Finite Group Databasebeta

← #34053950

Subgroups — #34053950

Order 19072.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14558737C211yesmaximal, characteristic
2C2×#18011943C212yesmaximal, complemented
3C22⋊Q8×D149C211yesmaximal, characteristic, complemented
4#23749790C212yesmaximal, complemented
5#23749786C212yesmaximal, complemented
6#23749781C211yesmaximal, characteristic, complemented
7#23749778C211yesmaximal, characteristic, complemented
8#23749775C211yesmaximal, characteristic, complemented
9#23749625C212yesmaximal, complemented
10#23749587C211yesmaximal, characteristic, complemented
11#451×C149C211yesmaximal, characteristic, complemented
12C2×#14558737—28no—
13C2×#14558737—21no—
14C2×#14558737—21no—
15C2×#14558737C2212yes—
16C2×#14558737C2212yes—
17C2×Q8×D149C2211yescharacteristic, complemented
18C22×D596—21no—
19C22×D596C2211yescharacteristic
20C22×C4×D149—21no—
21C22×C4×D149C2211yescharacteristic
22#18011943—28no—
23C2×#14558726C2212yescomplemented
24C2×#14558725C2212yescomplemented
25C2×#14558723C2212yescomplemented
26C2×#14558719C2212yescomplemented
27#18011891C2211yescharacteristic, complemented
28#18011890C2211yescharacteristic
29#18011889—21no—
30#18011888—21no—
31#18011887C2211yescharacteristic, complemented
32C4⋊C4×D149—21no—
33C4⋊C4×D149C2211yescharacteristic, complemented
34C22⋊C4×D149—21no—
35#18011835C2212yescomplemented
36#18011833C2212yescomplemented
37#18011831C2212yescomplemented
38C22⋊#11924746C2211yescharacteristic
39#18011823C2212yescomplemented
40C22×Q8×C149C2211yescharacteristic
41C2×Q16×C149C2212yescomplemented
42C22⋊Q8×C149C2211yescharacteristic
43#103×C149C2212yescomplemented
44C22⋊C8×C149C2211yescharacteristic
45C23×D149—21no—
46C23×D149D411yescharacteristic
47#14558737—28no—
48#14558737—28no—
49#14558737—24no—
50#14558737—24no—
51#14558737—24no—
52#14558737D418yes—
53C2×Dic298—21no—
54C2×Dic298C2311yescharacteristic
55Q8×D149—24no—
56C2×D596—28no—
57C2×D596—22no—
58C2×D596—22no—
59C2×D596—24no—
60C2×D596—21no—
61C2×D596C2312yes—
62C22×Dic149D411yescharacteristic
63C2×C4×D149—28no—
64C2×C4×D149—22no—
65C2×C4×D149—22no—
66C2×C4×D149—24no—
67C2×C4×D149—21no—
68C2×C4×D149—21no—
69C2×C4×D149D411yescharacteristic
70C2×C4×D149D411yescharacteristic
71C2×C4×D149C2311yescharacteristic
72C2×C4×D149C2311yescharacteristic
73#14558726—24no—
74#14558725—24no—
75#14558723—24no—
76#14558719—24no—
77C2×#11924746C2312yes—
78C4⋊Dic149—21no—
79C4⋊Dic149C2311yescharacteristic
80#14558715—21no—
81#14558715—21no—
82#14558715—21no—
83C22⋊Dic149—21no—
84#14558713—21no—
85#14558713—21no—
86C2×Q8×C149—24no—
87C2×Q8×C149—21no—
88C2×Q8×C149C2311yescharacteristic
89C2×Q8×C149C2312yes—
90C22×C4×C149—21no—
91C22×C4×C149C2311yescharacteristic
92Q16×C149—24no—
93C2×C8×C149C2312yes—
94C4⋊C4×C149—21no—
95C4⋊C4×C149C2311yescharacteristic
96C22⋊C4×C149—21no—
97C22×D149—24no—
98C22×D149—22no—
99C22×D149—22no—
100C22×D149—22no—
101C22×D149—41no—
102C22×D149—42no—
103C22×D149—42no—
104C22×D149C2×D411yescharacteristic
105C22×D149C2×D411yescharacteristic
106Dic298—22no—
107Dic298—42no—
108D596—28no—
109D596—24no—
110D596—28no—
111D596—22no—
112D596C2×D414yes—
113C2×Dic149—22no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149C2×D411yescharacteristic
117C2×Dic149C2×D411yescharacteristic
118C4×D149—28no—
119C4×D149—22no—
120C4×D149—22no—
121C4×D149—28no—
122C4×D149—22no—
123C4×D149—42no—
124C4×D149—42no—
125C4×D149C2×D414yes—
126#11924746—22no—
127C23×C149C2×D411yescharacteristic
128Q8×C149—24no—
129Q8×C149—22no—
130Q8×C149—22no—
131Q8×C149C2×D414yes—
132C2×C4×C149—24no—
133C2×C4×C149—22no—
134C2×C4×C149—22no—
135C2×C4×C149—21no—
136C2×C4×C149—21no—
137C2×C4×C149C2×D411yescharacteristic
138C2×C4×C149C2411yescharacteristic
139C8×C149—22no—
140C22×C149—21no—
141C22×C149—21no—
142C22×C149#13712yescomplemented
143C22×C149C22×D411yescharacteristic
144C2×D149—21no—
145C2×D149—21no—
146C2×D149—21no—
147C2×D149—24no—
148C2×D149—21no—
149C2×D149—21no—
150C2×D149—44no—
151C2×D149—48no—
152C2×D149—42no—
153C2×D149—42no—
154C2×D149#12012yes—
155C4×C149—24no—
156C4×C149—21no—
157C4×C149—41no—
158C4×C149—41no—
159C4×C149C22×D412yes—
160Dic149—21no—
161Dic149—41no—
162Dic149—41no—
163Dic149#12012yes—
164C2×C149—22no—
165C2×C149C2×#12011yescharacteristic
166C2×C149C2×#13711yescharacteristic
167C2×C149#57511yescharacteristic
168D149—22no—
169D149—41no—
170D149—44no—
171C149#300711yescharacteristic, complemented, Sylow
172#3007—1491nomaximal, Sylow
173C22×#41—1491no—
174C2×#137—1492no—
175C2×C22⋊Q8—1491no—
176#452—1492no—
177#452—1492no—
178#451—1491no—
179#451D14911yescharacteristic, complemented
180#450—1491no—
181#450—1491no—
182#417—1492no—
183#407—1491no—
184C2×#41—2988no—
185C2×#41—2981no—
186C2×#41—2981no—
187C2×#41—1492no—
188C2×#41—1492no—
189C22×Q8—1491no—
190C22×Q8C2×D14911yescharacteristic
191C22×D4—2981no—
192C22×D4—1491no—
193C23×C4—2981no—
194C23×C4—1491no—
195#137—2988no—
196C2×Q16—1492no—
197C2×Q16C2×D14912yescomplemented
198C2×SD16—1492no—
199C2×SD16—1492no—
200C2×M16—1492no—
201C22⋊Q8—2981no—
202C22⋊Q8—2981no—
203C22⋊Q8—1491no—
204C22⋊Q8—1491no—
205C22⋊Q8—1491no—
206C22⋊Q8C2×D14911yescharacteristic
207C2×C4⋊C4—2981no—
208C2×C4⋊C4—1491no—
209C2×C22⋊C4—2981no—
210#103—1492no—
211#103C2×D14912yescomplemented
212#102—1492no—
213#102—1492no—
214C22⋊C8—1491no—
215C22⋊C8—1492no—
216C22⋊C8C2×D14911yescharacteristic
217C24—2981no—
218C24—1491no—
219#41—2988no—
220#41—2988no—
221#41—2984no—
222#41—2984no—
223#41—2984no—
224#41—1498no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—2981no—
228C2×Q8—2984no—
229C2×Q8—1491no—
230C2×Q8C22×D14911yescharacteristic
231C2×Q8C22×D14912yes—
232C2×D4—2988no—
233C2×D4—2982no—
234C2×D4—2982no—
235C2×D4—2984no—
236C2×D4—2981no—
237C2×D4—1492no—
238C22×C4—21no—
239C22×C4—2988no—
240C22×C4—2982no—
241C22×C4—2982no—
242C22×C4—2984no—
243C22×C4—2981no—
244C22×C4—2981no—
245C22×C4—1491no—
246C22×C4—1491no—
247C22×C4—1491no—
248C22×C4—1491no—
249C22×C4—1491no—
250C22×C4C22×D14911yescharacteristic
251Q16—24no—
252Q16—2984no—
253SD16—2984no—
254SD16—2984no—
255M16—2984no—
256C2×C8—1492no—
257C2×C8C22×D14912yes—
258C4⋊C4—21no—
259C4⋊C4—2981no—
260C4⋊C4—2981no—
261C4⋊C4—2981no—
262C4⋊C4—2981no—
263C4⋊C4—1491no—
264C4⋊C4C22×D14911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2981no—
267C22⋊C4—2981no—
268C22⋊C4—2981no—
269C23—2984no—
270C23—2982no—
271C23—2982no—
272C23—2982no—
273C23—5961no—
274C23—5962no—
275C23—5962no—
276C23—1491no—
277C23—1491no—
278C23D4×D14911yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—2982no—
283Q8—5962no—
284Q8D4×D14914yes—
285D4—2988no—
286D4—2984no—
287D4—2988no—
288D4—2982no—
289D4—1494no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—2988no—
296C2×C4—2982no—
297C2×C4—2988no—
298C2×C4—2982no—
299C2×C4—2982no—
300C2×C4—2982no—
301C2×C4—2981no—
302C2×C4—2981no—
303C2×C4—5962no—
304C2×C4—5962no—
305C2×C4—1491no—
306C2×C4—1491no—
307C2×C4—1494no—
308C2×C4D4×D14911yescharacteristic
309C2×C4C23×D14911yescharacteristic
310C8—22no—
311C8—2982no—
312C22—21no—
313C22—21no—
314C22—2981no—
315C22—2981no—
316C22—2981no—
317C22—2981no—
318C22—2981no—
319C22—2984no—
320C22—5964no—
321C22—5968no—
322C22—5962no—
323C22—5962no—
324C22—1492no—
325C22#1801194312yescomplemented
326C22C2×D4×D14911yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—2981no—
332C4—5961no—
333C4—5961no—
334C4—1492no—
335C4C2×D4×D14912yes—
336C2—22no—
337C2—2982no—
338C2—5961no—
339C2—5964no—
340C2#120×D14911yescharacteristic
341C2C2×#1801194311yescharacteristic
342C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#451×C149C2
socleC22×C149C22×D4
prime radicalC22×#14558737C2
hypercenter#451D149
hypocenterC149#3007
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#18011943C2semidirect
3. C22⋊Q8×D149C2semidirect
4. #23749790C2semidirect
5. #23749786C2semidirect
6. #23749781C2semidirect
7. #23749778C2semidirect
8. #23749775C2semidirect
9. #23749625C2semidirect
10. #23749587C2semidirect
11. #451×C149C2semidirect
17. C2×Q8×D149C22semidirect
23. C2×#14558726C22semidirect
24. C2×#14558725C22semidirect
25. C2×#14558723C22semidirect
26. C2×#14558719C22semidirect
27. #18011891C22semidirect
31. #18011887C22semidirect
33. C4⋊C4×D149C22semidirect
35. #18011835C22semidirect
36. #18011833C22semidirect
37. #18011831C22semidirect
39. #18011823C22semidirect
41. C2×Q16×C149C22semidirect
43. #103×C149C22semidirect
142. C22×C149#137semidirect
171. C149#3007semidirect
179. #451D149semidirect
197. C2×Q16C2×D149semidirect
211. #103C2×D149semidirect
325. C22#18011943semidirect

Maximal subgroups

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

subgroupindex
C22×#145587372
C2×#180119432
C22⋊Q8×D1492
#237497902
#237497862
#237497812
#237497782
#237497752
#237496252
#237495872
#451×C1492
#3007149