Finite Group Databasebeta

← #34109371

Subgroups — #34109371

Order 20096.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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×Q8×D157C211yesmaximal, characteristic, complemented
2C2×#19097803C212yesmaximal, complemented
3#23757786C211yesmaximal, characteristic
4#23757428C211yesmaximal, characteristic, complemented
5#23757427C212yesmaximal, complemented
6#23757414C211yesmaximal, characteristic, complemented
7#23757413C212yesmaximal, complemented
8#23757412C211yesmaximal, characteristic, complemented
9#23757280C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic, complemented
11#451×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2211yescharacteristic
13C22×Dic314—21no—
14C22×Dic314C2211yescharacteristic
15C2×Q8×D157—28no—
16C2×Q8×D157—21no—
17C2×Q8×D157—21no—
18C2×Q8×D157C2212yes—
19C2×Q8×D157C2212yes—
20C22×C4×D157—21no—
21C22×C4×D157C2211yescharacteristic
22#19097803—28no—
23C2×#14578944C2212yescomplemented
24C2×#14578943C2212yescomplemented
25C2×#14578941C2212yescomplemented
26C2×#14578937C2212yes—
27#19097756—21no—
28#19097747C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097741—21no—
32#19097733—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic
35#19097696C2212yescomplemented
36#19097695C2212yescomplemented
37#19097694C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic
39#19097683C2212yes—
40C22×Q8×C157C2211yescharacteristic
41C2×Q16×C157C2212yescomplemented
42C22⋊Q8×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic
45C23×D157D411yescharacteristic
46#14578955—24no—
47C2×Dic314—28no—
48C2×Dic314—24no—
49C2×Dic314—22no—
50C2×Dic314—22no—
51C2×Dic314—21no—
52C2×Dic314C2312yes—
53Q8×D157—28no—
54Q8×D157—28no—
55Q8×D157—24no—
56Q8×D157—24no—
57Q8×D157—24no—
58Q8×D157D418yescomplemented
59C2×D628—21no—
60C2×D628—21no—
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C22×Dic157—21no—
64C22×Dic157D411yescharacteristic
65C2×C4×D157—28no—
66C2×C4×D157—22no—
67C2×C4×D157—22no—
68C2×C4×D157—24no—
69C2×C4×D157—21no—
70C2×C4×D157D411yescharacteristic
71C2×C4×D157D411yescharacteristic
72C2×C4×D157C2311yescharacteristic
73C2×C4×D157C2311yescharacteristic
74#14578944—24no—
75#14578943—24no—
76#14578941—24no—
77#14578937—24no—
78C2×#11929434C2312yes—
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578931—21no—
82#14578931—21no—
83#14578931—21no—
84#14578931—21no—
85C4×Dic157—21no—
86C2×Q8×C157—24no—
87C2×Q8×C157—21no—
88C2×Q8×C157C2311yescharacteristic
89C2×Q8×C157C2312yes—
90C22×C4×C157—21no—
91C22×C4×C157C2311yescharacteristic
92Q16×C157—24no—
93C2×C8×C157C2312yes—
94C4⋊C4×C157—21no—
95C4⋊C4×C157C2311yescharacteristic
96C22⋊C4×C157—21no—
97C22×D157—22no—
98C22×D157—22no—
99C22×D157—22no—
100C22×D157—21no—
101C22×D157C2×D411yescharacteristic
102C22×D157C2×D411yescharacteristic
103Dic314—24no—
104Dic314—28no—
105Dic314—22no—
106Dic314—28no—
107Dic314C2×D414yes—
108D628—22no—
109D628—42no—
110D628—42no—
111C22⋊D157—42no—
112C2×Dic157—24no—
113C2×Dic157—22no—
114C2×Dic157—22no—
115C2×Dic157—21no—
116C2×Dic157C2×D411yescharacteristic
117C2×Dic157C2×D411yescharacteristic
118C4×D157—28no—
119C4×D157—28no—
120C4×D157—22no—
121C4×D157—22no—
122C4×D157—22no—
123C4×D157—42no—
124C4×D157C2×D414yes—
125#11929434—22no—
126C23×C157C2×D411yescharacteristic
127Q8×C157—24no—
128Q8×C157—22no—
129Q8×C157—22no—
130Q8×C157C2×D414yes—
131C2×C4×C157—24no—
132C2×C4×C157—22no—
133C2×C4×C157—22no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157C2×D411yescharacteristic
137C2×C4×C157C2411yescharacteristic
138C8×C157—22no—
139C22×C157—21no—
140C22×C157—21no—
141C22×C157#13712yescomplemented
142C22×C157C22×D411yescharacteristic
143C2×D157—28no—
144C2×D157—21no—
145C2×D157—21no—
146C2×D157—21no—
147C2×D157—21no—
148C2×D157—41no—
149C2×D157—41no—
150C2×D157—41no—
151C2×D157#12012yes—
152C2×D157#13712yescomplemented
153C4×C157—24no—
154C4×C157—21no—
155C4×C157—41no—
156C4×C157—41no—
157C4×C157C22×D412yes—
158Dic157—24no—
159Dic157—21no—
160Dic157—41no—
161Dic157#12012yes—
162C2×C157—22no—
163C2×C157C2×#12011yescharacteristic
164C2×C157C2×#13711yescharacteristic
165C2×C157C2×#13711yescharacteristic
166D157—22no—
167D157—22no—
168D157—81no—
169C157#300411yescharacteristic, complemented, Sylow
170#3004—1571nomaximal, Sylow
171C23×Q8—1571no—
172C2×#137—1572no—
173#525—1571no—
174#451—1572no—
175#451—1571no—
176#451D15711yescharacteristic, complemented
177#448—1571no—
178#448—1572no—
179#448—1571no—
180#419—1572no—
181#407—1571no—
182C2×#41—1571no—
183C22×Q8—3141no—
184C22×Q8—3148no—
185C22×Q8—3141no—
186C22×Q8—3141no—
187C22×Q8—1572no—
188C22×Q8—1572no—
189C22×Q8—1571no—
190C22×Q8C2×D15711yescharacteristic
191C23×C4—3141no—
192C23×C4—1571no—
193#137—3148no—
194C2×Q16—1572no—
195C2×Q16C2×D15712yescomplemented
196C2×SD16—1572no—
197C2×SD16—1572no—
198C2×M16—1572no—
199#123—3141no—
200C22⋊Q8—1571no—
201C22⋊Q8C2×D15711yescharacteristic
202#121—1571no—
203#121—1571no—
204#120—3141no—
205C4×D4—3141no—
206C4×D4—3141no—
207#117—1571no—
208#103—1572no—
209#103—1572no—
210#103—1572no—
211#103C2×D15712yescomplemented
212C22⋊C8—1571no—
213C22⋊C8—1572no—
214C22⋊C8C2×D15711yescharacteristic
215C24—1571no—
216#41—3144no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—3148no—
220C2×Q8—3144no—
221C2×Q8—3142no—
222C2×Q8—3142no—
223C2×Q8—3148no—
224C2×Q8—3148no—
225C2×Q8—3144no—
226C2×Q8—3144no—
227C2×Q8—3144no—
228C2×Q8—3141no—
229C2×Q8—1578no—
230C2×Q8—1572no—
231C2×Q8C22×D15711yescharacteristic
232C2×Q8C22×D15712yes—
233C2×D4—3141no—
234C2×D4—3141no—
235C2×D4—3141no—
236C2×D4—1571no—
237C22×C4—21no—
238C22×C4—3148no—
239C22×C4—3142no—
240C22×C4—3142no—
241C22×C4—3144no—
242C22×C4—3141no—
243C22×C4—3141no—
244C22×C4—1571no—
245C22×C4—1571no—
246C22×C4—1571no—
247C22×C4—1571no—
248C22×C4—1571no—
249C22×C4C22×D15711yescharacteristic
250Q16—24no—
251Q16—3144no—
252SD16—3144no—
253SD16—3144no—
254M16—3144no—
255C2×C8—1572no—
256C2×C8C22×D15712yes—
257C4⋊C4—21no—
258C4⋊C4—3141no—
259C4⋊C4—1571no—
260C4⋊C4C22×D15711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—3141no—
263C22⋊C4—3141no—
264C22⋊C4—3141no—
265C22⋊C4—3141no—
266C42—3141no—
267C23—3142no—
268C23—3142no—
269C23—3142no—
270C23—3141no—
271C23—1571no—
272C23—1571no—
273C23D4×D15711yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8—22no—
277Q8—3148no—
278Q8—3144no—
279Q8—3148no—
280Q8—3142no—
281Q8—1574no—
282Q8D4×D15714yes—
283D4—3142no—
284D4—6282no—
285D4—6282no—
286D4—6282no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—3148no—
293C2×C4—3148no—
294C2×C4—3142no—
295C2×C4—3142no—
296C2×C4—3144no—
297C2×C4—3142no—
298C2×C4—3142no—
299C2×C4—3142no—
300C2×C4—3141no—
301C2×C4—6282no—
302C2×C4—1571no—
303C2×C4—1571no—
304C2×C4—1574no—
305C2×C4D4×D15711yescharacteristic
306C2×C4C23×D15711yescharacteristic
307C8—22no—
308C8—3142no—
309C22—21no—
310C22—21no—
311C22—3148no—
312C22—3141no—
313C22—3141no—
314C22—3141no—
315C22—3141no—
316C22—6281no—
317C22—6281no—
318C22—6281no—
319C22—1572no—
320C22—1572no—
321C22#1909780312yescomplemented
322C22C2×D4×D15711yescharacteristic
323C4—24no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—3144no—
328C4—3141no—
329C4—6281no—
330C4—1572no—
331C4C2×D4×D15712yes—
332C2—22no—
333C2—3142no—
334C2—3142no—
335C2—12561no—
336C2#120×D15711yescharacteristic
337C2#137×D15711yescharacteristic
338C2C2×#1909780311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
prime radical#23757786C2
hypercenter#451D157
hypocenterC157#3004
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D157C2semidirect
2. C2×#19097803C2semidirect
4. #23757428C2semidirect
5. #23757427C2semidirect
6. #23757414C2semidirect
7. #23757413C2semidirect
8. #23757412C2semidirect
9. #23757280C2semidirect
10. #23757235C2semidirect
11. #451×C157C2semidirect
23. C2×#14578944C22semidirect
24. C2×#14578943C22semidirect
25. C2×#14578941C22semidirect
35. #19097696C22semidirect
36. #19097695C22semidirect
37. #19097694C22semidirect
41. C2×Q16×C157C22semidirect
43. #103×C157C22semidirect
58. Q8×D157D4semidirect
141. C22×C157#137semidirect
152. C2×D157#137semidirect
169. C157#3004semidirect
176. #451D157semidirect
195. C2×Q16C2×D157semidirect
211. #103C2×D157semidirect
321. C22#19097803semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1572
C2×#190978032
#237577862
#237574282
#237574272
#237574142
#237574132
#237574122
#237572802
#237572352
#451×C1572
#3004157