Finite Group Databasebeta

← C23×#12226 #19159852

Subgroups — C23×#12226 #19159852

Order 5888.

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

normal subgroups only

This group has 510 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
2C23×#2062C211yesmaximal, characteristic, complemented
3C23×#1614C211yesmaximal, characteristic, complemented
4C23×#1596C211yesmaximal, characteristic
5C23×#1352C211yesmaximal, characteristic, complemented
6C23×#1343C211yesmaximal, characteristic
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#543C2211yescharacteristic
9C2×C23×#123—21no—
10C2×C4×D4×C23—21no—
11C23×#484—21no—
12C23×#450—21no—
13C23×#399C2211yescharacteristic
14C23×#394—21no—
15C23×#392C2211yescharacteristic
16C23×#389—21no—
17C23×#388—21no—
18C23×#385—21no—
19C23×#385—21no—
20C23×#385C2211yescharacteristic
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—44no—
33C23×C22⋊Q8—21no—
34C23×C22⋊Q8—21no—
35C23×C22⋊Q8—41no—
36C23×C22⋊Q8—41no—
37C4×D4×C23—44no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—41no—
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—21no—
51C2×C22⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23C2311yescharacteristic
53C2×C42×C23—21no—
54C23×#107—41no—
55C23×#103—41no—
56C23×#102—41no—
57C23×C22⋊C8—21no—
58C23×C22⋊C8—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—42no—
69C2×D4×C23—41no—
70C2×D4×C23—42no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—42no—
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—42no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23—42no—
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—42no—
98C4⋊C4×C23—42no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—82no—
102C22⋊C4×C23—24no—
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—41no—
111C22⋊C4×C23—42no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C42×C23—42no—
115#12226C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—22no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23#12011yescharacteristic
128C23×C23#13511yescharacteristic
129C23×C23#13711yescharacteristic
130Q8×C23—81no—
131Q8×C23—81no—
132D4×C23—42no—
133D4×C23—44no—
134D4×C23—44no—
135D4×C23—81no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—22no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
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—42no—
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—41no—
159C2×C4×C23—81no—
160C2×C4×C23—84no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C8×C23—81no—
166C8×C23—81no—
167#2455C2×C2311yescharacteristic
168#2062C2×C2311yescharacteristic, complemented
169#1614C2×C2311yescharacteristic, complemented
170#1596C2×C2311yescharacteristic
171#1352C2×C2311yescharacteristic, complemented
172#1343C2×C2311yescharacteristic
173#1326C2×C2311yescharacteristic, complemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—42no—
181C22×C23—42no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23—42no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23#45211yescharacteristic
191C22×C23#45611yescharacteristic
192C22×C23B(4,𝔽2)11yescharacteristic
193C4×C23—42no—
194C4×C23—42no—
195C4×C23—41no—
196C4×C23—42no—
197C4×C23—41no—
198C4×C23—81no—
199C4×C23—82no—
200C4×C23—81no—
201C4×C23—81no—
202C4×C23—81no—
203#543C22×C2311yescharacteristic
204C2×#123—21no—
205C2×C4×D4—21no—
206#484—21no—
207#450—21no—
208#399C22×C2311yescharacteristic
209#394—21no—
210#392C22×C2311yescharacteristic
211#389—21no—
212#388—21no—
213#385—21no—
214#385—21no—
215#385C22×C2311yescharacteristic
216#342—21no—
217#342C22×C2311yescharacteristic
218#328—21no—
219#328—21no—
220#323C22×C2311yescharacteristic
221#323C22×C2311yescharacteristic
222C2×C23—21no—
223C2×C23—21no—
224C2×C23—41no—
225C2×C23—41no—
226C2×C23—82no—
227C2×C23#160311yescharacteristic
228C2×C23#219611yescharacteristic
229C2×C23#219811yescharacteristic
230C22×D4—21no—
231C22×D4D4×C2311yescharacteristic
232C23×C4—21no—
233C23×C4D4×C2311yescharacteristic
234C2×SD16—41no—
235#123—44no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240C4×D4—44no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4D4×C2311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4D4×C2311yescharacteristic
255C2×C22⋊C4C23×C2311yescharacteristic
256C2×C42—21no—
257#107—41no—
258#103—41no—
259#102—41no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95D4×C2311yescharacteristic
266#95D4×C2311yescharacteristic
267C23#1222611yescharacteristic, direct product factor, Sylow
268C24—21no—
269C24C2×D4×C2311yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4C2×D4×C2311yescharacteristic
292C22×C4C2×D4×C2311yescharacteristic
293SD16—82no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—42no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C23×#12011yescharacteristic
331C23C23×#13511yescharacteristic
332C23C23×#13711yescharacteristic
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—44no—
337D4—44no—
338D4—81no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C8—81no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C23×#45211yescharacteristic
387C22C23×#45611yescharacteristic
388C22C23×B(4,𝔽2)11yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2C23×#160311yescharacteristic
405C2C23×#219611yescharacteristic
406C2C23×#219811yescharacteristic
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×#1614C2semidirect
5. C23×#1352C2semidirect
7. C23×#1326C2semidirect
115. #12226C23direct (N × H)
168. #2062C2×C23semidirect
169. #1614C2×C23semidirect
171. #1352C2×C23semidirect
173. #1326C2×C23semidirect
267. C23#12226direct (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×#20622
C23×#16142
C23×#15962
C23×#13522
C23×#13432
C23×#13262
#1222623