Finite Group Databasebeta

← C23×#12133 #19159759

Subgroups — C23×#12133 #19159759

Order 5888.

554 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2463C211yesmaximal, characteristic, complemented
2C23×#2044C211yesmaximal, characteristic, complemented
3C23×#1601C211yesmaximal, characteristic, complemented
4C23×#1595C211yesmaximal, characteristic
5C23×#1351C211yesmaximal, characteristic, complemented
6C23×#1343C211yesmaximal, characteristic, complemented
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#534C2211yescharacteristic
9C2×C23×C22⋊Q8—21no—
10C2×C4×D4×C23—21no—
11C23×#448—21no—
12C23×#447—21no—
13C2×C23×#99—21no—
14C23×#394—21no—
15C23×#393C2211yescharacteristic
16C23×#392C2211yescharacteristic
17C23×#388—21no—
18C23×#387—21no—
19C23×#386—21no—
20C23×#386—21no—
21C23×#385C2211yescharacteristic
22C23×#342C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×Q8×C23D411yescharacteristic
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C22×D4×C23D411yescharacteristic
31C23×C4×C23—21no—
32C23×C4×C23D411yescharacteristic
33C2×SD16×C23—41no—
34C2×D8×C23—41no—
35C23×#123—41no—
36C23×C22⋊Q8—44no—
37C23×#121—21no—
38C23×#121—21no—
39C23×#120—41no—
40C4×D4×C23—44no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—41no—
51C2×C22⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23C2311yescharacteristic
53C2×C42×C23—21no—
54C23×#103—41no—
55C23×#102—41no—
56C23×#99—42no—
57C23×C22⋊C8—21no—
58C23×C22⋊C8—21no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—41no—
63C23×#95D411yescharacteristic
64C23×#95D411yescharacteristic
65C24×C23—21no—
66C24×C23—21no—
67C24×C23C2×D411yescharacteristic
68C2×Q8×C23—42no—
69C2×Q8×C23—41no—
70C2×D4×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—42no—
73C2×D4×C23—41no—
74C2×D4×C23—42no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C2×D4×C23—42no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23—21no—
90C22×C4×C23—42no—
91C22×C4×C23—42no—
92C22×C4×C23—42no—
93C22×C4×C23—41no—
94C22×C4×C23—41no—
95C22×C4×C23—81no—
96C22×C4×C23C2×D411yescharacteristic
97C22×C4×C23C2×D411yescharacteristic
98SD16×C23—82no—
99D8×C23—82no—
100C2×C8×C23—41no—
101C2×C8×C23—41no—
102C4⋊C4×C23—24no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—42no—
106C4⋊C4×C23—42no—
107C22⋊C4×C23—21no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—42no—
111C22⋊C4×C23—42no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23—42no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—42no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—82no—
120C42×C23—42no—
121#12133C2311yesmaximal, characteristic, direct product factor, Sylow
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—22no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—81no—
137C23×C23—81no—
138C23×C23#12011yescharacteristic
139C23×C23#13511yescharacteristic
140C23×C23AGL(1,ℤ/8)11yescharacteristic
141Q8×C23—42no—
142Q8×C23—81no—
143D4×C23—42no—
144D4×C23—44no—
145D4×C23—44no—
146D4×C23—81no—
147D4×C23—81no—
148D4×C23—81no—
149D4×C23—81no—
150D4×C23—81no—
151D4×C23—81no—
152D4×C23—81no—
153C2×C4×C23—22no—
154C2×C4×C23—22no—
155C2×C4×C23—22no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—42no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—81no—
177C2×C4×C23—84no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23—81no—
181C8×C23—81no—
182C8×C23—81no—
183#2463C2×C2311yescharacteristic, complemented
184#2044C2×C2311yescharacteristic, complemented
185#1601C2×C2311yescharacteristic, complemented
186#1595C2×C2311yescharacteristic
187#1351C2×C2311yescharacteristic, complemented
188#1343C2×C2311yescharacteristic, complemented
189#1326C2×C2311yescharacteristic, complemented
190C22×C23—21no—
191C22×C23—21no—
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—41no—
195C22×C23—41no—
196C22×C23—42no—
197C22×C23—42no—
198C22×C23—41no—
199C22×C23—41no—
200C22×C23—42no—
201C22×C23—81no—
202C22×C23—81no—
203C22×C23—81no—
204C22×C23—82no—
205C22×C23—81no—
206C22×C23—81no—
207C22×C23—81no—
208C22×C23—81no—
209C22×C23—81no—
210C22×C23—161no—
211C22×C23#44911yescharacteristic
212C22×C23#45411yescharacteristic
213C22×C23B(4,𝔽2)11yescharacteristic
214C4×C23—42no—
215C4×C23—41no—
216C4×C23—42no—
217C4×C23—42no—
218C4×C23—41no—
219C4×C23—81no—
220C4×C23—81no—
221C4×C23—81no—
222C4×C23—161no—
223#534C22×C2311yescharacteristic
224C2×C22⋊Q8—21no—
225C2×C4×D4—21no—
226#448—21no—
227#447—21no—
228C2×#99—21no—
229#394—21no—
230#393C22×C2311yescharacteristic
231#392C22×C2311yescharacteristic
232#388—21no—
233#387—21no—
234#386—21no—
235#386—21no—
236#385C22×C2311yescharacteristic
237#342C22×C2311yescharacteristic
238#327—21no—
239#327—21no—
240#323C22×C2311yescharacteristic
241#323C22×C2311yescharacteristic
242C2×C23—21no—
243C2×C23—21no—
244C2×C23—41no—
245C2×C23—41no—
246C2×C23—82no—
247C2×C23—161no—
248C2×C23#159711yescharacteristic
249C2×C23#219611yescharacteristic
250C2×C23#219711yescharacteristic
251C22×Q8D4×C2311yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4D4×C2311yescharacteristic
255C23×C4—21no—
256C23×C4D4×C2311yescharacteristic
257C2×SD16—41no—
258C2×D8—41no—
259#123—41no—
260C22⋊Q8—44no—
261#121—21no—
262#121—21no—
263#120—41no—
264C4×D4—44no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4D4×C2311yescharacteristic
276C2×C22⋊C4C23×C2311yescharacteristic
277C2×C42—21no—
278#103—41no—
279#102—41no—
280#99—42no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287#95D4×C2311yescharacteristic
288#95D4×C2311yescharacteristic
289C23#1213311yescharacteristic, direct product factor, Sylow
290C24—21no—
291C24—21no—
292C24C2×D4×C2311yescharacteristic
293C2×Q8—42no—
294C2×Q8—41no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4C2×D4×C2311yescharacteristic
322C22×C4C2×D4×C2311yescharacteristic
323SD16—82no—
324D8—82no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—82no—
345C42—42no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—81no—
361C23—81no—
362C23C23×#12011yescharacteristic
363C23C23×#13511yescharacteristic
364C23C23×AGL(1,ℤ/8)11yescharacteristic
365Q8—42no—
366Q8—81no—
367D4—42no—
368D4—44no—
369D4—44no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—84no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C8—81no—
406C8—81no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—161no—
428C22C23×#44911yescharacteristic
429C22C23×#45411yescharacteristic
430C22C23×B(4,𝔽2)11yescharacteristic
431C4—42no—
432C4—41no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—161no—
440C2—21no—
441C2—21no—
442C2—41no—
443C2—41no—
444C2—82no—
445C2—161no—
446C2C23×#159711yescharacteristic
447C2C23×#219611yescharacteristic
448C2C23×#219711yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#1595C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2463C2semidirect
2. C23×#2044C2semidirect
3. C23×#1601C2semidirect
5. C23×#1351C2semidirect
6. C23×#1343C2semidirect
7. C23×#1326C2semidirect
121. #12133C23direct (N × H)
183. #2463C2×C23semidirect
184. #2044C2×C23semidirect
185. #1601C2×C23semidirect
187. #1351C2×C23semidirect
188. #1343C2×C23semidirect
189. #1326C2×C23semidirect
289. C23#12133direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24632
C23×#20442
C23×#16012
C23×#15952
C23×#13512
C23×#13432
C23×#13262
#1213323