Finite Group Databasebeta

← C23×#9688 #19157314

Subgroups — C23×#9688 #19157314

Order 5888.

482 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2861C211yesmaximal, characteristic
2C23×#1767C211yesmaximal, characteristic, complemented
3C23×#1745C211yesmaximal, characteristic, complemented
4C2×C23×#330C212yesmaximal, complemented
5C2×C23×#329C212yesmaximal
6C2×C23×#125C2211yescharacteristic
7C2×C23×#124C2211yescharacteristic
8C2×C23×C4⋊C8—21no—
9C2×C23×#97C2212yes—
10C2×C23×#97C2212yes—
11C23×#391C411yescharacteristic
12C23×#391C411yescharacteristic
13C23×#390C411yescharacteristic
14C23×#390C411yescharacteristic
15C23×#388—21no—
16C23×#388—21no—
17C23×#388—21no—
18C23×#388—21no—
19C43×C23C2211yescharacteristic
20C23×#336—21no—
21C23×#336—21no—
22C23×#336—21no—
23C23×#330—24no—
24C23×#329—24no—
25C22×Q8×C23C2×C411yescharacteristic
26C22×D4×C23C2×C411yescharacteristic
27C22×C8×C23—22no—
28C22×C8×C23—22no—
29C23×#125—22no—
30C23×#125—22no—
31C23×#124—22no—
32C23×#124—22no—
33C2×C4⋊C4×C23—21no—
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×C23C2×C411yescharacteristic
44C2×C42×C23C2×C411yescharacteristic
45C2×C42×C23D411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C2×C42×C23D411yescharacteristic
48C2×C42×C23D411yescharacteristic
49C2×C42×C23C2311yescharacteristic
50C23×C4⋊C8—24no—
51C23×#97—22no—
52C23×#97—22no—
53C23×#97—22no—
54C23×#97D414yes—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C24×C23—21no—
60C2×Q8×C23—22no—
61C2×Q8×C23—41no—
62C2×Q8×C23—41no—
63C2×D4×C23—22no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22⋊C411yescharacteristic
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22×C411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76C2×C8×C23—24no—
77C2×C8×C23—24no—
78C2×C8×C23—42no—
79C2×C8×C23—42no—
80C2×C8×C23—42no—
81C2×C8×C23—42no—
82C4⋊C4×C23—22no—
83C4⋊C4×C23—22no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—41no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—42no—
89C22⋊C4×C23—42no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—41no—
93C22⋊C4×C23—41no—
94C42×C23—22no—
95C42×C23—41no—
96C42×C23—41no—
97C42×C23—41no—
98C42×C23—41no—
99C42×C23—41no—
100C42×C23C2×D412yes—
101C42×C23#4112yes—
102#9688C2311yesmaximal, characteristic, direct product factor, Sylow
103C23×C23—41no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23C2×C22⋊C411yescharacteristic
111Q8×C23—82no—
112D4×C23—82no—
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—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—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—41no—
138C2×C4×C23—41no—
139C2×C4×C23#10412yes—
140C2×C4×C23C4×D412yes—
141C2×C4×C23#12112yes—
142C2×C4×C23#12411yescharacteristic
143C2×C4×C23#12711yescharacteristic
144C8×C23—44no—
145C8×C23—44no—
146#2861C2×C2311yescharacteristic
147#1767C2×C2311yescharacteristic, complemented
148#1745C2×C2311yescharacteristic, complemented
149C2×#330C2×C2312yescomplemented
150C2×#329C2×C2312yes—
151C22×C23—81no—
152C22×C23—81no—
153C22×C23—81no—
154C22×C23—81no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23—81no—
158C22×C23#39011yescharacteristic
159C22×C23#41311yescharacteristic
160C22×C23#41311yescharacteristic
161C22×C23C2×#10411yescharacteristic
162C22×C23#42111yescharacteristic
163C22×C23#42111yescharacteristic
164C22×C23#42111yescharacteristic
165C4×C23—22no—
166C4×C23—22no—
167C4×C23—21no—
168C4×C23—21no—
169C4×C23—22no—
170C4×C23—41no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—81no—
174C4×C23—81no—
175C4×C23—81no—
176C2×#125C22×C2311yescharacteristic
177C2×#124C22×C2311yescharacteristic
178C2×C4⋊C8—21no—
179C2×#97C22×C2312yes—
180C2×#97C22×C2312yes—
181#391C4×C2311yescharacteristic
182#391C4×C2311yescharacteristic
183#390C4×C2311yescharacteristic
184#390C4×C2311yescharacteristic
185#388—21no—
186#388—21no—
187#388—21no—
188#388—21no—
189C43C22×C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#330—24no—
194#329—24no—
195C2×C23—161no—
196C2×C23#152911yescharacteristic
197C2×C23#153211yescharacteristic
198C2×C23#153511yescharacteristic
199C2×C23#153511yescharacteristic
200C2×C23#196311yescharacteristic
201C2×C23#196711yescharacteristic
202C2×C23#198011yescharacteristic
203C22×Q8C2×C4×C2311yescharacteristic
204C22×D4C2×C4×C2311yescharacteristic
205C22×C8—22no—
206C22×C8—22no—
207#125—22no—
208#125—22no—
209#124—22no—
210#124—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
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×C22⋊C4—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—24no—
229#97—22no—
230#97—22no—
231#97—22no—
232#97D4×C2314yes—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C23#968811yescharacteristic, direct product factor, Sylow
238C24—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C22⋊C4×C2311yescharacteristic
249C22×C4C22⋊C4×C2311yescharacteristic
250C22×C4C22⋊C4×C2311yescharacteristic
251C22×C4C22⋊C4×C2311yescharacteristic
252C22×C4C22×C4×C2311yescharacteristic
253C22×C4C2×D4×C2311yescharacteristic
254C22×C4C2×D4×C2311yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42—22no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42C2×D4×C2312yes—
280C42#41×C2312yes—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊C4×C2311yescharacteristic
289Q8—82no—
290D4—82no—
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—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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×C4—41no—
317C2×C4C23×#10412yes—
318C2×C4C4×D4×C2312yes—
319C2×C4C23×#12112yes—
320C2×C4C23×#12411yescharacteristic
321C2×C4C23×#12711yescharacteristic
322C8—44no—
323C8—44no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22C23×#39011yescharacteristic
332C22C23×#41311yescharacteristic
333C22C23×#41311yescharacteristic
334C22C2×C23×#10411yescharacteristic
335C22C23×#42111yescharacteristic
336C22C23×#42111yescharacteristic
337C22C23×#42111yescharacteristic
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—161no—
350C2C23×#152911yescharacteristic
351C2C23×#153211yescharacteristic
352C2C23×#153511yescharacteristic
353C2C23×#153511yescharacteristic
354C2C23×#196311yescharacteristic
355C2C23×#196711yescharacteristic
356C2C23×#198011yescharacteristic
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×#1767C2semidirect
3. C23×#1745C2semidirect
4. C2×C23×#330C2semidirect
102. #9688C23direct (N × H)
147. #1767C2×C23semidirect
148. #1745C2×C23semidirect
149. C2×#330C2×C23semidirect
237. C23#9688direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28612
C23×#17672
C23×#17452
C2×C23×#3302
C2×C23×#3292
#968823