Finite Group Databasebeta

← C23×#12132 #19159758

Subgroups — C23×#12132 #19159758

Order 5888.

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

normal subgroups only

This group has 544 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×#2455C211yesmaximal, characteristic, complemented
2C23×#2044C211yesmaximal, characteristic, complemented
3C23×#1613C211yesmaximal, characteristic, complemented
4C23×#1595C211yesmaximal, characteristic
5C23×#1352C211yesmaximal, characteristic, complemented
6C23×#1343C211yesmaximal, characteristic, complemented
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#534C2211yescharacteristic
9C2×C23×#123—21no—
10C2×C4×D4×C23—21no—
11C23×#481—21no—
12C23×#447—21no—
13C2×C23×#99—21no—
14C23×#399C2211yescharacteristic
15C23×#394—21no—
16C23×#392C2211yescharacteristic
17C23×#389—21no—
18C23×#388—21no—
19C23×#386—21no—
20C23×#385—21no—
21C23×#385C2211yescharacteristic
22C23×#342C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×D4×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C23×C4×C23D411yescharacteristic
32C2×D8×C23—41no—
33C23×#123—44no—
34C23×#123—41no—
35C23×#121—21no—
36C23×#121—21no—
37C23×#120—41no—
38C4×D4×C23—44no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23D411yescharacteristic
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×#106—41no—
55C23×#102—41no—
56C23×#102—41no—
57C23×#99—42no—
58C23×C22⋊C8—21no—
59C23×C22⋊C8—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—41no—
63C23×#95D411yescharacteristic
64C23×#95D411yescharacteristic
65C24×C23—21no—
66C24×C23—21no—
67C24×C23C2×D411yescharacteristic
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23—42no—
71C2×D4×C23—41no—
72C2×D4×C23—42no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23—42no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—42no—
89C22×C4×C23—42no—
90C22×C4×C23—41no—
91C22×C4×C23—42no—
92C22×C4×C23—41no—
93C22×C4×C23—81no—
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96D8×C23—82no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—42no—
103C4⋊C4×C23—42no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23—42no—
106C22⋊C4×C23—24no—
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—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—42no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—82no—
119C42×C23—42no—
120#12132C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—22no—
124C23×C23—41no—
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—81no—
136C23×C23—81no—
137C23×C23#12011yescharacteristic
138C23×C23#13511yescharacteristic
139C23×C23AGL(1,ℤ/8)11yescharacteristic
140D4×C23—42no—
141D4×C23—44no—
142D4×C23—44no—
143D4×C23—81no—
144D4×C23—81no—
145D4×C23—81no—
146D4×C23—81no—
147D4×C23—81no—
148D4×C23—81no—
149D4×C23—81no—
150C2×C4×C23—22no—
151C2×C4×C23—22no—
152C2×C4×C23—22no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—42no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
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—81no—
174C2×C4×C23—84no—
175C2×C4×C23—81no—
176C2×C4×C23—81no—
177C2×C4×C23—81no—
178C8×C23—81no—
179C8×C23—81no—
180#2455C2×C2311yescharacteristic, complemented
181#2044C2×C2311yescharacteristic, complemented
182#1613C2×C2311yescharacteristic, complemented
183#1595C2×C2311yescharacteristic
184#1352C2×C2311yescharacteristic, complemented
185#1343C2×C2311yescharacteristic, complemented
186#1326C2×C2311yescharacteristic, complemented
187C22×C23—21no—
188C22×C23—21no—
189C22×C23—21no—
190C22×C23—21no—
191C22×C23—41no—
192C22×C23—41no—
193C22×C23—42no—
194C22×C23—42no—
195C22×C23—41no—
196C22×C23—41no—
197C22×C23—42no—
198C22×C23—81no—
199C22×C23—81no—
200C22×C23—81no—
201C22×C23—82no—
202C22×C23—81no—
203C22×C23—81no—
204C22×C23—81no—
205C22×C23—81no—
206C22×C23—81no—
207C22×C23—161no—
208C22×C23#44911yescharacteristic
209C22×C23#45411yescharacteristic
210C22×C23B(4,𝔽2)11yescharacteristic
211C4×C23—42no—
212C4×C23—42no—
213C4×C23—41no—
214C4×C23—42no—
215C4×C23—41no—
216C4×C23—81no—
217C4×C23—81no—
218C4×C23—81no—
219C4×C23—161no—
220#534C22×C2311yescharacteristic
221C2×#123—21no—
222C2×C4×D4—21no—
223#481—21no—
224#447—21no—
225C2×#99—21no—
226#399C22×C2311yescharacteristic
227#394—21no—
228#392C22×C2311yescharacteristic
229#389—21no—
230#388—21no—
231#386—21no—
232#385—21no—
233#385C22×C2311yescharacteristic
234#342C22×C2311yescharacteristic
235#327—21no—
236#327—21no—
237#323C22×C2311yescharacteristic
238#323C22×C2311yescharacteristic
239C2×C23—21no—
240C2×C23—21no—
241C2×C23—41no—
242C2×C23—41no—
243C2×C23—82no—
244C2×C23—161no—
245C2×C23#159711yescharacteristic
246C2×C23#219611yescharacteristic
247C2×C23#219811yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4D4×C2311yescharacteristic
251C23×C4—21no—
252C23×C4D4×C2311yescharacteristic
253C2×D8—41no—
254#123—44no—
255#123—41no—
256#121—21no—
257#121—21no—
258#120—41no—
259C4×D4—44no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4D4×C2311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4D4×C2311yescharacteristic
273C2×C22⋊C4C23×C2311yescharacteristic
274C2×C42—21no—
275#106—41no—
276#102—41no—
277#102—41no—
278#99—42no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281#95—21no—
282#95—21no—
283#95—41no—
284#95D4×C2311yescharacteristic
285#95D4×C2311yescharacteristic
286C23#1213211yescharacteristic, direct product factor, Sylow
287C24—21no—
288C24—21no—
289C24C2×D4×C2311yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—81no—
316C22×C4C2×D4×C2311yescharacteristic
317C22×C4C2×D4×C2311yescharacteristic
318D8—82no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C22⋊C4—24no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—41no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—82no—
341C42—42no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23—81no—
358C23C23×#12011yescharacteristic
359C23C23×#13511yescharacteristic
360C23C23×AGL(1,ℤ/8)11yescharacteristic
361D4—42no—
362D4—44no—
363D4—44no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C8—81no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—82no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—161no—
422C22C23×#44911yescharacteristic
423C22C23×#45411yescharacteristic
424C22C23×B(4,𝔽2)11yescharacteristic
425C4—42no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4—41no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—161no—
434C2—21no—
435C2—21no—
436C2—41no—
437C2—41no—
438C2—82no—
439C2—161no—
440C2C23×#159711yescharacteristic
441C2C23×#219611yescharacteristic
442C2C23×#219811yescharacteristic
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×#2455C2semidirect
2. C23×#2044C2semidirect
3. C23×#1613C2semidirect
5. C23×#1352C2semidirect
6. C23×#1343C2semidirect
7. C23×#1326C2semidirect
120. #12132C23direct (N × H)
180. #2455C2×C23semidirect
181. #2044C2×C23semidirect
182. #1613C2×C23semidirect
184. #1352C2×C23semidirect
185. #1343C2×C23semidirect
186. #1326C2×C23semidirect
286. C23#12132direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24552
C23×#20442
C23×#16132
C23×#15952
C23×#13522
C23×#13432
C23×#13262
#1213223