22no—
197C4×C23—22no—
198C4×C23—21no—
199C4×C23—22no—
200C4×C23—21no—
201C4×C23—41no—
202C4×C23—42no—
203C4×C23—81no—
204C4×C23—81no—
205C2×#121C4×C2311yescharacteristic
206C2×C4×D4—21no—
207C2×C4×D4C4×C2311yescharacteristic
208C22×C42C22×C2311yescharacteristic
209C2×C4⋊C8—21no—
210C2×C22⋊C8—21no—
211C2×C22⋊C8—21no—
212#390C22×C2311yescharacteristic
213#389C22×C2311yescharacteristic
214#388—21no—
215#386—21no—
216#380C4×C2311yescharacteristic
217C4×C4⋊C4C4×C2311yescharacteristic
218C4×C22⋊C4—21no—
219#336—21no—
220#336C22×C2312yes—
221#336C22×C2312yes—
222C2×C23—42no—
223C2×C23—82no—
224C2×C23#146511yescharacteristic
225C2×C23#148611yescharacteristic
226C2×C23#148811yescharacteristic
227C2×C23#151611yescharacteristic
228C2×C23#151911yescharacteristic
229C2×C23#153511yescharacteristic
230C2×C23#154011yescharacteristic
231C22×D4—21no—
232C22×D4C2×C4×C2311yescharacteristic
233C23×C4—21no—
234C23×C4C2×C4×C2311yescharacteristic
235C23×C4D4×C2311yescharacteristic
236C23×C4D4×C2311yescharacteristic
237C22×C8—22no—
238C22×C8—22no—
239#121—22no—
240#121—22no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×C4×C2311yescharacteristic
246C2×C4⋊C4C2×C4×C2311yescharacteristic
247C2×C4⋊C4C2×C4×C2311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C42—21no—
254C2×C42—42no—
255C2×C42C2×C4×C2311yescharacteristic
256C2×C42D4×C2311yescharacteristic
257C2×C42D4×C2311yescharacteristic
258C2×C42C23×C2311yescharacteristic
259C4⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262#95—21no—
263#95—21no—
264#95—21no—
265C23#764711yescharacteristic, direct product factor, Sylow
266C24—21no—
267C24C22⋊C4×C2311yescharacteristic
268C2×D4—24no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4M16×C2312yes—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊C4×C2311yescharacteristic
291C22×C4C22⋊C4×C2311yescharacteristic
292C22×C4C22⋊C4×C2311yescharacteristic
293C22×C4D8×C2312yes—
294C22×C4SD16×C2312yes—
295C22×C4C22×C4×C2311yescharacteristic
296C22×C4C2×D4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4M16×C2312yes—
310C22⋊C4—24no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C42—22no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42—42no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23C23×#10212yes—
335C23C2×C22⋊C4×C2311yescharacteristic
336D4—42no—
337D4—42no—
338D4—48no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4C23×#9912yes—
368C2×C4C23×#10212yes—
369C2×C4C2×M16×C2311yescharacteristic
370C2×C4C2×M16×C2311yescharacteristic
371C2×C4C2×D8×C2311yescharacteristic
372C2×C4C2×SD16×C2311yescharacteristic
373C2×C4C23×#13512yes—
374C8—82no—
375C8—82no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C23×#40711yescharacteristic
391C22C2×C23×#9911yescharacteristic
392C22C23×#41311yescharacteristic
393C22C2×C23×#10211yescharacteristic
394C22C23×#41611yescharacteristic
395C22C23×#42111yescharacteristic
396C22C23×#42111yescharacteristic
397C4—21no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—42no—
406C4—81no—
407C4—81no—
408C2—42no—
409C2—82no—
410C2C23×#146511yescharacteristic
411C2C23×#148611yescharacteristic
412C2C23×#148811yescharacteristic
413C2C23×#151611yescharacteristic
414C2C23×#151911yescharacteristic
415C2C23×#153511yescharacteristic
416C2C23×#154011yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1817C2semidirect
3. C23×#1797C2semidirect
4. C23×#1271C2semidirect
5. C23×#1270C2semidirect
112. #7647C23direct (N × H)
169. #1817C2×C23semidirect
170. #1797C2×C23semidirect
171. #1271C2×C23semidirect
172. #1270C2×C23semidirect
265. C23#7647direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C23×#1212
C23×#18172
C23×#17972
C23×#12712
C23×#12702
#764723