Finite Group Databasebeta

← C23×#12243 #19159869

Subgroups — C23×#12243 #19159869

Order 5888.

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2693C211yesmaximal, characteristic
2C23×#2062C211yesmaximal, characteristic, complemented
3C23×#1608C211yesmaximal, characteristic, complemented
4C23×#1596C211yesmaximal, characteristic
5C23×#1344C211yesmaximal, characteristic, complemented
6C23×#1343C211yesmaximal, characteristic
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#543C2211yescharacteristic
9C2×C23×#123—21no—
10C23×#483—21no—
11C23×#450—21no—
12C23×#400—21no—
13C23×#397C2211yescharacteristic
14C23×#396—21no—
15C23×#392C2211yescharacteristic
16C23×#386—21no—
17C23×#386—21no—
18C23×#385—21no—
19C23×#385C2211yescharacteristic
20C23×#380—21no—
21C23×#342—21no—
22C23×#342C2211yescharacteristic
23C23×#328—21no—
24C23×#328—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×D4×C23—21no—
28C22×D4×C23D411yescharacteristic
29C23×C4×C23—21no—
30C23×C4×C23D411yescharacteristic
31C2×SD16×C23—41no—
32C23×#123—42no—
33C23×#123—42no—
34C23×C22⋊Q8—21no—
35C23×C22⋊Q8—21no—
36C23×C22⋊Q8—41no—
37C23×C22⋊Q8—41no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—41no—
42C2×C4⋊C4×C23D411yescharacteristic
43C2×C22⋊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×C23D411yescharacteristic
51C2×C22⋊C4×C23C2311yescharacteristic
52C23×#106—41no—
53C23×#103—41no—
54C23×#102—41no—
55C23×C22⋊C8—21no—
56C23×C22⋊C8—21no—
57C23×#95—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×D4×C23—41no—
69C2×D4×C23—42no—
70C2×D4×C23—42no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—81no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—42no—
82C22×C4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23—42no—
85C22×C4×C23—81no—
86C22×C4×C23—81no—
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89SD16×C23—82no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23—21no—
94C4⋊C4×C23—41no—
95C4⋊C4×C23—42no—
96C4⋊C4×C23—42no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—81no—
100C4⋊C4×C23—82no—
101C22⋊C4×C23—21no—
102C22⋊C4×C23—22no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—41no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—81no—
111C22⋊C4×C23—81no—
112C22⋊C4×C23—81no—
113C22⋊C4×C23SD1612yescomplemented
114#12243C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—22no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23#12011yescharacteristic
127C23×C23C2×SD1611yescharacteristic
128C23×C23AGL(1,ℤ/8)11yescharacteristic
129Q8×C23—81no—
130Q8×C23—81no—
131D4×C23—42no—
132D4×C23—81no—
133D4×C23—82no—
134D4×C23—82no—
135C2×C4×C23—22no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—84no—
159C2×C4×C23—81no—
160C2×C4×C23—81no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C8×C23—81no—
165C8×C23—81no—
166#2693C2×C2311yescharacteristic
167#2062C2×C2311yescharacteristic, complemented
168#1608C2×C2311yescharacteristic, complemented
169#1596C2×C2311yescharacteristic
170#1344C2×C2311yescharacteristic, complemented
171#1343C2×C2311yescharacteristic
172#1326C2×C2311yescharacteristic, complemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—42no—
180C22×C23—42no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—42no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23#45011yescharacteristic
190C22×C23#45611yescharacteristic
191C22×C23B(4,𝔽2)11yescharacteristic
192C4×C23—42no—
193C4×C23—41no—
194C4×C23—41no—
195C4×C23—81no—
196C4×C23—81no—
197C4×C23—82no—
198C4×C23—81no—
199C4×C23—81no—
200C4×C23—81no—
201C4×C23—81no—
202#543C22×C2311yescharacteristic
203C2×#123—21no—
204#483—21no—
205#450—21no—
206#400—21no—
207#397C22×C2311yescharacteristic
208#396—21no—
209#392C22×C2311yescharacteristic
210#386—21no—
211#386—21no—
212#385—21no—
213#385C22×C2311yescharacteristic
214#380—21no—
215#342—21no—
216#342C22×C2311yescharacteristic
217#328—21no—
218#328—21no—
219#323C22×C2311yescharacteristic
220#323C22×C2311yescharacteristic
221C2×C23—21no—
222C2×C23—21no—
223C2×C23—41no—
224C2×C23—41no—
225C2×C23—82no—
226C2×C23#160411yescharacteristic
227C2×C23#219711yescharacteristic
228C2×C23#220011yescharacteristic
229C22×D4—21no—
230C22×D4D4×C2311yescharacteristic
231C23×C4—21no—
232C23×C4D4×C2311yescharacteristic
233C2×SD16—41no—
234#123—42no—
235#123—42no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4D4×C2311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4D4×C2311yescharacteristic
253C2×C22⋊C4C23×C2311yescharacteristic
254#106—41no—
255#103—41no—
256#102—41no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95D4×C2311yescharacteristic
265#95D4×C2311yescharacteristic
266C23#1224311yescharacteristic, direct product factor, Sylow
267C24—21no—
268C24C2×D4×C2311yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×D4×C2311yescharacteristic
291C22×C4C2×D4×C2311yescharacteristic
292SD16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4SD16×C2312yescomplemented
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23C23×#12011yescharacteristic
329C23C2×SD16×C2311yescharacteristic
330C23C23×AGL(1,ℤ/8)11yescharacteristic
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—81no—
335D4—82no—
336D4—82no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C23×#45011yescharacteristic
385C22C23×#45611yescharacteristic
386C22C23×B(4,𝔽2)11yescharacteristic
387C4—42no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—82no—
402C2C23×#160411yescharacteristic
403C2C23×#219711yescharacteristic
404C2C23×#220011yescharacteristic
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×#1608C2semidirect
5. C23×#1344C2semidirect
7. C23×#1326C2semidirect
113. C22⋊C4×C23SD16semidirect
114. #12243C23direct (N × H)
167. #2062C2×C23semidirect
168. #1608C2×C23semidirect
170. #1344C2×C23semidirect
172. #1326C2×C23semidirect
266. C23#12243direct (N × H)
316. C22⋊C4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#26932
C23×#20622
C23×#16082
C23×#15962
C23×#13442
C23×#13432
C23×#13262
#1224323