Finite Group Databasebeta

← C23×#7642 #19155268

Subgroups — C23×#7642 #19155268

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2293C211yesmaximal, characteristic
2C23×#1839C211yesmaximal, characteristic, complemented
3C23×#1831C211yesmaximal, characteristic, complemented
4C23×#1271C212yesmaximal, complemented
5C23×#1270C212yesmaximal
6C2×C4×D4×C23C411yescharacteristic
7C2×C23×#117—21no—
8C2×C23×C4⋊C8—21no—
9C23×#390C411yescharacteristic
10C23×#390C2211yescharacteristic
11C23×#389C411yescharacteristic
12C23×#389C2211yescharacteristic
13C23×#388—21no—
14C23×#388—21no—
15C4×C4⋊C4×C23C411yescharacteristic
16C4×C22⋊C4×C23—21no—
17C43×C23C2211yescharacteristic
18C23×#336—21no—
19C23×#336—21no—
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×D4×C23C2×C411yescharacteristic
24C23×C4×C23—21no—
25C22×C8×C23—22no—
26C22×C8×C23—22no—
27C4×D4×C23—22no—
28C4×D4×C23—22no—
29C23×#117—42no—
30C23×#117—42no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23C2×C411yescharacteristic
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C42×C23—21no—
42C2×C42×C23C2×C411yescharacteristic
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23D411yescharacteristic
45C2×C42×C23D411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C2×C42×C23D411yescharacteristic
48C2×C42×C23C2311yescharacteristic
49C23×C4⋊C8—42no—
50C23×#95—21no—
51C23×#95—21no—
52C23×#95—21no—
53C24×C23—21no—
54C2×D4×C23—21no—
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C2×D4×C23M1612yes—
58C22×C4×C23—24no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23C22⋊C411yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22⋊C411yescharacteristic
68C22×C4×C23C22×C411yescharacteristic
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23C2×D411yescharacteristic
71C2×C8×C23—42no—
72C2×C8×C23—42no—
73C2×C8×C23—42no—
74C2×C8×C23—42no—
75C2×C8×C23—42no—
76C2×C8×C23—42no—
77C4⋊C4×C23—21no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23M1612yes—
84C22⋊C4×C23—24no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—41no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—42no—
90C42×C23—21no—
91C42×C23—21no—
92C42×C23—22no—
93C42×C23—41no—
94C42×C23—41no—
95C42×C23—41no—
96C42×C23—41no—
97C42×C23—42no—
98C42×C23D812yes—
99C42×C23SD1612yes—
100#7642C2311yesmaximal, characteristic, direct product factor, Sylow
101C23×C23—22no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23C2×C22⋊C411yescharacteristic
109D4×C23—42no—
110D4×C23—42no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—22no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—48no—
129C2×C4×C23—41no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23#10212yes—
138C2×C4×C23#10212yes—
139C2×C4×C23C2×M1611yescharacteristic
140C2×C4×C23C2×M1611yescharacteristic
141C2×C4×C23C2×D811yescharacteristic
142C2×C4×C23C2×SD1611yescharacteristic
143C2×C4×C23#13512yes—
144C8×C23—82no—
145C8×C23—82no—
146#2293C2×C2311yescharacteristic
147#1839C2×C2311yescharacteristic, complemented
148#1831C2×C2311yescharacteristic, complemented
149#1271C2×C2312yescomplemented
150#1270C2×C2312yes—
151C22×C23—42no—
152C22×C23—42no—
153C22×C23—42no—
154C22×C23—81no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23—81no—
158C22×C23#40711yescharacteristic
159C22×C23#41011yescharacteristic
160C22×C23#41311yescharacteristic
161C22×C23C2×#10211yescharacteristic
162C22×C23#41611yescharacteristic
163C22×C23#42111yescharacteristic
164C22×C23#42111yescharacteristic
165C4×C23—22no—
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—22no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—22no—
172C4×C23—42no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23—81no—
176C4×C23—81no—
177C2×C4×D4C4×C2311yescharacteristic
178C2×#117—21no—
179C2×C4⋊C8—21no—
180#390C4×C2311yescharacteristic
181#390C22×C2311yescharacteristic
182#389C4×C2311yescharacteristic
183#389C22×C2311yescharacteristic
184#388—21no—
185#388—21no—
186C4×C4⋊C4C4×C2311yescharacteristic
187C4×C22⋊C4—21no—
188C43C22×C2311yescharacteristic
189#336—21no—
190#336—21no—
191#336—21no—
192#336C22×C2312yes—
193#336C22×C2312yes—
194C2×C23—82no—
195C2×C23#146411yescharacteristic
196C2×C23#148611yescharacteristic
197C2×C23#148811yescharacteristic
198C2×C23#151711yescharacteristic
199C2×C23#152111yescharacteristic
200C2×C23#153511yescharacteristic
201C2×C23#154011yescharacteristic
202C22×D4C2×C4×C2311yescharacteristic
203C23×C4—21no—
204C22×C8—22no—
205C22×C8—22no—
206C4×D4—22no—
207C4×D4—22no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C2311yescharacteristic
213C2×C4⋊C4C2×C4×C2311yescharacteristic
214C2×C4⋊C4C2×C4×C2311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42C2×C4×C2311yescharacteristic
222C2×C42C2×C4×C2311yescharacteristic
223C2×C42D4×C2311yescharacteristic
224C2×C42D4×C2311yescharacteristic
225C2×C42D4×C2311yescharacteristic
226C2×C42D4×C2311yescharacteristic
227C2×C42C23×C2311yescharacteristic
228C4⋊C8—42no—
229#95—21no—
230#95—21no—
231#95—21no—
232C23#764211yescharacteristic, direct product factor, Sylow
233C24—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4M16×C2312yes—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊C4×C2311yescharacteristic
245C22×C4C22⋊C4×C2311yescharacteristic
246C22×C4C22⋊C4×C2311yescharacteristic
247C22×C4C22⋊C4×C2311yescharacteristic
248C22×C4C22×C4×C2311yescharacteristic
249C22×C4C2×D4×C2311yescharacteristic
250C22×C4C2×D4×C2311yescharacteristic
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4M16×C2312yes—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42D8×C2312yes—
279C42SD16×C2312yes—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊C4×C2311yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4C23×#10212yes—
317C2×C4C23×#10212yes—
318C2×C4C2×M16×C2311yescharacteristic
319C2×C4C2×M16×C2311yescharacteristic
320C2×C4C2×D8×C2311yescharacteristic
321C2×C4C2×SD16×C2311yescharacteristic
322C2×C4C23×#13512yes—
323C8—82no—
324C8—82no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C23×#40711yescharacteristic
333C22C23×#41011yescharacteristic
334C22C23×#41311yescharacteristic
335C22C2×C23×#10211yescharacteristic
336C22C23×#41611yescharacteristic
337C22C23×#42111yescharacteristic
338C22C23×#42111yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C2—82no—
352C2C23×#146411yescharacteristic
353C2C23×#148611yescharacteristic
354C2C23×#148811yescharacteristic
355C2C23×#151711yescharacteristic
356C2C23×#152111yescharacteristic
357C2C23×#153511yescharacteristic
358C2C23×#154011yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C43×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1839C2semidirect
3. C23×#1831C2semidirect
4. C23×#1271C2semidirect
100. #7642C23direct (N × H)
147. #1839C2×C23semidirect
148. #1831C2×C23semidirect
149. #1271C2×C23semidirect
232. C23#7642direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22932
C23×#18392
C23×#18312
C23×#12712
C23×#12702
#764223