Finite Group Databasebeta

← #451×D157 #34109327

Subgroups — #451×D157 #34109327

Order 20096.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
2C2×Q16×D157C212yesmaximal, complemented
3C22⋊Q8×D157C211yesmaximal, characteristic
4#23757431C211yesmaximal, characteristic, complemented
5#23757430C212yesmaximal, complemented
6#23757429C211yesmaximal, characteristic, complemented
7#23757428C211yesmaximal, characteristic, complemented
8#23757427C212yesmaximal, complemented
9#103×D157C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic
11#451×C157C211yesmaximal, characteristic, complemented
12C22×Dic314—21no—
13C22×Dic314C2211yescharacteristic
14C2×Q8×D157—28no—
15C2×Q8×D157—21no—
16C2×Q8×D157—21no—
17C2×Q8×D157C2212yes—
18C2×Q8×D157C2211yescharacteristic
19C2×Q8×D157C2212yes—
20C22×C4×D157—21no—
21C22×C4×D157C2211yescharacteristic
22C2×Dic628C2212yescomplemented
23C2×#14578944C2212yescomplemented
24C2×#14578944C2212yescomplemented
25Q16×D157—28no—
26C2×C8×D157C2212yes—
27#19097751C2211yescharacteristic
28#19097750C2211yescharacteristic
29#19097749—21no—
30#19097748—21no—
31#19097747C2211yescharacteristic
32C4⋊C4×D157—21no—
33C4⋊C4×D157C2211yescharacteristic
34C22⋊C4×D157—21no—
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
46C2×Dic314—28no—
47C2×Dic314—24no—
48C2×Dic314—22no—
49C2×Dic314—22no—
50C2×Dic314—21no—
51C2×Dic314—21no—
52C2×Dic314C2311yescharacteristic
53C2×Dic314C2312yes—
54Q8×D157—28no—
55Q8×D157—28no—
56Q8×D157—24no—
57Q8×D157—24no—
58Q8×D157—24no—
59Q8×D157—24no—
60Q8×D157D418yes—
61C22×Dic157—21no—
62C22×Dic157D411yescharacteristic
63C2×C4×D157—28no—
64C2×C4×D157—22no—
65C2×C4×D157—22no—
66C2×C4×D157—24no—
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157D411yescharacteristic
70C2×C4×D157D411yescharacteristic
71C2×C4×D157C2311yescharacteristic
72C2×C4×D157C2311yescharacteristic
73Dic628—24no—
74#14578944—24no—
75#14578944—24no—
76C2×#11929434C2312yes—
77C8×D157—24no—
78C4⋊Dic157—21no—
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—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×D157Q1614yescomplemented
100C22×D157C2×D411yescharacteristic
101C22×D157C2×D411yescharacteristic
102Dic314—24no—
103Dic314—28no—
104Dic314—22no—
105Dic314—28no—
106Dic314—22no—
107Dic314—42no—
108Dic314C2×D414yes—
109C2×Dic157—24no—
110C2×Dic157—22no—
111C2×Dic157—22no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157C2×D411yescharacteristic
115C2×Dic157C2×D411yescharacteristic
116C4×D157—28no—
117C4×D157—28no—
118C4×D157—22no—
119C4×D157—22no—
120C4×D157—22no—
121C4×D157—42no—
122C4×D157—42no—
123C4×D157C2×D414yes—
124#11929434—22no—
125C23×C157C2×D411yescharacteristic
126Q8×C157—24no—
127Q8×C157—22no—
128Q8×C157—22no—
129Q8×C157C2×D414yes—
130C2×C4×C157—24no—
131C2×C4×C157—22no—
132C2×C4×C157—22no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157C2×D411yescharacteristic
136C2×C4×C157C2411yescharacteristic
137C8×C157—22no—
138C22×C157—21no—
139C22×C157—21no—
140C22×C157C2×Q1612yescomplemented
141C22×C157C22×D411yescharacteristic
142C2×D157—21no—
143C2×D157—21no—
144C2×D157—28no—
145C2×D157#12012yes—
146C2×D157C2×Q1612yes—
147C2×D157C2×Q1612yescomplemented
148C2×D157#13712yes—
149C4×C157—24no—
150C4×C157—21no—
151C4×C157—41no—
152C4×C157—41no—
153C4×C157C22×D412yes—
154Dic157—24no—
155Dic157—21no—
156Dic157—41no—
157Dic157—41no—
158Dic157#12012yes—
159C2×C157—22no—
160C2×C157C2×#12011yescharacteristic
161C2×C157C22×Q1611yescharacteristic
162C2×C157C2×#13711yescharacteristic
163D157—22no—
164D157#45114yesdirect product factor
165C157C2×#45111yescharacteristic, complemented, Sylow
166C2×#451—1571nomaximal, Sylow
167C23×Q8—1571no—
168C22×Q16—1572no—
169C2×C22⋊Q8—1571no—
170#451—1571no—
171#451—1572no—
172#451—1571no—
173#451—1572no—
174#451—1571no—
175#451D15711yescharacteristic, direct product factor
176C2×#103—1572no—
177C2×C22⋊C8—1571no—
178C22×Q8—3141no—
179C22×Q8—3148no—
180C22×Q8—3141no—
181C22×Q8—3141no—
182C22×Q8—1571no—
183C22×Q8—1572no—
184C22×Q8—1572no—
185C22×Q8—1571no—
186C22×Q8C2×D15711yescharacteristic
187C23×C4—3141no—
188C23×C4—1571no—
189C2×Q16—3148no—
190C2×Q16—1572no—
191C2×Q16—1572no—
192C2×Q16—1572no—
193C2×Q16C2×D15712yescomplemented
194C22×C8—1572no—
195C22⋊Q8—3141no—
196C22⋊Q8—3141no—
197C22⋊Q8—1571no—
198C22⋊Q8—1571no—
199C22⋊Q8—1571no—
200C22⋊Q8C2×D15711yescharacteristic
201C2×C4⋊C4—3141no—
202C2×C4⋊C4—1571no—
203C2×C22⋊C4—3141no—
204#103—1572no—
205#103—1572no—
206#103—1572no—
207#103C2×D15712yescomplemented
208C22⋊C8—1571no—
209C22⋊C8—1572no—
210C22⋊C8C2×D15711yescharacteristic
211C24—1571no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—3148no—
215C2×Q8—3144no—
216C2×Q8—3142no—
217C2×Q8—3142no—
218C2×Q8—3141no—
219C2×Q8—3148no—
220C2×Q8—3148no—
221C2×Q8—3144no—
222C2×Q8—3144no—
223C2×Q8—3144no—
224C2×Q8—3141no—
225C2×Q8—3144no—
226C2×Q8—1578no—
227C2×Q8—1571no—
228C2×Q8—1572no—
229C2×Q8C22×D15711yescharacteristic
230C2×Q8C22×D15712yes—
231C22×C4—21no—
232C22×C4—3148no—
233C22×C4—3142no—
234C22×C4—3142no—
235C22×C4—3144no—
236C22×C4—3141no—
237C22×C4—3141no—
238C22×C4—3141no—
239C22×C4—1571no—
240C22×C4—1571no—
241C22×C4—1571no—
242C22×C4—1571no—
243C22×C4—1571no—
244C22×C4C22×D15711yescharacteristic
245Q16—24no—
246Q16—3144no—
247Q16—3144no—
248Q16—3144no—
249C2×C8—3144no—
250C2×C8—1572no—
251C2×C8C22×D15712yes—
252C4⋊C4—21no—
253C4⋊C4—3141no—
254C4⋊C4—3141no—
255C4⋊C4—3141no—
256C4⋊C4—3141no—
257C4⋊C4—1571no—
258C4⋊C4C22×D15711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—3141no—
261C22⋊C4—3141no—
262C22⋊C4—3141no—
263C23—3142no—
264C23—3142no—
265C23—1574no—
266C23—1571no—
267C23—1571no—
268C23D4×D15711yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—3148no—
273Q8—3144no—
274Q8—3148no—
275Q8—3142no—
276Q8—3142no—
277Q8—6282no—
278Q8—1574no—
279Q8D4×D15714yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3148no—
286C2×C4—3148no—
287C2×C4—3142no—
288C2×C4—3142no—
289C2×C4—3144no—
290C2×C4—3142no—
291C2×C4—3142no—
292C2×C4—3142no—
293C2×C4—3141no—
294C2×C4—3141no—
295C2×C4—6282no—
296C2×C4—6282no—
297C2×C4—1571no—
298C2×C4—1571no—
299C2×C4—1574no—
300C2×C4D4×D15711yescharacteristic
301C2×C4C23×D15711yescharacteristic
302C8—22no—
303C8—3142no—
304C22—21no—
305C22—21no—
306C22—3148no—
307C22—3141no—
308C22—3141no—
309C22—1572no—
310C22—1572no—
311C22—1572no—
312C22—1572no—
313C22Q16×D15712yescomplemented
314C22C2×D4×D15711yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—3144no—
320C4—3141no—
321C4—6281no—
322C4—6281no—
323C4—1572no—
324C4C2×D4×D15712yes—
325C2—22no—
326C2—3142no—
327C2—1574no—
328C2#120×D15711yescharacteristic
329C2C2×Q16×D15711yescharacteristic
330C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#451D157
hypocenterC157C2×#451
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
2. C2×Q16×D157C2semidirect
4. #23757431C2semidirect
5. #23757430C2semidirect
6. #23757429C2semidirect
7. #23757428C2semidirect
8. #23757427C2semidirect
9. #103×D157C2semidirect
11. #451×C157C2semidirect
22. C2×Dic628C22semidirect
23. C2×#14578944C22semidirect
24. C2×#14578944C22semidirect
35. #19097696C22semidirect
36. #19097695C22semidirect
37. #19097694C22semidirect
41. C2×Q16×C157C22semidirect
43. #103×C157C22semidirect
99. C22×D157Q16semidirect
140. C22×C157C2×Q16semidirect
147. C2×D157C2×Q16semidirect
164. D157#451direct (N × H)
165. C157C2×#451semidirect
175. #451D157direct (N × H)
193. C2×Q16C2×D157semidirect
207. #103C2×D157semidirect
313. C22Q16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1572
C2×Q16×D1572
C22⋊Q8×D1572
#237574312
#237574302
#237574292
#237574282
#237574272
#103×D1572
C22⋊C8×D1572
#451×C1572
C2×#451157