Finite Group Databasebeta

← #34109599

Subgroups — #34109599

Order 20096.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19097799C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3#23757819C211yesmaximal, characteristic, complemented
4C2×#19097733C211yesmaximal, characteristic, complemented
5#23757491C212yesmaximal, complemented
6#23757487C212yesmaximal, complemented
7#23757477C212yesmaximal, complemented
8#23757467C212yesmaximal, complemented
9#23757296C211yesmaximal, characteristic, complemented
10#23757272C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628C2211yescharacteristic
14C22×C4×D157—21no—
15#19097799—24no—
16#19097796—24no—
17C2×#14578943C2212yes—
18C2×#14578941C2212yes—
19C2×D1256C2212yes—
20C2×#14578939C2212yes—
21C2×#14578938C2211yescharacteristic
22#19097769C2211yescharacteristic, complemented
23#19097767C2211yescharacteristic, complemented
24#19097760C2211yescharacteristic, complemented
25#19097758C2211yescharacteristic, complemented
26#19097743—21no—
27#19097743—21no—
28#19097733—21no—
29#19097733—21no—
30#19097733C2212yes—
31#19097733C2212yes—
32C4×C22⋊D157—21no—
33C4×C22⋊D157—21no—
34C2×#14578931—21no—
35C2×C4×Dic157C2211yescharacteristic
36#19097700C2212yescomplemented
37#19097700C2212yescomplemented
38#19097694C2212yescomplemented
39#19097691C2212yescomplemented
40C2×#41×C157C2211yescharacteristic
41C2×M16×C157C2211yescharacteristic
42C2×C4⋊C4×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44#102×C157C2212yescomplemented
45C23×D157—21no—
46#14578954—22no—
47#14578954—22no—
48C2×Dic314C2311yescharacteristic
49C2×D628—21no—
50C2×D628—22no—
51C2×D628D414yescomplemented
52C2×D628C2311yescharacteristic
53C2×D628C2312yes—
54C2×C22⋊D157—21no—
55C2×C22⋊D157—21no—
56C2×C22⋊D157—21no—
57C22×Dic157D411yescharacteristic
58C22×Dic157D411yescharacteristic
59C2×C4×D157—22no—
60C2×C4×D157—22no—
61C2×C4×D157—21no—
62C2×C4×D157—44no—
63#14578943—24no—
64#14578941—24no—
65D1256—24no—
66#14578939—24no—
67#14578938—22no—
68C2×#11929434C2312yes—
69#14578933—21no—
70#14578933—21no—
71C22⋊Dic157—21no—
72C22⋊Dic157—21no—
73#14578931—22no—
74#14578931—22no—
75#14578931—21no—
76#14578931—21no—
77C4×Dic157D411yescharacteristic
78C4×Dic157D411yescharacteristic
79C4×Dic157C2311yescharacteristic
80C4×Dic157C2311yescharacteristic
81#41×C157—22no—
82#41×C157—22no—
83C2×Q8×C157C2311yescharacteristic
84C2×D4×C157—21no—
85C2×D4×C157C2311yescharacteristic
86C22×C4×C157—21no—
87C22×C4×C157—21no—
88C22×C4×C157C2311yescharacteristic
89M16×C157—22no—
90C2×C8×C157C2312yes—
91C4⋊C4×C157—21no—
92C4⋊C4×C157C2312yes—
93C22×D157—24no—
94C22×D157—22no—
95C22×D157—21no—
96C22×D157—42no—
97C22×D157—42no—
98Dic314—21no—
99Dic314—21no—
100D628—24no—
101D628—21no—
102D628—22no—
103D628—21no—
104D628C2×D414yes—
105C22⋊D157—42no—
106C22⋊D157—42no—
107C22⋊D157—42no—
108C2×Dic157—22no—
109C2×Dic157—21no—
110C2×Dic157—41no—
111C2×Dic157C2×D411yescharacteristic
112C2×Dic157C2×D411yescharacteristic
113C2×Dic157C2×D411yescharacteristic
114C2×Dic157C2×D411yescharacteristic
115C4×D157—44no—
116C4×D157—42no—
117C4×D157—44no—
118#11929434—22no—
119C23×C157—21no—
120C23×C157C2×D411yescharacteristic
121Q8×C157—21no—
122Q8×C157—21no—
123D4×C157—21no—
124D4×C157—21no—
125D4×C157—42no—
126C2×C4×C157—22no—
127C2×C4×C157—21no—
128C2×C4×C157—42no—
129C2×C4×C157—42no—
130C2×C4×C157C2×D411yescharacteristic
131C2×C4×C157C2×D412yes—
132C2×C4×C157#4112yes—
133C2×C4×C157C2411yescharacteristic
134C8×C157—22no—
135C22×C157—21no—
136C22×C157—21no—
137C22×C157—41no—
138C22×C157—41no—
139C22×C157—41no—
140C22×C157#12112yes—
141C22×C157C22×D411yescharacteristic
142C2×D157—24no—
143C2×D157—48no—
144C2×D157—42no—
145C2×D157—42no—
146C2×D157—41no—
147C2×D157—41no—
148C2×D157—41no—
149C4×C157—41no—
150C4×C157—42no—
151C4×C157#12112yes—
152C4×C157C22×D411yescharacteristic
153C4×C157C2×#4111yescharacteristic
154Dic157—21no—
155Dic157—21no—
156Dic157—21no—
157Dic157—41no—
158Dic157AGL(1,ℤ/8)12yescomplemented
159C2×C157—22no—
160C2×C157—81no—
161C2×C157C2×#12111yescharacteristic
162C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C157#57511yescharacteristic
164D157—44no—
165D157—81no—
166C157#303611yescharacteristic, complemented, Sylow
167#3036—1571nomaximal, Sylow
168C2×AGL(1,ℤ/8)—1571no—
169C2×AGL(1,ℤ/8)—1571no—
170#532—1571no—
171C2×C4×D4—1571no—
172#463—1572no—
173#463—1572no—
174#461—1572no—
175#459—1572no—
176#423—1571no—
177#417—1571no—
178#417D15711yescharacteristic, complemented
179C2×#41—1571no—
180C2×#41C2×D15711yescharacteristic
181C22×D4—1571no—
182C23×C4—3141no—
183AGL(1,ℤ/8)—3144no—
184AGL(1,ℤ/8)—3144no—
185C2×SD16—1572no—
186C2×SD16—1572no—
187C2×D8—1572no—
188C2×D8—1572no—
189C2×M16—1571no—
190C2×M16C2×D15711yescharacteristic
191C4⋊Q8—1571no—
192#127—1571no—
193#124—1571no—
194#124—1571no—
195#121—3141no—
196#121—3141no—
197C4×D4—3141no—
198C4×D4—3141no—
199C4×D4—3141no—
200C4×D4—3141no—
201C4×D4—1572no—
202C4×D4—1572no—
203C2×C4⋊C4C2×D15711yescharacteristic
204C2×C22⋊C4—3141no—
205C2×C42—1571no—
206C4⋊C8—1572no—
207C4⋊C8—1572no—
208#103—1572no—
209#103C2×D15712yescomplemented
210#102—1572no—
211#102C2×D15712yescomplemented
212C24—3141no—
213#41—22no—
214#41—22no—
215#41—3142no—
216#41—3142no—
217C2×Q8—1571no—
218C2×Q8C22×D15711yescharacteristic
219C2×D4—21no—
220C2×D4—3141no—
221C2×D4—3141no—
222C2×D4—3141no—
223C2×D4—3141no—
224C2×D4—3142no—
225C2×D4—1574no—
226C2×D4—1571no—
227C2×D4—1572no—
228C2×D4C22×D15711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—3142no—
232C22×C4—3142no—
233C22×C4—3141no—
234C22×C4—6284no—
235C22×C4—1571no—
236C22×C4—1571no—
237C22×C4C22×D15711yescharacteristic
238SD16—3144no—
239SD16—3144no—
240D8—3144no—
241D8—3144no—
242M16—22no—
243M16—3142no—
244C2×C8—1572no—
245C2×C8C22×D15712yes—
246C4⋊C4—21no—
247C4⋊C4—3141no—
248C4⋊C4—3141no—
249C4⋊C4C22×D15712yes—
250C22⋊C4—3142no—
251C22⋊C4—3142no—
252C22⋊C4—3141no—
253C22⋊C4—3141no—
254C22⋊C4—3141no—
255C22⋊C4—3141no—
256C42—1571no—
257C42—1571no—
258C42—1571no—
259C42—1571no—
260C23—21no—
261C23—3144no—
262C23—3142no—
263C23—3141no—
264C23—6282no—
265C23—6282no—
266C23D4×D15711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—3141no—
270Q8—3141no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—3144no—
275D4—3141no—
276D4—3142no—
277D4—3141no—
278D4—6282no—
279D4—6282no—
280D4—6282no—
281D4—1574no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—3142no—
287C2×C4—3141no—
288C2×C4—6284no—
289C2×C4—6284no—
290C2×C4—6281no—
291C2×C4—6282no—
292C2×C4—1571no—
293C2×C4—1571no—
294C2×C4—1571no—
295C2×C4—1571no—
296C2×C4D4×D15711yescharacteristic
297C2×C4D4×D15712yes—
298C2×C4#1457895412yes—
299C2×C4C23×D15711yescharacteristic
300C8—22no—
301C8—3142no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—3144no—
308C22—6288no—
309C22—6282no—
310C22—6282no—
311C22—6281no—
312C22—6281no—
313C22—6281no—
314C22#1909774412yes—
315C22C2×D4×D15711yescharacteristic
316C4—41no—
317C4—42no—
318C4—3141no—
319C4—3141no—
320C4—3141no—
321C4—6281no—
322C4—1572no—
323C4#1909774412yes—
324C4C2×D4×D15711yescharacteristic
325C4C2×#1457895411yescharacteristic
326C2—22no—
327C2—81no—
328C2—6284no—
329C2—12561no—
330C2C2×#1909774411yescharacteristic
331C2AGL(1,ℤ/8)×D15711yescharacteristic
332C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
hypercenter#417D157
hypocenterC157#3036
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×#19097796C2semidirect
3. #23757819C2semidirect
4. C2×#19097733C2semidirect
5. #23757491C2semidirect
6. #23757487C2semidirect
7. #23757477C2semidirect
8. #23757467C2semidirect
9. #23757296C2semidirect
10. #23757272C2semidirect
11. #417×C157C2semidirect
22. #19097769C22semidirect
23. #19097767C22semidirect
24. #19097760C22semidirect
25. #19097758C22semidirect
36. #19097700C22semidirect
37. #19097700C22semidirect
38. #19097694C22semidirect
39. #19097691C22semidirect
43. #103×C157C22semidirect
44. #102×C157C22semidirect
51. C2×D628D4semidirect
158. Dic157AGL(1,ℤ/8)semidirect
166. C157#3036semidirect
178. #417D157semidirect
209. #103C2×D157semidirect
211. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×#190977962
#237578192
C2×#190977332
#237574912
#237574872
#237574772
#237574672
#237572962
#237572722
#417×C1572
#3036157