Finite Group Databasebeta

← #28349607

Subgroups — #28349607

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#20293441C211yesmaximal, characteristic, complemented
2#20293428C211yesmaximal, characteristic, complemented
3#20293070C211yesmaximal, characteristic, complemented
4#20293066C212yesmaximal, complemented
5#20293061C212yesmaximal, complemented
6#20293057C211yesmaximal, characteristic, complemented
7#20293056C212yesmaximal, complemented
8C2×#16812053C211yesmaximal, characteristic, complemented
9SD32×D107C212yesmaximal, complemented
10#20293045C211yesmaximal, characteristic, complemented
11#509×C107C211yesmaximal, characteristic, complemented
12#16812156—21no—
13#16812154—21no—
14#16812139—21no—
15#16812138—22no—
16#16812135—21no—
17#16812133—22no—
18#16812131C2211yescharacteristic
19#16812128C2212yes—
20#16812128C2212yescomplemented
21C2×Dic428C2211yescharacteristic, complemented
22Q16×D107C2212yescomplemented
23D8×D107C2212yes—
24#16812115C2211yescharacteristic
25Dic856C2212yescomplemented
26#16812055C2211yescharacteristic, complemented
27#16812054C2212yescomplemented
28#16812053C2212yescomplemented
29#16812053C2212yescomplemented
30#16812053C2211yescharacteristic, complemented
31#16812052C2211yescharacteristic, complemented
32#16812050C2211yescharacteristic, complemented
33#16812048C2212yescomplemented
34C2×C107⋊C16C2211yescharacteristic, complemented
35C16×D107C2212yescomplemented
36#135×C107C2211yescharacteristic, complemented
37C2×D8×C107C2211yescharacteristic
38SD32×C107C2212yescomplemented
39D16×C107C2212yescomplemented
40M32×C107C2211yescharacteristic, complemented
41#13133818—21no—
42#13133818—41no—
43#13133818D411yescharacteristic
44#13133817—22no—
45#13133817—22no—
46#13133817—42no—
47#13133817—41no—
48C2×Dic214—21no—
49C2×Dic214—41no—
50Q8×D107—22no—
51C2×C22⋊D107—42no—
52D4×D107—22no—
53D4×D107—41no—
54Dic428—21no—
55Dic428C2312yes—
56Dic428C2311yescharacteristic
57#13133808—22no—
58#13133807—21no—
59#13133806—22no—
60#13133806—22no—
61D856C2311yescharacteristic
62#13133803—22no—
63#13133802D411yescharacteristic
64#13133801D412yes—
65C8×D107C2312yes—
66C107⋊C16C2311yescharacteristic
67C107⋊C16C2311yescharacteristic
68#41×C107—21no—
69C2×D4×C107—21no—
70Q16×C107C2311yescharacteristic
71SD16×C107—21no—
72D8×C107—21no—
73D8×C107C2311yescharacteristic
74D8×C107C2312yes—
75C2×C8×C107C2311yescharacteristic
76C16×C107C2312yes—
77C23×C107—41no—
78C22×D107—42no—
79Q8×C107—21no—
80D4×C107—22no—
81D4×C107—21no—
82D4×C107—41no—
83D4×C107—41no—
84C2×C4×C107—41no—
85C2×C4×C107C2×D411yescharacteristic
86C22⋊D107—41no—
87C22⋊D107—42no—
88C22⋊D107—42no—
89C22⋊D107—42no—
90C22⋊D107D812yes—
91C2×Dic107—41no—
92C2×Dic107—42no—
93C2×Dic107—42no—
94D428—21no—
95D428D811yescharacteristic
96C4×D107—42no—
97C4×D107—41no—
98C4×D107C2×D412yes—
99Dic214—21no—
100Dic214—22no—
101Dic214—41no—
102Dic214—42no—
103Dic214D811yescharacteristic
104C8×C107C2×D411yescharacteristic
105C8×C107C2411yescharacteristic
106C107⋊C8C2×D412yes—
107C22×C107—41no—
108C22×C107—42no—
109C22×C107—161no—
110C22×C107C2×D811yescharacteristic
111C2×D107—41no—
112C2×D107—162no—
113C2×D107C2×D812yes—
114C4×C107—41no—
115C4×C107C2×D811yescharacteristic
116C4×C107C22×D411yescharacteristic
117Dic107—42no—
118Dic107—41no—
119Dic107C2×D812yes—
120C2×C107—21no—
121C2×C107—82no—
122C2×C107—81no—
123C2×C107C22×D811yescharacteristic
124D107—22no—
125D107—81no—
126#3416—1071nomaximal, Sylow
127C107#341611yescharacteristic, complemented, Sylow
128#578—1071no—
129#576—1071no—
130#510—1072no—
131#510—1071no—
132#509—1072no—
133#509D10711yescharacteristic, complemented
134#508—1071no—
135#508—1072no—
136C2×SD32—1071no—
137C2×SD32—1072no—
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#135C2×D10711yescharacteristic, complemented
149C2×Q16—1072no—
150C2×Q16—1071no—
151C2×D8—1072no—
152C2×D8C2×D10711yescharacteristic
153#131—1071no—
154Q32—1071no—
155Q32—1072no—
156SD32—1071no—
157SD32—1072no—
158SD32—1072no—
159SD32—1071no—
160SD32—1072no—
161SD32C2×D10712yescomplemented
162D16—1071no—
163D16C2×D10712yescomplemented
164M32—1072no—
165M32C2×D10711yescharacteristic, complemented
166C2×C16—1071no—
167C2×C16—1072no—
168#41—21no—
169#41—1071no—
170#41—4281no—
171#41—4282no—
172#41—4281no—
173#41—2142no—
174#41—2142no—
175#41—2141no—
176C2×Q8—4281no—
177C2×Q8—2142no—
178C2×Q8—2141no—
179C2×D4—21no—
180C2×D4—4281no—
181C2×D4—4282no—
182C2×D4—2142no—
183Q16—1072no—
184Q16—1071no—
185Q16—2142no—
186Q16—2141no—
187Q16C22×D10711yescharacteristic
188SD16—21no—
189SD16—2142no—
190SD16—2142no—
191SD16—2141no—
192D8—21no—
193D8—1071no—
194D8—2142no—
195D8C22×D10711yescharacteristic
196D8C22×D10712yes—
197M16—1071no—
198M16—1072no—
199C2×C8—1072no—
200C2×C8C22×D10711yescharacteristic
201C16—1071no—
202C16—1071no—
203C16C22×D10712yes—
204C23—41no—
205C23—4282no—
206Q8—21no—
207Q8—1071no—
208Q8—4282no—
209Q8—4281no—
210Q8—2141no—
211Q8—2142no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—1072no—
217D4—1071no—
218D4—4282no—
219D4—4282no—
220D4—4282no—
221D4—4281no—
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—41no—
235C22—42no—
236C22—161no—
237C22—1072no—
238C22—4281no—
239C22—17122no—
240C22D8×D10711yescharacteristic
241C4—41no—
242C4—1072no—
243C4—4281no—
244C4—4282no—
245C4D8×D10711yescharacteristic
246C4C2×D4×D10711yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—8561no—
251C2—2142no—
252C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#509×C107C2
socleC2×C107C22×D8
hypercenter#509D107
hypocenterC107#3416
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20293441C2semidirect
2. #20293428C2semidirect
3. #20293070C2semidirect
4. #20293066C2semidirect
5. #20293061C2semidirect
6. #20293057C2semidirect
7. #20293056C2semidirect
8. C2×#16812053C2semidirect
9. SD32×D107C2semidirect
10. #20293045C2semidirect
11. #509×C107C2semidirect
20. #16812128C22semidirect
21. C2×Dic428C22semidirect
22. Q16×D107C22semidirect
25. Dic856C22semidirect
26. #16812055C22semidirect
27. #16812054C22semidirect
28. #16812053C22semidirect
29. #16812053C22semidirect
30. #16812053C22semidirect
31. #16812052C22semidirect
32. #16812050C22semidirect
33. #16812048C22semidirect
34. C2×C107⋊C16C22semidirect
35. C16×D107C22semidirect
36. #135×C107C22semidirect
38. SD32×C107C22semidirect
39. D16×C107C22semidirect
40. M32×C107C22semidirect
127. C107#3416semidirect
133. #509D107semidirect
148. #135C2×D107semidirect
161. SD32C2×D107semidirect
163. D16C2×D107semidirect
165. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934412
#202934282
#202930702
#202930662
#202930612
#202930572
#202930562
C2×#168120532
SD32×D1072
#202930452
#509×C1072
#3416107