Finite Group Databasebeta

← #34078469

Subgroups — #34078469

Order 19328.

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×#18012785C211yesmaximal, characteristic, complemented
2#23753012C211yesmaximal, characteristic, complemented
3#23753010C211yesmaximal, characteristic, complemented
4#23752781C211yesmaximal, characteristic, complemented
5#23752391C214yesmaximal, complemented
6#23752381C214yesmaximal, complemented
7#23752230C211yesmaximal, characteristic, complemented
8C2×#18012655C211yesmaximal, characteristic, complemented
9#421×C151C211yesmaximal, characteristic, complemented
10#18012785—22no—
11#18012785—22no—
12#18012785—22no—
13#18012785C2214yes—
14C2×#14559005—21no—
15C2×#14559004—21no—
16C2×#14559004C2211yescharacteristic
17C2×#14559004C2211yescharacteristic, complemented
18C22×D604—21no—
19C22×D604C2211yescharacteristic
20C2×D4×D151—21no—
21C2×D4×D151—22no—
22#18012772—21no—
23#18012772C2212yescomplemented
24#18012770—21no—
25#18012769C2212yescomplemented
26#18012767—21no—
27#18012767C2212yescomplemented
28#18012765—21no—
29#18012764C2212yescomplemented
30#18012761—22no—
31#18012759—22no—
32C2×#14558988C2211yescharacteristic, complemented
33#18012736C2212yescomplemented
34#18012727C2212yescomplemented
35#18012714—22no—
36#18012709—22no—
37#18012667C2214yescomplemented
38#18012655C2214yescomplemented
39C2×#41×C151C2211yescharacteristic
40C2×M16×C151C2211yescharacteristic, complemented
41#117×C151C2211yescharacteristic, complemented
42#104×C151C2214yescomplemented
43C23×D151—22no—
44C23×D151D412yes—
45#14559005—24no—
46#14559005—24no—
47#14559004—24no—
48#14559004—24no—
49#14559004—21no—
50#14559004—21no—
51#14559004—21no—
52#14559004—21no—
53#14559004—21no—
54#14559004—21no—
55#14559004D412yescomplemented
56#14559004C2312yes—
57C2×Dic302D411yescharacteristic, complemented
58C2×Dic302C2311yescharacteristic, complemented
59C2×D604—22no—
60C2×D604—24no—
61C2×D604—24no—
62C2×D604—24no—
63C2×D604—21no—
64C2×D604—21no—
65C2×D604—21no—
66C2×D604—21no—
67C2×D604D412yes—
68C2×D604D411yescharacteristic, complemented
69C2×D604C2312yes—
70C2×D604C2311yescharacteristic, complemented
71C2×C22⋊D151—22no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74D4×D151—28no—
75D4×D151—24no—
76D4×D151—28no—
77D4×D151—24no—
78C2×C4×D151—22no—
79C2×C4×D151—21no—
80C2×C4×D151—21no—
81Dic604—22no—
82#14558994—22no—
83#14558993—22no—
84#14558993—22no—
85#14558992—22no—
86#14558991—22no—
87D1208—22no—
88#14558989—22no—
89#14558988—21no—
90#14558988C2312yescomplemented
91C2×C151⋊C8—21no—
92(C2×D151)⋊C4—24no—
93#41×C151—21no—
94#41×C151—21no—
95#41×C151D412yescomplemented
96#41×C151C2312yes—
97C2×Q8×C151D411yescharacteristic, complemented
98C2×D4×C151—21no—
99C2×D4×C151D411yescharacteristic, complemented
100C22×C4×C151—21no—
101C22×C4×C151C2311yescharacteristic
102M16×C151—21no—
103M16×C151C2312yescomplemented
104C2×C8×C151—21no—
105C4⋊C4×C151—21no—
106C22⋊C4×C151—21no—
107C42×C151C2312yescomplemented
108C22×D151—22no—
109C22×D151—21no—
110C22×D151—28no—
111C22×D151—22no—
112C22×D151—21no—
113C22×D151—82no—
114C22×D151—84no—
115C22×D151C2×D414yes—
116Dic302—21no—
117Dic302—21no—
118Dic302—21no—
119Dic302C2×D412yes—
120D604—28no—
121D604—22no—
122D604—22no—
123D604—21no—
124D604—22no—
125D604—22no—
126D604—21no—
127D604—21no—
128D604—84no—
129D604C2×D412yes—
130C22⋊D151—28no—
131C22⋊D151—22no—
132C22⋊D151—22no—
133C22⋊D151—41no—
134C22⋊D151—41no—
135C2×Dic151—21no—
136C2×Dic151—21no—
137C4×D151—28no—
138C4×D151—22no—
139C4×D151—22no—
140C4×D151—41no—
141C4×D151—41no—
142C151⋊C8—22no—
143C23×C151—21no—
144C23×C151C2×D411yescharacteristic
145Q8×C151—21no—
146Q8×C151C2×D412yes—
147D4×C151—22no—
148D4×C151—22no—
149D4×C151—21no—
150D4×C151C2×D412yes—
151C2×C4×C151—21no—
152C2×C4×C151—22no—
153C2×C4×C151—22no—
154C2×C4×C151—22no—
155C2×C4×C151C2×D411yescharacteristic
156C2×C4×C151C2×D411yescharacteristic
157C2×C4×C151C2×D411yescharacteristic
158C2×C4×C151C2×D411yescharacteristic
159C2×C4×C151C2×D411yescharacteristic
160C2×C4×C151C2411yescharacteristic
161C8×C151—22no—
162C22×C151—22no—
163C22×C151—41no—
164C22×C151—42no—
165C22×C151#12011yescharacteristic
166C22×C151#12011yescharacteristic
167C22×C151C22×D411yescharacteristic
168C2×D151—24no—
169C2×D151—22no—
170C2×D151—44no—
171C2×D151—42no—
172C2×D151—44no—
173C2×D151—42no—
174C2×D151—41no—
175C2×D151—82no—
176C2×D151—88no—
177C4×C151—22no—
178C4×C151—42no—
179C4×C151#12011yescharacteristic
180C4×C151#12011yescharacteristic
181C4×C151C22×D411yescharacteristic
182C4×C151C22×D411yescharacteristic
183Dic151—22no—
184Dic151—41no—
185C2×C151—21no—
186C2×C151—21no—
187C2×C151—21no—
188C2×C151—42no—
189C2×C151C2×#12011yescharacteristic
190D151—44no—
191D151—42no—
192D151—81no—
193C151#301911yescharacteristic, complemented, Sylow
194#3019—1511nomaximal, Sylow
195C2×H(5,𝔽2)—1511no—
196#575—1511no—
197#575—1511no—
198#535—1511no—
199#455—1514no—
200#453—1514no—
201#421—1511no—
202#421D15111yescharacteristic, complemented
203C2×#100—1511no—
204H(5,𝔽2)—3022no—
205H(5,𝔽2)—3022no—
206H(5,𝔽2)—3022no—
207H(5,𝔽2)—1514no—
208C2×#41—3021no—
209C2×#41—3021no—
210C2×#41—1511no—
211C2×#41—1511no—
212C2×#41C2×D15111yescharacteristic
213C22×D4—3021no—
214C22×D4—3022no—
215C22×D4—3021no—
216C22×D4—1511no—
217#137—3021no—
218#137—3021no—
219#137—1512no—
220#137—1512no—
221AGL(1,ℤ/8)—3021no—
222AGL(1,ℤ/8)—3021no—
223AGL(1,ℤ/8)—1512no—
224AGL(1,ℤ/8)—1512no—
225#135—3022no—
226#135—3022no—
227C2×M16—1511no—
228C2×M16C2×D15111yescharacteristic, complemented
229#127—1512no—
230#124—1512no—
231#121—3022no—
232#120—3022no—
233#117C2×D15111yescharacteristic, complemented
234#104—1514no—
235#104C2×D15114yescomplemented
236#100—1514no—
237C24—3022no—
238C24—1512no—
239#41—21no—
240#41—21no—
241#41—3024no—
242#41—3024no—
243#41—3024no—
244#41—3024no—
245#41—3021no—
246#41—3021no—
247#41—3021no—
248#41—3021no—
249#41—3021no—
250#41—3021no—
251#41—1512no—
252#41—1512no—
253#41D60412yescomplemented
254#41C22×D15112yes—
255C2×Q8—1511no—
256C2×Q8—1511no—
257C2×Q8D60411yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—3022no—
260C2×D4—3024no—
261C2×D4—3024no—
262C2×D4—3024no—
263C2×D4—3028no—
264C2×D4—3024no—
265C2×D4—3028no—
266C2×D4—3022no—
267C2×D4—3021no—
268C2×D4—3024no—
269C2×D4—3021no—
270C2×D4—3021no—
271C2×D4—3021no—
272C2×D4—3021no—
273C2×D4—3021no—
274C2×D4—1512no—
275C2×D4—1511no—
276C2×D4—1511no—
277C2×D4—1512no—
278C2×D4D60411yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—3022no—
281C22×C4—3021no—
282C22×C4—3021no—
283C22×C4C22×D15111yescharacteristic
284Q16—3022no—
285Q16—3022no—
286SD16—3022no—
287SD16—3022no—
288SD16—3022no—
289SD16—3022no—
290D8—3022no—
291D8—3022no—
292M16—21no—
293M16—3021no—
294M16—1512no—
295M16C22×D15112yescomplemented
296C2×C8—21no—
297C2×C8—3021no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3024no—
301C42C22×D15112yescomplemented
302C23—21no—
303C23—3028no—
304C23—3022no—
305C23—3021no—
306C23—3022no—
307C23—3021no—
308C23—1514no—
309C23—12082no—
310C23—12084no—
311C23D4×D15111yescharacteristic
312Q8—21no—
313Q8—3021no—
314Q8—3021no—
315Q8—3021no—
316Q8—1512no—
317Q8C2×D60412yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—3028no—
322D4—3028no—
323D4—3022no—
324D4—3022no—
325D4—3022no—
326D4—3021no—
327D4—3022no—
328D4—3022no—
329D4—3021no—
330D4—3022no—
331D4—3021no—
332D4—6041no—
333D4—6041no—
334D4—1512no—
335D4—12084no—
336D4C2×D60412yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—3028no—
342C2×C4—3021no—
343C2×C4—3022no—
344C2×C4—3022no—
345C2×C4—3021no—
346C2×C4—6041no—
347C2×C4—6041no—
348C2×C4D4×D15111yescharacteristic
349C2×C4D4×D15111yescharacteristic
350C2×C4D4×D15111yescharacteristic
351C2×C4C2×D60411yescharacteristic
352C2×C4C2×D60411yescharacteristic
353C2×C4C23×D15111yescharacteristic
354C8—22no—
355C8—3022no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—3024no—
360C22—3022no—
361C22—6044no—
362C22—6044no—
363C22—6042no—
364C22—6042no—
365C22—6041no—
366C22—12082no—
367C22—12088no—
368C22#1801270911yescharacteristic
369C22#1801270911yescharacteristic
370C22C2×D4×D15111yescharacteristic
371C4—22no—
372C4—42no—
373C4—3022no—
374C4—6041no—
375C4#1801270911yescharacteristic
376C4#1801270911yescharacteristic
377C4C2×D4×D15111yescharacteristic
378C4C22×D60411yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—6044no—
384C2—6042no—
385C2—12081no—
386C2C2×#1801270911yescharacteristic
subgroupG/H
centerC2C2×#18012709
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#421×C151C2
socleC2×C151C2×#120
hypercenter#421D151
hypocenterC151#3019
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#18012785C2semidirect
2. #23753012C2semidirect
3. #23753010C2semidirect
4. #23752781C2semidirect
5. #23752391C2semidirect
6. #23752381C2semidirect
7. #23752230C2semidirect
8. C2×#18012655C2semidirect
9. #421×C151C2semidirect
17. C2×#14559004C22semidirect
23. #18012772C22semidirect
25. #18012769C22semidirect
27. #18012767C22semidirect
29. #18012764C22semidirect
32. C2×#14558988C22semidirect
33. #18012736C22semidirect
34. #18012727C22semidirect
37. #18012667C22semidirect
38. #18012655C22semidirect
40. C2×M16×C151C22semidirect
41. #117×C151C22semidirect
42. #104×C151C22semidirect
55. #14559004D4semidirect
57. C2×Dic302D4semidirect
58. C2×Dic302C23semidirect
68. C2×D604D4semidirect
70. C2×D604C23semidirect
90. #14558988C23semidirect
95. #41×C151D4semidirect
97. C2×Q8×C151D4semidirect
99. C2×D4×C151D4semidirect
103. M16×C151C23semidirect
107. C42×C151C23semidirect
193. C151#3019semidirect
202. #421D151semidirect
228. C2×M16C2×D151semidirect
233. #117C2×D151semidirect
235. #104C2×D151semidirect
253. #41D604semidirect
257. C2×Q8D604semidirect
278. C2×D4D604semidirect
295. M16C22×D151semidirect
301. C42C22×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127852
#237530122
#237530102
#237527812
#237523912
#237523812
#237522302
C2×#180126552
#421×C1512
#3019151