Finite Group Databasebeta

← C23×#9479 #19157105

Subgroups — C23×#9479 #19157105

Order 5888.

838 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2404C211yesmaximal, characteristic
2C23×#2312C211yesmaximal, characteristic, complemented
3C23×#1789C212yesmaximal, complemented
4C23×#1462C211yesmaximal, characteristic, complemented
5C23×#1284C212yesmaximal, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×#120—21no—
8C22×C22⋊C4×C23C2211yescharacteristic
9C2×C23×#104—24no—
10C23×#407C2212yes—
11C23×#393C412yescomplemented
12C23×#392C412yescomplemented
13C23×#388—24no—
14C23×#386—21no—
15C23×#386—22no—
16C23×#381C2211yescharacteristic
17C23×#380—21no—
18C23×#379—21no—
19C4×C22⋊C4×C23C2212yes—
20C23×#323—22no—
21C25×C23D411yescharacteristic
22C2×#41×C23—22no—
23C2×#41×C23—22no—
24C2×#41×C23—44no—
25C22×Q8×C23C2×C411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23—22no—
28C22×D4×C23C2×C411yescharacteristic
29C23×C4×C23—21no—
30C23×C4×C23D411yescharacteristic
31C23×C4×C23C2311yescharacteristic
32C2×M16×C23—22no—
33C23×#120—44no—
34C2×C4⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23C2×C412yes—
44C2×C22⋊C4×C23D412yes—
45C2×C42×C23D414yescomplemented
46C23×#104—48no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—22no—
49C23×#95—22no—
50C23×#95—22no—
51C23×#95—22no—
52C23×#95—21no—
53C23×#95C2×C412yes—
54C24×C23—21no—
55C24×C23—21no—
56C24×C23—21no—
57C24×C23—42no—
58C24×C23—42no—
59C24×C23—81no—
60C24×C23C22⋊C412yes—
61C24×C23C2×D411yescharacteristic
62#41×C23—28no—
63#41×C23—44no—
64#41×C23—44no—
65#41×C23—44no—
66C2×Q8×C23—22no—
67C2×Q8×C23—42no—
68C2×D4×C23—24no—
69C2×D4×C23—22no—
70C2×D4×C23—42no—
71C2×D4×C23—42no—
72C2×D4×C23—41no—
73C2×D4×C23—41no—
74C2×D4×C23—44no—
75C2×D4×C23—44no—
76C2×D4×C23—82no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—22no—
81C22×C4×C23—24no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23—81no—
87C22×C4×C23C22⋊C412yes—
88C22×C4×C23C22×C411yescharacteristic
89C22×C4×C23C2×D412yes—
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23#4112yes—
92M16×C23—44no—
93C2×C8×C23—42no—
94C4⋊C4×C23—82no—
95C22⋊C4×C23—24no—
96C22⋊C4×C23—24no—
97C22⋊C4×C23—44no—
98C22⋊C4×C23—44no—
99C22⋊C4×C23—42no—
100C22⋊C4×C23—82no—
101C22⋊C4×C23—82no—
102C22⋊C4×C23—82no—
103C42×C23—44no—
104#9479C2311yesmaximal, characteristic, direct product factor, Sylow
105C23×C23—22no—
106C23×C23—22no—
107C23×C23—22no—
108C23×C23—22no—
109C23×C23—22no—
110C23×C23—22no—
111C23×C23—41no—
112C23×C23—42no—
113C23×C23—42no—
114C23×C23—42no—
115C23×C23—42no—
116C23×C23—41no—
117C23×C23—42no—
118C23×C23—42no—
119C23×C23—41no—
120C23×C23—42no—
121C23×C23—42no—
122C23×C23—81no—
123C23×C23—81no—
124C23×C23—88no—
125C23×C23C2×C22⋊C411yescharacteristic
126C23×C23#12011yescharacteristic
127C23×C23#12311yescharacteristic
128Q8×C23—24no—
129Q8×C23—42no—
130D4×C23—24no—
131D4×C23—42no—
132D4×C23—44no—
133D4×C23—44no—
134D4×C23—88no—
135C2×C4×C23—24no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—48no—
142C2×C4×C23—42no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—44no—
148C2×C4×C23—44no—
149C2×C4×C23—42no—
150C2×C4×C23—42no—
151C2×C4×C23—44no—
152C2×C4×C23—81no—
153C2×C4×C23C4×D412yes—
154C2×C4×C23#12112yes—
155C8×C23—82no—
156#2404C2×C2311yescharacteristic
157#2312C2×C2311yescharacteristic, complemented
158#1789C2×C2312yescomplemented
159#1462C2×C2311yescharacteristic, complemented
160#1284C2×C2312yescomplemented
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—21no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—44no—
169C22×C23—44no—
170C22×C23—44no—
171C22×C23—42no—
172C22×C23—42no—
173C22×C23—42no—
174C22×C23—42no—
175C22×C23—82no—
176C22×C23—82no—
177C22×C23—84no—
178C22×C23—84no—
179C22×C23—81no—
180C22×C23—81no—
181C22×C23#38611yescharacteristic
182C22×C23#42112yes—
183C22×C23B(4,𝔽2)12yes—
184C4×C23—22no—
185C4×C23—22no—
186C4×C23—42no—
187C4×C23—42no—
188C4×C23—84no—
189C4×C23—82no—
190C22×#41C22×C2311yescharacteristic
191C2×#120—21no—
192C22×C22⋊C4C22×C2311yescharacteristic
193C2×#104—24no—
194#407C22×C2312yes—
195#393C4×C2312yescomplemented
196#392C4×C2312yescomplemented
197#388—24no—
198#386—21no—
199#386—22no—
200#381C22×C2311yescharacteristic
201#380—21no—
202#379—21no—
203C4×C22⋊C4C22×C2312yes—
204#323—22no—
205C2×C23—22no—
206C2×C23—22no—
207C2×C23—41no—
208C2×C23—82no—
209C2×C23—82no—
210C2×C23#188911yescharacteristic
211C2×C23#189711yescharacteristic
212C2×C23#191111yescharacteristic
213C25D4×C2311yescharacteristic
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—44no—
217C22×Q8C2×C4×C2311yescharacteristic
218C22×D4—21no—
219C22×D4—22no—
220C22×D4C2×C4×C2311yescharacteristic
221C23×C4—21no—
222C23×C4D4×C2311yescharacteristic
223C23×C4C23×C2311yescharacteristic
224C2×M16—22no—
225#120—44no—
226C2×C4⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C2×C4×C2312yes—
236C2×C22⋊C4D4×C2312yes—
237C2×C42D4×C2314yescomplemented
238#104—48no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95C2×C4×C2312yes—
246C23#947911yescharacteristic, direct product factor, Sylow
247C24—21no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—81no—
253C24C22⋊C4×C2312yes—
254C24C2×D4×C2311yescharacteristic
255#41—28no—
256#41—44no—
257#41—44no—
258#41—44no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—81no—
280C22×C4C22⋊C4×C2312yes—
281C22×C4C22×C4×C2311yescharacteristic
282C22×C4C2×D4×C2312yes—
283C22×C4C2×D4×C2311yescharacteristic
284C22×C4#41×C2312yes—
285M16—44no—
286C2×C8—42no—
287C4⋊C4—82no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—44no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—88no—
317C23C2×C22⋊C4×C2311yescharacteristic
318C23C23×#12011yescharacteristic
319C23C23×#12311yescharacteristic
320Q8—24no—
321Q8—42no—
322D4—24no—
323D4—42no—
324D4—44no—
325D4—44no—
326D4—88no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—81no—
345C2×C4C4×D4×C2312yes—
346C2×C4C23×#12112yes—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—84no—
364C22—82no—
365C22—84no—
366C22—81no—
367C22—81no—
368C22C23×#38611yescharacteristic
369C22C23×#42112yes—
370C22C23×B(4,𝔽2)12yes—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—84no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—82no—
381C2—82no—
382C2C23×#188911yescharacteristic
383C2C23×#189711yescharacteristic
384C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2404C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#2312C2semidirect
3. C23×#1789C2semidirect
4. C23×#1462C2semidirect
5. C23×#1284C2semidirect
11. C23×#393C4semidirect
12. C23×#392C4semidirect
45. C2×C42×C23D4semidirect
104. #9479C23direct (N × H)
157. #2312C2×C23semidirect
158. #1789C2×C23semidirect
159. #1462C2×C23semidirect
160. #1284C2×C23semidirect
195. #393C4×C23semidirect
196. #392C4×C23semidirect
237. C2×C42D4×C23semidirect
246. C23#9479direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24042
C23×#23122
C23×#17892
C23×#14622
C23×#12842
#947923