Finite Group Databasebeta

← C23×#9056 #19156682

Subgroups — C23×#9056 #19156682

Order 5888.

432 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×#3170C211yesmaximal, characteristic
2C23×#2962C211yesmaximal, characteristic
3C23×#1986C211yesmaximal, characteristic
4C23×#1503C211yesmaximal, characteristic
5C23×#1481C211yesmaximal, characteristic
6C23×#1279C211yesmaximal, characteristic
7C23×#1278C211yesmaximal, characteristic
8C23×#533C2211yescharacteristic
9C23×#533C2211yescharacteristic
10C23×#500C411yescharacteristic, complemented
11C23×#494C411yescharacteristic, complemented
12C23×#487C411yescharacteristic, complemented
13C23×#487C411yescharacteristic, complemented
14C23×#467—22no—
15C23×#446—22no—
16C23×#446—21no—
17C8×Q8×C23—21no—
18C4×Q16×C23—22no—
19C23×#431C2211yescharacteristic
20C23×#424—21no—
21C2×C23×C4⋊C8C2211yescharacteristic
22C23×#405—21no—
23C4×M16×C23C2211yescharacteristic
24C2×C4×C8×C23C2211yescharacteristic
25C4×C4⋊C4×C23C2211yescharacteristic
26C23×#339—21no—
27C23×#339—21no—
28C23×#328—22no—
29C23×Q8⋊C8—21no—
30C23×Q8⋊C8—21no—
31C2×Q16×C23—22no—
32C2×M16×C23D411yescharacteristic
33C2×M16×C23D411yescharacteristic
34C22×C8×C23—21no—
35C22×C8×C23—21no—
36C23×C4⋊Q8C2×C411yescharacteristic
37C23×C4⋊Q8C2×C411yescharacteristic
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×#125—21no—
41C23×#125—21no—
42C23×C22⋊Q8—21no—
43C23×C22⋊Q8—21no—
44C23×C22⋊Q8—21no—
45C23×C22⋊Q8D412yes—
46C4×Q8×C23—21no—
47C4×Q8×C23—21no—
48C4×Q8×C23—21no—
49C4×Q8×C23D412yes—
50C23×#117—21no—
51C23×#117—21no—
52C2×C4⋊C4×C23—21no—
53C2×C42×C23—21no—
54C2×C42×C23C2311yescharacteristic
55C23×#107—22no—
56C23×C4⋊C8—22no—
57C23×C4⋊C8—22no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8D411yescharacteristic
61C23×C4⋊C8D411yescharacteristic
62C23×#103—22no—
63C23×#103—22no—
64C23×C22⋊C8—21no—
65C23×#97—22no—
66C23×#97—21no—
67C4×C8×C23—22no—
68C4×C8×C23—21no—
69C4×C8×C23—21no—
70C4×C8×C23—21no—
71C4×C8×C23C2×C411yescharacteristic
72C4×C8×C23C2×C411yescharacteristic
73C23×#95—21no—
74C2×Q8×C23—21no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23C22⋊C412yes—
78C22×C4×C23—41no—
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23C2×D411yescharacteristic
81C22×C4×C23#4111yescharacteristic
82Q16×C23—44no—
83M16×C23—21no—
84M16×C23—21no—
85M16×C23—21no—
86M16×C23Q1612yescomplemented
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—21no—
93C2×C8×C23—42no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23—41no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—22no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—41no—
108C4⋊C4×C23—41no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23C22⋊C412yes—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—41no—
113C42×C23—22no—
114C42×C23—41no—
115C42×C23—41no—
116C42×C23—41no—
117C42×C23C22×C411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23C2×D411yescharacteristic
120C42×C23#4111yescharacteristic
121#9056C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23#12111yescharacteristic
123Q8×C23—42no—
124Q8×C23—42no—
125Q8×C23—41no—
126Q8×C23—41no—
127Q8×C23—41no—
128Q8×C23—41no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—22no—
132C2×C4×C23—22no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—81no—
138C2×C4×C23—81no—
139C2×C4×C23C2×C22⋊C411yescharacteristic
140C2×C4×C23C4×D411yescharacteristic
141C2×C4×C23C4×D411yescharacteristic
142C2×C4×C23#12011yescharacteristic
143C2×C4×C23#12111yescharacteristic
144C2×C4×C23#12311yescharacteristic
145C2×C4×C23C2×Q1611yescharacteristic
146C2×C4×C23#13511yescharacteristic
147C2×C4×C23#13711yescharacteristic
148C2×C4×C23#13711yescharacteristic
149C8×C23—22no—
150C8×C23—22no—
151C8×C23—41no—
152C8×C23—41no—
153C8×C23—41no—
154C8×C23—41no—
155C8×C23—81no—
156#3170C2×C2311yescharacteristic
157#2962C2×C2311yescharacteristic
158#1986C2×C2311yescharacteristic
159#1503C2×C2311yescharacteristic
160#1481C2×C2311yescharacteristic
161#1279C2×C2311yescharacteristic
162#1278C2×C2311yescharacteristic
163C22×C23—21no—
164C22×C23—21no—
165C22×C23#38611yescharacteristic
166C22×C23#46211yescharacteristic
167C22×C23#46411yescharacteristic
168C4×C23—21no—
169C4×C23—21no—
170C4×C23—21no—
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—82no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23C4×Q1611yescharacteristic
177C4×C23#44111yescharacteristic
178C4×C23#44311yescharacteristic
179C4×C23#44411yescharacteristic
180C4×C23#45111yescharacteristic
181C4×C23#45211yescharacteristic
182#533C22×C2311yescharacteristic
183#533C22×C2311yescharacteristic
184#500C4×C2311yescharacteristic, complemented
185#494C4×C2311yescharacteristic, complemented
186#487C4×C2311yescharacteristic, complemented
187#487C4×C2311yescharacteristic, complemented
188#467—22no—
189#446—22no—
190#446—21no—
191C8×Q8—21no—
192C4×Q16—22no—
193#431C22×C2311yescharacteristic
194#424—21no—
195C2×C4⋊C8C22×C2311yescharacteristic
196#405—21no—
197C4×M16C22×C2311yescharacteristic
198C2×C4×C8C22×C2311yescharacteristic
199C4×C4⋊C4C22×C2311yescharacteristic
200#339—21no—
201#339—21no—
202#328—22no—
203Q8⋊C8—21no—
204Q8⋊C8—21no—
205C2×C23—21no—
206C2×C23—21no—
207C2×C23#187611yescharacteristic
208C2×C23#187811yescharacteristic
209C2×C23#188411yescharacteristic
210C2×Q16—22no—
211C2×M16D4×C2311yescharacteristic
212C2×M16D4×C2311yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C4⋊Q8C2×C4×C2311yescharacteristic
216C4⋊Q8C2×C4×C2311yescharacteristic
217C4⋊Q8C2×C4×C2311yescharacteristic
218C4⋊Q8C2×C4×C2311yescharacteristic
219#125—21no—
220#125—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8D4×C2312yes—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8D4×C2312yes—
229#117—21no—
230#117—21no—
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42C23×C2311yescharacteristic
234#107—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8D4×C2311yescharacteristic
240C4⋊C8D4×C2311yescharacteristic
241#103—22no—
242#103—22no—
243C22⋊C8—21no—
244#97—22no—
245#97—21no—
246C4×C8—22no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C2×C4×C2311yescharacteristic
251C4×C8C2×C4×C2311yescharacteristic
252#95—21no—
253C23#905611yescharacteristic, direct product factor, Sylow
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊C4×C2312yes—
258C22×C4—41no—
259C22×C4C2×D4×C2311yescharacteristic
260C22×C4C2×D4×C2311yescharacteristic
261C22×C4#41×C2311yescharacteristic
262Q16—44no—
263M16—21no—
264M16—21no—
265M16—21no—
266M16Q16×C2312yescomplemented
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4C22⋊C4×C2312yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42C22×C4×C2311yescharacteristic
298C42C2×D4×C2311yescharacteristic
299C42C2×D4×C2311yescharacteristic
300C42#41×C2311yescharacteristic
301C23C23×#12111yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4C2×C22⋊C4×C2311yescharacteristic
319C2×C4C4×D4×C2311yescharacteristic
320C2×C4C4×D4×C2311yescharacteristic
321C2×C4C23×#12011yescharacteristic
322C2×C4C23×#12111yescharacteristic
323C2×C4C23×#12311yescharacteristic
324C2×C4C2×Q16×C2311yescharacteristic
325C2×C4C23×#13511yescharacteristic
326C2×C4C23×#13711yescharacteristic
327C2×C4C23×#13711yescharacteristic
328C8—22no—
329C8—22no—
330C8—41no—
331C8—41no—
332C8—41no—
333C8—41no—
334C8—81no—
335C22—21no—
336C22—21no—
337C22C23×#38611yescharacteristic
338C22C23×#46211yescharacteristic
339C22C23×#46411yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4C4×Q16×C2311yescharacteristic
349C4C23×#44111yescharacteristic
350C4C23×#44311yescharacteristic
351C4C23×#44411yescharacteristic
352C4C23×#45111yescharacteristic
353C4C23×#45211yescharacteristic
354C2—21no—
355C2—21no—
356C2C23×#187611yescharacteristic
357C2C23×#187811yescharacteristic
358C2C23×#188411yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×C23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
10. C23×#500C4semidirect
11. C23×#494C4semidirect
12. C23×#487C4semidirect
13. C23×#487C4semidirect
86. M16×C23Q16semidirect
121. #9056C23direct (N × H)
184. #500C4×C23semidirect
185. #494C4×C23semidirect
186. #487C4×C23semidirect
187. #487C4×C23semidirect
253. C23#9056direct (N × H)
266. M16Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31702
C23×#29622
C23×#19862
C23×#15032
C23×#14812
C23×#12792
C23×#12782
#905623