Finite Group Databasebeta

← C23×#12722 #19160348

Subgroups — C23×#12722 #19160348

Order 5888.

790 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3508C211yesmaximal, characteristic
2C23×#2134C211yesmaximal, characteristic, complemented
3C23×#2128C211yesmaximal, characteristic, complemented
4C23×#2122C212yesmaximal, complemented
5C23×#1827C212yesmaximal, complemented
6C23×#559—22no—
7C23×#550—22no—
8C23×#546—24no—
9C23×#540—22no—
10C23×#535C2211yescharacteristic
11C23×#535C2211yescharacteristic
12C23×#533C411yescharacteristic, complemented
13C23×#532—22no—
14C2×C23×#127C411yescharacteristic, complemented
15C23×#518C411yescharacteristic, complemented
16C23×#518C411yescharacteristic, complemented
17C4×#41×C23C2211yescharacteristic
18C23×#413C2212yes—
19C23×#410C2212yes—
20C23×#355—22no—
21C23×#353—22no—
22C23×AΣL(1,𝔽16)—24no—
23C23×#339—22no—
24C23×H(3,ℤ/4)—22no—
25C2×#41×C23—24no—
26C2×#41×C23C2311yescharacteristic
27C22×D4×C23—22no—
28C22×D4×C23C2×C411yescharacteristic
29C22×D4×C23C2×C411yescharacteristic
30C2×M16×C23—22no—
31C23×C4⋊Q8—21no—
32C23×#127—22no—
33C23×#127—21no—
34C23×#127—21no—
35C23×#127—21no—
36C23×#127D412yes—
37C23×#125—21no—
38C23×#124—24no—
39C23×#124—21no—
40C23×#124D412yes—
41C23×#123—24no—
42C23×#123—24no—
43C23×C22⋊Q8—22no—
44C23×#121—24no—
45C23×#121—24no—
46C23×#121—24no—
47C23×#121—22no—
48C23×#121—22no—
49C23×#120—22no—
50C23×#120D414yes—
51C4×Q8×C23—21no—
52C4×Q8×C23—22no—
53C4×D4×C23—24no—
54C4×D4×C23—22no—
55C4×D4×C23—21no—
56C4×D4×C23—22no—
57C4×D4×C23—22no—
58C4×D4×C23—22no—
59C4×D4×C23—22no—
60C23×#117—22no—
61C23×#117C2×C411yescharacteristic
62C23×#117C2×C411yescharacteristic
63C23×#117C2×C411yescharacteristic
64C23×#117D412yes—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—22no—
67C2×C42×C23C2×C411yescharacteristic
68C2×C42×C23D412yes—
69C23×#101—22no—
70C23×#100—22no—
71C23×#99—22no—
72C23×#99—22no—
73C24×C23—21no—
74C24×C23C22⋊C412yes—
75#41×C23—42no—
76#41×C23—88no—
77C2×Q8×C23—22no—
78C2×Q8×C23C2×D411yescharacteristic
79C2×D4×C23—24no—
80C2×D4×C23—24no—
81C2×D4×C23—24no—
82C2×D4×C23—24no—
83C2×D4×C23—21no—
84C2×D4×C23—21no—
85C2×D4×C23—21no—
86C2×D4×C23—42no—
87C2×D4×C23—42no—
88C2×D4×C23—84no—
89C2×D4×C23C22⋊C412yes—
90C2×D4×C23C2×D411yescharacteristic
91C2×D4×C23C2×D412yes—
92C22×C4×C23—21no—
93C22×C4×C23—22no—
94C22×C4×C23—24no—
95C22×C4×C23—24no—
96C22×C4×C23C22×C411yescharacteristic
97C22×C4×C23C2×D412yes—
98M16×C23—42no—
99M16×C23—42no—
100C2×C8×C23—42no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—22no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—42no—
109C22⋊C4×C23—24no—
110C22⋊C4×C23—24no—
111C22⋊C4×C23—21no—
112C22⋊C4×C23—21no—
113C22⋊C4×C23—22no—
114C22⋊C4×C23—24no—
115C22⋊C4×C23—24no—
116C22⋊C4×C23—42no—
117C22⋊C4×C23C22⋊C412yes—
118C42×C23—21no—
119C42×C23—21no—
120C42×C23—22no—
121C42×C23—42no—
122C42×C23C22⋊C412yes—
123C42×C23C22⋊C411yescharacteristic
124C42×C23C22⋊C411yescharacteristic
125C42×C23C22⋊C412yes—
126#12722C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—22no—
128C23×C23—22no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—82no—
132C23×C23—82no—
133C23×C23C2×C22⋊C411yescharacteristic
134C23×C23#12012yes—
135Q8×C23—41no—
136Q8×C23—44no—
137D4×C23—42no—
138D4×C23—41no—
139D4×C23—42no—
140D4×C23—42no—
141D4×C23—48no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23—84no—
145D4×C23—84no—
146D4×C23—84no—
147C2×C4×C23—21no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—22no—
152C2×C4×C23—21no—
153C2×C4×C23—22no—
154C2×C4×C23—22no—
155C2×C4×C23—42no—
156C2×C4×C23—44no—
157C2×C4×C23—84no—
158C2×C4×C23—84no—
159C2×C4×C23—82no—
160C2×C4×C23C2×C22⋊C411yescharacteristic
161C2×C4×C23C2×C22⋊C411yescharacteristic
162C2×C4×C23#12012yes—
163C8×C23—82no—
164#3508C2×C2311yescharacteristic
165#2134C2×C2311yescharacteristic, complemented
166#2128C2×C2311yescharacteristic, complemented
167#2122C2×C2312yescomplemented
168#1827C2×C2312yescomplemented
169C22×C23—22no—
170C22×C23—21no—
171C22×C23—42no—
172C22×C23—41no—
173C22×C23—44no—
174C22×C23—44no—
175C22×C23—42no—
176C22×C23—42no—
177C22×C23—162no—
178C22×C23—162no—
179C22×C23#37911yescharacteristic
180C4×C23—22no—
181C4×C23—22no—
182C4×C23—21no—
183C4×C23—42no—
184C4×C23—41no—
185C4×C23—42no—
186C4×C23—42no—
187C4×C23—162no—
188C4×C23#41012yes—
189#559—22no—
190#550—22no—
191#546—24no—
192#540—22no—
193#535C22×C2311yescharacteristic
194#535C22×C2311yescharacteristic
195#533C4×C2311yescharacteristic, complemented
196#532—22no—
197C2×#127C4×C2311yescharacteristic, complemented
198#518C4×C2311yescharacteristic, complemented
199#518C4×C2311yescharacteristic, complemented
200C4×#41C22×C2311yescharacteristic
201#413C22×C2312yes—
202#410C22×C2312yes—
203#355—22no—
204#353—22no—
205AΣL(1,𝔽16)—24no—
206#339—22no—
207H(3,ℤ/4)—22no—
208C2×C23—21no—
209C2×C23—42no—
210C2×C23—41no—
211C2×C23—82no—
212C2×C23—82no—
213C2×C23#178211yescharacteristic
214C2×#41—24no—
215C2×#41C23×C2311yescharacteristic
216C22×D4—22no—
217C22×D4C2×C4×C2311yescharacteristic
218C22×D4C2×C4×C2311yescharacteristic
219C2×M16—22no—
220C4⋊Q8—21no—
221#127—22no—
222#127—21no—
223#127—21no—
224#127—21no—
225#127D4×C2312yes—
226#125—21no—
227#124—24no—
228#124—21no—
229#124D4×C2312yes—
230#123—24no—
231#123—24no—
232C22⋊Q8—22no—
233#121—24no—
234#121—24no—
235#121—24no—
236#121—22no—
237#121—22no—
238#120—22no—
239#120D4×C2314yes—
240C4×Q8—21no—
241C4×Q8—22no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—21no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249#117—22no—
250#117C2×C4×C2311yescharacteristic
251#117C2×C4×C2311yescharacteristic
252#117C2×C4×C2311yescharacteristic
253#117D4×C2312yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42C2×C4×C2311yescharacteristic
257C2×C42D4×C2312yes—
258#101—22no—
259#100—22no—
260#99—22no—
261#99—22no—
262C23#1272211yescharacteristic, direct product factor, Sylow
263C24—21no—
264C24C22⋊C4×C2312yes—
265#41—42no—
266#41—88no—
267C2×Q8—22no—
268C2×Q8C2×D4×C2311yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4C22⋊C4×C2312yes—
280C2×D4C2×D4×C2311yescharacteristic
281C2×D4C2×D4×C2312yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C22×C4×C2311yescharacteristic
287C22×C4C2×D4×C2312yes—
288M16—42no—
289M16—42no—
290C2×C8—42no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4C22⋊C4×C2312yes—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—42no—
312C42C22⋊C4×C2311yescharacteristic
313C42C22⋊C4×C2311yescharacteristic
314C42C22⋊C4×C2312yes—
315C42C22⋊C4×C2312yes—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23C2×C22⋊C4×C2311yescharacteristic
323C23C23×#12012yes—
324Q8—41no—
325Q8—44no—
326D4—42no—
327D4—41no—
328D4—42no—
329D4—42no—
330D4—48no—
331D4—42no—
332D4—42no—
333D4—84no—
334D4—84no—
335D4—84no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4C2×C22⋊C4×C2311yescharacteristic
350C2×C4C2×C22⋊C4×C2311yescharacteristic
351C2×C4C23×#12012yes—
352C8—82no—
353C22—22no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—162no—
362C22—162no—
363C22C23×#37911yescharacteristic
364C4—22no—
365C4—22no—
366C4—21no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—162no—
372C4C23×#41012yes—
373C2—21no—
374C2—42no—
375C2—41no—
376C2—82no—
377C2—82no—
378C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C4×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2134C2semidirect
3. C23×#2128C2semidirect
4. C23×#2122C2semidirect
5. C23×#1827C2semidirect
12. C23×#533C4semidirect
14. C2×C23×#127C4semidirect
15. C23×#518C4semidirect
16. C23×#518C4semidirect
126. #12722C23direct (N × H)
165. #2134C2×C23semidirect
166. #2128C2×C23semidirect
167. #2122C2×C23semidirect
168. #1827C2×C23semidirect
195. #533C4×C23semidirect
197. C2×#127C4×C23semidirect
198. #518C4×C23semidirect
199. #518C4×C23semidirect
262. C23#12722direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35082
C23×#21342
C23×#21282
C23×#21222
C23×#18272
#1272223