Finite Group Databasebeta

← C23×#12125 #19159751

Subgroups — C23×#12125 #19159751

Order 5888.

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

normal subgroups only

This group has 440 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×#2841C211yesmaximal, characteristic, complemented
2C23×#2022C211yesmaximal, characteristic
3C23×#1613C211yesmaximal, characteristic
4C23×#1601C211yesmaximal, characteristic
5C23×#1352C211yesmaximal, characteristic, complemented
6C23×#1351C211yesmaximal, characteristic, complemented
7C23×#1317C211yesmaximal, characteristic, complemented
8C23×#534C2211yescharacteristic
9C2×C23×#123—21no—
10C23×#481—21no—
11C23×#448—21no—
12C2×C23×#99—21no—
13C23×#400—21no—
14C23×#399C2211yescharacteristic
15C23×#398—21no—
16C23×#398—21no—
17C23×#396—21no—
18C23×#395—21no—
19C23×#393C2211yescharacteristic
20C23×#381C2211yescharacteristic
21C23×#380—21no—
22C23×#342C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#323C2211yescharacteristic
26C23×#323C2211yescharacteristic
27C22×Q8×C23D411yescharacteristic
28C22×D4×C23—21no—
29C23×C4×C23D411yescharacteristic
30C2×SD16×C23—41no—
31C23×#123—42no—
32C23×#123—41no—
33C23×#123—42no—
34C23×#121—21no—
35C23×#121—21no—
36C23×#120—41no—
37C2×C4⋊C4×C23—21no—
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—41no—
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
51C23×#106—41no—
52C23×#103—41no—
53C23×#102—41no—
54C23×#99—42no—
55C23×C22⋊C8—21no—
56C23×C22⋊C8—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—21no—
63C23×#95D411yescharacteristic
64C23×#95D411yescharacteristic
65C24×C23—21no—
66C24×C23C2×D411yescharacteristic
67C2×Q8×C23—41no—
68C2×Q8×C23—81no—
69C2×D4×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
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—21no—
84C22×C4×C23—21no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23SD1612yescomplemented
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91SD16×C23—82no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—44no—
98C4⋊C4×C23—81no—
99C4⋊C4×C23—81no—
100C4⋊C4×C23—81no—
101C4⋊C4×C23—81no—
102C4⋊C4×C23—81no—
103C4⋊C4×C23—81no—
104C22⋊C4×C23—22no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—44no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—81no—
112C22⋊C4×C23—81no—
113#12125C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—81no—
122C23×C23—81no—
123C23×C23#12011yescharacteristic
124C23×C23C2×SD1611yescharacteristic
125C23×C23#13711yescharacteristic
126Q8×C23—42no—
127Q8×C23—81no—
128D4×C23—81no—
129D4×C23—81no—
130D4×C23—81no—
131D4×C23—81no—
132D4×C23—81no—
133D4×C23—81no—
134C2×C4×C23—22no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—42no—
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—41no—
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—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C8×C23—81no—
165C8×C23—81no—
166#2841C2×C2311yescharacteristic, complemented
167#2022C2×C2311yescharacteristic
168#1613C2×C2311yescharacteristic
169#1601C2×C2311yescharacteristic
170#1352C2×C2311yescharacteristic, complemented
171#1351C2×C2311yescharacteristic, complemented
172#1317C2×C2311yescharacteristic, complemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—81no—
182C22×C23—82no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—161no—
187C22×C23#44811yescharacteristic
188C22×C23#45311yescharacteristic
189C22×C23B(4,𝔽2)11yescharacteristic
190C4×C23—42no—
191C4×C23—41no—
192C4×C23—41no—
193C4×C23—81no—
194C4×C23—81no—
195C4×C23—81no—
196C4×C23—82no—
197C4×C23—81no—
198C4×C23—81no—
199C4×C23—81no—
200#534C22×C2311yescharacteristic
201C2×#123—21no—
202#481—21no—
203#448—21no—
204C2×#99—21no—
205#400—21no—
206#399C22×C2311yescharacteristic
207#398—21no—
208#398—21no—
209#396—21no—
210#395—21no—
211#393C22×C2311yescharacteristic
212#381C22×C2311yescharacteristic
213#380—21no—
214#342C22×C2311yescharacteristic
215#327—21no—
216#327—21no—
217#323C22×C2311yescharacteristic
218#323C22×C2311yescharacteristic
219C2×C23—21no—
220C2×C23—21no—
221C2×C23—41no—
222C2×C23—41no—
223C2×C23—161no—
224C2×C23#159611yescharacteristic
225C2×C23#220011yescharacteristic
226C2×C23#220111yescharacteristic
227C22×Q8D4×C2311yescharacteristic
228C22×D4—21no—
229C23×C4D4×C2311yescharacteristic
230C2×SD16—41no—
231#123—42no—
232#123—41no—
233#123—42no—
234#121—21no—
235#121—21no—
236#120—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
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—41no—
249C2×C22⋊C4D4×C2311yescharacteristic
250C2×C22⋊C4C23×C2311yescharacteristic
251#106—41no—
252#103—41no—
253#102—41no—
254#99—42no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95D4×C2311yescharacteristic
264#95D4×C2311yescharacteristic
265C23#1212511yescharacteristic, direct product factor, Sylow
266C24—21no—
267C24C2×D4×C2311yescharacteristic
268C2×Q8—41no—
269C2×Q8—81no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4SD16×C2312yescomplemented
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—44no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—81no—
322C23—81no—
323C23C23×#12011yescharacteristic
324C23C2×SD16×C2311yescharacteristic
325C23C23×#13711yescharacteristic
326Q8—42no—
327Q8—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
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—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C8—81no—
365C8—81no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—161no—
380C22C23×#44811yescharacteristic
381C22C23×#45311yescharacteristic
382C22C23×B(4,𝔽2)11yescharacteristic
383C4—41no—
384C4—41no—
385C4—42no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—81no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—161no—
398C2C23×#159611yescharacteristic
399C2C23×#220011yescharacteristic
400C2C23×#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×#2841C2semidirect
5. C23×#1352C2semidirect
6. C23×#1351C2semidirect
7. C23×#1317C2semidirect
88. C22×C4×C23SD16semidirect
113. #12125C23direct (N × H)
166. #2841C2×C23semidirect
170. #1352C2×C23semidirect
171. #1351C2×C23semidirect
172. #1317C2×C23semidirect
265. C23#12125direct (N × H)
289. C22×C4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28412
C23×#20222
C23×#16132
C23×#16012
C23×#13522
C23×#13512
C23×#13172
#1212523