Finite Group Databasebeta

← C23×#12176 #19159802

Subgroups — C23×#12176 #19159802

Order 5888.

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

normal subgroups only

This group has 486 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×#2677C211yesmaximal, characteristic, complemented
2C23×#2044C211yesmaximal, characteristic
3C23×#1613C211yesmaximal, characteristic
4C23×#1601C211yesmaximal, characteristic
5C23×#1352C211yesmaximal, characteristic, complemented
6C23×#1351C211yesmaximal, characteristic, complemented
7C23×#1326C211yesmaximal, characteristic, complemented
8C23×#534C2211yescharacteristic
9C2×C23×C4⋊Q8—21no—
10C2×C23×C22⋊Q8—21no—
11C23×#481—21no—
12C23×#448—21no—
13C2×C23×#99—21no—
14C23×#399C2211yescharacteristic
15C23×#398—21no—
16C23×#395—21no—
17C23×#393C2211yescharacteristic
18C23×#388—21no—
19C23×#387—21no—
20C23×#386—21no—
21C23×#385C2211yescharacteristic
22C23×#342C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×Q8×C23—21no—
28C22×Q8×C23D411yescharacteristic
29C22×D4×C23—21no—
30C23×C4×C23D411yescharacteristic
31C2×SD16×C23—41no—
32C23×C4⋊Q8—44no—
33C23×#123—41no—
34C23×C22⋊Q8—44no—
35C23×#121—21no—
36C23×#121—21no—
37C23×#120—41no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23D411yescharacteristic
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—21no—
48C2×C22⋊C4×C23—41no—
49C2×C22⋊C4×C23D411yescharacteristic
50C2×C22⋊C4×C23C2311yescharacteristic
51C2×C42×C23—21no—
52C23×#106—41no—
53C23×#103—41no—
54C23×#102—41no—
55C23×#99—42no—
56C23×C22⋊C8—21no—
57C23×C22⋊C8—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—41no—
63C23×#95D411yescharacteristic
64C23×#95D411yescharacteristic
65C24×C23—21no—
66C24×C23C2×D411yescharacteristic
67C2×Q8×C23—42no—
68C2×Q8×C23—42no—
69C2×Q8×C23—42no—
70C2×Q8×C23—41no—
71C2×Q8×C23—81no—
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—42no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
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—41no—
99C4⋊C4×C23—42no—
100C4⋊C4×C23—42no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23—42no—
103C4⋊C4×C23—42no—
104C22⋊C4×C23—21no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—41no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—82no—
113C22⋊C4×C23SD1612yescomplemented
114C42×C23—42no—
115#12176C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—81no—
124C23×C23—81no—
125C23×C23#12011yescharacteristic
126C23×C23C2×SD1611yescharacteristic
127C23×C23#13711yescharacteristic
128Q8×C23—42no—
129Q8×C23—81no—
130Q8×C23—84no—
131D4×C23—81no—
132D4×C23—81no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—81no—
136D4×C23—81no—
137C2×C4×C23—22no—
138C2×C4×C23—22no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
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—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—41no—
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—81no—
164C2×C4×C23—81no—
165C2×C4×C23—81no—
166C2×C4×C23—81no—
167C8×C23—81no—
168C8×C23—81no—
169#2677C2×C2311yescharacteristic, complemented
170#2044C2×C2311yescharacteristic
171#1613C2×C2311yescharacteristic
172#1601C2×C2311yescharacteristic
173#1352C2×C2311yescharacteristic, complemented
174#1351C2×C2311yescharacteristic, complemented
175#1326C2×C2311yescharacteristic, complemented
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23—81no—
185C22×C23—82no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—161no—
190C22×C23#44811yescharacteristic
191C22×C23#45411yescharacteristic
192C22×C23B(4,𝔽2)11yescharacteristic
193C4×C23—42no—
194C4×C23—42no—
195C4×C23—41no—
196C4×C23—42no—
197C4×C23—42no—
198C4×C23—41no—
199C4×C23—81no—
200C4×C23—81no—
201C4×C23—81no—
202C4×C23—161no—
203#534C22×C2311yescharacteristic
204C2×C4⋊Q8—21no—
205C2×C22⋊Q8—21no—
206#481—21no—
207#448—21no—
208C2×#99—21no—
209#399C22×C2311yescharacteristic
210#398—21no—
211#395—21no—
212#393C22×C2311yescharacteristic
213#388—21no—
214#387—21no—
215#386—21no—
216#385C22×C2311yescharacteristic
217#342C22×C2311yescharacteristic
218#327—21no—
219#327—21no—
220#323C22×C2311yescharacteristic
221#323C22×C2311yescharacteristic
222C2×C23—21no—
223C2×C23—21no—
224C2×C23—41no—
225C2×C23—41no—
226C2×C23—161no—
227C2×C23#159911yescharacteristic
228C2×C23#219611yescharacteristic
229C2×C23#220111yescharacteristic
230C22×Q8—21no—
231C22×Q8D4×C2311yescharacteristic
232C22×D4—21no—
233C23×C4D4×C2311yescharacteristic
234C2×SD16—41no—
235C4⋊Q8—44no—
236#123—41no—
237C22⋊Q8—44no—
238#121—21no—
239#121—21no—
240#120—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4D4×C2311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4D4×C2311yescharacteristic
253C2×C22⋊C4C23×C2311yescharacteristic
254C2×C42—21no—
255#106—41no—
256#103—41no—
257#102—41no—
258#99—42no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95D4×C2311yescharacteristic
267#95D4×C2311yescharacteristic
268C23#1217611yescharacteristic, direct product factor, Sylow
269C24—21no—
270C24C2×D4×C2311yescharacteristic
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C2311yescharacteristic
295C22×C4C2×D4×C2311yescharacteristic
296SD16—82no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—82no—
317C22⋊C4SD16×C2312yescomplemented
318C42—42no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—81no—
327C23—81no—
328C23C23×#12011yescharacteristic
329C23C2×SD16×C2311yescharacteristic
330C23C23×#13711yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—84no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C8—81no—
371C8—81no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—161no—
386C22C23×#44811yescharacteristic
387C22C23×#45411yescharacteristic
388C22C23×B(4,𝔽2)11yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—161no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—161no—
404C2C23×#159911yescharacteristic
405C2C23×#219611yescharacteristic
406C2C23×#220111yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC23×#534C22
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×#2677C2semidirect
5. C23×#1352C2semidirect
6. C23×#1351C2semidirect
7. C23×#1326C2semidirect
113. C22⋊C4×C23SD16semidirect
115. #12176C23direct (N × H)
169. #2677C2×C23semidirect
173. #1352C2×C23semidirect
174. #1351C2×C23semidirect
175. #1326C2×C23semidirect
268. C23#12176direct (N × H)
317. C22⋊C4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#26772
C23×#20442
C23×#16132
C23×#16012
C23×#13522
C23×#13512
C23×#13262
#1217623