Finite Group Databasebeta

← C23×#9446 #19157072

Subgroups — C23×#9446 #19157072

Order 5888.

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic, complemented
2C23×#2500C211yesmaximal, characteristic
3C23×#1820C211yesmaximal, characteristic, complemented
4C23×#1510C212yesmaximal
5C23×#1284C212yesmaximal, complemented
6C2×C23×#121C2211yescharacteristic
7C2×C23×#117C2211yescharacteristic
8C23×#433—24no—
9C23×#423—21no—
10C23×#407C2212yes—
11C4×M16×C23—21no—
12C23×#394—21no—
13C23×#394—21no—
14C23×#390C411yescharacteristic, complemented
15C23×#390C411yescharacteristic, complemented
16C23×#388—21no—
17C23×#388—21no—
18C23×#384C411yescharacteristic, complemented
19C23×#382C411yescharacteristic, complemented
20C23×#381C2212yes—
21C4×C22⋊C4×C23C2211yescharacteristic
22C23×#342—21no—
23C23×#342—21no—
24C23×#327—24no—
25C22×D4×C23—21no—
26C22×D4×C23C2×C411yescharacteristic
27C23×C4×C23C2311yescharacteristic
28C2×M16×C23—22no—
29C23×#121—22no—
30C23×#121—22no—
31C23×#117—22no—
32C23×#117—22no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23D411yescharacteristic
46C2×C22⋊C4×C23D411yescharacteristic
47C2×C42×C23C2×C411yescharacteristic
48C2×C42×C23C2×C411yescharacteristic
49C2×C42×C23D411yescharacteristic
50C2×C42×C23D411yescharacteristic
51C23×C4⋊C8—24no—
52C23×C22⋊C8—22no—
53C23×C22⋊C8D414yes—
54C23×#97—22no—
55C4×C8×C23—22no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—22no—
60C23×#95—22no—
61C24×C23—21no—
62C24×C23C2×D411yescharacteristic
63C2×D4×C23—22no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23—42no—
67C2×D4×C23—81no—
68C2×D4×C23—81no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—41no—
73C22×C4×C23—41no—
74C22×C4×C23—42no—
75C22×C4×C23C22⋊C411yescharacteristic
76C22×C4×C23C22⋊C411yescharacteristic
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22×C411yescharacteristic
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23C2×D412yes—
82C22×C4×C23#4112yes—
83M16×C23—42no—
84M16×C23—42no—
85C2×C8×C23—24no—
86C4⋊C4×C23—22no—
87C4⋊C4×C23—22no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—81no—
90C4⋊C4×C23—81no—
91C22⋊C4×C23—22no—
92C22⋊C4×C23—22no—
93C22⋊C4×C23—42no—
94C22⋊C4×C23—42no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—42no—
99C22⋊C4×C23—42no—
100C22⋊C4×C23—81no—
101C22⋊C4×C23—81no—
102C42×C23—21no—
103C42×C23—21no—
104C42×C23—21no—
105C42×C23—21no—
106C42×C23—41no—
107C42×C23—41no—
108#9446C2311yesmaximal, characteristic, direct product factor, Sylow
109C23×C23—21no—
110C23×C23—21no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23C2×C22⋊C411yescharacteristic
121C23×C23#12112yes—
122D4×C23—82no—
123D4×C23—84no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—22no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
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—82no—
148C2×C4×C23—82no—
149C2×C4×C23—82no—
150C2×C4×C23C4×D412yes—
151C2×C4×C23#12011yescharacteristic
152C2×C4×C23#12311yescharacteristic
153C8×C23—44no—
154#2921C2×C2311yescharacteristic, complemented
155#2500C2×C2311yescharacteristic
156#1820C2×C2311yescharacteristic, complemented
157#1510C2×C2312yes—
158#1284C2×C2312yescomplemented
159C22×C23—21no—
160C22×C23—21no—
161C22×C23—22no—
162C22×C23—22no—
163C22×C23—22no—
164C22×C23—81no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23—81no—
171C22×C23—81no—
172C22×C23—81no—
173C22×C23#38611yescharacteristic
174C22×C23#41011yescharacteristic
175C22×C23#41011yescharacteristic
176C4×C23—21no—
177C4×C23—22no—
178C4×C23—21no—
179C4×C23—42no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184C4×C23—81no—
185C4×C23—81no—
186C4×C23—162no—
187C2×#121C22×C2311yescharacteristic
188C2×#117C22×C2311yescharacteristic
189#433—24no—
190#423—21no—
191#407C22×C2312yes—
192C4×M16—21no—
193#394—21no—
194#394—21no—
195#390C4×C2311yescharacteristic, complemented
196#390C4×C2311yescharacteristic, complemented
197#388—21no—
198#388—21no—
199#384C4×C2311yescharacteristic, complemented
200#382C4×C2311yescharacteristic, complemented
201#381C22×C2312yes—
202C4×C22⋊C4C22×C2311yescharacteristic
203#342—21no—
204#342—21no—
205#327—24no—
206C2×C23—21no—
207C2×C23—21no—
208C2×C23—42no—
209C2×C23—161no—
210C2×C23#188411yescharacteristic
211C2×C23#189911yescharacteristic
212C2×C23#191011yescharacteristic
213C22×D4—21no—
214C22×D4C2×C4×C2311yescharacteristic
215C23×C4C23×C2311yescharacteristic
216C2×M16—22no—
217#121—22no—
218#121—22no—
219#117—22no—
220#117—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×C4×C2311yescharacteristic
224C2×C4⋊C4C2×C4×C2311yescharacteristic
225C2×C4⋊C4C2×C4×C2311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4D4×C2311yescharacteristic
234C2×C22⋊C4D4×C2311yescharacteristic
235C2×C42C2×C4×C2311yescharacteristic
236C2×C42C2×C4×C2311yescharacteristic
237C2×C42D4×C2311yescharacteristic
238C2×C42D4×C2311yescharacteristic
239C4⋊C8—24no—
240C22⋊C8—22no—
241C22⋊C8D4×C2314yes—
242#97—22no—
243C4×C8—22no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249C23#944611yescharacteristic, direct product factor, Sylow
250C24—21no—
251C24C2×D4×C2311yescharacteristic
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4C22⋊C4×C2311yescharacteristic
265C22×C4C22⋊C4×C2311yescharacteristic
266C22×C4C22⋊C4×C2311yescharacteristic
267C22×C4C22⋊C4×C2311yescharacteristic
268C22×C4C22×C4×C2311yescharacteristic
269C22×C4C2×D4×C2311yescharacteristic
270C22×C4C2×D4×C2312yes—
271C22×C4#41×C2312yes—
272M16—42no—
273M16—42no—
274C2×C8—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—41no—
296C42—41no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊C4×C2311yescharacteristic
309C23C23×#12112yes—
310D4—82no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4C4×D4×C2312yes—
339C2×C4C23×#12011yescharacteristic
340C2×C4C23×#12311yescharacteristic
341C8—44no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C23×#38611yescharacteristic
357C22C23×#41011yescharacteristic
358C22C23×#41011yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—162no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2C23×#188411yescharacteristic
375C2C23×#189911yescharacteristic
376C2C23×#191011yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2921C2semidirect
3. C23×#1820C2semidirect
5. C23×#1284C2semidirect
14. C23×#390C4semidirect
15. C23×#390C4semidirect
18. C23×#384C4semidirect
19. C23×#382C4semidirect
108. #9446C23direct (N × H)
154. #2921C2×C23semidirect
156. #1820C2×C23semidirect
158. #1284C2×C23semidirect
195. #390C4×C23semidirect
196. #390C4×C23semidirect
199. #384C4×C23semidirect
200. #382C4×C23semidirect
249. C23#9446direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29212
C23×#25002
C23×#18202
C23×#15102
C23×#12842
#944623