Finite Group Databasebeta

← #28349603

Subgroups — #28349603

Order 13696.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293440C211yesmaximal, characteristic, complemented
2#20293428C211yesmaximal, characteristic, complemented
3#20293068C211yesmaximal, characteristic, complemented
4#20293066C212yesmaximal, complemented
5#20293064C212yesmaximal, complemented
6#20293063C211yesmaximal, characteristic, complemented
7#20293059C211yesmaximal, characteristic, complemented
8#20293058C212yesmaximal, complemented
9SD32×D107C212yesmaximal, complemented
10#20293043C211yesmaximal, characteristic, complemented
11C2×SD32×C107C211yesmaximal, characteristic, complemented
12#16812157—21no—
13#16812154—21no—
14#16812141—22no—
15#16812140—21no—
16#16812135—21no—
17#16812133—22no—
18#16812132C2212yescomplemented
19#16812131C2211yescharacteristic
20#16812131C2211yescharacteristic, complemented
21#16812128C2212yes—
22Q16×D107C2212yescomplemented
23D8×D107C2212yes—
24#16812115C2211yescharacteristic
25Dic856C2211yescharacteristic, complemented
26#16812055C2212yescomplemented
27#16812054C2211yescharacteristic, complemented
28#16812054C2211yescharacteristic, complemented
29#16812053C2212yescomplemented
30#16812052C2212yescomplemented
31D1712C2211yescharacteristic, complemented
32#16812050C2212yescomplemented
33#16812049C2211yescharacteristic, complemented
34#16812048C2212yescomplemented
35C16×D107C2212yescomplemented
36C2×Q16×C107C2211yescharacteristic, complemented
37C2×D8×C107C2211yescharacteristic
38SD32×C107C2212yescomplemented
39SD32×C107C2212yescomplemented
40C2×C16×C107C2211yescharacteristic, complemented
41#13133819—22no—
42#13133819—41no—
43#13133818—21no—
44#13133818—21no—
45#13133818—42no—
46#13133818D411yescharacteristic
47#13133817—22no—
48#13133817—41no—
49Q8×D107—22no—
50Q8×D107—41no—
51C2×C22⋊D107—42no—
52D4×D107—22no—
53D4×D107—41no—
54Dic428C2311yescharacteristic
55Dic428C2311yescharacteristic
56#13133808—22no—
57#13133807—21no—
58#13133807—21no—
59#13133806—22no—
60#13133805—22no—
61D856C2311yescharacteristic
62D856C2311yescharacteristic
63#13133803—22no—
64#13133802D411yescharacteristic
65#13133801D412yes—
66C8×D107C2312yes—
67C107⋊C16C2312yes—
68C2×Q8×C107—21no—
69C2×D4×C107—21no—
70Q16×C107—21no—
71Q16×C107C2312yes—
72D8×C107—21no—
73D8×C107C2312yes—
74C2×C8×C107C2311yescharacteristic
75C16×C107C2311yescharacteristic
76C16×C107C2311yescharacteristic
77C23×C107—41no—
78C22×D107—42no—
79Q8×C107—22no—
80Q8×C107—41no—
81D4×C107—22no—
82D4×C107—41no—
83C2×C4×C107—41no—
84C2×C4×C107C2×D411yescharacteristic
85C22⋊D107—41no—
86C22⋊D107—41no—
87C22⋊D107—42no—
88C22⋊D107—42no—
89C22⋊D107D812yes—
90C2×Dic107—42no—
91D428—21no—
92D428—21no—
93D428—42no—
94D428D811yescharacteristic
95C4×D107—42no—
96C4×D107—41no—
97C4×D107—41no—
98C4×D107—42no—
99C4×D107C2×D412yes—
100Dic214—21no—
101Dic214—21no—
102Dic214—42no—
103Dic214D811yescharacteristic
104C8×C107C2×D411yescharacteristic
105C8×C107C2411yescharacteristic
106C107⋊C8C2×D412yes—
107C22×C107—42no—
108C22×C107—161no—
109C22×C107C2×D811yescharacteristic
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—162no—
113C2×D107C2×D812yes—
114C4×C107—42no—
115C4×C107C2×D811yescharacteristic
116C4×C107C22×D411yescharacteristic
117Dic107—41no—
118Dic107—41no—
119Dic107C2×D812yes—
120C2×C107—21no—
121C2×C107—82no—
122C2×C107C22×D811yescharacteristic
123D107—22no—
124D107—81no—
125D107—81no—
126#3416—1071nomaximal, Sylow
127C107#341611yescharacteristic, complemented, Sylow
128#578—1071no—
129#576—1071no—
130#510—1071no—
131#510—1072no—
132#509—1071no—
133#509—1072no—
134#508—1072no—
135#508—1071no—
136C2×SD32—1072no—
137C2×SD32D10711yescharacteristic, complemented
138#504—1071no—
1392−1+4—2141no—
140H(5,𝔽2)—2141no—
141#137—2142no—
142#137—2141no—
143AGL(1,ℤ/8)—2142no—
144AGL(1,ℤ/8)—2141no—
145#135—1072no—
146#135—1072no—
147#135—1071no—
148#135—1071no—
149C2×Q16—1072no—
150C2×Q16C2×D10711yescharacteristic, complemented
151C2×D8—1072no—
152C2×D8C2×D10711yescharacteristic
153#131—1071no—
154Q32—1072no—
155Q32—1071no—
156SD32—1072no—
157SD32—1072no—
158SD32—1071no—
159SD32—1071no—
160SD32C2×D10712yescomplemented
161SD32C2×D10712yescomplemented
162D16—1072no—
163D16—1071no—
164M32—1071no—
165M32—1072no—
166C2×C16—1072no—
167C2×C16C2×D10711yescharacteristic, complemented
168#41—1071no—
169#41—4282no—
170#41—4281no—
171#41—4281no—
172#41—2142no—
173#41—2142no—
174#41—2141no—
175#41—2141no—
176C2×Q8—21no—
177C2×Q8—4281no—
178C2×Q8—2142no—
179C2×D4—21no—
180C2×D4—4281no—
181C2×D4—4282no—
182C2×D4—2142no—
183Q16—21no—
184Q16—1071no—
185Q16—1071no—
186Q16—2142no—
187Q16C22×D10712yes—
188SD16—2142no—
189SD16—2142no—
190SD16—2141no—
191SD16—2141no—
192D8—21no—
193D8—1071no—
194D8—1071no—
195D8—2142no—
196D8C22×D10712yes—
197M16—1071no—
198M16—1072no—
199C2×C8—1072no—
200C2×C8C22×D10711yescharacteristic
201C16—1072no—
202C16C22×D10711yescharacteristic
203C16C22×D10711yescharacteristic
204C23—41no—
205C23—4282no—
206Q8—22no—
207Q8—41no—
208Q8—1071no—
209Q8—4282no—
210Q8—2141no—
211Q8—2141no—
212D4—22no—
213D4—41no—
214D4—1072no—
215D4—1071no—
216D4—4282no—
217D4—4282no—
218D4—4282no—
219D4—4281no—
220D4—4281no—
221D4—2141no—
222D4—2141no—
223C2×C4—41no—
224C2×C4—1072no—
225C2×C4—4282no—
226C2×C4—4282no—
227C2×C4—4282no—
228C2×C4—4281no—
229C2×C4—4281no—
230C2×C4D4×D10711yescharacteristic
231C8—1072no—
232C8D4×D10711yescharacteristic
233C8C23×D10711yescharacteristic
234C22—42no—
235C22—161no—
236C22—1072no—
237C22—4281no—
238C22—4281no—
239C22—17122no—
240C22D8×D10711yescharacteristic
241C4—42no—
242C4—1072no—
243C4—4281no—
244C4—4281no—
245C4D8×D10711yescharacteristic
246C4C2×D4×D10711yescharacteristic
247C2—21no—
248C2—82no—
249C2—8561no—
250C2—8561no—
251C2—2142no—
252C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
FittingC2×SD32×C107C2
socleC2×C107C22×D8
hypercenterC2×SD32D107
hypocenterC107#3416
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #20293440C2semidirect
2. #20293428C2semidirect
3. #20293068C2semidirect
4. #20293066C2semidirect
5. #20293064C2semidirect
6. #20293063C2semidirect
7. #20293059C2semidirect
8. #20293058C2semidirect
9. SD32×D107C2semidirect
10. #20293043C2semidirect
11. C2×SD32×C107C2semidirect
18. #16812132C22semidirect
20. #16812131C22semidirect
22. Q16×D107C22semidirect
25. Dic856C22semidirect
26. #16812055C22semidirect
27. #16812054C22semidirect
28. #16812054C22semidirect
29. #16812053C22semidirect
30. #16812052C22semidirect
31. D1712C22semidirect
32. #16812050C22semidirect
33. #16812049C22semidirect
34. #16812048C22semidirect
35. C16×D107C22semidirect
36. C2×Q16×C107C22semidirect
38. SD32×C107C22semidirect
39. SD32×C107C22semidirect
40. C2×C16×C107C22semidirect
127. C107#3416semidirect
137. C2×SD32D107semidirect
150. C2×Q16C2×D107semidirect
160. SD32C2×D107semidirect
161. SD32C2×D107semidirect
167. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934402
#202934282
#202930682
#202930662
#202930642
#202930632
#202930592
#202930582
SD32×D1072
#202930432
C2×SD32×C1072
#3416107