Finite Group Databasebeta

← C23×#10540 #19158166

Subgroups — C23×#10540 #19158166

Order 5888.

506 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#486C211yesmaximal, characteristic
2C23×#1982C211yesmaximal, characteristic
3C23×#1981C211yesmaximal, characteristic, complemented
4C23×#1939C211yesmaximal, characteristic, complemented
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C2×C23×#331C211yesmaximal, characteristic
7C23×#1277C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C2×C23×#127C2211yescharacteristic
10C23×#486—22no—
11C23×#486—22no—
12C2×C23×C4⋊C8—21no—
13C2×C23×C4⋊C8C2211yescharacteristic
14C2×C23×C4⋊C8C2211yescharacteristic
15C2×C23×C4⋊C8C2211yescharacteristic
16C2×C23×#103—21no—
17C2×C23×#103—21no—
18C2×C23×#102—21no—
19C2×C23×#102—21no—
20C2×C23×#102—21no—
21C2×C23×#102—21no—
22C2×C4×C8×C23C2211yescharacteristic
23C4×C4⋊C4×C23C2211yescharacteristic
24C23×#336—21no—
25C23×#332—22no—
26C23×#331—22no—
27C22×Q8×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C22×D4×C23—21no—
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C22×C8×C23D411yescharacteristic
35C22×C8×C23D411yescharacteristic
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8D412yes—
40C23×#127—21no—
41C23×#127—21no—
42C23×#127—21no—
43C23×#127D412yes—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23D411yescharacteristic
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C42×C23—21no—
49C2×C42×C23C2311yescharacteristic
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—42no—
57C23×#103—42no—
58C23×#103—42no—
59C23×#102—24no—
60C23×#102—24no—
61C23×#102—42no—
62C23×#102—42no—
63C4×C8×C23—21no—
64C4×C8×C23—21no—
65C23×#95—21no—
66C24×C23—41no—
67C2×Q8×C23—22no—
68C2×Q8×C23—41no—
69C2×Q8×C23—41no—
70C2×Q8×C23—41no—
71C2×Q8×C23—41no—
72C2×Q8×C23—41no—
73C2×D4×C23—22no—
74C2×D4×C23—22no—
75C2×D4×C23—22no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—81no—
88C2×D4×C23—81no—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—41no—
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23#4111yescharacteristic
95C2×C8×C23—21no—
96C2×C8×C23—21no—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—41no—
105C2×C8×C23—41no—
106C2×C8×C23—41no—
107C2×C8×C23—41no—
108C2×C8×C23—41no—
109C2×C8×C23—41no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—22no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23D812yes—
118C42×C23—41no—
119C42×C23—41no—
120C42×C23C2×D411yescharacteristic
121C42×C23C2×D411yescharacteristic
122C42×C23#4111yescharacteristic
123C42×C23#4111yescharacteristic
124#10540C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—42no—
126C23×C23—81no—
127C23×C23—81no—
128C23×C23—81no—
129C23×C23—81no—
130C23×C23—81no—
131C23×C23—81no—
132C23×C23#12111yescharacteristic
133Q8×C23—42no—
134Q8×C23—42no—
135Q8×C23—81no—
136Q8×C23—81no—
137D4×C23—42no—
138D4×C23—42no—
139D4×C23—42no—
140D4×C23—42no—
141D4×C23—82no—
142D4×C23—82no—
143D4×C23—81no—
144D4×C23—81no—
145D4×C23—81no—
146D4×C23—81no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—22no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—81no—
157C2×C4×C23—81no—
158C2×C4×C23#12011yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C2×C4×C23#12111yescharacteristic
161C2×C4×C23#12311yescharacteristic
162C2×C4×C23#12311yescharacteristic
163C2×C4×C23#12411yescharacteristic
164C2×C4×C23C2×D811yescharacteristic
165C2×C4×C23#13511yescharacteristic
166C2×C4×C23#13511yescharacteristic
167C2×C4×C23#13511yescharacteristic
168C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C23#13711yescharacteristic
172C8×C23—41no—
173C8×C23—41no—
174C8×C23—81no—
175C8×C23—81no—
176C8×C23—81no—
177C2×#486C2×C2311yescharacteristic
178#1982C2×C2311yescharacteristic
179#1981C2×C2311yescharacteristic, complemented
180#1939C2×C2311yescharacteristic, complemented
181C2×#332C2×C2311yescharacteristic, complemented
182C2×#331C2×C2311yescharacteristic
183#1277C2×C2311yescharacteristic, complemented
184C22×C23—82no—
185C22×C23—82no—
186C22×C23—82no—
187C22×C23—161no—
188C22×C23—161no—
189C22×C23—161no—
190C22×C23—161no—
191C22×C23#39411yescharacteristic
192C22×C23#46311yescharacteristic
193C22×C23#46411yescharacteristic
194C22×C23#46611yescharacteristic
195C22×C23#46911yescharacteristic
196C22×C23#47111yescharacteristic
197C22×C23#47111yescharacteristic
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—42no—
201C4×C23—82no—
202C4×C23—81no—
203C4×C23#44911yescharacteristic
204C4×C23#45211yescharacteristic
205C4×C23#45911yescharacteristic
206C4×C23#45911yescharacteristic
207C4×C23#46311yescharacteristic
208C4×C23#46311yescharacteristic
209C4×C23#48811yescharacteristic
210C4×C23#48911yescharacteristic
211C2×C4⋊Q8C22×C2311yescharacteristic
212C2×#127C22×C2311yescharacteristic
213#486—22no—
214#486—22no—
215C2×C4⋊C8—21no—
216C2×C4⋊C8C22×C2311yescharacteristic
217C2×C4⋊C8C22×C2311yescharacteristic
218C2×C4⋊C8C22×C2311yescharacteristic
219C2×#103—21no—
220C2×#103—21no—
221C2×#102—21no—
222C2×#102—21no—
223C2×#102—21no—
224C2×#102—21no—
225C2×C4×C8C22×C2311yescharacteristic
226C4×C4⋊C4C22×C2311yescharacteristic
227#336—21no—
228#332—22no—
229#331—22no—
230C2×C23—162no—
231C2×C23#168111yescharacteristic
232C2×C23#168211yescharacteristic
233C2×C23#170111yescharacteristic
234C2×C23#170211yescharacteristic
235C2×C23#204211yescharacteristic
236C2×C23#204611yescharacteristic
237C2×C23#205411yescharacteristic
238C22×Q8—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8D4×C2311yescharacteristic
246C22×C8D4×C2311yescharacteristic
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8D4×C2312yes—
251#127—21no—
252#127—21no—
253#127—21no—
254#127D4×C2312yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C2311yescharacteristic
258C2×C4⋊C4D4×C2311yescharacteristic
259C2×C42—21no—
260C2×C42C23×C2311yescharacteristic
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—42no—
268#103—42no—
269#103—42no—
270#102—24no—
271#102—24no—
272#102—42no—
273#102—42no—
274C4×C8—21no—
275C4×C8—21no—
276#95—21no—
277C23#1054011yescharacteristic, direct product factor, Sylow
278C24—41no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4C2×D4×C2311yescharacteristic
305C22×C4C2×D4×C2311yescharacteristic
306C22×C4#41×C2311yescharacteristic
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4D8×C2312yes—
330C42—41no—
331C42—41no—
332C42C2×D4×C2311yescharacteristic
333C42C2×D4×C2311yescharacteristic
334C42#41×C2311yescharacteristic
335C42#41×C2311yescharacteristic
336C23—42no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23C23×#12111yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—81no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—82no—
353D4—82no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4C23×#12011yescharacteristic
370C2×C4C23×#12111yescharacteristic
371C2×C4C23×#12111yescharacteristic
372C2×C4C23×#12311yescharacteristic
373C2×C4C23×#12311yescharacteristic
374C2×C4C23×#12411yescharacteristic
375C2×C4C2×D8×C2311yescharacteristic
376C2×C4C23×#13511yescharacteristic
377C2×C4C23×#13511yescharacteristic
378C2×C4C23×#13511yescharacteristic
379C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
382C2×C4C23×#13711yescharacteristic
383C8—41no—
384C8—41no—
385C8—81no—
386C8—81no—
387C8—81no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22C23×#39411yescharacteristic
396C22C23×#46311yescharacteristic
397C22C23×#46411yescharacteristic
398C22C23×#46611yescharacteristic
399C22C23×#46911yescharacteristic
400C22C23×#47111yescharacteristic
401C22C23×#47111yescharacteristic
402C4—21no—
403C4—21no—
404C4—42no—
405C4—81no—
406C4—82no—
407C4C23×#44911yescharacteristic
408C4C23×#45211yescharacteristic
409C4C23×#45911yescharacteristic
410C4C23×#45911yescharacteristic
411C4C23×#46311yescharacteristic
412C4C23×#46311yescharacteristic
413C4C23×#48811yescharacteristic
414C4C23×#48911yescharacteristic
415C2—162no—
416C2C23×#168111yescharacteristic
417C2C23×#168211yescharacteristic
418C2C23×#170111yescharacteristic
419C2C23×#170211yescharacteristic
420C2C23×#204211yescharacteristic
421C2C23×#204611yescharacteristic
422C2C23×#205411yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
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
3. C23×#1981C2semidirect
4. C23×#1939C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1277C2semidirect
124. #10540C23direct (N × H)
179. #1981C2×C23semidirect
180. #1939C2×C23semidirect
181. C2×#332C2×C23semidirect
183. #1277C2×C23semidirect
277. C23#10540direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4862
C23×#19822
C23×#19812
C23×#19392
C2×C23×#3322
C2×C23×#3312
C23×#12772
#1054023