Finite Group Databasebeta

← C23×#11563 #19159189

Subgroups — C23×#11563 #19159189

Order 5888.

616 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2541C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic, complemented
3C23×#1459C211yesmaximal, characteristic, complemented
4C23×#1318C214yesmaximal, complemented
5C2×C23×#127C411yescharacteristic
6C2×C23×#127C412yes—
7C2×C23×#121—21no—
8C2×C23×#121—21no—
9C2×C4×D4×C23C2211yescharacteristic
10C23×#407—21no—
11C2×C23×C22⋊C8—21no—
12C23×#392—21no—
13C23×#392—21no—
14C23×#390C2212yes—
15C4×C4⋊C4×C23C411yescharacteristic
16C23×#324C2212yes—
17C23×#324C2212yes—
18C23×#323—21no—
19C23×#323—21no—
20C22×D4×C23—22no—
21C22×D4×C23—22no—
22C22×D4×C23—21no—
23C22×D4×C23—21no—
24C22×D4×C23C2×C411yescharacteristic
25C22×D4×C23C2×C412yes—
26C23×C4×C23D411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C2×M16×C23—41no—
29C22×C8×C23—41no—
30C23×#127—41no—
31C23×#127—41no—
32C23×#127—44no—
33C23×#121—44no—
34C23×#121—44no—
35C4×D4×C23—41no—
36C4×D4×C23—41no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23C2311yescharacteristic
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23C2×C412yes—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—41no—
48C23×C22⋊C8—41no—
49C23×#95—21no—
50C24×C23—22no—
51C24×C23C22⋊C411yescharacteristic
52C24×C23C22⋊C411yescharacteristic
53C2×D4×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—24no—
56C2×D4×C23—41no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23—42no—
60C2×D4×C23—42no—
61C2×D4×C23—44no—
62C2×D4×C23—44no—
63C2×D4×C23—42no—
64C2×D4×C23—44no—
65C2×D4×C23—44no—
66C2×D4×C23—41no—
67C2×D4×C23—41no—
68C2×D4×C23—82no—
69C2×D4×C23—82no—
70C2×D4×C23—82no—
71C22×C4×C23—22no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—42no—
77C22×C4×C23—42no—
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C2×D411yescharacteristic
82C22×C4×C23C2×D411yescharacteristic
83M16×C23—82no—
84C2×C8×C23—42no—
85C2×C8×C23—42no—
86C2×C8×C23—82no—
87C4⋊C4×C23—24no—
88C4⋊C4×C23—41no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—44no—
92C22⋊C4×C23—44no—
93C42×C23—21no—
94C42×C23—21no—
95C42×C23—42no—
96#11563C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—42no—
107C23×C23—42no—
108C23×C23—42no—
109C23×C23—42no—
110C23×C23—42no—
111C23×C23—42no—
112C23×C23—42no—
113C23×C23#10012yes—
114C23×C23C2×C22⋊C411yescharacteristic
115D4×C23—44no—
116D4×C23—84no—
117D4×C23—82no—
118D4×C23—88no—
119D4×C23—88no—
120D4×C23—84no—
121D4×C23—82no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—22no—
125C2×C4×C23—24no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—81no—
137C2×C4×C23—81no—
138C2×C4×C23#9911yescharacteristic
139C2×C4×C23#9911yescharacteristic
140C2×C4×C23#13112yes—
141C8×C23—82no—
142C8×C23—82no—
143#2541C2×C2311yescharacteristic
144#1465C2×C2311yescharacteristic, complemented
145#1459C2×C2311yescharacteristic, complemented
146#1318C2×C2314yescomplemented
147C22×C23—21no—
148C22×C23—21no—
149C22×C23—22no—
150C22×C23—42no—
151C22×C23—42no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23—41no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—81no—
158C22×C23—81no—
159C22×C23—82no—
160C22×C23—82no—
161C22×C23—82no—
162C22×C23—82no—
163C22×C23—82no—
164C22×C23—82no—
165C22×C23—82no—
166C22×C23#40811yescharacteristic
167C22×C23C2×#9911yescharacteristic
168C22×C23C2×#10011yescharacteristic
169C4×C23—21no—
170C4×C23—22no—
171C4×C23—44no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23—41no—
175C4×C23#35312yes—
176C2×#127C4×C2311yescharacteristic
177C2×#127C4×C2312yes—
178C2×#121—21no—
179C2×#121—21no—
180C2×C4×D4C22×C2311yescharacteristic
181#407—21no—
182C2×C22⋊C8—21no—
183#392—21no—
184#392—21no—
185#390C22×C2312yes—
186C4×C4⋊C4C4×C2311yescharacteristic
187#324C22×C2312yes—
188#324C22×C2312yes—
189#323—21no—
190#323—21no—
191C2×C23—21no—
192C2×C23—21no—
193C2×C23—42no—
194C2×C23—81no—
195C2×C23—162no—
196C2×C23#147111yescharacteristic
197C2×C23C2×#35311yescharacteristic
198C2×C23#213411yescharacteristic
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C2×C4×C2311yescharacteristic
204C22×D4C2×C4×C2312yes—
205C23×C4D4×C2311yescharacteristic
206C23×C4D4×C2311yescharacteristic
207C2×M16—41no—
208C22×C8—41no—
209#127—41no—
210#127—41no—
211#127—44no—
212#121—44no—
213#121—44no—
214C4×D4—41no—
215C4×D4—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C23×C2311yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4D4×C2311yescharacteristic
221C2×C22⋊C4D4×C2311yescharacteristic
222C2×C42C2×C4×C2311yescharacteristic
223C2×C42C2×C4×C2312yes—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228#95—21no—
229C23#1156311yescharacteristic, direct product factor, Sylow
230C24—22no—
231C24C22⋊C4×C2311yescharacteristic
232C24C22⋊C4×C2311yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C22⋊C4×C2311yescharacteristic
259C22×C4C22⋊C4×C2311yescharacteristic
260C22×C4C22×C4×C2311yescharacteristic
261C22×C4C2×D4×C2311yescharacteristic
262C22×C4C2×D4×C2311yescharacteristic
263M16—82no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—82no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—21no—
274C42—21no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23C23×#10012yes—
293C23C2×C22⋊C4×C2311yescharacteristic
294D4—44no—
295D4—84no—
296D4—82no—
297D4—88no—
298D4—88no—
299D4—84no—
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C23×#9911yescharacteristic
318C2×C4C23×#9911yescharacteristic
319C2×C4C23×#13112yes—
320C8—82no—
321C8—82no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22C23×#40811yescharacteristic
342C22C2×C23×#9911yescharacteristic
343C22C2×C23×#10011yescharacteristic
344C4—22no—
345C4—21no—
346C4—44no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4C23×#35312yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—162no—
356C2C23×#147111yescharacteristic
357C2C2×C23×#35311yescharacteristic
358C2C23×#213411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#2541C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1465C2semidirect
3. C23×#1459C2semidirect
4. C23×#1318C2semidirect
96. #11563C23direct (N × H)
144. #1465C2×C23semidirect
145. #1459C2×C23semidirect
146. #1318C2×C23semidirect
229. C23#11563direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25412
C23×#14652
C23×#14592
C23×#13182
#1156323