Finite Group Databasebeta

← #510×D107 #28349587

Subgroups — #510×D107 #28349587

Order 13696.

360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D107C211yesmaximal, characteristic
2C2×Q16×D107C211yesmaximal, characteristic, complemented
3#20293070C211yesmaximal, characteristic, complemented
4#20293069C212yesmaximal, complemented
5#20293068C211yesmaximal, characteristic, complemented
6#20293067C211yesmaximal, characteristic, complemented
7#20293066C212yesmaximal, complemented
8Q32×D107C212yesmaximal, complemented
9SD32×D107C212yesmaximal, complemented
10M32×D107C211yesmaximal, characteristic, complemented
11#510×C107C211yesmaximal, characteristic, complemented
12#41×D107—21no—
13C2×Q8×D107—21no—
14#16812132C2211yescharacteristic
15#16812131C2211yescharacteristic
16#16812130—21no—
17#16812129—21no—
18#16812128C2211yescharacteristic
19C2×Dic428C2211yescharacteristic, complemented
20C2×#13133808—21no—
21Q16×D107—21no—
22Q16×D107—21no—
23Q16×D107C2212yes—
24Q16×D107C2211yescharacteristic
25Q16×D107C2212yescomplemented
26SD16×D107—21no—
27D8×D107C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29Dic856C2212yescomplemented
30#16812055C2212yescomplemented
31#16812055C2212yescomplemented
32#16812054C2212yescomplemented
33#16812053C2212yescomplemented
34#16812052C2212yescomplemented
35#16812049C2211yescharacteristic, complemented
36#16812048C2212yescomplemented
37C16×D107C2212yes—
38#135×C107C2211yescharacteristic
39C2×Q16×C107C2211yescharacteristic, complemented
40Q32×C107C2212yescomplemented
41SD32×C107C2212yescomplemented
42M32×C107C2211yescharacteristic, complemented
43#13133819—21no—
44#13133818—21no—
45#13133818—41no—
46#13133817—21no—
47#13133817—41no—
48C2×Dic214—21no—
49C2×Dic214—41no—
50Q8×D107—21no—
51Q8×D107—22no—
52Q8×D107—22no—
53Q8×D107—41no—
54Q8×D107—41no—
55D4×D107—21no—
56D4×D107—41no—
57C2×C4×D107—41no—
58C2×C4×D107—41no—
59C2×C4×D107D411yescharacteristic
60Dic428—21no—
61Dic428C2312yes—
62Dic428C2311yescharacteristic
63#13133808—22no—
64#13133808—22no—
65#13133808—21no—
66#13133807—21no—
67#13133806—21no—
68#13133805—21no—
69D856C2311yescharacteristic
70#13133803—21no—
71C2×C107⋊C8D411yescharacteristic
72C8×D107D411yescharacteristic
73C8×D107D411yescharacteristic
74C8×D107C2311yescharacteristic
75C8×D107C2311yescharacteristic
76C107⋊C16C2312yes—
77#41×C107—21no—
78C2×Q8×C107—21no—
79Q16×C107—21no—
80Q16×C107C2312yes—
81Q16×C107C2311yescharacteristic
82SD16×C107—21no—
83D8×C107C2311yescharacteristic
84C2×C8×C107C2311yescharacteristic
85C16×C107C2312yes—
86C22×D107—41no—
87C22×D107D811yescharacteristic
88Q8×C107—21no—
89Q8×C107—22no—
90Q8×C107—41no—
91D4×C107—21no—
92D4×C107—41no—
93C2×C4×C107—41no—
94C2×C4×C107—41no—
95C2×C4×C107C2×D411yescharacteristic
96C22⋊D107—41no—
97C22⋊D107—41no—
98C22⋊D107—41no—
99C2×Dic107—41no—
100C2×Dic107—41no—
101C2×Dic107D811yescharacteristic
102D428—21no—
103D428—41no—
104C4×D107—41no—
105C4×D107—42no—
106C4×D107—42no—
107C4×D107—41no—
108C4×D107—41no—
109C4×D107D811yescharacteristic
110C4×D107D811yescharacteristic
111C4×D107C2×D411yescharacteristic
112C4×D107C2×D411yescharacteristic
113Dic214—21no—
114Dic214—22no—
115Dic214—41no—
116Dic214—42no—
117Dic214—42no—
118Dic214—41no—
119C8×C107C2×D411yescharacteristic
120C8×C107C2411yescharacteristic
121C107⋊C8C2×D411yescharacteristic
122C107⋊C8C2×D411yescharacteristic
123C22×C107—41no—
124C22×C107C2×D811yescharacteristic
125C2×D107—22no—
126C2×D107—41no—
127C2×D107—82no—
128C2×D107C2×D811yescharacteristic
129C2×D107C2×D811yescharacteristic
130C4×C107—42no—
131C4×C107—41no—
132C4×C107C2×D811yescharacteristic
133C4×C107C22×D411yescharacteristic
134Dic107—42no—
135Dic107—41no—
136Dic107C2×D811yescharacteristic
137Dic107C2×D811yescharacteristic
138C2×C107—21no—
139C2×C107—81no—
140C2×C107C22×D811yescharacteristic
141D107—21no—
142D107—81no—
143D107#51012yesdirect product factor
144C2×#510—1071nomaximal, Sylow
145C107C2×#51011yescharacteristic, complemented, Sylow
146C2×#135—1071no—
147C22×Q16—1071no—
148#510—1071no—
149#510—1071no—
150#510—1072no—
151#510—1072no—
152#510—1071no—
153#510D10711yescharacteristic, direct product factor
154C2×Q32—1072no—
155C2×SD32—1072no—
156C2×M32—1071no—
157C2×#41—2141no—
158C22×Q8—2141no—
159#135—1071no—
160#135—1071no—
161#135—1071no—
162#135—2141no—
163#135—2141no—
164#135C2×D10711yescharacteristic
165C2×Q16—1072no—
166C2×Q16—1072no—
167C2×Q16—1071no—
168C2×Q16—1071no—
169C2×Q16—2141no—
170C2×Q16—2141no—
171C2×Q16—2141no—
172C2×Q16C2×D10711yescharacteristic, complemented
173C2×SD16—2141no—
174C2×D8—1071no—
175C22×C8—1071no—
176Q32—1072no—
177Q32—1072no—
178Q32—1072no—
179Q32C2×D10712yescomplemented
180SD32—1072no—
181SD32—1072no—
182SD32—1072no—
183SD32C2×D10712yescomplemented
184M32—1071no—
185M32—1072no—
186M32C2×D10711yescharacteristic, complemented
187C2×C16—1072no—
188#41—21no—
189#41—4281no—
190#41—4281no—
191#41—2141no—
192#41—2141no—
193#41—2141no—
194C2×Q8—21no—
195C2×Q8—4281no—
196C2×Q8—4281no—
197C2×Q8—4281no—
198C2×Q8—2141no—
199C2×Q8—2142no—
200C2×Q8—2142no—
201C2×Q8—2141no—
202C2×D4—4281no—
203C2×D4—2141no—
204C22×C4—1071no—
205C22×C4—4281no—
206C22×C4—4281no—
207Q16—21no—
208Q16—1072no—
209Q16—1071no—
210Q16—2142no—
211Q16—2142no—
212Q16—2141no—
213Q16—2141no—
214Q16C22×D10712yes—
215Q16C22×D10711yescharacteristic
216SD16—21no—
217SD16—2141no—
218SD16—2141no—
219SD16—2141no—
220D8—1071no—
221D8—2141no—
222D8C22×D10711yescharacteristic
223C2×C8—1071no—
224C2×C8—1071no—
225C2×C8—1071no—
226C2×C8—1071no—
227C2×C8—1071no—
228C2×C8C22×D10711yescharacteristic
229C16—1072no—
230C16C22×D10712yes—
231C23—1071no—
232C23—4281no—
233Q8—21no—
234Q8—22no—
235Q8—41no—
236Q8—4282no—
237Q8—4282no—
238Q8—4281no—
239Q8—4281no—
240Q8—2141no—
241Q8—2142no—
242D4—21no—
243D4—41no—
244D4—4281no—
245D4—4281no—
246D4—4281no—
247D4—4281no—
248D4—2141no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—1071no—
252C2×C4—1071no—
253C2×C4—1071no—
254C2×C4—1071no—
255C2×C4—1071no—
256C2×C4—4281no—
257C2×C4—4281no—
258C2×C4—4282no—
259C2×C4—4282no—
260C2×C4—4281no—
261C2×C4—4281no—
262C2×C4—4281no—
263C2×C4D4×D10711yescharacteristic
264C8—1071no—
265C8—1071no—
266C8D4×D10711yescharacteristic
267C8C23×D10711yescharacteristic
268C22—41no—
269C22—8562no—
270C22—1071no—
271C22—1071no—
272C22—4281no—
273C22—2142no—
274C22D8×D10711yescharacteristic
275C4—41no—
276C4—42no—
277C4—1071no—
278C4—1071no—
279C4—4281no—
280C4—4282no—
281C4D8×D10711yescharacteristic
282C4C2×D4×D10711yescharacteristic
283C2—21no—
284C2—81no—
285C2—8561no—
286C2—1072no—
287C2—2141no—
288C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#510×C107C2
socleC2×C107C22×D8
prime radical#135×D107C2
hypercenter#510D107
hypocenterC107C2×#510
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D107C2semidirect
3. #20293070C2semidirect
4. #20293069C2semidirect
5. #20293068C2semidirect
6. #20293067C2semidirect
7. #20293066C2semidirect
8. Q32×D107C2semidirect
9. SD32×D107C2semidirect
10. M32×D107C2semidirect
11. #510×C107C2semidirect
19. C2×Dic428C22semidirect
25. Q16×D107C22semidirect
29. Dic856C22semidirect
30. #16812055C22semidirect
31. #16812055C22semidirect
32. #16812054C22semidirect
33. #16812053C22semidirect
34. #16812052C22semidirect
35. #16812049C22semidirect
36. #16812048C22semidirect
39. C2×Q16×C107C22semidirect
40. Q32×C107C22semidirect
41. SD32×C107C22semidirect
42. M32×C107C22semidirect
143. D107#510direct (N × H)
145. C107C2×#510semidirect
153. #510D107direct (N × H)
172. C2×Q16C2×D107semidirect
179. Q32C2×D107semidirect
183. SD32C2×D107semidirect
186. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#135×D1072
C2×Q16×D1072
#202930702
#202930692
#202930682
#202930672
#202930662
Q32×D1072
SD32×D1072
M32×D1072
#510×C1072
C2×#510107