Finite Group Databasebeta

← C23×#12779 #19160405

Subgroups — C23×#12779 #19160405

Order 5888.

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

normal subgroups only

This group has 732 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×#3550C211yesmaximal, characteristic
2C23×#2129C212yesmaximal, complemented
3C23×#2126C211yesmaximal, characteristic
4C23×#1909C212yesmaximal, complemented
5C23×#1884C211yesmaximal, characteristic
6C23×#568C2211yescharacteristic
7C23×#563—21no—
8C23×#563C412yescomplemented
9C23×#559—21no—
10Q82×C23C412yescomplemented
11C23×#556—22no—
12D4×Q8×C23—22no—
13C23×#542—24no—
14C23×#536—21no—
15C23×#536C2212yes—
16C23×#533—21no—
17C23×#532C2211yescharacteristic
18C23×#421—21no—
19C23×#419—22no—
20C23×#410C2212yes—
21C23×#410C2211yescharacteristic
22C23×#354—21no—
23C23×#354—24no—
24C23×#353—21no—
25C23×#339—21no—
26C23×H(3,ℤ/4)—22no—
27C2×#41×C23D411yescharacteristic
28C2×#41×C23C2311yescharacteristic
29C22×Q8×C23—21no—
30C22×Q8×C23D412yes—
31C23×#131—42no—
32C2×M16×C23—22no—
33C2×M16×C23—21no—
34C23×C4⋊Q8—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—24no—
37C23×C4⋊Q8—22no—
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8C2×C414yes—
40C23×#127D411yescharacteristic
41C23×#126—24no—
42C23×#126—24no—
43C23×#125—22no—
44C23×#124—24no—
45C23×#124—22no—
46C23×#124C2×C411yescharacteristic
47C23×#124D411yescharacteristic
48C23×#124D414yes—
49C23×#123—22no—
50C23×#123—22no—
51C23×#123—22no—
52C23×#123—22no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8—24no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—24no—
57C23×C22⋊Q8—22no—
58C23×C22⋊Q8—22no—
59C23×#121—22no—
60C4×Q8×C23—22no—
61C4×Q8×C23—24no—
62C4×Q8×C23—22no—
63C4×D4×C23—24no—
64C4×D4×C23—22no—
65C23×#117—21no—
66C23×#117—22no—
67C23×#117—22no—
68C23×#117—21no—
69C23×#117D412yes—
70C2×C42×C23D411yescharacteristic
71C23×#104—42no—
72C23×#103—44no—
73C23×#99—22no—
74C23×#99—22no—
75C23×#99—22no—
76#41×C23—41no—
77#41×C23—41no—
78#41×C23—41no—
79#41×C23—41no—
80C2×Q8×C23—22no—
81C2×Q8×C23—22no—
82C2×Q8×C23—22no—
83C2×Q8×C23—42no—
84C2×Q8×C23—42no—
85C2×Q8×C23—81no—
86C2×Q8×C23—81no—
87C2×Q8×C23C22⋊C414yes—
88C2×Q8×C23C22⋊C411yescharacteristic
89C2×Q8×C23C22×C411yescharacteristic
90C2×D4×C23—22no—
91C2×D4×C23—21no—
92C2×D4×C23C22⋊C411yescharacteristic
93C2×D4×C23C2×D412yes—
94C2×D4×C23C2×D411yescharacteristic
95C22×C4×C23—21no—
96C22×C4×C23—22no—
97C22×C4×C23—22no—
98C22×C4×C23—21no—
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23C2×D412yes—
101M16×C23—22no—
102M16×C23—42no—
103M16×C23—41no—
104C2×C8×C23—24no—
105C2×C8×C23—41no—
106C4⋊C4×C23—22no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—24no—
110C4⋊C4×C23—24no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—24no—
113C4⋊C4×C23—24no—
114C4⋊C4×C23—22no—
115C4⋊C4×C23—42no—
116C4⋊C4×C23—41no—
117C22⋊C4×C23—22no—
118C22⋊C4×C23—24no—
119C22⋊C4×C23—24no—
120C22⋊C4×C23—22no—
121C22⋊C4×C23—22no—
122C22⋊C4×C23—24no—
123C22⋊C4×C23—22no—
124C22⋊C4×C23—42no—
125C22⋊C4×C23—41no—
126C42×C23—24no—
127C42×C23—22no—
128C42×C23—21no—
129C42×C23—41no—
130C42×C23—42no—
131C42×C23C22⋊C411yescharacteristic
132C42×C23C22⋊C411yescharacteristic
133C42×C23C22⋊C414yes—
134#12779C2311yesmaximal, characteristic, direct product factor, Sylow
135C23×C23—21no—
136C23×C23#12012yes—
137C23×C23#12011yescharacteristic
138Q8×C23—41no—
139Q8×C23—44no—
140Q8×C23—41no—
141Q8×C23—44no—
142Q8×C23—42no—
143Q8×C23—42no—
144Q8×C23—82no—
145D4×C23—41no—
146D4×C23—42no—
147D4×C23—41no—
148D4×C23—44no—
149D4×C23—81no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—22no—
153C2×C4×C23—22no—
154C2×C4×C23—21no—
155C2×C4×C23—24no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—24no—
159C2×C4×C23—42no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—82no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165C2×C4×C23—81no—
166C2×C4×C23C2×C22⋊C412yes—
167C2×C4×C23C2×C22⋊C411yescharacteristic
168C2×C4×C23#12011yescharacteristic
169C8×C23—42no—
170#3550C2×C2311yescharacteristic
171#2129C2×C2312yescomplemented
172#2126C2×C2311yescharacteristic
173#1909C2×C2312yescomplemented
174#1884C2×C2311yescharacteristic
175C22×C23—22no—
176C22×C23—21no—
177C22×C23—41no—
178C22×C23—42no—
179C22×C23—42no—
180C22×C23—41no—
181C22×C23#37911yescharacteristic
182C4×C23—22no—
183C4×C23—21no—
184C4×C23—22no—
185C4×C23—41no—
186C4×C23—44no—
187C4×C23—41no—
188C4×C23—44no—
189C4×C23—162no—
190C4×C23—161no—
191C4×C23B(4,𝔽2)11yescharacteristic
192C4×C23#45811yescharacteristic
193#568C22×C2311yescharacteristic
194#563—21no—
195#563C4×C2312yescomplemented
196#559—21no—
197Q82C4×C2312yescomplemented
198#556—22no—
199D4×Q8—22no—
200#542—24no—
201#536—21no—
202#536C22×C2312yes—
203#533—21no—
204#532C22×C2311yescharacteristic
205#421—21no—
206#419—22no—
207#410C22×C2312yes—
208#410C22×C2311yescharacteristic
209#354—21no—
210#354—24no—
211#353—21no—
212#339—21no—
213H(3,ℤ/4)—22no—
214C2×C23—21no—
215C2×C23—42no—
216C2×C23—41no—
217C2×C23—81no—
218C2×C23#179011yescharacteristic
219C2×#41D4×C2311yescharacteristic
220C2×#41C23×C2311yescharacteristic
221C22×Q8—21no—
222C22×Q8D4×C2312yes—
223#131—42no—
224C2×M16—22no—
225C2×M16—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—22no—
229C4⋊Q8—24no—
230C4⋊Q8C2×C4×C2311yescharacteristic
231C4⋊Q8C2×C4×C2314yes—
232#127D4×C2311yescharacteristic
233#126—24no—
234#126—24no—
235#125—22no—
236#124—22no—
237#124—24no—
238#124C2×C4×C2311yescharacteristic
239#124D4×C2311yescharacteristic
240#124D4×C2314yes—
241#123—22no—
242#123—22no—
243#123—22no—
244#123—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—24no—
255C4×D4—24no—
256C4×D4—22no—
257#117—22no—
258#117—21no—
259#117—22no—
260#117—21no—
261#117D4×C2312yes—
262C2×C42D4×C2311yescharacteristic
263#104—42no—
264#103—44no—
265#99—22no—
266#99—22no—
267#99—22no—
268C23#1277911yescharacteristic, direct product factor, Sylow
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×Q8C22⋊C4×C2314yes—
281C2×Q8C22⋊C4×C2311yescharacteristic
282C2×Q8C22×C4×C2311yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C22⋊C4×C2311yescharacteristic
286C2×D4C2×D4×C2312yes—
287C2×D4C2×D4×C2311yescharacteristic
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4C2×D4×C2311yescharacteristic
293C22×C4C2×D4×C2312yes—
294M16—22no—
295M16—42no—
296M16—41no—
297C2×C8—24no—
298C2×C8—41no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C22⋊C4—22no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C42—24no—
320C42—22no—
321C42—21no—
322C42—41no—
323C42—42no—
324C42C22⋊C4×C2311yescharacteristic
325C42C22⋊C4×C2311yescharacteristic
326C42C22⋊C4×C2314yes—
327C23—21no—
328C23C23×#12012yes—
329C23C23×#12011yescharacteristic
330Q8—41no—
331Q8—44no—
332Q8—41no—
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—82no—
337D4—41no—
338D4—42no—
339D4—41no—
340D4—44no—
341D4—81no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4C2×C22⋊C4×C2312yes—
359C2×C4C2×C22⋊C4×C2311yescharacteristic
360C2×C4C23×#12011yescharacteristic
361C8—42no—
362C22—22no—
363C22—21no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22C23×#37911yescharacteristic
369C4—22no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—44no—
374C4—41no—
375C4—44no—
376C4—162no—
377C4—161no—
378C4C23×B(4,𝔽2)11yescharacteristic
379C4C23×#45811yescharacteristic
380C2—21no—
381C2—42no—
382C2—41no—
383C2—81no—
384C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×Q8C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#532C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2129C2semidirect
4. C23×#1909C2semidirect
8. C23×#563C4semidirect
10. Q82×C23C4semidirect
134. #12779C23direct (N × H)
171. #2129C2×C23semidirect
173. #1909C2×C23semidirect
195. #563C4×C23semidirect
197. Q82C4×C23semidirect
268. C23#12779direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35502
C23×#21292
C23×#21262
C23×#19092
C23×#18842
#1277923