Finite Group Databasebeta

← C23×#11879 #19159505

Subgroups — C23×#11879 #19159505

Order 5888.

440 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2536C211yesmaximal, characteristic
2C23×#1801C211yesmaximal, characteristic, complemented
3C23×#1461C211yesmaximal, characteristic, complemented
4C23×#1346C212yesmaximal, complemented
5C23×#1345C212yesmaximal
6C2×C23×C22⋊Q8—21no—
7C2×C4×Q8×C23C2211yescharacteristic
8C2×C4×D4×C23C411yescharacteristic
9C23×#431—21no—
10C23×#423—21no—
11C2×C23×#101—21no—
12C23×#398C411yescharacteristic
13C23×#398C2211yescharacteristic
14C23×#394C411yescharacteristic
15C23×#394C2211yescharacteristic
16C23×#388—21no—
17C23×#388—21no—
18C23×#382C411yescharacteristic
19C23×#380—21no—
20C23×#343C2212yes—
21C23×#324—21no—
22C23×#324C2212yes—
23C22×Q8×C23D411yescharacteristic
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23—21no—
26C2×M16×C23—22no—
27C23×C22⋊Q8—42no—
28C23×C22⋊Q8—42no—
29C4×Q8×C23—41no—
30C4×Q8×C23—41no—
31C4×D4×C23—41no—
32C4×D4×C23—41no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C4⋊C4×C23C2311yescharacteristic
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×C23C2×C411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C2×C42×C23D411yescharacteristic
48C23×C4⋊C8—41no—
49C23×C4⋊C8—41no—
50C23×C4⋊C8—41no—
51C23×#101—42no—
52C23×C22⋊C8—22no—
53C4×C8×C23—41no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95C2×C411yescharacteristic
58C23×#95C2×C411yescharacteristic
59C24×C23—21no—
60C2×Q8×C23—41no—
61C2×Q8×C23—41no—
62C2×Q8×C23D812yes—
63C2×Q8×C23SD1612yes—
64C2×D4×C23—41no—
65C2×D4×C23—42no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23—42no—
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22⋊C411yescharacteristic
74C22×C4×C23C22⋊C411yescharacteristic
75C22×C4×C23C22⋊C411yescharacteristic
76C22×C4×C23C22×C411yescharacteristic
77C22×C4×C23C2×D411yescharacteristic
78C22×C4×C23C2×D411yescharacteristic
79M16×C23—42no—
80M16×C23—42no—
81C2×C8×C23—42no—
82C2×C8×C23—42no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—41no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—41no—
87C4⋊C4×C23—42no—
88C4⋊C4×C23—42no—
89C4⋊C4×C23—41no—
90C4⋊C4×C23—41no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—41no—
94C22⋊C4×C23—41no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C42×C23—21no—
99C42×C23—21no—
100C42×C23—21no—
101C42×C23—21no—
102C42×C23—21no—
103C42×C23—21no—
104#11879C2311yesmaximal, characteristic, direct product factor, Sylow
105C23×C23—22no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23—41no—
111C23×C23—41no—
112C23×C23C2×C22⋊C411yescharacteristic
113Q8×C23—42no—
114Q8×C23—42no—
115D4×C23—42no—
116D4×C23—42no—
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—41no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
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—41no—
138C2×C4×C23—41no—
139C2×C4×C23—84no—
140C2×C4×C23#9911yescharacteristic
141C2×C4×C23#9911yescharacteristic
142C2×C4×C23#10212yes—
143C2×C4×C23#10212yes—
144C2×C4×C23C2×D811yescharacteristic
145C2×C4×C23C2×SD1611yescharacteristic
146C8×C23—82no—
147C8×C23—82no—
148#2536C2×C2311yescharacteristic
149#1801C2×C2311yescharacteristic, complemented
150#1461C2×C2311yescharacteristic, complemented
151#1346C2×C2312yescomplemented
152#1345C2×C2312yes—
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—42no—
156C22×C23—42no—
157C22×C23—42no—
158C22×C23—81no—
159C22×C23—81no—
160C22×C23—81no—
161C22×C23—81no—
162C22×C23C2×#9911yescharacteristic
163C22×C23C2×#10211yescharacteristic
164C22×C23#42111yescharacteristic
165C4×C23—21no—
166C4×C23—21no—
167C4×C23—21no—
168C4×C23—22no—
169C4×C23—22no—
170C4×C23—21no—
171C4×C23—41no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23—81no—
177C2×C22⋊Q8—21no—
178C2×C4×Q8C22×C2311yescharacteristic
179C2×C4×D4C4×C2311yescharacteristic
180#431—21no—
181#423—21no—
182C2×#101—21no—
183#398C4×C2311yescharacteristic
184#398C22×C2311yescharacteristic
185#394C4×C2311yescharacteristic
186#394C22×C2311yescharacteristic
187#388—21no—
188#388—21no—
189#382C4×C2311yescharacteristic
190#380—21no—
191#343C22×C2312yes—
192#324—21no—
193#324C22×C2312yes—
194C2×C23—21no—
195C2×C23—21no—
196C2×C23—82no—
197C2×C23#151911yescharacteristic
198C2×C23#212011yescharacteristic
199C2×C23#213311yescharacteristic
200C22×Q8D4×C2311yescharacteristic
201C22×D4C2×C4×C2311yescharacteristic
202C23×C4—21no—
203C2×M16—22no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×D4—41no—
209C4×D4—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×C4×C2311yescharacteristic
214C2×C4⋊C4C2×C4×C2311yescharacteristic
215C2×C4⋊C4D4×C2311yescharacteristic
216C2×C4⋊C4C23×C2311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42C2×C4×C2311yescharacteristic
223C2×C42D4×C2311yescharacteristic
224C2×C42D4×C2311yescharacteristic
225C4⋊C8—41no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228#101—42no—
229C22⋊C8—22no—
230C4×C8—41no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C2×C4×C2311yescharacteristic
235#95C2×C4×C2311yescharacteristic
236C23#1187911yescharacteristic, direct product factor, Sylow
237C24—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8D8×C2312yes—
241C2×Q8SD16×C2312yes—
242C2×D4—41no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊C4×C2311yescharacteristic
251C22×C4C22⋊C4×C2311yescharacteristic
252C22×C4C22⋊C4×C2311yescharacteristic
253C22×C4C22⋊C4×C2311yescharacteristic
254C22×C4C22×C4×C2311yescharacteristic
255C22×C4C2×D4×C2311yescharacteristic
256C22×C4C2×D4×C2311yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊C4×C2311yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—42no—
293D4—42no—
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—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
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—84no—
317C2×C4C23×#9911yescharacteristic
318C2×C4C23×#9911yescharacteristic
319C2×C4C23×#10212yes—
320C2×C4C23×#10212yes—
321C2×C4C2×D8×C2311yescharacteristic
322C2×C4C2×SD16×C2311yescharacteristic
323C8—82no—
324C8—82no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×C23×#9911yescharacteristic
335C22C2×C23×#10211yescharacteristic
336C22C23×#42111yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—82no—
352C2C23×#151911yescharacteristic
353C2C23×#212011yescharacteristic
354C2C23×#213311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1801C2semidirect
3. C23×#1461C2semidirect
4. C23×#1346C2semidirect
104. #11879C23direct (N × H)
149. #1801C2×C23semidirect
150. #1461C2×C23semidirect
151. #1346C2×C23semidirect
236. C23#11879direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25362
C23×#18012
C23×#14612
C23×#13462
C23×#13452
#1187923