Finite Group Databasebeta

← #34109679

Subgroups — #34109679

Order 20096.

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

normal subgroups only

This group has 504 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
1C2×#19097791C211yesmaximal, characteristic, complemented
2#23757840C211yesmaximal, characteristic, complemented
3#23757756C211yesmaximal, characteristic, complemented
4#23757493C212yesmaximal, complemented
5#23757486C212yesmaximal, complemented
6#23757471C212yesmaximal, complemented
7#23757466C212yesmaximal, complemented
8#23757301C211yesmaximal, characteristic, complemented
9#23757275C211yesmaximal, characteristic, complemented
10C2×#19097692C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13#19097791—24no—
14C2×#14578944C2211yescharacteristic, complemented
15C2×#14578942C2211yescharacteristic, complemented
16C2×#14578941C2211yescharacteristic, complemented
17C2×#14578939C2211yescharacteristic, complemented
18C22×#11929434C2211yescharacteristic
19#19097767C2212yes—
20#19097758C2212yes—
21#19097742—21no—
22#19097742—21no—
23#19097740—22no—
24Q8×Dic157C2212yescomplemented
25D4×Dic157—22no—
26D4×Dic157C2212yescomplemented
27#19097730—21no—
28#19097730C2211yescharacteristic
29C2×C4⋊Dic157—21no—
30C2×C4⋊Dic157C2211yescharacteristic
31#19097700C2212yescomplemented
32#19097700C2212yescomplemented
33#19097696C2212yescomplemented
34#19097693C2212yescomplemented
35#19097692C2212yes—
36#19097692C2212yes—
37C2×#41×C157C2211yescharacteristic
38C22×D4×C157C2211yescharacteristic
39AGL(1,ℤ/8)×C157—24no—
40C2×SD16×C157C2212yes—
41C2×D8×C157C2212yes—
42C2×M16×C157C2211yescharacteristic
43#14578954—22no—
44#14578954—22no—
45C2×Dic314C2311yescharacteristic
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—22no—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—21no—
50C22×Dic157—22no—
51C22×Dic157—21no—
52C2×C4×D157—21no—
53#14578944—22no—
54#14578942—22no—
55#14578941—22no—
56#14578939—22no—
57C2×#11929434—22no—
58C2×#11929434C2311yescharacteristic
59C2×#11929434C2311yescharacteristic
60C4⋊Dic157—22no—
61C4⋊Dic157—21no—
62C4⋊Dic157D411yescharacteristic
63C4⋊Dic157D411yescharacteristic
64C4⋊Dic157C2312yes—
65C22⋊Dic157—22no—
66C22⋊Dic157—22no—
67C22⋊Dic157—22no—
68C22⋊Dic157D412yes—
69C4×Dic157—22no—
70C4×Dic157C2312yes—
71C24×C157—21no—
72#41×C157—22no—
73#41×C157D414yescomplemented
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157—21no—
76C2×D4×C157—22no—
77C2×D4×C157—21no—
78C2×D4×C157—22no—
79C2×D4×C157C2312yes—
80C2×D4×C157C2311yescharacteristic
81C22×C4×C157—21no—
82C22×C4×C157C2311yescharacteristic
83SD16×C157—24no—
84D8×C157—24no—
85M16×C157—22no—
86C2×C8×C157C2312yes—
87C22×D157—21no—
88Dic314—21no—
89Dic314—21no—
90D628—21no—
91D628—21no—
92C22⋊D157—42no—
93C22⋊D157—82no—
94C22⋊D157—82no—
95C2×Dic157—21no—
96C2×Dic157—22no—
97C2×Dic157—82no—
98C2×Dic157—81no—
99C2×Dic157C2×D412yes—
100C2×Dic157C2×D412yes—
101C4×D157—42no—
102#11929434—21no—
103#11929434—21no—
104C23×C157—21no—
105C23×C157—22no—
106C23×C157—42no—
107C23×C157—81no—
108C23×C157—81no—
109C23×C157C2×D411yescharacteristic
110Q8×C157—21no—
111Q8×C157C2×D412yes—
112D4×C157—24no—
113D4×C157—22no—
114D4×C157—21no—
115D4×C157—22no—
116D4×C157—42no—
117D4×C157C2×D412yes—
118C2×C4×C157—21no—
119C2×C4×C157—24no—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D412yes—
122C2×C4×C157#4112yes—
123C2×C4×C157C2411yescharacteristic
124C8×C157—22no—
125C22×C157—21no—
126C22×C157—21no—
127C22×C157—22no—
128C22×C157—41no—
129C22×C157—41no—
130C22×C157—42no—
131C22×C157—42no—
132C22×C157—42no—
133C22×C157—84no—
134C22×C157#12112yes—
135C22×C157C22×D411yescharacteristic
136C2×D157—41no—
137C2×D157—41no—
138C2×D157—41no—
139C4×C157—22no—
140C4×C157#12112yes—
141C4×C157C22×D411yescharacteristic
142C4×C157C2×#4111yescharacteristic
143Dic157—22no—
144Dic157—22no—
145Dic157—41no—
146Dic157—42no—
147C2×C157—22no—
148C2×C157—42no—
149C2×C157—82no—
150C2×C157C2×#12111yescharacteristic
151C2×C157#57611yescharacteristic
152C2×C157#57711yescharacteristic
153D157—81no—
154C157#304511yescharacteristic, complemented, Sylow
155#3045—1571nomaximal, Sylow
156C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
157C2×#135—1571no—
158#535—1571no—
159#520—1571no—
160#464—1572no—
161#463—1572no—
162#460—1572no—
163#459—1572no—
164#424—1571no—
165#417—1571no—
166C2×#102—1571no—
167C2×#41—1571no—
168C2×#41C2×D15711yescharacteristic
169C22×D4C2×D15711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—3144no—
172C2×Q16—1571no—
173C2×SD16—1571no—
174C2×SD16—1571no—
175C2×SD16C2×D15712yes—
176C2×D8—1571no—
177C2×D8C2×D15712yes—
178C2×M16C2×D15711yescharacteristic
179C22×C8—1571no—
180#127—1572no—
181#124—1572no—
182#121—3141no—
183#121—3141no—
184#120—3142no—
185C4×Q8—1572no—
186C4×D4—3142no—
187C4×D4—1572no—
188#117—3141no—
189#117—1571no—
190C2×C4⋊C4—3141no—
191C2×C4⋊C4—1571no—
192C4⋊C8—1572no—
193C4⋊C8—1572no—
194#103—1572no—
195#102—1572no—
196#102—1572no—
197#102—1572no—
198C24—21no—
199#41—22no—
200#41—3142no—
201#41—3142no—
202#41C22⋊D15714yescomplemented
203C2×Q8—1571no—
204C2×Q8C22×D15711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—3141no—
210C2×D4—3142no—
211C2×D4—3142no—
212C2×D4—1571no—
213C2×D4C22×D15712yes—
214C2×D4C22×D15711yescharacteristic
215C22×C4—21no—
216C22×C4—3141no—
217C22×C4—3141no—
218C22×C4—3142no—
219C22×C4C22×D15711yescharacteristic
220Q16—3142no—
221SD16—24no—
222SD16—3142no—
223SD16—3142no—
224D8—24no—
225D8—3142no—
226M16—22no—
227C2×C8—3142no—
228C2×C8—1571no—
229C2×C8—1571no—
230C2×C8C22×D15712yes—
231C4⋊C4—3142no—
232C4⋊C4—3141no—
233C4⋊C4—1572no—
234C4⋊C4—1571no—
235C4⋊C4—1571no—
236C22⋊C4—3142no—
237C22⋊C4—3142no—
238C22⋊C4—3142no—
239C22⋊C4—1572no—
240C42—3142no—
241C42—1572no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—3141no—
248C23D4×D15711yescharacteristic
249Q8—21no—
250Q8—3141no—
251Q8—3141no—
252Q8C2×C22⋊D15712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—3141no—
259D4—3141no—
260D4—6282no—
261D4—12562no—
262D4—12562no—
263D4C2×C22⋊D15712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—3142no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—1572no—
270C2×C4—1572no—
271C2×C4—12562no—
272C2×C4—12561no—
273C2×C4D4×D15711yescharacteristic
274C2×C4C2×C22⋊D15712yes—
275C2×C4#1457895312yes—
276C2×C4C23×D15711yescharacteristic
277C8—22no—
278C8—3141no—
279C8—3141no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—6281no—
290C22—6281no—
291C22—6281no—
292C22#1909774312yes—
293C22C2×D4×D15711yescharacteristic
294C4—22no—
295C4—3142no—
296C4—3142no—
297C4—6281no—
298C4—6282no—
299C4#1909774312yes—
300C4C22×C22⋊D15711yescharacteristic
301C4C2×#1457895311yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—12561no—
306C2C2×#1909774311yescharacteristic
307C2#2375807711yescharacteristic
308C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3045
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097791C2semidirect
2. #23757840C2semidirect
3. #23757756C2semidirect
4. #23757493C2semidirect
5. #23757486C2semidirect
6. #23757471C2semidirect
7. #23757466C2semidirect
8. #23757301C2semidirect
9. #23757275C2semidirect
10. C2×#19097692C2semidirect
11. C2×AGL(1,ℤ/8)×C157C2semidirect
14. C2×#14578944C22semidirect
15. C2×#14578942C22semidirect
16. C2×#14578941C22semidirect
17. C2×#14578939C22semidirect
24. Q8×Dic157C22semidirect
26. D4×Dic157C22semidirect
31. #19097700C22semidirect
32. #19097700C22semidirect
33. #19097696C22semidirect
34. #19097693C22semidirect
73. #41×C157D4semidirect
154. C157#3045semidirect
156. C2×AGL(1,ℤ/8)D157semidirect
202. #41C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977912
#237578402
#237577562
#237574932
#237574862
#237574712
#237574662
#237573012
#237572752
C2×#190976922
C2×AGL(1,ℤ/8)×C1572
#3045157