Finite Group Databasebeta

← C23×#12227 #19159853

Subgroups — C23×#12227 #19159853

Order 5888.

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

normal subgroups only

This group has 520 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
2C23×#2062C211yesmaximal, characteristic, complemented
3C23×#1602C211yesmaximal, characteristic, complemented
4C23×#1596C211yesmaximal, characteristic
5C23×#1351C211yesmaximal, characteristic, complemented
6C23×#1343C211yesmaximal, characteristic
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#543C2211yescharacteristic
9C2×C23×C22⋊Q8—21no—
10C2×C4×D4×C23—21no—
11C23×#451—21no—
12C23×#450—21no—
13C23×#394—21no—
14C23×#393C2211yescharacteristic
15C23×#392C2211yescharacteristic
16C23×#388—21no—
17C23×#387—21no—
18C23×#386—21no—
19C23×#385—21no—
20C23×#385C2211yescharacteristic
21C23×#342—21no—
22C23×#342C2211yescharacteristic
23C23×#328—21no—
24C23×#328—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×Q8×C23D411yescharacteristic
28C22×D4×C23—21no—
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C23×C4×C23D411yescharacteristic
32C2×Q16×C23—41no—
33C2×SD16×C23—41no—
34C23×C22⋊Q8—21no—
35C23×C22⋊Q8—21no—
36C23×C22⋊Q8—41no—
37C23×C22⋊Q8—41no—
38C23×C22⋊Q8—44no—
39C4×D4×C23—44no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—41no—
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—21no—
51C2×C22⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23C2311yescharacteristic
53C2×C42×C23—21no—
54C23×#103—41no—
55C23×#102—41no—
56C23×C22⋊C8—21no—
57C23×C22⋊C8—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—41no—
62C23×#95D411yescharacteristic
63C23×#95D411yescharacteristic
64C24×C23—21no—
65C24×C23C2×D411yescharacteristic
66C2×Q8×C23—21no—
67C2×Q8×C23—21no—
68C2×Q8×C23—42no—
69C2×Q8×C23—41no—
70C2×D4×C23—42no—
71C2×D4×C23—41no—
72C2×D4×C23—42no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23—42no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23—81no—
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91Q16×C23—82no—
92SD16×C23—82no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—21no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—42no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—82no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—21no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—42no—
109C22⋊C4×C23—42no—
110C22⋊C4×C23—42no—
111C22⋊C4×C23—42no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C42×C23—42no—
116#12227C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—22no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23#12011yescharacteristic
129C23×C23#13511yescharacteristic
130C23×C23#13711yescharacteristic
131Q8×C23—42no—
132Q8×C23—81no—
133Q8×C23—81no—
134Q8×C23—81no—
135D4×C23—42no—
136D4×C23—44no—
137D4×C23—44no—
138D4×C23—81no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—42no—
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—81no—
163C2×C4×C23—84no—
164C2×C4×C23—81no—
165C2×C4×C23—81no—
166C2×C4×C23—81no—
167C2×C4×C23—81no—
168C8×C23—81no—
169C8×C23—81no—
170#2463C2×C2311yescharacteristic
171#2062C2×C2311yescharacteristic, complemented
172#1602C2×C2311yescharacteristic, complemented
173#1596C2×C2311yescharacteristic
174#1351C2×C2311yescharacteristic, complemented
175#1343C2×C2311yescharacteristic
176#1326C2×C2311yescharacteristic, complemented
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—21no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—42no—
184C22×C23—42no—
185C22×C23—41no—
186C22×C23—41no—
187C22×C23—42no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23#45211yescharacteristic
194C22×C23#45611yescharacteristic
195C22×C23B(4,𝔽2)11yescharacteristic
196C4×C23—42no—
197C4×C23—41no—
198C4×C23—42no—
199C4×C23—42no—
200C4×C23—41no—
201C4×C23—81no—
202C4×C23—81no—
203C4×C23—82no—
204C4×C23—81no—
205C4×C23—81no—
206#543C22×C2311yescharacteristic
207C2×C22⋊Q8—21no—
208C2×C4×D4—21no—
209#451—21no—
210#450—21no—
211#394—21no—
212#393C22×C2311yescharacteristic
213#392C22×C2311yescharacteristic
214#388—21no—
215#387—21no—
216#386—21no—
217#385—21no—
218#385C22×C2311yescharacteristic
219#342—21no—
220#342C22×C2311yescharacteristic
221#328—21no—
222#328—21no—
223#323C22×C2311yescharacteristic
224#323C22×C2311yescharacteristic
225C2×C23—21no—
226C2×C23—21no—
227C2×C23—41no—
228C2×C23—41no—
229C2×C23—82no—
230C2×C23#160311yescharacteristic
231C2×C23#219611yescharacteristic
232C2×C23#219711yescharacteristic
233C22×Q8D4×C2311yescharacteristic
234C22×D4—21no—
235C22×D4D4×C2311yescharacteristic
236C23×C4—21no—
237C23×C4D4×C2311yescharacteristic
238C2×Q16—41no—
239C2×SD16—41no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C22⋊Q8—44no—
245C4×D4—44no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—41no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4D4×C2311yescharacteristic
258C2×C22⋊C4C23×C2311yescharacteristic
259C2×C42—21no—
260#103—41no—
261#102—41no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95D4×C2311yescharacteristic
269#95D4×C2311yescharacteristic
270C23#1222711yescharacteristic, direct product factor, Sylow
271C24—21no—
272C24C2×D4×C2311yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—81no—
296C22×C4C2×D4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298Q16—82no—
299SD16—82no—
300C2×C8—41no—
301C2×C8—41no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C22⋊C4—22no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C42—42no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23C23×#12011yescharacteristic
335C23C23×#13511yescharacteristic
336C23C23×#13711yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—81no—
340Q8—81no—
341D4—42no—
342D4—44no—
343D4—44no—
344D4—81no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C8—81no—
375C8—81no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C23×#45211yescharacteristic
393C22C23×#45611yescharacteristic
394C22C23×B(4,𝔽2)11yescharacteristic
395C4—42no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4—81no—
405C2—21no—
406C2—21no—
407C2—41no—
408C2—41no—
409C2—82no—
410C2C23×#160311yescharacteristic
411C2C23×#219611yescharacteristic
412C2C23×#219711yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#392C22
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
2. C23×#2062C2semidirect
3. C23×#1602C2semidirect
5. C23×#1351C2semidirect
7. C23×#1326C2semidirect
116. #12227C23direct (N × H)
171. #2062C2×C23semidirect
172. #1602C2×C23semidirect
174. #1351C2×C23semidirect
176. #1326C2×C23semidirect
270. C23#12227direct (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×#20622
C23×#16022
C23×#15962
C23×#13512
C23×#13432
C23×#13262
#1222723