Finite Group Databasebeta

← C23×#8904 #19156530

Subgroups — C23×#8904 #19156530

Order 5888.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2906C211yesmaximal, characteristic
2C23×#1806C211yesmaximal, characteristic
3C23×#1570C211yesmaximal, characteristic, complemented
4C23×#1566C211yesmaximal, characteristic, complemented
5C23×#1476C211yesmaximal, characteristic
6C23×#1303C211yesmaximal, characteristic, complemented
7C23×#1279C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C4×#41×C23C2211yescharacteristic
10C23×#431C2211yescharacteristic
11C23×#424—21no—
12C23×#424—22no—
13C23×#423—21no—
14C23×#423C2211yescharacteristic
15C23×#423C2211yescharacteristic
16C2×C23×C4⋊C8—21no—
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×#104—21no—
19C2×C4×C8×C23C2211yescharacteristic
20C4×C4⋊C4×C23C2211yescharacteristic
21C23×#339—21no—
22C23×#334—22no—
23C23×#334—21no—
24C23×#334—21no—
25C23×Q8⋊C8C411yescharacteristic, complemented
26C23×Q8⋊C8C411yescharacteristic, complemented
27C23×D4⋊C8C411yescharacteristic, complemented
28C23×D4⋊C8C411yescharacteristic, complemented
29C2×#41×C23D411yescharacteristic
30C23×#131—42no—
31C23×#131—42no—
32C2×M16×C23—22no—
33C2×M16×C23—21no—
34C2×M16×C23—21no—
35C22×C8×C23—22no—
36C22×C8×C23—21no—
37C22×C8×C23—21no—
38C4×Q8×C23C2×C411yescharacteristic
39C4×Q8×C23Q811yescharacteristic
40C4×D4×C23—21no—
41C4×D4×C23—21no—
42C4×D4×C23C2×C411yescharacteristic
43C4×D4×C23Q811yescharacteristic
44C23×#117—21no—
45C23×#117D411yescharacteristic
46C2×C4⋊C4×C23—21no—
47C2×C42×C23—21no—
48C2×C42×C23—21no—
49C2×C42×C23C2311yescharacteristic
50C23×C4⋊C8—22no—
51C23×C4⋊C8—22no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8C2×C411yescharacteristic
57C23×C4⋊C8C2×C411yescharacteristic
58C23×C4⋊C8D411yescharacteristic
59C23×C4⋊C8D411yescharacteristic
60C23×C4⋊C8D412yes—
61C23×#104—42no—
62C23×C22⋊C8—21no—
63C4×C8×C23—21no—
64C4×C8×C23—21no—
65C4×C8×C23C2×C411yescharacteristic
66C4×C8×C23C2×C411yescharacteristic
67C23×#95—21no—
68#41×C23—21no—
69#41×C23—21no—
70#41×C23SD1612yes—
71#41×C23SD1612yes—
72C2×Q8×C23C4⋊C411yescharacteristic
73C2×D4×C23—21no—
74C2×D4×C23C4⋊C411yescharacteristic
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—41no—
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23#4111yescharacteristic
81C22×C4×C23#4111yescharacteristic
82M16×C23—44no—
83M16×C23—41no—
84M16×C23—41no—
85M16×C23—41no—
86M16×C23—41no—
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—21no—
91C2×C8×C23—21no—
92C2×C8×C23—22no—
93C2×C8×C23—21no—
94C2×C8×C23—42no—
95C2×C8×C23—44no—
96C2×C8×C23—41no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23C4⋊C411yescharacteristic
104C4⋊C4×C23C4⋊C411yescharacteristic
105C22⋊C4×C23—21no—
106C22⋊C4×C23—21no—
107C22⋊C4×C23—21no—
108C42×C23—21no—
109C42×C23—21no—
110C42×C23—21no—
111C42×C23—41no—
112C42×C23—41no—
113C42×C23C22×C411yescharacteristic
114C42×C23C2×D411yescharacteristic
115C42×C23C2×D411yescharacteristic
116C42×C23C2×Q811yescharacteristic
117#8904C2311yesmaximal, characteristic, direct product factor, Sylow
118C23×C23—21no—
119C23×C23#12311yescharacteristic
120Q8×C23—21no—
121Q8×C23#10612yes—
122D4×C23—22no—
123D4×C23—22no—
124D4×C23—21no—
125D4×C23#10612yes—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—22no—
130C2×C4×C23—22no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—42no—
137C2×C4×C23—81no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23C2×C4⋊C411yescharacteristic
141C2×C4×C23C4×D411yescharacteristic
142C2×C4×C23C4×D411yescharacteristic
143C2×C4×C23#12011yescharacteristic
144C2×C4×C23C22⋊Q811yescharacteristic
145C2×C4×C23C22⋊Q811yescharacteristic
146C2×C4×C23C2×SD1611yescharacteristic
147C2×C4×C23C2×SD1611yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C23#13711yescharacteristic
150C8×C23—22no—
151C8×C23—41no—
152C8×C23—41no—
153C8×C23—41no—
154C8×C23—41no—
155C8×C23—42no—
156C8×C23—41no—
157C8×C23—81no—
158#2906C2×C2311yescharacteristic
159#1806C2×C2311yescharacteristic
160#1570C2×C2311yescharacteristic, complemented
161#1566C2×C2311yescharacteristic, complemented
162#1476C2×C2311yescharacteristic
163#1303C2×C2311yescharacteristic, complemented
164#1279C2×C2311yescharacteristic, complemented
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—22no—
168C22×C23—41no—
169C22×C23—41no—
170C22×C23#38511yescharacteristic
171C22×C23#48111yescharacteristic
172C22×C23#48311yescharacteristic
173C4×C23—22no—
174C4×C23—21no—
175C4×C23—21no—
176C4×C23—21no—
177C4×C23—41no—
178C4×C23—41no—
179C4×C23—82no—
180C4×C23C2×#10611yescharacteristic
181C4×C23#42811yescharacteristic
182C4×C23#44311yescharacteristic
183C4×C23#44411yescharacteristic
184C4×C23#44811yescharacteristic
185C4×C23#45011yescharacteristic
186C2×#131—21no—
187C4×#41C22×C2311yescharacteristic
188#431C22×C2311yescharacteristic
189#424—21no—
190#424—22no—
191#423—21no—
192#423C22×C2311yescharacteristic
193#423C22×C2311yescharacteristic
194C2×C4⋊C8—21no—
195C2×C4⋊C8C22×C2311yescharacteristic
196C2×#104—21no—
197C2×C4×C8C22×C2311yescharacteristic
198C4×C4⋊C4C22×C2311yescharacteristic
199#339—21no—
200#334—22no—
201#334—21no—
202#334—21no—
203Q8⋊C8C4×C2311yescharacteristic, complemented
204Q8⋊C8C4×C2311yescharacteristic, complemented
205D4⋊C8C4×C2311yescharacteristic, complemented
206D4⋊C8C4×C2311yescharacteristic, complemented
207C2×C23—21no—
208C2×C23—21no—
209C2×C23—42no—
210C2×C23#185311yescharacteristic
211C2×C23#186111yescharacteristic
212C2×C23#186611yescharacteristic
213C2×#41D4×C2311yescharacteristic
214#131—42no—
215#131—42no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C4×Q8C2×C4×C2311yescharacteristic
223C4×Q8Q8×C2311yescharacteristic
224C4×D4—21no—
225C4×D4—21no—
226C4×D4C2×C4×C2311yescharacteristic
227C4×D4Q8×C2311yescharacteristic
228#117—21no—
229#117D4×C2311yescharacteristic
230C2×C4⋊C4—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C23×C2311yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8C2×C4×C2311yescharacteristic
241C4⋊C8C2×C4×C2311yescharacteristic
242C4⋊C8D4×C2311yescharacteristic
243C4⋊C8D4×C2311yescharacteristic
244C4⋊C8D4×C2312yes—
245#104—42no—
246C22⋊C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8C2×C4×C2311yescharacteristic
250C4×C8C2×C4×C2311yescharacteristic
251#95—21no—
252C23#890411yescharacteristic, direct product factor, Sylow
253#41—21no—
254#41—21no—
255#41SD16×C2312yes—
256#41SD16×C2312yes—
257C2×Q8C4⋊C4×C2311yescharacteristic
258C2×D4—21no—
259C2×D4C4⋊C4×C2311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4C2×D4×C2311yescharacteristic
265C22×C4#41×C2311yescharacteristic
266C22×C4#41×C2311yescharacteristic
267M16—44no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—42no—
280C2×C8—44no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4C4⋊C4×C2311yescharacteristic
289C4⋊C4C4⋊C4×C2311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C22×C4×C2311yescharacteristic
299C42C2×D4×C2311yescharacteristic
300C42C2×D4×C2311yescharacteristic
301C42C2×Q8×C2311yescharacteristic
302C23—21no—
303C23C23×#12311yescharacteristic
304Q8—21no—
305Q8C23×#10612yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4C23×#10612yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×C4⋊C4×C2311yescharacteristic
325C2×C4C4×D4×C2311yescharacteristic
326C2×C4C4×D4×C2311yescharacteristic
327C2×C4C23×#12011yescharacteristic
328C2×C4C23×C22⋊Q811yescharacteristic
329C2×C4C23×C22⋊Q811yescharacteristic
330C2×C4C2×SD16×C2311yescharacteristic
331C2×C4C2×SD16×C2311yescharacteristic
332C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
333C2×C4C23×#13711yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C8—42no—
340C8—41no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22C23×#38511yescharacteristic
348C22C23×#48111yescharacteristic
349C22C23×#48311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×C23×#10611yescharacteristic
358C4C23×#42811yescharacteristic
359C4C23×#44311yescharacteristic
360C4C23×#44411yescharacteristic
361C4C23×#44811yescharacteristic
362C4C23×#45011yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2C23×#185311yescharacteristic
367C2C23×#186111yescharacteristic
368C2C23×#186611yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#385
prime radicalC2×#41×C23D4
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
3. C23×#1570C2semidirect
4. C23×#1566C2semidirect
6. C23×#1303C2semidirect
7. C23×#1279C2semidirect
25. C23×Q8⋊C8C4semidirect
26. C23×Q8⋊C8C4semidirect
27. C23×D4⋊C8C4semidirect
28. C23×D4⋊C8C4semidirect
117. #8904C23direct (N × H)
160. #1570C2×C23semidirect
161. #1566C2×C23semidirect
163. #1303C2×C23semidirect
164. #1279C2×C23semidirect
203. Q8⋊C8C4×C23semidirect
204. Q8⋊C8C4×C23semidirect
205. D4⋊C8C4×C23semidirect
206. D4⋊C8C4×C23semidirect
252. C23#8904direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29062
C23×#18062
C23×#15702
C23×#15662
C23×#14762
C23×#13032
C23×#12792
#890423