Finite Group Databasebeta

← C23×#7670 #19155296

Subgroups — C23×#7670 #19155296

Order 5888.

538 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2850C211yesmaximal, characteristic
2C23×#1817C211yesmaximal, characteristic, complemented
3C4×C23×C22⋊C8C211yesmaximal, characteristic, complemented
4C23×#1272C212yesmaximal, complemented
5C23×#1270C212yesmaximal, complemented
6C2×C23×#124C411yescharacteristic
7C2×C23×C22⋊Q8C411yescharacteristic
8C2×C23×#121C411yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C2×C23×C22⋊C8—21no—
11C2×C23×C22⋊C8—21no—
12C2×C4×C8×C23—21no—
13C23×#391C2211yescharacteristic
14C23×#390C2211yescharacteristic
15C23×#388—21no—
16C23×#388—21no—
17C23×#386—21no—
18C23×#386—21no—
19C23×#382C411yescharacteristic
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×Q8×C23C2×C411yescharacteristic
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C22×C8×C23—22no—
30C22×C8×C23—22no—
31C23×#124—42no—
32C23×C22⋊Q8—42no—
33C23×#121—42no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C42×C23—42no—
44C2×C42×C23C2×C411yescharacteristic
45C2×C42×C23D411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C2×C42×C23C2311yescharacteristic
48C23×C22⋊C8—24no—
49C23×C22⋊C8—42no—
50C4×C8×C23—42no—
51C23×#95—21no—
52C23×#95—21no—
53C23×#95—21no—
54C23×#95—21no—
55C24×C23—21no—
56C24×C23C22⋊C411yescharacteristic
57C2×Q8×C23—22no—
58C2×Q8×C23—41no—
59C2×Q8×C23—41no—
60C2×D4×C23—22no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23—42no—
64C2×D4×C23—42no—
65C2×D4×C23—42no—
66C22×C4×C23—22no—
67C22×C4×C23—22no—
68C22×C4×C23—22no—
69C22×C4×C23—22no—
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23—41no—
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22⋊C411yescharacteristic
81C22×C4×C23C22×C411yescharacteristic
82C22×C4×C23C2×D411yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84C2×C8×C23—24no—
85C2×C8×C23—42no—
86C2×C8×C23—42no—
87C2×C8×C23—42no—
88C2×C8×C23—42no—
89C2×C8×C23—42no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—41no—
92C4⋊C4×C23—42no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—42no—
98C22⋊C4×C23—42no—
99C42×C23—22no—
100C42×C23—22no—
101C42×C23—41no—
102C42×C23—41no—
103#7670C2311yesmaximal, characteristic, direct product factor, Sylow
104C23×C23—21no—
105C23×C23—21no—
106C23×C23—21no—
107C23×C23—21no—
108C23×C23—21no—
109C23×C23—21no—
110C23×C23—41no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—41no—
117C23×C23#10412yes—
118C23×C23C2×C22⋊C411yescharacteristic
119Q8×C23—82no—
120D4×C23—82no—
121D4×C23—84no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—44no—
136C2×C4×C23—44no—
137C2×C4×C23—44no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23#9912yes—
148C2×C4×C23#10012yes—
149C2×C4×C23#13112yes—
150C2×C4×C23#13512yes—
151C2×C4×C23#13512yes—
152C8×C23—44no—
153C8×C23—82no—
154#2850C2×C2311yescharacteristic
155#1817C2×C2311yescharacteristic, complemented
156C4×C22⋊C8C2×C2311yescharacteristic, complemented
157#1272C2×C2312yescomplemented
158#1270C2×C2312yescomplemented
159C22×C23—22no—
160C22×C23—22no—
161C22×C23—22no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23—41no—
165C22×C23—41no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23—81no—
173C22×C23#40811yescharacteristic
174C22×C23C2×#9911yescharacteristic
175C22×C23C2×#10011yescharacteristic
176C22×C23#41611yescharacteristic
177C22×C23#41611yescharacteristic
178C22×C23C2×#10411yescharacteristic
179C22×C23#42111yescharacteristic
180C4×C23—22no—
181C4×C23—22no—
182C4×C23—22no—
183C4×C23—22no—
184C4×C23—22no—
185C4×C23—41no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23—81no—
189C2×#124C4×C2311yescharacteristic
190C2×C22⋊Q8C4×C2311yescharacteristic
191C2×#121C4×C2311yescharacteristic
192C22×C42C22×C2311yescharacteristic
193C2×C22⋊C8—21no—
194C2×C22⋊C8—21no—
195C2×C4×C8—21no—
196#391C22×C2311yescharacteristic
197#390C22×C2311yescharacteristic
198#388—21no—
199#388—21no—
200#386—21no—
201#386—21no—
202#382C4×C2311yescharacteristic
203#336—21no—
204#336C22×C2312yes—
205#336C22×C2312yes—
206C2×C23—42no—
207C2×C23—161no—
208C2×C23#147111yescharacteristic
209C2×C23#148211yescharacteristic
210C2×C23#149411yescharacteristic
211C2×C23#150111yescharacteristic
212C2×C23#151611yescharacteristic
213C2×C23#152711yescharacteristic
214C2×C23#154311yescharacteristic
215C22×Q8C2×C4×C2311yescharacteristic
216C22×D4—21no—
217C22×D4C2×C4×C2311yescharacteristic
218C23×C4C2×C4×C2311yescharacteristic
219C23×C4D4×C2311yescharacteristic
220C23×C4D4×C2311yescharacteristic
221C22×C8—22no—
222C22×C8—22no—
223#124—42no—
224C22⋊Q8—42no—
225#121—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C2×C4×C2311yescharacteristic
228C2×C4⋊C4C2×C4×C2311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—42no—
236C2×C42C2×C4×C2311yescharacteristic
237C2×C42D4×C2311yescharacteristic
238C2×C42D4×C2311yescharacteristic
239C2×C42C23×C2311yescharacteristic
240C22⋊C8—24no—
241C22⋊C8—42no—
242C4×C8—42no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247C23#767011yescharacteristic, direct product factor, Sylow
248C24—21no—
249C24C22⋊C4×C2311yescharacteristic
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C22⋊C4×C2311yescharacteristic
272C22×C4C22⋊C4×C2311yescharacteristic
273C22×C4C22⋊C4×C2311yescharacteristic
274C22×C4C22×C4×C2311yescharacteristic
275C22×C4C2×D4×C2311yescharacteristic
276C22×C4C2×D4×C2311yescharacteristic
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—22no—
293C42—22no—
294C42—41no—
295C42—41no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23C23×#10412yes—
310C23C2×C22⋊C4×C2311yescharacteristic
311Q8—82no—
312D4—82no—
313D4—84no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4C23×#9912yes—
340C2×C4C23×#10012yes—
341C2×C4C23×#13112yes—
342C2×C4C23×#13512yes—
343C2×C4C23×#13512yes—
344C8—44no—
345C8—82no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C23×#40811yescharacteristic
361C22C2×C23×#9911yescharacteristic
362C22C2×C23×#10011yescharacteristic
363C22C23×#41611yescharacteristic
364C22C23×#41611yescharacteristic
365C22C2×C23×#10411yescharacteristic
366C22C23×#42111yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C2—42no—
377C2—161no—
378C2C23×#147111yescharacteristic
379C2C23×#148211yescharacteristic
380C2C23×#149411yescharacteristic
381C2C23×#150111yescharacteristic
382C2C23×#151611yescharacteristic
383C2C23×#152711yescharacteristic
384C2C23×#154311yescharacteristic
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. C4×C23×C22⋊C8C2semidirect
4. C23×#1272C2semidirect
5. C23×#1270C2semidirect
103. #7670C23direct (N × H)
155. #1817C2×C23semidirect
156. C4×C22⋊C8C2×C23semidirect
157. #1272C2×C23semidirect
158. #1270C2×C23semidirect
247. C23#7670direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28502
C23×#18172
C4×C23×C22⋊C82
C23×#12722
C23×#12702
#767023