Finite Group Databasebeta

← #34054050

Subgroups — #34054050

Order 19072.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#18011957C211yesmaximal, characteristic, complemented
2#23750425C211yesmaximal, characteristic, complemented
3#23750423C211yesmaximal, characteristic, complemented
4#23750194C211yesmaximal, characteristic, complemented
5#23749804C214yesmaximal, complemented
6#23749794C214yesmaximal, complemented
7#23749643C211yesmaximal, characteristic, complemented
8C2×#18011827C211yesmaximal, characteristic, complemented
9#421×C149C211yesmaximal, characteristic, complemented
10#18011957—22no—
11#18011957—22no—
12#18011957—22no—
13#18011957C2214yes—
14C2×#14558737—21no—
15C2×#14558736—21no—
16C2×#14558736C2211yescharacteristic
17C2×#14558736C2211yescharacteristic, complemented
18C22×D596—21no—
19C22×D596C2211yescharacteristic
20C2×D4×D149—21no—
21C2×D4×D149—22no—
22#18011944—21no—
23#18011944C2212yescomplemented
24#18011942—21no—
25#18011941C2212yescomplemented
26#18011939—21no—
27#18011939C2212yescomplemented
28#18011937—21no—
29#18011936C2212yescomplemented
30#18011933—22no—
31#18011931—22no—
32C2×#14558720C2211yescharacteristic, complemented
33#18011908C2212yescomplemented
34#18011899C2212yescomplemented
35#18011886—22no—
36#18011881—22no—
37#18011839C2214yescomplemented
38#18011827C2214yescomplemented
39C2×#41×C149C2211yescharacteristic
40C2×M16×C149C2211yescharacteristic, complemented
41#117×C149C2211yescharacteristic, complemented
42#104×C149C2214yescomplemented
43C23×D149—22no—
44C23×D149D412yes—
45#14558737—24no—
46#14558737—24no—
47#14558736—24no—
48#14558736—24no—
49#14558736—21no—
50#14558736—21no—
51#14558736—21no—
52#14558736—21no—
53#14558736—21no—
54#14558736—21no—
55#14558736D412yescomplemented
56#14558736C2312yes—
57C2×Dic298D411yescharacteristic, complemented
58C2×Dic298C2311yescharacteristic, complemented
59C2×D596—22no—
60C2×D596—24no—
61C2×D596—24no—
62C2×D596—24no—
63C2×D596—21no—
64C2×D596—21no—
65C2×D596—21no—
66C2×D596—21no—
67C2×D596D412yes—
68C2×D596D411yescharacteristic, complemented
69C2×D596C2312yes—
70C2×D596C2311yescharacteristic, complemented
71C2×C22⋊D149—22no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74D4×D149—28no—
75D4×D149—24no—
76D4×D149—28no—
77D4×D149—24no—
78C2×C4×D149—22no—
79C2×C4×D149—21no—
80C2×C4×D149—21no—
81Dic596—22no—
82#14558726—22no—
83#14558725—22no—
84#14558725—22no—
85#14558724—22no—
86#14558723—22no—
87D1192—22no—
88#14558721—22no—
89#14558720—21no—
90#14558720C2312yescomplemented
91C2×#11924746—21no—
92#14558713—24no—
93#41×C149—21no—
94#41×C149—21no—
95#41×C149D412yescomplemented
96#41×C149C2312yes—
97C2×Q8×C149D411yescharacteristic, complemented
98C2×D4×C149—21no—
99C2×D4×C149D411yescharacteristic, complemented
100C22×C4×C149—21no—
101C22×C4×C149C2311yescharacteristic
102M16×C149—21no—
103M16×C149C2312yescomplemented
104C2×C8×C149—21no—
105C4⋊C4×C149—21no—
106C22⋊C4×C149—21no—
107C42×C149C2312yescomplemented
108C22×D149—22no—
109C22×D149—21no—
110C22×D149—28no—
111C22×D149—22no—
112C22×D149—21no—
113C22×D149—82no—
114C22×D149—84no—
115C22×D149C2×D414yes—
116Dic298—21no—
117Dic298—21no—
118Dic298—21no—
119Dic298C2×D412yes—
120D596—28no—
121D596—22no—
122D596—22no—
123D596—21no—
124D596—22no—
125D596—22no—
126D596—21no—
127D596—21no—
128D596—84no—
129D596C2×D412yes—
130C22⋊D149—28no—
131C22⋊D149—22no—
132C22⋊D149—22no—
133C22⋊D149—41no—
134C22⋊D149—41no—
135C2×Dic149—21no—
136C2×Dic149—21no—
137C4×D149—28no—
138C4×D149—22no—
139C4×D149—22no—
140C4×D149—41no—
141C4×D149—41no—
142#11924746—22no—
143C23×C149—21no—
144C23×C149C2×D411yescharacteristic
145Q8×C149—21no—
146Q8×C149C2×D412yes—
147D4×C149—22no—
148D4×C149—22no—
149D4×C149—21no—
150D4×C149C2×D412yes—
151C2×C4×C149—21no—
152C2×C4×C149—22no—
153C2×C4×C149—22no—
154C2×C4×C149—22no—
155C2×C4×C149C2×D411yescharacteristic
156C2×C4×C149C2×D411yescharacteristic
157C2×C4×C149C2×D411yescharacteristic
158C2×C4×C149C2×D411yescharacteristic
159C2×C4×C149C2×D411yescharacteristic
160C2×C4×C149C2411yescharacteristic
161C8×C149—22no—
162C22×C149—22no—
163C22×C149—41no—
164C22×C149—42no—
165C22×C149#12011yescharacteristic
166C22×C149#12011yescharacteristic
167C22×C149C22×D411yescharacteristic
168C2×D149—24no—
169C2×D149—22no—
170C2×D149—44no—
171C2×D149—42no—
172C2×D149—44no—
173C2×D149—42no—
174C2×D149—41no—
175C2×D149—82no—
176C2×D149—88no—
177C4×C149—22no—
178C4×C149—42no—
179C4×C149#12011yescharacteristic
180C4×C149#12011yescharacteristic
181C4×C149C22×D411yescharacteristic
182C4×C149C22×D411yescharacteristic
183Dic149—22no—
184Dic149—41no—
185C2×C149—21no—
186C2×C149—21no—
187C2×C149—21no—
188C2×C149—42no—
189C2×C149C2×#12011yescharacteristic
190D149—44no—
191D149—42no—
192D149—81no—
193C149#301911yescharacteristic, complemented, Sylow
194#3019—1491nomaximal, Sylow
195C2×H(5,𝔽2)—1491no—
196#575—1491no—
197#575—1491no—
198#535—1491no—
199#455—1494no—
200#453—1494no—
201#421—1491no—
202#421D14911yescharacteristic, complemented
203C2×#100—1491no—
204H(5,𝔽2)—2982no—
205H(5,𝔽2)—2982no—
206H(5,𝔽2)—2982no—
207H(5,𝔽2)—1494no—
208C2×#41—2981no—
209C2×#41—2981no—
210C2×#41—1491no—
211C2×#41—1491no—
212C2×#41C2×D14911yescharacteristic
213C22×D4—2981no—
214C22×D4—2982no—
215C22×D4—2981no—
216C22×D4—1491no—
217#137—2981no—
218#137—2981no—
219#137—1492no—
220#137—1492no—
221AGL(1,ℤ/8)—2981no—
222AGL(1,ℤ/8)—2981no—
223AGL(1,ℤ/8)—1492no—
224AGL(1,ℤ/8)—1492no—
225#135—2982no—
226#135—2982no—
227C2×M16—1491no—
228C2×M16C2×D14911yescharacteristic, complemented
229#127—1492no—
230#124—1492no—
231#121—2982no—
232#120—2982no—
233#117C2×D14911yescharacteristic, complemented
234#104—1494no—
235#104C2×D14914yescomplemented
236#100—1494no—
237C24—2982no—
238C24—1492no—
239#41—21no—
240#41—21no—
241#41—2984no—
242#41—2984no—
243#41—2984no—
244#41—2984no—
245#41—2981no—
246#41—2981no—
247#41—2981no—
248#41—2981no—
249#41—2981no—
250#41—2981no—
251#41—1492no—
252#41—1492no—
253#41D59612yescomplemented
254#41C22×D14912yes—
255C2×Q8—1491no—
256C2×Q8—1491no—
257C2×Q8D59611yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—2982no—
260C2×D4—2984no—
261C2×D4—2984no—
262C2×D4—2984no—
263C2×D4—2988no—
264C2×D4—2984no—
265C2×D4—2988no—
266C2×D4—2982no—
267C2×D4—2981no—
268C2×D4—2984no—
269C2×D4—2981no—
270C2×D4—2981no—
271C2×D4—2981no—
272C2×D4—2981no—
273C2×D4—2981no—
274C2×D4—1492no—
275C2×D4—1491no—
276C2×D4—1491no—
277C2×D4—1492no—
278C2×D4D59611yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—2982no—
281C22×C4—2981no—
282C22×C4—2981no—
283C22×C4C22×D14911yescharacteristic
284Q16—2982no—
285Q16—2982no—
286SD16—2982no—
287SD16—2982no—
288SD16—2982no—
289SD16—2982no—
290D8—2982no—
291D8—2982no—
292M16—21no—
293M16—2981no—
294M16—1492no—
295M16C22×D14912yescomplemented
296C2×C8—21no—
297C2×C8—2981no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2984no—
301C42C22×D14912yescomplemented
302C23—21no—
303C23—2988no—
304C23—2982no—
305C23—2981no—
306C23—2982no—
307C23—2981no—
308C23—1494no—
309C23—11922no—
310C23—11924no—
311C23D4×D14911yescharacteristic
312Q8—21no—
313Q8—2981no—
314Q8—2981no—
315Q8—2981no—
316Q8—1492no—
317Q8C2×D59612yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—2988no—
322D4—2988no—
323D4—2982no—
324D4—2982no—
325D4—2982no—
326D4—2981no—
327D4—2982no—
328D4—2981no—
329D4—2982no—
330D4—2981no—
331D4—2982no—
332D4—5961no—
333D4—5961no—
334D4—1492no—
335D4—11924no—
336D4C2×D59612yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—2988no—
342C2×C4—2981no—
343C2×C4—2982no—
344C2×C4—2982no—
345C2×C4—2981no—
346C2×C4—5961no—
347C2×C4—5961no—
348C2×C4D4×D14911yescharacteristic
349C2×C4D4×D14911yescharacteristic
350C2×C4D4×D14911yescharacteristic
351C2×C4C2×D59611yescharacteristic
352C2×C4C2×D59611yescharacteristic
353C2×C4C23×D14911yescharacteristic
354C8—22no—
355C8—2982no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—2984no—
360C22—2982no—
361C22—5964no—
362C22—5964no—
363C22—5962no—
364C22—5962no—
365C22—5961no—
366C22—11922no—
367C22—11928no—
368C22#1801188111yescharacteristic
369C22#1801188111yescharacteristic
370C22C2×D4×D14911yescharacteristic
371C4—22no—
372C4—42no—
373C4—2982no—
374C4—5961no—
375C4#1801188111yescharacteristic
376C4#1801188111yescharacteristic
377C4C2×D4×D14911yescharacteristic
378C4C22×D59611yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—5964no—
384C2—5962no—
385C2—11921no—
386C2C2×#1801188111yescharacteristic
subgroupG/H
centerC2C2×#18011881
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#421×C149C2
socleC2×C149C2×#120
hypercenter#421D149
hypocenterC149#3019
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#18011957C2semidirect
2. #23750425C2semidirect
3. #23750423C2semidirect
4. #23750194C2semidirect
5. #23749804C2semidirect
6. #23749794C2semidirect
7. #23749643C2semidirect
8. C2×#18011827C2semidirect
9. #421×C149C2semidirect
17. C2×#14558736C22semidirect
23. #18011944C22semidirect
25. #18011941C22semidirect
27. #18011939C22semidirect
29. #18011936C22semidirect
32. C2×#14558720C22semidirect
33. #18011908C22semidirect
34. #18011899C22semidirect
37. #18011839C22semidirect
38. #18011827C22semidirect
40. C2×M16×C149C22semidirect
41. #117×C149C22semidirect
42. #104×C149C22semidirect
55. #14558736D4semidirect
57. C2×Dic298D4semidirect
58. C2×Dic298C23semidirect
68. C2×D596D4semidirect
70. C2×D596C23semidirect
90. #14558720C23semidirect
95. #41×C149D4semidirect
97. C2×Q8×C149D4semidirect
99. C2×D4×C149D4semidirect
103. M16×C149C23semidirect
107. C42×C149C23semidirect
193. C149#3019semidirect
202. #421D149semidirect
228. C2×M16C2×D149semidirect
233. #117C2×D149semidirect
235. #104C2×D149semidirect
253. #41D596semidirect
257. C2×Q8D596semidirect
278. C2×D4D596semidirect
295. M16C22×D149semidirect
301. C42C22×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
#237504252
#237504232
#237501942
#237498042
#237497942
#237496432
C2×#180118272
#421×C1492
#3019149