Finite Group Databasebeta

← #15682772

Subgroups — #15682772

Order 2944.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#13093252C211yesmaximal, characteristic
2#137×D23C212yesmaximal, complemented
3#13092987C211yesmaximal, characteristic, complemented
4#13092601C212yesmaximal, complemented
5#13092598C212yesmaximal, complemented
6#13092597C211yesmaximal, characteristic, complemented
7#13092595C211yesmaximal, characteristic, complemented
8#13092594C211yesmaximal, characteristic, complemented
9#13092432C212yesmaximal, complemented
10#13092398C211yesmaximal, characteristic, complemented
11C23×#455C211yesmaximal, characteristic, complemented
12C23×H(5,𝔽2)C2211yescharacteristic
13#10625161—22no—
14#10625161C2212yes—
15#41×D23—22no—
16#41×D23C2212yes—
17C2×Q8×D23C2211yescharacteristic, complemented
18#10625155—22no—
19#10625155—22no—
20#10625155C2211yescharacteristic
21C2×#67290—22no—
22C2×#67290—22no—
23C2×#67290C2211yescharacteristic
24C23×#137C2212yescomplemented
25C23×#124C2211yescharacteristic, complemented
26#10625122C2212yescomplemented
27#10625115C2211yescharacteristic, complemented
28#10625113C2212yescomplemented
29#10625107C2211yescharacteristic, complemented
30#10625089—22no—
31Q16×D23—22no—
32#10625086—22no—
33SD16×D23—22no—
34#10625080C2212yescomplemented
35M16×D23C2212yescomplemented
36#10625054—22no—
37#10625049—22no—
38#10625043C2211yescharacteristic, complemented
39#10625040C2211yescharacteristic, complemented
40C23×#104C2212yescomplemented
41C23×#100C2211yescharacteristic, complemented
42#10625019C2212yescomplemented
43#10625015C2211yescharacteristic, complemented
44#10625007C2212yescomplemented
45#10625006C2212yescomplemented
46#10624987C2212yescomplemented
47#41×C23—22no—
48#41×C23C2312yes—
49C2×Q8×C23C2311yescharacteristic
50C2×D4×C23—22no—
51C2×D4×C23—22no—
52C2×D4×C23C2311yescharacteristic
53C2×C22⋊D23—24no—
54C2×C22⋊D23—22no—
55C2×C22⋊D23—22no—
56C2×C22⋊D23D412yes—
57C22×Dic23—22no—
58C22×Dic23D412yes—
59#67292D412yes—
60Q8×D23—22no—
61Q8×D23—22no—
62Q8×D23—22no—
63Q8×D23D412yes—
64#67290—24no—
65#67290—24no—
66#67290—24no—
67#67290—22no—
68#67290—22no—
69#67290—24no—
70#67290—22no—
71#67290—22no—
72#67290—22no—
73#67290D412yes—
74D4×D23—24no—
75D4×D23—22no—
76D4×D23—22no—
77D4×D23D412yes—
78#67288—24no—
79#67288—22no—
80#67288—22no—
81#67288C2312yes—
82C2×C4×D23—22no—
83C2×C4×D23—22no—
84C2×C4×D23—21no—
85C2×C4×D23C2311yescharacteristic
86C2×Dic46—22no—
87C2×Dic46—22no—
88C2×Dic46—21no—
89C2×Dic46C2311yescharacteristic
90C2×Dic46C2311yescharacteristic
91Q16×C23—22no—
92SD16×C23—22no—
93M16×C23C2312yes—
94C22⋊C4×C23—22no—
95C42×C23C2311yescharacteristic
96C22⋊Dic23—22no—
97#67275—22no—
98#67275—22no—
99#67274—22no—
100#67273—22no—
101(C2×D23)⋊C4—21no—
102C4⋊Dic23—22no—
103Dic23⋊C4—21no—
104Dic23⋊C4—22no—
105C4×Dic23C2311yescharacteristic
106#67267C2312yes—
107Dic92—22no—
108#67263—22no—
109#67262—22no—
110C8×D23—22no—
111C23×C23—22no—
112C23×C23C2×D412yes—
113C22×D23—22no—
114C22×D23—22no—
115C22×D23C2×D411yescharacteristic
116Q8×C23—22no—
117Q8×C23C2×D412yes—
118D4×C23—24no—
119D4×C23—22no—
120D4×C23—22no—
121D4×C23C2×D412yes—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23C2411yescharacteristic
127C22⋊D23—24no—
128C22⋊D23—24no—
129C22⋊D23—22no—
130C22⋊D23—24no—
131C22⋊D23—22no—
132C22⋊D23—22no—
133C2×Dic23—24no—
134C2×Dic23—22no—
135C2×Dic23—22no—
136C2×Dic23—22no—
137C2×Dic23—22no—
138C2×Dic23—21no—
139C2×Dic23—21no—
140C2×Dic23C2×D412yes—
141C2×Dic23C2×D411yescharacteristic
142D92—22no—
143D92C2×D412yes—
144C4×D23—24no—
145C4×D23—22no—
146C4×D23—22no—
147C4×D23—22no—
148C4×D23—42no—
149C4×D23C2×D412yes—
150C4×D23C2×D412yes—
151Dic46—24no—
152Dic46—22no—
153Dic46—22no—
154Dic46—22no—
155Dic46—41no—
156Dic46—41no—
157Dic46C2×D412yes—
158C8×C23—22no—
159C23⋊C8—22no—
160#3022—231nomaximal, Sylow
161C22×C23—22no—
162C22×C23—22no—
163C22×C23—42no—
164C22×C23—82no—
165C22×C23C22×D411yescharacteristic
166C2×D23—22no—
167C2×D23—41no—
168C2×D23—82no—
169C2×D23—82no—
170C2×D23#12012yes—
171C4×C23—22no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23C22×D412yes—
175Dic23—22no—
176Dic23—22no—
177Dic23—41no—
178Dic23—41no—
179Dic23#12011yescharacteristic
180Dic23#12011yescharacteristic
181#585—231no—
182C2×#137—232no—
183#536—231no—
184#456—232no—
185#456—232no—
186#455—231no—
187#455—231no—
188#455—231no—
189#455D2311yescharacteristic, complemented
190#421—232no—
191#413—231no—
192C2×C23—21no—
193C2×C23—42no—
194C2×C23—42no—
195C2×C23C2×#12011yescharacteristic
196D23—22no—
197D23—42no—
1982−1+4—232no—
1992−1+4—462no—
200H(5,𝔽2)—231no—
201H(5,𝔽2)—462no—
202H(5,𝔽2)—462no—
203H(5,𝔽2)C2×D2311yescharacteristic
204C2×#41—231no—
205C2×#41—232no—
206C2×#41—462no—
207C2×#41—462no—
208C2×#41—462no—
209C22×Q8—231no—
210#137—232no—
211#137—232no—
212#137—232no—
213#137—462no—
214#137—462no—
215#137C2×D2312yescomplemented
216C2×Q16—462no—
217C2×SD16—462no—
218C2×M16—232no—
219C4⋊Q8—231no—
220C4⋊Q8—231no—
221#124—231no—
222#124C2×D2311yescharacteristic, complemented
223#123—462no—
224C22⋊Q8—462no—
225#117—231no—
226#104—232no—
227#104—232no—
228#104—232no—
229#104C2×D2312yescomplemented
230#101—232no—
231#100—231no—
232#100C2×D2311yescharacteristic, complemented
233C23#302211yescharacteristic, complemented, Sylow
234#41—22no—
235#41—232no—
236#41—232no—
237#41—232no—
238#41—462no—
239#41—462no—
240#41—464no—
241#41—462no—
242#41—462no—
243#41—462no—
244#41—464no—
245#41—462no—
246#41—464no—
247#41—464no—
248#41—464no—
249#41—462no—
250#41C22×D2312yes—
251C2×Q8—231no—
252C2×Q8—232no—
253C2×Q8—231no—
254C2×Q8—461no—
255C2×Q8—462no—
256C2×Q8—462no—
257C2×Q8—462no—
258C2×Q8—462no—
259C2×Q8—462no—
260C2×Q8C22×D2311yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—232no—
264C2×D4—232no—
265C2×D4—462no—
266C2×D4—462no—
267C2×D4—464no—
268C2×D4—462no—
269C2×D4—464no—
270C2×D4—462no—
271C2×D4C22×D2311yescharacteristic
272C22×C4—231no—
273C22×C4—232no—
274C22×C4—462no—
275C22×C4—462no—
276C22×C4—461no—
277C22×C4—462no—
278Q16—22no—
279Q16—462no—
280Q16—462no—
281Q16—462no—
282SD16—22no—
283SD16—462no—
284SD16—462no—
285SD16—462no—
286M16—232no—
287M16—462no—
288M16C22×D2312yes—
289C2×C8—462no—
290C4⋊C4—461no—
291C4⋊C4—462no—
292C4⋊C4—462no—
293C22⋊C4—22no—
294C22⋊C4—461no—
295C22⋊C4—462no—
296C42—231no—
297C42C22×D2311yescharacteristic
298C23—22no—
299C23—231no—
300C23—462no—
301C23—462no—
302C23D4×D2312yes—
303Q8—22no—
304Q8—232no—
305Q8—462no—
306Q8—464no—
307Q8—462no—
308Q8—462no—
309Q8—921no—
310Q8—921no—
311Q8D4×D2312yes—
312D4—24no—
313D4—22no—
314D4—22no—
315D4—232no—
316D4—462no—
317D4—462no—
318D4—464no—
319D4—462no—
320D4—464no—
321D4—464no—
322D4—462no—
323D4D4×D2312yes—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—231no—
329C2×C4—232no—
330C2×C4—232no—
331C2×C4—232no—
332C2×C4—462no—
333C2×C4—462no—
334C2×C4—464no—
335C2×C4—462no—
336C2×C4—462no—
337C2×C4—462no—
338C2×C4—464no—
339C2×C4—462no—
340C2×C4—462no—
341C2×C4—461no—
342C2×C4—461no—
343C2×C4—922no—
344C2×C4C23×D2311yescharacteristic
345C8—22no—
346C8—462no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—82no—
351C22—232no—
352C22—462no—
353C22—921no—
354C22—1842no—
355C22—1842no—
356C22C2×D4×D2311yescharacteristic
357C4—22no—
358C4—41no—
359C4—41no—
360C4—231no—
361C4—231no—
362C4—462no—
363C4—462no—
364C4—921no—
365C4—921no—
366C4C2×D4×D2312yes—
367C2—21no—
368C2—42no—
369C2—42no—
370C2—462no—
371C2—922no—
372C2#120×D2311yescharacteristic
subgroupG/H
centerC2#120×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#455C2
socleC2×C23C2×#120
prime radical#13093252C2
hypercenter#455D23
hypocenterC23#3022
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #137×D23C2semidirect
3. #13092987C2semidirect
4. #13092601C2semidirect
5. #13092598C2semidirect
6. #13092597C2semidirect
7. #13092595C2semidirect
8. #13092594C2semidirect
9. #13092432C2semidirect
10. #13092398C2semidirect
11. C23×#455C2semidirect
17. C2×Q8×D23C22semidirect
24. C23×#137C22semidirect
25. C23×#124C22semidirect
26. #10625122C22semidirect
27. #10625115C22semidirect
28. #10625113C22semidirect
29. #10625107C22semidirect
34. #10625080C22semidirect
35. M16×D23C22semidirect
38. #10625043C22semidirect
39. #10625040C22semidirect
40. C23×#104C22semidirect
41. C23×#100C22semidirect
42. #10625019C22semidirect
43. #10625015C22semidirect
44. #10625007C22semidirect
45. #10625006C22semidirect
46. #10624987C22semidirect
189. #455D23semidirect
215. #137C2×D23semidirect
222. #124C2×D23semidirect
229. #104C2×D23semidirect
232. #100C2×D23semidirect
233. C23#3022semidirect

Maximal subgroups

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

subgroupindex
#130932522
#137×D232
#130929872
#130926012
#130925982
#130925972
#130925952
#130925942
#130924322
#130923982
C23×#4552
#302223