Finite Group Databasebeta

← D31×#455 #16855038

Subgroups — D31×#455 #16855038

Order 3968.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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
1H(5,𝔽2)×D31C211yesmaximal, characteristic
2#137×D31C212yesmaximal, complemented
3#124×D31C211yesmaximal, characteristic, complemented
4#14521229C211yesmaximal, characteristic, complemented
5#14521228C212yesmaximal, complemented
6#14521227C211yesmaximal, characteristic, complemented
7#14521226C211yesmaximal, characteristic, complemented
8#14521225C212yesmaximal, complemented
9#104×D31C212yesmaximal, complemented
10#100×D31C211yesmaximal, characteristic, complemented
11C31×#455C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic
13#11758566—22no—
14#11758566C2212yes—
15#41×D31—22no—
16#41×D31C2212yes—
17C2×Q8×D31C2211yescharacteristic, complemented
18#11758561—22no—
19#11758561—22no—
20#11758561C2211yescharacteristic
21C2×D4×D31—22no—
22C2×D4×D31—22no—
23C2×D4×D31C2211yescharacteristic
24C31×#137C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758524C2211yescharacteristic, complemented
28#11758519C2212yescomplemented
29#11758513C2211yescharacteristic, complemented
30#11758495—22no—
31Q16×D31—22no—
32#11758492—22no—
33SD16×D31—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758462—22no—
37C22⋊C4×D31—22no—
38#11758452C2211yescharacteristic, complemented
39C42×D31C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—22no—
48C23×D31D412yes—
49#41×C31—22no—
50#41×C31C2312yes—
51C2×Q8×C31C2311yescharacteristic, complemented
52C2×D4×C31—22no—
53C2×D4×C31—22no—
54C2×D4×C31C2311yescharacteristic
55C2×C22⋊D31—22no—
56C2×C22⋊D31—24no—
57C2×C22⋊D31—22no—
58C2×C22⋊D31D412yes—
59#92499—22no—
60#92499D412yes—
61Q8×D31—22no—
62Q8×D31—22no—
63Q8×D31D412yes—
64#92497—24no—
65#92497—22no—
66#92497—22no—
67#92497D412yes—
68D4×D31—24no—
69D4×D31—24no—
70D4×D31—24no—
71D4×D31—22no—
72D4×D31—24no—
73D4×D31—22no—
74D4×D31—22no—
75D4×D31—22no—
76D4×D31—22no—
77D4×D31D412yes—
78#92495—24no—
79#92495—22no—
80#92495—22no—
81#92495C2312yes—
82C2×D124—22no—
83C2×D124—22no—
84C2×D124C2311yescharacteristic
85C2×C4×D31—22no—
86C2×C4×D31—22no—
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×Dic62—21no—
91C2×Dic62C2311yescharacteristic, complemented
92Q16×C31—22no—
93SD16×C31—22no—
94M16×C31C2312yescomplemented
95C22⋊C4×C31—22no—
96C42×C31C2311yescharacteristic, complemented
97C22⋊Dic31—22no—
98#92482—22no—
99#92482—22no—
100#92481—22no—
101#92480—22no—
102(C2×D31)⋊C4—22no—
103(C2×D31)⋊C4—22no—
104C4×Dic31—21no—
105C4×Dic31C2311yescharacteristic, complemented
106#92474C2312yescomplemented
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—22no—
112C23×C31C2×D412yes—
113C22×D31—24no—
114C22×D31—22no—
115C22×D31—22no—
116C22×D31—22no—
117C22×D31—22no—
118C22×D31—44no—
119C22×D31—84no—
120C22×D31C2×D411yescharacteristic
121C22×D31C2×D412yes—
122Q8×C31—22no—
123Q8×C31C2×D412yes—
124D4×C31—24no—
125D4×C31—22no—
126D4×C31—22no—
127D4×C31C2×D412yes—
128C2×C4×C31—22no—
129C2×C4×C31—22no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31C2411yescharacteristic
133C22⋊D31—24no—
134C22⋊D31—24no—
135C22⋊D31—22no—
136C22⋊D31—22no—
137C22⋊D31—24no—
138C22⋊D31—22no—
139C2×Dic31—22no—
140C2×Dic31—22no—
141C2×Dic31—21no—
142C2×Dic31—21no—
143C2×Dic31C2×D411yescharacteristic
144D124—24no—
145D124—22no—
146D124—22no—
147D124C2×D412yes—
148C4×D31—24no—
149C4×D31—22no—
150C4×D31—22no—
151C4×D31—22no—
152C4×D31—42no—
153C4×D31—41no—
154C4×D31—41no—
155C4×D31C2×D412yes—
156C4×D31C2×D412yes—
157Dic62—22no—
158Dic62—22no—
159Dic62—42no—
160Dic62C2×D412yes—
161C8×C31—22no—
162C31⋊C8—22no—
163C2×#455—311nomaximal, Sylow
164C22×C31—22no—
165C22×C31—22no—
166C22×C31—42no—
167C22×C31—82no—
168C22×C31C22×D411yescharacteristic
169C2×D31—22no—
170C2×D31—22no—
171C2×D31—22no—
172C2×D31—44no—
173C2×D31—44no—
174C2×D31—42no—
175C2×D31—82no—
176C2×D31—84no—
177C2×D31—82no—
178C2×D31#12011yescharacteristic
179C2×D31#12011yescharacteristic
180C4×C31—22no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31C22×D412yes—
184Dic31—22no—
185Dic31—41no—
186Dic31—41no—
187Dic31#12012yes—
188C2×H(5,𝔽2)—311no—
189C2×#137—312no—
190C2×#124—311no—
191#455—311no—
192#455—312no—
193#455—311no—
194#455—312no—
195#455—311no—
196#455D3111yescharacteristic, direct product factor
197C2×#104—312no—
198C2×#100—311no—
199C2×C31—21no—
200C2×C31—42no—
201C2×C31—42no—
202C2×C31C2×#12011yescharacteristic
203D31—21no—
204D31—42no—
205D31—42no—
206D31#45512yesdirect product factor
207H(5,𝔽2)—312no—
208H(5,𝔽2)—311no—
209H(5,𝔽2)—622no—
210H(5,𝔽2)—622no—
211H(5,𝔽2)—622no—
212H(5,𝔽2)C2×D3111yescharacteristic
213C2×#41—312no—
214C2×#41—622no—
215C22×Q8—311no—
216C22×D4—311no—
217C22×D4—622no—
218C22×D4—622no—
219#137—312no—
220#137—312no—
221#137—312no—
222#137—622no—
223#137—622no—
224#137C2×D3112yescomplemented
225C2×Q16—622no—
226C2×SD16—622no—
227C2×M16—312no—
228#124—311no—
229#124—311no—
230#124—311no—
231#124—622no—
232#124C2×D3111yescharacteristic, complemented
233C2×C22⋊C4—622no—
234C2×C42—311no—
235#104—312no—
236#104—312no—
237#104—312no—
238#104C2×D3112yescomplemented
239#100—311no—
240#100—312no—
241#100C2×D3111yescharacteristic, complemented
242C31C2×#45511yescharacteristic, complemented, Sylow
243C24—312no—
244C24—622no—
245#41—22no—
246#41—312no—
247#41—312no—
248#41—312no—
249#41—622no—
250#41—622no—
251#41—624no—
252#41—622no—
253#41—622no—
254#41—624no—
255#41—622no—
256#41C22×D3112yes—
257C2×Q8—312no—
258C2×Q8—311no—
259C2×Q8—621no—
260C2×Q8—622no—
261C2×Q8—622no—
262C2×Q8C22×D3111yescharacteristic, complemented
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—312no—
266C2×D4—312no—
267C2×D4—311no—
268C2×D4—622no—
269C2×D4—622no—
270C2×D4—622no—
271C2×D4—622no—
272C2×D4—622no—
273C2×D4—622no—
274C2×D4—624no—
275C2×D4—624no—
276C2×D4—624no—
277C2×D4—624no—
278C2×D4—624no—
279C2×D4—622no—
280C2×D4—622no—
281C2×D4—622no—
282C2×D4C22×D3111yescharacteristic
283C22×C4—311no—
284C22×C4—622no—
285C22×C4—622no—
286C22×C4—621no—
287C22×C4—621no—
288Q16—22no—
289Q16—622no—
290Q16—622no—
291Q16—622no—
292SD16—22no—
293SD16—622no—
294SD16—622no—
295SD16—622no—
296M16—312no—
297M16—622no—
298M16C22×D3112yescomplemented
299C2×C8—622no—
300C22⋊C4—22no—
301C22⋊C4—622no—
302C22⋊C4—622no—
303C22⋊C4—622no—
304C42—311no—
305C42—621no—
306C42C22×D3111yescharacteristic, complemented
307C23—22no—
308C23—311no—
309C23—312no—
310C23—622no—
311C23—622no—
312C23—622no—
313C23—622no—
314C23—624no—
315C23—1244no—
316C23—2484no—
317C23D4×D3112yes—
318Q8—22no—
319Q8—312no—
320Q8—622no—
321Q8—622no—
322Q8—1242no—
323Q8D4×D3112yes—
324D4—24no—
325D4—22no—
326D4—22no—
327D4—312no—
328D4—622no—
329D4—624no—
330D4—624no—
331D4—622no—
332D4—622no—
333D4—624no—
334D4—624no—
335D4—622no—
336D4—622no—
337D4D4×D3112yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—311no—
343C2×C4—312no—
344C2×C4—312no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—624no—
348C2×C4—622no—
349C2×C4—622no—
350C2×C4—622no—
351C2×C4—621no—
352C2×C4—621no—
353C2×C4—1241no—
354C2×C4—1241no—
355C2×C4—1242no—
356C2×C4C23×D3111yescharacteristic
357C8—22no—
358C8—622no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—82no—
363C22—311no—
364C22—311no—
365C22—622no—
366C22—622no—
367C22—622no—
368C22—1242no—
369C22—1244no—
370C22—1244no—
371C22—2482no—
372C22—2482no—
373C22—2484no—
374C22C2×D4×D3111yescharacteristic
375C4—22no—
376C4—41no—
377C4—41no—
378C4—312no—
379C4—622no—
380C4—1241no—
381C4—1241no—
382C4C2×D4×D3112yes—
383C2—21no—
384C2—42no—
385C2—42no—
386C2—312no—
387C2—621no—
388C2—1242no—
389C2—1242no—
390C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#455C2
socleC2×C31C2×#120
prime radicalH(5,𝔽2)×D31C2
hypercenter#455D31
hypocenterC31C2×#455
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #137×D31C2semidirect
3. #124×D31C2semidirect
4. #14521229C2semidirect
5. #14521228C2semidirect
6. #14521227C2semidirect
7. #14521226C2semidirect
8. #14521225C2semidirect
9. #104×D31C2semidirect
10. #100×D31C2semidirect
11. C31×#455C2semidirect
17. C2×Q8×D31C22semidirect
24. C31×#137C22semidirect
25. C31×#124C22semidirect
26. #11758528C22semidirect
27. #11758524C22semidirect
28. #11758519C22semidirect
29. #11758513C22semidirect
34. #11758486C22semidirect
35. M16×D31C22semidirect
38. #11758452C22semidirect
39. C42×D31C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
51. C2×Q8×C31C23semidirect
91. C2×Dic62C23semidirect
94. M16×C31C23semidirect
96. C42×C31C23semidirect
105. C4×Dic31C23semidirect
106. #92474C23semidirect
196. #455D31direct (N × H)
206. D31#455direct (N × H)
224. #137C2×D31semidirect
232. #124C2×D31semidirect
238. #104C2×D31semidirect
241. #100C2×D31semidirect
242. C31C2×#455semidirect
262. C2×Q8C22×D31semidirect
298. M16C22×D31semidirect
306. C42C22×D31semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D312
#137×D312
#124×D312
#145212292
#145212282
#145212272
#145212262
#145212252
#104×D312
#100×D312
C31×#4552
C2×#45531