Finite Group Databasebeta

← SD16×ΓL(1,𝔽8)⋊D7 #72245192

Subgroups — SD16×ΓL(1,𝔽8)⋊D7 #72245192

Order 4704.

340 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#72245191C211yesmaximal, characteristic, complemented
2#72245190C211yesmaximal, characteristic, complemented
3#72245187C211yesmaximal, characteristic, complemented
4C7×SD16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
5Q8×ΓL(1,𝔽8)⋊D7C211yesmaximal, characteristic, complemented
6C8×ΓL(1,𝔽8)⋊D7C211yesmaximal, characteristic, complemented
7D4×ΓL(1,𝔽8)⋊D7C211yesmaximal, characteristic
8SD16×C7⋊D7C311yesmaximal, characteristic, complemented, Hall
9C22×ΓL(1,𝔽8)⋊D7—21no—
10C7×Q8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11C7×D4×ΓL(1,𝔽8)C2211yescharacteristic
12#11924202—21no—
13C4⋊(ΓL(1,𝔽8)⋊D7)C2211yescharacteristic
14C4×ΓL(1,𝔽8)⋊D7—21no—
15C4×ΓL(1,𝔽8)⋊D7C2211yescharacteristic
16#11924198—21no—
17#11924198C2211yescharacteristic, complemented
18C7×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
19ΓL(1,𝔽8)⋊(C7⋊C8)C2211yescharacteristic, complemented
20Q8×C7⋊D7C2×C311yescharacteristic, complemented
21D4×C7⋊D7C2×C311yescharacteristic
22C72×SD16C2×C311yescharacteristic, complemented
23#11716073C2×C311yescharacteristic, complemented
24#11716072C2×C311yescharacteristic, complemented
25#11716062C2×C311yescharacteristic, complemented
26C8×C7⋊D7C2×C311yescharacteristic, complemented
27C3×D7×SD16—71nomaximal
28SD16×AGL(1,𝔽7)—71nomaximal
29C22×C7×ΓL(1,𝔽8)—21no—
30C2×ΓL(1,𝔽8)⋊D7—21no—
31C2×ΓL(1,𝔽8)⋊D7—42no—
32C2×ΓL(1,𝔽8)⋊D7D411yescharacteristic
33C4×C7×ΓL(1,𝔽8)—21no—
34C4×C7×ΓL(1,𝔽8)C2311yescharacteristic
35ΓL(1,𝔽8)⋊Dic7—21no—
36ΓL(1,𝔽8)⋊Dic7D411yescharacteristic
37C22×C7⋊D7—21no—
38C72×Q8C22×C311yescharacteristic, complemented
39C72×D4C22×C311yescharacteristic
40C22⋊(C7⋊D7)—21no—
41C4⋊(C7⋊D7)C22×C311yescharacteristic
42C4×C7⋊D7—21no—
43C4×C7⋊D7C22×C311yescharacteristic
44#87648—21no—
45#87648C22×C311yescharacteristic, complemented
46C72×C8C22×C311yescharacteristic, complemented
47C7⋊(C7⋊C8)C22×C311yescharacteristic, complemented
48C3×Q8×D7—71no—
49C3×D4×D7—71no—
50Q8×AGL(1,𝔽7)—71no—
51D4×AGL(1,𝔽7)—71no—
52C3×C7×SD16—71no—
53C3×#1143—71no—
54C3×#1142—71no—
55C3×#1132—71no—
56C3×C8×D7—71no—
57SD16×ΓL(1,𝔽8)D711yescharacteristic, complemented
58#1143⋊C3—71no—
59#1142⋊C3—71no—
60#1132⋊C3—71no—
61C8×AGL(1,𝔽7)—71no—
62C2×C7×ΓL(1,𝔽8)—41no—
63C2×C7×ΓL(1,𝔽8)C2×D411yescharacteristic
64ΓL(1,𝔽8)⋊D7—41no—
65ΓL(1,𝔽8)⋊D7SD1612yesdirect product factor
66D7×SD16—71no—
67D7×SD16—71no—
68D7×SD16—212no—
69C22×C72—21no—
70C2×C7⋊D7—21no—
71C2×C7⋊D7—42no—
72C2×C7⋊D7C3×D411yescharacteristic
73C4×C72—21no—
74C4×C72C23×C311yescharacteristic
75C7⋊Dic7—21no—
76C7⋊Dic7C3×D411yescharacteristic
77C22×C3×D7—141no—
78C22×AGL(1,𝔽7)—141no—
79C3×C7×Q8—71no—
80C3×C7×D4—71no—
81C3×C22⋊D7—141no—
82C3×D28—71no—
83C3×C4×D7—71no—
84C3×C4×D7—141no—
85C3×Dic14—71no—
86C3×Dic14—141no—
87Q8×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
88D4×ΓL(1,𝔽8)C2×D711yescharacteristic
89(C22⋊D7)⋊C3—141no—
90C4⋊AGL(1,𝔽7)—71no—
91C4×AGL(1,𝔽7)—71no—
92C4×AGL(1,𝔽7)—141no—
93ΓL(1,𝔽8)⋊Q8—71no—
94ΓL(1,𝔽8)⋊Q8—141no—
95C3×C7×C8—71no—
96C3×C7⋊C8—71no—
97C8×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
98ΓL(1,𝔽8)⋊C8—71no—
99C7×ΓL(1,𝔽8)C2×SD1611yescharacteristic, complemented, Hall
100Q8×D7—71no—
101Q8×D7—71no—
102Q8×D7—212no—
103D4×D7—71no—
104D4×D7—71no—
105D4×D7—212no—
106C7×SD16—32no—
107C7×SD16AGL(1,𝔽7)11yescharacteristic, complemented
108C7×SD16C3×D711yescharacteristic, complemented
109#1143—71no—
110#1143—71no—
111#1143—212no—
112#1142—71no—
113#1142—71no—
114#1142—212no—
115#1132—71no—
116#1132—71no—
117#1132—212no—
118C8×D7—71no—
119C8×D7—71no—
120C8×D7—212no—
121C2×C72—41no—
122C2×C72C2×C3×D411yescharacteristic
123C7⋊D7—41no—
124C7⋊D7C3×SD1612yescomplemented
125C2×C3×SD16—491noHall
126C22×C3×C7—141no—
127C2×C3×D7—71no—
128C2×C3×D7—141no—
129C2×C3×D7—282no—
130C22×ΓL(1,𝔽8)—21no—
131C2×AGL(1,𝔽7)—71no—
132C2×AGL(1,𝔽7)—141no—
133C2×AGL(1,𝔽7)—282no—
134C3×C4×C7—71no—
135C3×C4×C7—141no—
136C3×Dic7—71no—
137C3×Dic7—141no—
138C4×ΓL(1,𝔽8)—21no—
139C4×ΓL(1,𝔽8)C22×D711yescharacteristic
140ΓL(1,𝔽8)⋊C4—71no—
141ΓL(1,𝔽8)⋊C4—141no—
142C22×D7—141no—
143C22×D7—141no—
144C22×D7—422no—
145C7×Q8—32no—
146C7×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
147C7×Q8C2×C3×D711yescharacteristic, complemented
148C7×D4—32no—
149C7×D4C2×AGL(1,𝔽7)11yescharacteristic
150C7×D4C2×C3×D711yescharacteristic
151C22⋊D7—141no—
152C22⋊D7—141no—
153C22⋊D7—422no—
154D28—71no—
155D28—71no—
156D28—212no—
157C4×D7—71no—
158C4×D7—71no—
159C4×D7—141no—
160C4×D7—141no—
161C4×D7—212no—
162C4×D7—422no—
163Dic14—71no—
164Dic14—71no—
165Dic14—141no—
166Dic14—141no—
167Dic14—212no—
168Dic14—422no—
169C7×C8—32no—
170C7×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
171C7×C8C2×C3×D711yescharacteristic, complemented
172C7⋊C8—71no—
173C7⋊C8—71no—
174C7⋊C8—212no—
175C72C2×C3×SD1611yescharacteristic, complemented, Sylow
176C2×C3×Q8—491no—
177C2×C3×D4—491no—
178C3×SD16—141no—
179C3×SD16—491no—
180C3×SD16—491no—
181C3×SD16—491no—
182C2×C3×C8—491no—
183C2×C3×C7—71no—
184C2×C3×C7—281no—
185C3×D7—72no—
186C3×D7—281no—
187C2×ΓL(1,𝔽8)—41no—
188C2×ΓL(1,𝔽8)D4×D711yescharacteristic
189AGL(1,𝔽7)—72no—
190AGL(1,𝔽7)—281no—
191C2×SD16—491noSylow
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—62no—
195C2×D7—71no—
196C2×D7—71no—
197C2×D7—141no—
198C2×D7—141no—
199C2×D7—212no—
200C2×D7—282no—
201C2×D7—282no—
202C2×D7—422no—
203C2×D7—844no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—32no—
207C4×C7—62no—
208C4×C7C22×AGL(1,𝔽7)11yescharacteristic
209C4×C7C22×C3×D711yescharacteristic
210Dic7—71no—
211Dic7—71no—
212Dic7—141no—
213Dic7—141no—
214Dic7—212no—
215Dic7—422no—
216C23×C3—981no—
217C3×Q8—141no—
218C3×Q8—491no—
219C3×Q8—981no—
220C3×D4—141no—
221C3×D4—491no—
222C3×D4—981no—
223C2×C3×C4—491no—
224C2×C3×C4—981no—
225C3×C8—141no—
226C3×C8—491no—
227C3×C7—71no—
228ΓL(1,𝔽8)D7×SD1611yescharacteristic, complemented
229C2×Q8—491no—
230C2×D4—491no—
231SD16—491no—
232SD16—491no—
233SD16—491no—
234SD16ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor
235C2×C8—491no—
236C2×C7—32no—
237C2×C7—41no—
238C2×C7—41no—
239C2×C7—122no—
240C2×C7D4×AGL(1,𝔽7)11yescharacteristic
241C2×C7C3×D4×D711yescharacteristic
242D7—72no—
243D7—72no—
244D7—214no—
245D7—281no—
246D7—281no—
247D7—842no—
248C22×C3—281no—
249C22×C3—491no—
250C22×C3—981no—
251C22×C3—1962no—
252C3×C4—141no—
253C3×C4—281no—
254C3×C4—491no—
255C3×C4—981no—
256C23—981no—
257Q8—491no—
258Q8—981no—
259Q8C2×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
260D4—491no—
261D4—981no—
262D4C2×ΓL(1,𝔽8)⋊D711yescharacteristic
263C2×C4—491no—
264C2×C4—981no—
265C8—491no—
266C8C2×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
267C7—32no—
268C7SD16×AGL(1,𝔽7)11yescharacteristic, complemented
269C7C3×D7×SD1611yescharacteristic, complemented
270C2×C3—141no—
271C2×C3—492no—
272C2×C3—561no—
273C2×C3—1961no—
274C22—21no—
275C22—491no—
276C22—981no—
277C22—1962no—
278C4—21no—
279C4—491no—
280C4—981no—
281C4C22×ΓL(1,𝔽8)⋊D711yescharacteristic
282C3—141noSylow
283C2—41no—
284C2—492no—
285C2—1961no—
286C2D4×ΓL(1,𝔽8)⋊D711yescharacteristic
subgroupG/H
centerC2D4×ΓL(1,𝔽8)⋊D7
commutator, cosocleC4×C72C23×C3
FrattiniC4C22×ΓL(1,𝔽8)⋊D7
FittingC72×SD16C2×C3
socleC2×C72C2×C3×D4
prime radicalD4×ΓL(1,𝔽8)⋊D7C2
hypercenterSD16ΓL(1,𝔽8)⋊D7
hypocenterC72C2×C3×SD16
LCSG ▷ C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #72245191C2semidirect
2. #72245190C2semidirect
3. #72245187C2semidirect
4. C7×SD16×ΓL(1,𝔽8)C2semidirect
5. Q8×ΓL(1,𝔽8)⋊D7C2semidirect
6. C8×ΓL(1,𝔽8)⋊D7C2semidirect
8. SD16×C7⋊D7C3semidirect
10. C7×Q8×ΓL(1,𝔽8)C22semidirect
17. #11924198C22semidirect
18. C7×C8×ΓL(1,𝔽8)C22semidirect
19. ΓL(1,𝔽8)⋊(C7⋊C8)C22semidirect
20. Q8×C7⋊D7C2×C3semidirect
22. C72×SD16C2×C3semidirect
23. #11716073C2×C3semidirect
24. #11716072C2×C3semidirect
25. #11716062C2×C3semidirect
26. C8×C7⋊D7C2×C3semidirect
38. C72×Q8C22×C3semidirect
45. #87648C22×C3semidirect
46. C72×C8C22×C3semidirect
47. C7⋊(C7⋊C8)C22×C3semidirect
57. SD16×ΓL(1,𝔽8)D7semidirect
65. ΓL(1,𝔽8)⋊D7SD16direct (N × H)
87. Q8×ΓL(1,𝔽8)C2×D7semidirect
97. C8×ΓL(1,𝔽8)C2×D7semidirect
99. C7×ΓL(1,𝔽8)C2×SD16semidirect
107. C7×SD16AGL(1,𝔽7)semidirect
108. C7×SD16C3×D7semidirect
124. C7⋊D7C3×SD16semidirect
146. C7×Q8C2×AGL(1,𝔽7)semidirect
147. C7×Q8C2×C3×D7semidirect
170. C7×C8C2×AGL(1,𝔽7)semidirect
171. C7×C8C2×C3×D7semidirect
175. C72C2×C3×SD16semidirect
228. ΓL(1,𝔽8)D7×SD16semidirect
234. SD16ΓL(1,𝔽8)⋊D7direct (N × H)
259. Q8C2×ΓL(1,𝔽8)⋊D7semidirect
266. C8C2×ΓL(1,𝔽8)⋊D7semidirect
268. C7SD16×AGL(1,𝔽7)semidirect
269. C7C3×D7×SD16semidirect

Maximal subgroups

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

subgroupindex
#722451912
#722451902
#722451872
C7×SD16×ΓL(1,𝔽8)2
Q8×ΓL(1,𝔽8)⋊D72
C8×ΓL(1,𝔽8)⋊D72
D4×ΓL(1,𝔽8)⋊D72
SD16×C7⋊D73
C3×D7×SD167
SD16×AGL(1,𝔽7)7