Finite Group Databasebeta

← C23×#10421 #19158047

Subgroups — C23×#10421 #19158047

Order 5888.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2542C211yesmaximal, characteristic
2C23×#1990C211yesmaximal, characteristic, complemented
3C23×#1947C211yesmaximal, characteristic, complemented
4C23×#1802C211yesmaximal, characteristic
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C2×C23×D4⋊C8C211yesmaximal, characteristic
7C23×#1279C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C2×C23×C22⋊Q8—21no—
10C2×C23×C22⋊Q8—21no—
11C2×C4×D4×C23C2211yescharacteristic
12C2×C23×#107—21no—
13C2×C23×C4⋊C8C2211yescharacteristic
14C2×C23×C4⋊C8C2211yescharacteristic
15C2×C23×#102—21no—
16C2×C4×C8×C23C2211yescharacteristic
17C23×#399—22no—
18C23×#390—21no—
19C23×#384—21no—
20C23×#384C2211yescharacteristic
21C4×C4⋊C4×C23C2211yescharacteristic
22C23×#340—21no—
23C23×#340—22no—
24C23×#340—21no—
25C23×#332—22no—
26C23×D4⋊C8—22no—
27C22×Q8×C23—21no—
28C22×D4×C23D411yescharacteristic
29C23×C4×C23—21no—
30C22×C8×C23—21no—
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C23×C4⋊Q8—21no—
34C23×C4⋊Q8—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—21no—
37C23×C22⋊Q8—42no—
38C23×C22⋊Q8—42no—
39C23×C22⋊Q8—42no—
40C23×C22⋊Q8—42no—
41C4×D4×C23—21no—
42C4×D4×C23—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C2×C4⋊C4×C23—22no—
46C2×C4⋊C4×C23—22no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23D411yescharacteristic
52C2×C4⋊C4×C23D412yes—
53C2×C4⋊C4×C23D411yescharacteristic
54C2×C4⋊C4×C23D411yescharacteristic
55C2×C22⋊C4×C23—22no—
56C2×C22⋊C4×C23—22no—
57C2×C22⋊C4×C23—21no—
58C2×C42×C23—21no—
59C2×C42×C23C2311yescharacteristic
60C23×#107—42no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8D412yes—
65C23×#102—42no—
66C4×C8×C23—21no—
67C4×C8×C23—21no—
68C23×#95—21no—
69C24×C23—21no—
70C2×Q8×C23—41no—
71C2×Q8×C23—41no—
72C2×Q8×C23—41no—
73C2×Q8×C23—41no—
74C2×Q8×C23—41no—
75C2×Q8×C23—41no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23Q1612yes—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—22no—
85C22×C4×C23—21no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96C2×C8×C23—22no—
97C2×C8×C23—22no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—44no—
112C4⋊C4×C23—44no—
113C4⋊C4×C23—42no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23—41no—
120C4⋊C4×C23D812yes—
121C4⋊C4×C23SD1612yes—
122C22⋊C4×C23—44no—
123C22⋊C4×C23—44no—
124C22⋊C4×C23—41no—
125C22⋊C4×C23—41no—
126C42×C23—41no—
127C42×C23—41no—
128C42×C23C2×D411yescharacteristic
129C42×C23#4111yescharacteristic
130C42×C23#4111yescharacteristic
131C42×C23#4111yescharacteristic
132#10421C2311yesmaximal, characteristic, direct product factor, Sylow
133C23×C23—22no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—41no—
137C23×C23—41no—
138C23×C23—41no—
139C23×C23—41no—
140C23×C23#12011yescharacteristic
141Q8×C23—81no—
142Q8×C23—81no—
143Q8×C23—81no—
144Q8×C23—81no—
145D4×C23—22no—
146D4×C23—22no—
147D4×C23—41no—
148D4×C23—41no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—22no—
152C2×C4×C23—22no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—42no—
159C2×C4×C23—42no—
160C2×C4×C23—41no—
161C2×C4×C23—42no—
162C2×C4×C23—41no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23—81no—
166C2×C4×C23—81no—
167C2×C4×C23—81no—
168C2×C4×C23—81no—
169C2×C4×C23—81no—
170C2×C4×C23#12111yescharacteristic
171C2×C4×C23#12111yescharacteristic
172C2×C4×C23#12111yescharacteristic
173C2×C4×C23#12311yescharacteristic
174C2×C4×C23#12311yescharacteristic
175C2×C4×C23#12411yescharacteristic
176C2×C4×C23C2×D811yescharacteristic
177C2×C4×C23C2×SD1611yescharacteristic
178C2×C4×C23C2×Q1611yescharacteristic
179C2×C4×C23#13511yescharacteristic
180C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C23#13711yescharacteristic
184C8×C23—42no—
185C8×C23—42no—
186C8×C23—81no—
187#2542C2×C2311yescharacteristic
188#1990C2×C2311yescharacteristic, complemented
189#1947C2×C2311yescharacteristic, complemented
190#1802C2×C2311yescharacteristic
191C2×#332C2×C2311yescharacteristic, complemented
192C2×D4⋊C8C2×C2311yescharacteristic
193#1279C2×C2311yescharacteristic, complemented
194C22×C23—42no—
195C22×C23—42no—
196C22×C23—42no—
197C22×C23—81no—
198C22×C23—81no—
199C22×C23—81no—
200C22×C23—81no—
201C22×C23#39411yescharacteristic
202C22×C23#44711yescharacteristic
203C22×C23#44911yescharacteristic
204C22×C23#45011yescharacteristic
205C22×C23#45111yescharacteristic
206C22×C23#45411yescharacteristic
207C22×C23#45611yescharacteristic
208C4×C23—21no—
209C4×C23—21no—
210C4×C23—42no—
211C4×C23—42no—
212C4×C23—81no—
213C4×C23—82no—
214C4×C23—161no—
215C4×C23#45911yescharacteristic
216C4×C23#46011yescharacteristic
217C4×C23#46711yescharacteristic
218C4×C23#46811yescharacteristic
219C4×C23#48411yescharacteristic
220C4×C23#48511yescharacteristic
221C4×C23#48611yescharacteristic
222C4×C23#49011yescharacteristic
223C2×C4⋊Q8C22×C2311yescharacteristic
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—21no—
226C2×C4×D4C22×C2311yescharacteristic
227C2×#107—21no—
228C2×C4⋊C8C22×C2311yescharacteristic
229C2×C4⋊C8C22×C2311yescharacteristic
230C2×#102—21no—
231C2×C4×C8C22×C2311yescharacteristic
232#399—22no—
233#390—21no—
234#384—21no—
235#384C22×C2311yescharacteristic
236C4×C4⋊C4C22×C2311yescharacteristic
237#340—21no—
238#340—21no—
239#340—22no—
240#332—22no—
241D4⋊C8—22no—
242C2×C23—82no—
243C2×C23#162511yescharacteristic
244C2×C23#163711yescharacteristic
245C2×C23#164911yescharacteristic
246C2×C23#165411yescharacteristic
247C2×C23#203611yescharacteristic
248C2×C23#203811yescharacteristic
249C2×C23#204111yescharacteristic
250C22×Q8—21no—
251C22×D4D4×C2311yescharacteristic
252C23×C4—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C22⋊Q8—42no—
261C22⋊Q8—42no—
262C22⋊Q8—42no—
263C22⋊Q8—42no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4D4×C2311yescharacteristic
275C2×C4⋊C4D4×C2312yes—
276C2×C4⋊C4D4×C2311yescharacteristic
277C2×C4⋊C4D4×C2311yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42C23×C2311yescharacteristic
283#107—42no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8D4×C2312yes—
288#102—42no—
289C4×C8—21no—
290C4×C8—21no—
291#95—21no—
292C23#1042111yescharacteristic, direct product factor, Sylow
293C24—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4Q16×C2312yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×D4×C2311yescharacteristic
318C22×C4C2×D4×C2311yescharacteristic
319C22×C4C2×D4×C2311yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4D8×C2312yes—
345C4⋊C4SD16×C2312yes—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—41no—
351C42—41no—
352C42C2×D4×C2311yescharacteristic
353C42#41×C2311yescharacteristic
354C42#41×C2311yescharacteristic
355C42#41×C2311yescharacteristic
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C23×#12011yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368D4—22no—
369D4—22no—
370D4—41no—
371D4—41no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C23×#12111yescharacteristic
394C2×C4C23×#12111yescharacteristic
395C2×C4C23×#12111yescharacteristic
396C2×C4C23×#12311yescharacteristic
397C2×C4C23×#12311yescharacteristic
398C2×C4C23×#12411yescharacteristic
399C2×C4C2×D8×C2311yescharacteristic
400C2×C4C2×SD16×C2311yescharacteristic
401C2×C4C2×Q16×C2311yescharacteristic
402C2×C4C23×#13511yescharacteristic
403C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
404C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C23×#13711yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22C23×#39411yescharacteristic
418C22C23×#44711yescharacteristic
419C22C23×#44911yescharacteristic
420C22C23×#45011yescharacteristic
421C22C23×#45111yescharacteristic
422C22C23×#45411yescharacteristic
423C22C23×#45611yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4—81no—
429C4—82no—
430C4—161no—
431C4C23×#45911yescharacteristic
432C4C23×#46011yescharacteristic
433C4C23×#46711yescharacteristic
434C4C23×#46811yescharacteristic
435C4C23×#48411yescharacteristic
436C4C23×#48511yescharacteristic
437C4C23×#48611yescharacteristic
438C4C23×#49011yescharacteristic
439C2—82no—
440C2C23×#162511yescharacteristic
441C2C23×#163711yescharacteristic
442C2C23×#164911yescharacteristic
443C2C23×#165411yescharacteristic
444C2C23×#203611yescharacteristic
445C2C23×#203811yescharacteristic
446C2C23×#204111yescharacteristic
subgroupG/H
center, socleC23×C23#120
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×#1990C2semidirect
3. C23×#1947C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1279C2semidirect
132. #10421C23direct (N × H)
188. #1990C2×C23semidirect
189. #1947C2×C23semidirect
191. C2×#332C2×C23semidirect
193. #1279C2×C23semidirect
292. C23#10421direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25422
C23×#19902
C23×#19472
C23×#18022
C2×C23×#3322
C2×C23×D4⋊C82
C23×#12792
#1042123