Finite Group Databasebeta

← C23×#10619 #19158245

Subgroups — C23×#10619 #19158245

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1954C211yesmaximal, characteristic
2C23×#1844C211yesmaximal, characteristic, complemented
3C23×#1802C211yesmaximal, characteristic
4C23×#1802C211yesmaximal, characteristic
5C23×#1296C211yesmaximal, characteristic, complemented
6C23×#1296C211yesmaximal, characteristic, complemented
7C23×#1277C211yesmaximal, characteristic, complemented
8C2×C4×D4×C23C2211yescharacteristic
9C2×C23×#106—21no—
10C2×C23×#106—21no—
11C2×C23×C4⋊C8C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C2×C23×C4⋊C8C2211yescharacteristic
14C2×C23×#102—21no—
15C2×C23×#102—21no—
16C2×C23×C22⋊C8—21no—
17C2×C23×C22⋊C8—21no—
18C2×C4×C8×C23C2211yescharacteristic
19C23×#384C2211yescharacteristic
20C23×#384C2211yescharacteristic
21C23×#340—21no—
22C23×#340—21no—
23C22×D4×C23D411yescharacteristic
24C23×C4×C23—21no—
25C22×C8×C23—21no—
26C22×C8×C23—21no—
27C22×C8×C23—21no—
28C22×C8×C23D411yescharacteristic
29C22×C8×C23D411yescharacteristic
30C4×D4×C23—21no—
31C4×D4×C23—21no—
32C4×D4×C23—21no—
33C4×D4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C42×C23C2311yescharacteristic
43C23×#106—42no—
44C23×#106—42no—
45C23×C4⋊C8—21no—
46C23×C4⋊C8—21no—
47C23×C4⋊C8—21no—
48C23×C4⋊C8—21no—
49C23×C4⋊C8—21no—
50C23×C4⋊C8—21no—
51C23×#102—42no—
52C23×#102—42no—
53C23×C22⋊C8—42no—
54C23×C22⋊C8—42no—
55C4×C8×C23—21no—
56C4×C8×C23—21no—
57C24×C23—21no—
58C2×D4×C23—21no—
59C2×D4×C23—21no—
60C2×D4×C23—21no—
61C2×D4×C23—21no—
62C2×D4×C23—21no—
63C2×D4×C23SD1612yes—
64C22×C4×C23—21no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23—41no—
68C22×C4×C23—41no—
69C22×C4×C23—41no—
70C22×C4×C23—41no—
71C22×C4×C23—41no—
72C22×C4×C23—41no—
73C22×C4×C23C2×D411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23#4111yescharacteristic
76C2×C8×C23—21no—
77C2×C8×C23—21no—
78C2×C8×C23—21no—
79C2×C8×C23—21no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23—41no—
83C2×C8×C23—41no—
84C2×C8×C23—41no—
85C2×C8×C23—41no—
86C2×C8×C23—41no—
87C2×C8×C23—41no—
88C2×C8×C23—41no—
89C2×C8×C23D812yes—
90C2×C8×C23SD1612yes—
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—41no—
94C4⋊C4×C23—41no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—81no—
102C4⋊C4×C23—81no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—41no—
105C42×C23#4111yescharacteristic
106C42×C23#4111yescharacteristic
107C42×C23#4111yescharacteristic
108C42×C23#4111yescharacteristic
109#10619C2311yesmaximal, characteristic, direct product factor, Sylow
110C23×C23—22no—
111C23×C23—41no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23—41no—
117C23×C23#12111yescharacteristic
118D4×C23—22no—
119D4×C23—22no—
120D4×C23—41no—
121D4×C23—41no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—81no—
128C2×C4×C23—81no—
129C2×C4×C23—81no—
130C2×C4×C23—81no—
131C2×C4×C23—81no—
132C2×C4×C23—81no—
133C2×C4×C23—81no—
134C2×C4×C23—81no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23#12311yescharacteristic
138C2×C4×C23#12311yescharacteristic
139C2×C4×C23#12311yescharacteristic
140C2×C4×C23#12411yescharacteristic
141C2×C4×C23#12611yescharacteristic
142C2×C4×C23#12611yescharacteristic
143C2×C4×C23C2×D811yescharacteristic
144C2×C4×C23C2×SD1611yescharacteristic
145C2×C4×C23C2×SD1611yescharacteristic
146C2×C4×C23#13511yescharacteristic
147C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C23#13711yescharacteristic
150C2×C4×C23#13711yescharacteristic
151C8×C23—22no—
152C8×C23—22no—
153C8×C23—81no—
154C8×C23—81no—
155C8×C23—81no—
156#1954C2×C2311yescharacteristic
157#1844C2×C2311yescharacteristic, complemented
158#1802C2×C2311yescharacteristic
159#1802C2×C2311yescharacteristic
160#1296C2×C2311yescharacteristic, complemented
161#1296C2×C2311yescharacteristic, complemented
162#1277C2×C2311yescharacteristic, complemented
163C22×C23—42no—
164C22×C23—42no—
165C22×C23—42no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23—81no—
169C22×C23—81no—
170C22×C23#39711yescharacteristic
171C22×C23#45911yescharacteristic
172C22×C23#46111yescharacteristic
173C22×C23#46511yescharacteristic
174C22×C23#46811yescharacteristic
175C22×C23#47111yescharacteristic
176C22×C23#47111yescharacteristic
177C4×C23—21no—
178C4×C23—21no—
179C4×C23—81no—
180C4×C23—161no—
181C4×C23—161no—
182C4×C23#48011yescharacteristic
183C4×C23#48111yescharacteristic
184C4×C23#48111yescharacteristic
185C4×C23#48211yescharacteristic
186C4×C23#48311yescharacteristic
187C4×C23#48511yescharacteristic
188C4×C23#48611yescharacteristic
189C4×C23#48911yescharacteristic
190C2×C4×D4C22×C2311yescharacteristic
191C2×#106—21no—
192C2×#106—21no—
193C2×C4⋊C8C22×C2311yescharacteristic
194C2×C4⋊C8C22×C2311yescharacteristic
195C2×C4⋊C8C22×C2311yescharacteristic
196C2×#102—21no—
197C2×#102—21no—
198C2×C22⋊C8—21no—
199C2×C22⋊C8—21no—
200C2×C4×C8C22×C2311yescharacteristic
201#384C22×C2311yescharacteristic
202#384C22×C2311yescharacteristic
203#340—21no—
204#340—21no—
205C2×C23—82no—
206C2×C23#166511yescharacteristic
207C2×C23#167111yescharacteristic
208C2×C23#168511yescharacteristic
209C2×C23#169111yescharacteristic
210C2×C23#207111yescharacteristic
211C2×C23#207711yescharacteristic
212C2×C23#207911yescharacteristic
213C22×D4D4×C2311yescharacteristic
214C23×C4—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8D4×C2311yescharacteristic
219C22×C8D4×C2311yescharacteristic
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C2311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C42C23×C2311yescharacteristic
233#106—42no—
234#106—42no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241#102—42no—
242#102—42no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C4×C8—21no—
246C4×C8—21no—
247C23#1061911yescharacteristic, direct product factor, Sylow
248C24—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4SD16×C2312yes—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C2×D4×C2311yescharacteristic
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4#41×C2311yescharacteristic
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8D8×C2312yes—
281C2×C8SD16×C2312yes—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42#41×C2311yescharacteristic
297C42#41×C2311yescharacteristic
298C42#41×C2311yescharacteristic
299C42#41×C2311yescharacteristic
300C23—22no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C23×#12111yescharacteristic
308D4—22no—
309D4—22no—
310D4—41no—
311D4—41no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C23×#12311yescharacteristic
328C2×C4C23×#12311yescharacteristic
329C2×C4C23×#12311yescharacteristic
330C2×C4C23×#12411yescharacteristic
331C2×C4C23×#12611yescharacteristic
332C2×C4C23×#12611yescharacteristic
333C2×C4C2×D8×C2311yescharacteristic
334C2×C4C2×SD16×C2311yescharacteristic
335C2×C4C2×SD16×C2311yescharacteristic
336C2×C4C23×#13511yescharacteristic
337C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
338C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
339C2×C4C23×#13711yescharacteristic
340C2×C4C23×#13711yescharacteristic
341C8—22no—
342C8—22no—
343C8—81no—
344C8—81no—
345C8—81no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22C23×#39711yescharacteristic
354C22C23×#45911yescharacteristic
355C22C23×#46111yescharacteristic
356C22C23×#46511yescharacteristic
357C22C23×#46811yescharacteristic
358C22C23×#47111yescharacteristic
359C22C23×#47111yescharacteristic
360C4—21no—
361C4—21no—
362C4—81no—
363C4—161no—
364C4—161no—
365C4C23×#48011yescharacteristic
366C4C23×#48111yescharacteristic
367C4C23×#48111yescharacteristic
368C4C23×#48211yescharacteristic
369C4C23×#48311yescharacteristic
370C4C23×#48511yescharacteristic
371C4C23×#48611yescharacteristic
372C4C23×#48911yescharacteristic
373C2—82no—
374C2C23×#166511yescharacteristic
375C2C23×#167111yescharacteristic
376C2C23×#168511yescharacteristic
377C2C23×#169111yescharacteristic
378C2C23×#207111yescharacteristic
379C2C23×#207711yescharacteristic
380C2C23×#207911yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1844C2semidirect
5. C23×#1296C2semidirect
6. C23×#1296C2semidirect
7. C23×#1277C2semidirect
109. #10619C23direct (N × H)
157. #1844C2×C23semidirect
160. #1296C2×C23semidirect
161. #1296C2×C23semidirect
162. #1277C2×C23semidirect
247. C23#10619direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#19542
C23×#18442
C23×#18022
C23×#12962
C23×#12772
#1061923