Finite Group Databasebeta

← #34078468

Subgroups — #34078468

Order 19328.

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

normal subgroups only

This group has 762 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
1H(5,𝔽2)×D151C211yesmaximal, characteristic, complemented
2#23753011C212yesmaximal, complemented
3#23752780C211yesmaximal, characteristic, complemented
4#23752389C211yesmaximal, characteristic, complemented
5#23752388C212yesmaximal, complemented
6#23752382C211yesmaximal, characteristic, complemented
7#23752381C212yesmaximal, complemented
8#23752380C211yesmaximal, characteristic, complemented
9#23752227C212yesmaximal, complemented
10#100×D151C211yesmaximal, characteristic, complemented
11#455×C151C211yesmaximal, characteristic, complemented
12#18012785—22no—
13#18012785C2212yes—
14#18012784—22no—
15#18012784—22no—
16#18012784C2211yescharacteristic
17C2×#14559005C2211yescharacteristic, complemented
18#41×D151—22no—
19#41×D151C2212yes—
20C2×D4×D151—22no—
21C2×D4×D151—22no—
22C2×D4×D151C2211yescharacteristic
23#18012771—22no—
24#18012770C2212yescomplemented
25#18012767C2212yescomplemented
26#18012766—22no—
27#18012764C2212yescomplemented
28#18012762—22no—
29#18012760—22no—
30M16×D151C2212yescomplemented
31#18012736C2211yescharacteristic, complemented
32#18012735C2211yescharacteristic, complemented
33#18012728C2211yescharacteristic, complemented
34#18012712—22no—
35#18012709—22no—
36#18012696C2211yescharacteristic, complemented
37#18012667C2212yescomplemented
38#18012666C2212yescomplemented
39#18012665C2212yescomplemented
40#18012656C2211yescharacteristic, complemented
41#18012655C2212yescomplemented
42H(5,𝔽2)×C151C2211yescharacteristic
43#137×C151C2212yescomplemented
44#124×C151C2211yescharacteristic, complemented
45#104×C151C2212yescomplemented
46#100×C151C2211yescharacteristic, complemented
47C23×D151—22no—
48C23×D151D412yes—
49#14559005—22no—
50#14559005—22no—
51#14559005—22no—
52#14559005D412yescomplemented
53#14559004—24no—
54#14559004—22no—
55#14559004—22no—
56#14559004C2312yes—
57#14559003—24no—
58#14559003—22no—
59#14559003—22no—
60#14559003D412yescomplemented
61Q8×D151D412yescomplemented
62C2×D604—22no—
63C2×D604—22no—
64C2×D604—21no—
65C2×D604—22no—
66C2×D604C2311yescharacteristic
67C2×D604C2311yescharacteristic, complemented
68C2×C22⋊D151—22no—
69C2×C22⋊D151—24no—
70C2×C22⋊D151—22no—
71C2×C22⋊D151—22no—
72C2×C22⋊D151D412yes—
73D4×D151—24no—
74D4×D151—24no—
75D4×D151—24no—
76D4×D151—22no—
77D4×D151—24no—
78D4×D151—22no—
79D4×D151—22no—
80D4×D151—22no—
81D4×D151—22no—
82D4×D151D412yescomplemented
83C2×C4×D151—22no—
84C2×C4×D151—22no—
85C2×C4×D151—21no—
86C2×C4×D151C2311yescharacteristic
87#14558994—22no—
88#14558993—22no—
89#14558991—22no—
90#14558991—22no—
91D1208—22no—
92#14558989—22no—
93#14558988C2312yescomplemented
94#14558987—22no—
95C8×D151—22no—
96Dic151⋊C4—21no—
97(C2×D151)⋊C4—21no—
98(C2×D151)⋊C4—22no—
99C4×Dic151C2311yescharacteristic, complemented
100#41×C151—22no—
101#41×C151C2312yes—
102C2×Q8×C151C2311yescharacteristic, complemented
103C2×D4×C151—22no—
104C2×D4×C151—22no—
105C2×D4×C151C2311yescharacteristic
106Q16×C151—22no—
107SD16×C151—22no—
108M16×C151C2312yescomplemented
109C22⋊C4×C151—22no—
110C42×C151C2311yescharacteristic, complemented
111C22×D151—24no—
112C22×D151—22no—
113C22×D151—22no—
114C22×D151—22no—
115C22×D151—22no—
116C22×D151—21no—
117C22×D151—82no—
118C22×D151—84no—
119C22×D151C2×D412yes—
120C22×D151C2×D411yescharacteristic
121Dic302—22no—
122Dic302C2×D412yes—
123D604—24no—
124D604—22no—
125D604—22no—
126D604—22no—
127D604—41no—
128D604—41no—
129D604—82no—
130D604C2×D412yes—
131C22⋊D151—24no—
132C22⋊D151—24no—
133C22⋊D151—22no—
134C22⋊D151—22no—
135C22⋊D151—24no—
136C22⋊D151—22no—
137C22⋊D151—82no—
138C2×Dic151—22no—
139C2×Dic151—22no—
140C2×Dic151—21no—
141C2×Dic151C2×D411yescharacteristic
142C4×D151—24no—
143C4×D151—22no—
144C4×D151—22no—
145C4×D151—22no—
146C4×D151—42no—
147C4×D151C2×D412yes—
148C4×D151C2×D412yes—
149C151⋊C8—22no—
150C23×C151—22no—
151C23×C151C2×D412yes—
152Q8×C151—22no—
153Q8×C151C2×D412yes—
154D4×C151—24no—
155D4×C151—22no—
156D4×C151—22no—
157D4×C151C2×D412yes—
158C2×C4×C151—22no—
159C2×C4×C151—22no—
160C2×C4×C151—21no—
161C2×C4×C151—21no—
162C2×C4×C151C2411yescharacteristic
163C8×C151—22no—
164C22×C151—22no—
165C22×C151—22no—
166C22×C151—42no—
167C22×C151—82no—
168C22×C151C22×D411yescharacteristic
169C2×D151—22no—
170C2×D151—22no—
171C2×D151—41no—
172C2×D151—44no—
173C2×D151—44no—
174C2×D151—42no—
175C2×D151—42no—
176C2×D151—41no—
177C2×D151—82no—
178C2×D151—84no—
179C2×D151—82no—
180C2×D151#12011yescharacteristic
181C2×D151#12011yescharacteristic
182C4×C151—22no—
183C4×C151—41no—
184C4×C151—41no—
185C4×C151C22×D412yes—
186Dic151—22no—
187Dic151—41no—
188Dic151#12012yes—
189C2×C151—21no—
190C2×C151—42no—
191C2×C151—42no—
192C2×C151C2×#12011yescharacteristic
193D151—21no—
194D151—21no—
195D151—42no—
196D151—42no—
197D151—81no—
198C151#301911yescharacteristic, complemented, Sylow
199#3019—1511nomaximal, Sylow
200C2×H(5,𝔽2)—1511no—
201#575—1512no—
202#535—1511no—
203#455—1512no—
204#455—1511no—
205#455D15111yescharacteristic, complemented
206#453—1511no—
207#453—1512no—
208#453—1511no—
209#421—1512no—
210C2×#100—1511no—
211H(5,𝔽2)—3022no—
212H(5,𝔽2)—3022no—
213H(5,𝔽2)—3022no—
214H(5,𝔽2)—1512no—
215H(5,𝔽2)—1511no—
216H(5,𝔽2)C2×D15111yescharacteristic
217C2×#41—3022no—
218C2×#41—1512no—
219C2×#41—1511no—
220C22×D4—3022no—
221C22×D4—3022no—
222C22×D4—1511no—
223#137—3022no—
224#137—1512no—
225#137C2×D15112yescomplemented
226AGL(1,ℤ/8)—3022no—
227AGL(1,ℤ/8)—1512no—
228AGL(1,ℤ/8)—1512no—
229#135—3022no—
230#135—3022no—
231C2×M16—1512no—
232#127—1511no—
233#127—1511no—
234#124—1511no—
235#124C2×D15111yescharacteristic, complemented
236#121—3022no—
237#120—3022no—
238#117—1511no—
239#104—1512no—
240#104—1512no—
241#104—1512no—
242#104C2×D15112yescomplemented
243#100—1511no—
244#100—1512no—
245#100C2×D15111yescharacteristic, complemented
246C24—3022no—
247C24—1512no—
248#41—22no—
249#41—3024no—
250#41—3022no—
251#41—3022no—
252#41—3024no—
253#41—3022no—
254#41—3022no—
255#41—3022no—
256#41—3022no—
257#41—3022no—
258#41—1512no—
259#41—1512no—
260#41—1512no—
261#41C22×D15112yes—
262C2×Q8—1512no—
263C2×Q8C22×D15111yescharacteristic, complemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—3022no—
267C2×D4—3022no—
268C2×D4—3021no—
269C2×D4—3022no—
270C2×D4—3024no—
271C2×D4—3024no—
272C2×D4—3024no—
273C2×D4—3022no—
274C2×D4—3024no—
275C2×D4—3022no—
276C2×D4—3022no—
277C2×D4—3024no—
278C2×D4—3022no—
279C2×D4—3022no—
280C2×D4—3022no—
281C2×D4—3022no—
282C2×D4—3022no—
283C2×D4—1512no—
284C2×D4—1512no—
285C2×D4—1511no—
286C2×D4—1511no—
287C2×D4C22×D15111yescharacteristic
288C22×C4—3022no—
289C22×C4—3022no—
290C22×C4—3021no—
291C22×C4—1511no—
292Q16—22no—
293Q16—3022no—
294SD16—22no—
295SD16—3022no—
296SD16—3022no—
297SD16—3022no—
298D8—3022no—
299D8—3022no—
300M16—3022no—
301M16—1512no—
302M16C22×D15112yescomplemented
303C2×C8—3022no—
304C4⋊C4—3021no—
305C22⋊C4—22no—
306C22⋊C4—3021no—
307C22⋊C4—3022no—
308C42—1511no—
309C42C22×D15111yescharacteristic, complemented
310C23—22no—
311C23—3022no—
312C23—3024no—
313C23—3022no—
314C23—3022no—
315C23—3022no—
316C23—3021no—
317C23—1512no—
318C23—1511no—
319C23—12082no—
320C23—12084no—
321C23D4×D15112yes—
322Q8—22no—
323Q8—3022no—
324Q8—1512no—
325Q8D4×D15112yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—3024no—
330D4—3022no—
331D4—3024no—
332D4—3024no—
333D4—3022no—
334D4—3022no—
335D4—3024no—
336D4—3022no—
337D4—3022no—
338D4—3022no—
339D4—6041no—
340D4—6041no—
341D4—1512no—
342D4—12082no—
343D4—12082no—
344D4D4×D15112yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—3024no—
350C2×C4—3022no—
351C2×C4—3022no—
352C2×C4—3022no—
353C2×C4—3022no—
354C2×C4—3022no—
355C2×C4—3021no—
356C2×C4—6042no—
357C2×C4—1511no—
358C2×C4—1512no—
359C2×C4—1512no—
360C2×C4C23×D15111yescharacteristic
361C8—22no—
362C8—3022no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—82no—
367C22—3022no—
368C22—3022no—
369C22—6041no—
370C22—6042no—
371C22—6042no—
372C22—6044no—
373C22—6044no—
374C22—6041no—
375C22—1511no—
376C22—1511no—
377C22—12082no—
378C22—12082no—
379C22—12084no—
380C22C2×D4×D15111yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—3022no—
385C4—6041no—
386C4—1512no—
387C4C2×D4×D15112yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—3021no—
392C2—3021no—
393C2—6042no—
394C2—6042no—
395C2—12081no—
396C2#120×D15111yescharacteristic
subgroupG/H
centerC2#120×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#455×C151C2
socleC2×C151C2×#120
hypercenter#455D151
hypocenterC151#3019
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×D151C2semidirect
2. #23753011C2semidirect
3. #23752780C2semidirect
4. #23752389C2semidirect
5. #23752388C2semidirect
6. #23752382C2semidirect
7. #23752381C2semidirect
8. #23752380C2semidirect
9. #23752227C2semidirect
10. #100×D151C2semidirect
11. #455×C151C2semidirect
17. C2×#14559005C22semidirect
24. #18012770C22semidirect
25. #18012767C22semidirect
27. #18012764C22semidirect
30. M16×D151C22semidirect
31. #18012736C22semidirect
32. #18012735C22semidirect
33. #18012728C22semidirect
36. #18012696C22semidirect
37. #18012667C22semidirect
38. #18012666C22semidirect
39. #18012665C22semidirect
40. #18012656C22semidirect
41. #18012655C22semidirect
43. #137×C151C22semidirect
44. #124×C151C22semidirect
45. #104×C151C22semidirect
46. #100×C151C22semidirect
52. #14559005D4semidirect
60. #14559003D4semidirect
61. Q8×D151D4semidirect
67. C2×D604C23semidirect
82. D4×D151D4semidirect
93. #14558988C23semidirect
99. C4×Dic151C23semidirect
102. C2×Q8×C151C23semidirect
108. M16×C151C23semidirect
110. C42×C151C23semidirect
198. C151#3019semidirect
205. #455D151semidirect
225. #137C2×D151semidirect
235. #124C2×D151semidirect
242. #104C2×D151semidirect
245. #100C2×D151semidirect
263. C2×Q8C22×D151semidirect
302. M16C22×D151semidirect
309. C42C22×D151semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D1512
#237530112
#237527802
#237523892
#237523882
#237523822
#237523812
#237523802
#237522272
#100×D1512
#455×C1512
#3019151