Finite Group Databasebeta

← C23×#8153 #19155779

Subgroups — C23×#8153 #19155779

Order 5888.

698 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊C4×C23C211yesmaximal, characteristic
2C23×#1749C212yesmaximal
3C2×C23×#324C212yesmaximal
4C23×#1280C212yesmaximal
5C22×C4⋊C4×C23C411yescharacteristic
6C22×C4⋊C4×C23C2211yescharacteristic
7C22×C42×C23C411yescharacteristic
8C22×C42×C23C411yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C2×C23×C22⋊C8C2214yes—
11C2×C23×#97—22no—
12C4×C4⋊C4×C23—22no—
13C4×C4⋊C4×C23—22no—
14C2×C23×#95C411yescharacteristic
15C2×C23×#95C2211yescharacteristic
16C23×#336—22no—
17C23×#324C418yescomplemented
18C23×C4×C23C2×C411yescharacteristic
19C23×C4×C23C2×C411yescharacteristic
20C23×C4×C23C2×C411yescharacteristic
21C23×C4×C23C2×C411yescharacteristic
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C22×C8×C23—24no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23C2×C414yes—
32C2×C42×C23—21no—
33C2×C42×C23—21no—
34C2×C42×C23—22no—
35C2×C42×C23—22no—
36C2×C42×C23—21no—
37C2×C42×C23—21no—
38C2×C42×C23—42no—
39C2×C42×C23—42no—
40C2×C42×C23D412yes—
41C2×C42×C23D412yes—
42C23×C22⋊C8—24no—
43C23×C22⋊C8C2×C418yes—
44C23×#97—44no—
45C23×#95—22no—
46C23×#95—22no—
47C24×C23C22×C411yescharacteristic
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—21no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—22no—
55C22×C4×C23—22no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23—41no—
68C22×C4×C23—41no—
69C22×C4×C23—41no—
70C22×C4×C23—41no—
71C22×C4×C23—41no—
72C22×C4×C23—41no—
73C22×C4×C23—41no—
74C22×C4×C23C4212yes—
75C22×C4×C23C4212yes—
76C22×C4×C23C22⋊C412yes—
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23M1612yes—
80C22×C4×C23M1612yes—
81C22×C4×C23M1612yes—
82C22×C4×C23M1612yes—
83C22×C4×C23C22×C412yes—
84C22×C4×C23C2×D412yes—
85C22×C4×C23#4112yes—
86C2×C8×C23—28no—
87C2×C8×C23—44no—
88C2×C8×C23—44no—
89C4⋊C4×C23—28no—
90C4⋊C4×C23—28no—
91C42×C23—24no—
92C42×C23—24no—
93C42×C23—24no—
94C42×C23—42no—
95C42×C23—42no—
96C42×C23—42no—
97#8153C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23C2×C4211yescharacteristic
103C23×C23C2×C22⋊C411yescharacteristic
104C23×C23#11711yescharacteristic
105C23×C23C2×M1611yescharacteristic
106C23×C23C2×M1611yescharacteristic
107C23×C23C2×M1611yescharacteristic
108C23×C23C2×M1611yescharacteristic
109C2×C4×C23—24no—
110C2×C4×C23—24no—
111C2×C4×C23—22no—
112C2×C4×C23—22no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—22no—
116C2×C4×C23—22no—
117C2×C4×C23—22no—
118C2×C4×C23—21no—
119C2×C4×C23—22no—
120C2×C4×C23—21no—
121C2×C4×C23—22no—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23—22no—
125C2×C4×C23—22no—
126C2×C4×C23—22no—
127C2×C4×C23—22no—
128C2×C4×C23—22no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—42no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—42no—
145C2×C4×C23#9714yescomplemented
146C2×C4×C23#9912yes—
147C2×C4×C23#10112yes—
148C2×C4×C23C4×D414yes—
149C8×C23—48no—
150C2×C4×C4⋊C4C2×C2311yescharacteristic
151#1749C2×C2312yes—
152C2×#324C2×C2312yes—
153#1280C2×C2312yes—
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—21no—
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23C4×C22⋊C411yescharacteristic
167C22×C23C2×#9711yescharacteristic
168C22×C23C4×M1611yescharacteristic
169C22×C23#40711yescharacteristic
170C22×C23#40711yescharacteristic
171C22×C23C2×#9911yescharacteristic
172C22×C23#41011yescharacteristic
173C22×C23C2×#10111yescharacteristic
174C22×C23#41311yescharacteristic
175C22×C23#43211yescharacteristic
176C22×C23#43211yescharacteristic
177C4×C23—22no—
178C4×C23—22no—
179C4×C23—24no—
180C4×C23—24no—
181C4×C23—42no—
182C4×C23—42no—
183C4×C23—42no—
184C4×C23—42no—
185C22×C4⋊C4C4×C2311yescharacteristic
186C22×C4⋊C4C22×C2311yescharacteristic
187C22×C42C4×C2311yescharacteristic
188C22×C42C4×C2311yescharacteristic
189C22×C42C22×C2311yescharacteristic
190C2×C22⋊C8C22×C2314yes—
191C2×#97—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C2×#95C4×C2311yescharacteristic
195C2×#95C22×C2311yescharacteristic
196#336—22no—
197#324C4×C2318yescomplemented
198C2×C23—21no—
199C2×C23—21no—
200C2×C23—21no—
201C2×C23—21no—
202C2×C23#146311yescharacteristic
203C2×C23#146411yescharacteristic
204C2×C23#146511yescharacteristic
205C2×C23#146711yescharacteristic
206C2×C23#174911yescharacteristic
207C2×C23C4×#9911yescharacteristic
208C2×C23C4×#10111yescharacteristic
209C23×C4C2×C4×C2311yescharacteristic
210C23×C4C2×C4×C2311yescharacteristic
211C23×C4C2×C4×C2311yescharacteristic
212C23×C4C2×C4×C2311yescharacteristic
213C23×C4C2×C4×C2311yescharacteristic
214C23×C4C2×C4×C2311yescharacteristic
215C23×C4C23×C2311yescharacteristic
216C22×C8—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4C2×C4×C2314yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42D4×C2312yes—
232C2×C42D4×C2312yes—
233C22⋊C8—24no—
234C22⋊C8C2×C4×C2318yes—
235#97—44no—
236#95—22no—
237#95—22no—
238C23#815311yescharacteristic, direct product factor, Sylow
239C24C22×C4×C2311yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C42×C2312yes—
267C22×C4C42×C2312yes—
268C22×C4C22⋊C4×C2312yes—
269C22×C4C22⋊C4×C2311yescharacteristic
270C22×C4C22⋊C4×C2311yescharacteristic
271C22×C4M16×C2312yes—
272C22×C4M16×C2312yes—
273C22×C4M16×C2312yes—
274C22×C4M16×C2312yes—
275C22×C4C22×C4×C2312yes—
276C22×C4C2×D4×C2312yes—
277C22×C4#41×C2312yes—
278C2×C8—28no—
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×C42×C2311yescharacteristic
294C23C2×C22⋊C4×C2311yescharacteristic
295C23C23×#11711yescharacteristic
296C23C2×M16×C2311yescharacteristic
297C23C2×M16×C2311yescharacteristic
298C23C2×M16×C2311yescharacteristic
299C23C2×M16×C2311yescharacteristic
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4C23×#9714yescomplemented
337C2×C4C23×#9912yes—
338C2×C4C23×#10112yes—
339C2×C4C4×D4×C2314yes—
340C8—48no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C22⋊C4×C2311yescharacteristic
354C22C2×C23×#9711yescharacteristic
355C22C4×M16×C2311yescharacteristic
356C22C23×#40711yescharacteristic
357C22C23×#40711yescharacteristic
358C22C2×C23×#9911yescharacteristic
359C22C23×#41011yescharacteristic
360C22C2×C23×#10111yescharacteristic
361C22C23×#41311yescharacteristic
362C22C23×#43211yescharacteristic
363C22C23×#43211yescharacteristic
364C4—22no—
365C4—22no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2C23×#146311yescharacteristic
377C2C23×#146411yescharacteristic
378C2C23×#146511yescharacteristic
379C2C23×#146711yescharacteristic
380C2C23×#174911yescharacteristic
381C2C4×C23×#9911yescharacteristic
382C2C4×C23×#10111yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC23C2×C42×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
17. C23×#324C4semidirect
97. #8153C23direct (N × H)
145. C2×C4×C23#97semidirect
197. #324C4×C23semidirect
238. C23#8153direct (N × H)
336. C2×C4C23×#97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C4×C232
C23×#17492
C2×C23×#3242
C23×#12802
#815323