Finite Group Databasebeta

← #28349604

Subgroups — #28349604

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#20293429C211yesmaximal, characteristic, complemented
3#20293069C212yesmaximal, complemented
4#20293067C211yesmaximal, characteristic, complemented
5#20293062C211yesmaximal, characteristic, complemented
6#20293061C212yesmaximal, complemented
7#20293058C212yesmaximal, complemented
8C2×#16812054C211yesmaximal, characteristic, complemented
9SD32×D107C212yesmaximal, complemented
10#20293043C211yesmaximal, characteristic, complemented
11#508×C107C211yesmaximal, characteristic, complemented
12#16812156—21no—
13#16812155—21no—
14#16812139—21no—
15#16812138—22no—
16#16812136—22no—
17#16812134—21no—
18#16812132C2212yes—
19#16812128C2212yescomplemented
20C2×Dic428C2211yescharacteristic, complemented
21Q16×D107C2212yescomplemented
22C2×D856C2211yescharacteristic
23D8×D107C2212yes—
24#16812115C2211yescharacteristic
25#16812055C2212yescomplemented
26#16812054C2212yescomplemented
27#16812054C2212yescomplemented
28#16812053C2212yescomplemented
29#16812052C2212yescomplemented
30#16812050C2212yescomplemented
31#16812049C2211yescharacteristic, complemented
32#16812048C2212yescomplemented
33C16×D107C2212yescomplemented
34#135×C107C2211yescharacteristic
35#135×C107C2211yescharacteristic, complemented
36Q32×C107C2211yescharacteristic, complemented
37SD32×C107C2211yescharacteristic, complemented
38SD32×C107C2211yescharacteristic, complemented
39D16×C107C2211yescharacteristic, complemented
40C2×C16×C107C2211yescharacteristic, complemented
41#13133819—22no—
42#13133818—41no—
43#13133818—41no—
44#13133818D411yescharacteristic
45#13133817—22no—
46#13133817—42no—
47C2×Dic214—21no—
48C2×Dic214—41no—
49Q8×D107—22no—
50C2×D428—21no—
51C2×D428—41no—
52D4×D107—22no—
53D4×D107—42no—
54Dic428—21no—
55Dic428C2312yes—
56#13133808—22no—
57#13133806—22no—
58#13133805—22no—
59D856—21no—
60D856C2312yes—
61#13133803—22no—
62#13133802D411yescharacteristic
63#13133801D412yes—
64C8×D107C2312yes—
65C107⋊C16C2312yes—
66#41×C107—21no—
67#41×C107—21no—
68Q16×C107C2311yescharacteristic
69Q16×C107C2311yescharacteristic
70SD16×C107—21no—
71SD16×C107—21no—
72D8×C107C2311yescharacteristic
73D8×C107C2311yescharacteristic
74C2×C8×C107C2311yescharacteristic
75C16×C107C2311yescharacteristic
76C16×C107C2311yescharacteristic
77C22×D107—41no—
78C22×D107—42no—
79Q8×C107—21no—
80Q8×C107—21no—
81D4×C107—21no—
82D4×C107—21no—
83D4×C107—41no—
84D4×C107—41no—
85C2×C4×C107—41no—
86C2×C4×C107—41no—
87C2×C4×C107C2×D411yescharacteristic
88C22⋊D107—42no—
89C22⋊D107—42no—
90C22⋊D107D812yes—
91C2×Dic107—41no—
92C2×Dic107—42no—
93D428—22no—
94D428—42no—
95D428—41no—
96D428D811yescharacteristic
97C4×D107—42no—
98C4×D107—42no—
99C4×D107C2×D412yes—
100Dic214—22no—
101Dic214—41no—
102Dic214—42no—
103Dic214D811yescharacteristic
104C8×C107C2×D411yescharacteristic
105C8×C107C2411yescharacteristic
106C107⋊C8C2×D412yes—
107C22×C107—41no—
108C22×C107—41no—
109C22×C107C2×D811yescharacteristic
110C2×D107—42no—
111C2×D107—162no—
112C2×D107—161no—
113C2×D107C2×D812yes—
114C4×C107—41no—
115C4×C107—41no—
116C4×C107C2×D811yescharacteristic
117C4×C107C22×D411yescharacteristic
118Dic107—42no—
119Dic107C2×D812yes—
120C2×C107—21no—
121C2×C107—81no—
122C2×C107—81no—
123C2×C107C22×D811yescharacteristic
124D107—22no—
125D107—82no—
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#508D10711yescharacteristic, complemented
136C2×SD32—1072no—
137C2×SD32—1071no—
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#135C2×D10711yescharacteristic
148#135C2×D10711yescharacteristic, complemented
149C2×Q16—1072no—
150C2×Q16—1071no—
151C2×D8—1072no—
152C2×D8—1071no—
153#131—1071no—
154Q32—1072no—
155Q32C2×D10711yescharacteristic, complemented
156SD32—1072no—
157SD32—1072no—
158SD32—1072no—
159SD32—1072no—
160SD32C2×D10711yescharacteristic, complemented
161SD32C2×D10711yescharacteristic, complemented
162D16—1072no—
163D16C2×D10711yescharacteristic, complemented
164M32—1071no—
165M32—1072no—
166C2×C16—1072no—
167C2×C16C2×D10711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—1071no—
171#41—4281no—
172#41—4281no—
173#41—4282no—
174#41—2142no—
175#41—2142no—
176C2×Q8—4281no—
177C2×Q8—2142no—
178C2×Q8—2141no—
179C2×D4—4281no—
180C2×D4—4282no—
181C2×D4—2142no—
182C2×D4—2141no—
183Q16—1072no—
184Q16—2142no—
185Q16—2141no—
186Q16C22×D10711yescharacteristic
187Q16C22×D10711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—2142no—
191SD16—2142no—
192D8—1072no—
193D8—2142no—
194D8—2141no—
195D8C22×D10711yescharacteristic
196D8C22×D10711yescharacteristic
197M16—1071no—
198M16—1072no—
199C2×C8—1072no—
200C2×C8C22×D10711yescharacteristic
201C16—1072no—
202C16C22×D10711yescharacteristic
203C16C22×D10711yescharacteristic
204C23—4282no—
205C23—4281no—
206Q8—21no—
207Q8—21no—
208Q8—1071no—
209Q8—4282no—
210Q8—4281no—
211Q8—2142no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—1072no—
217D4—1071no—
218D4—4282no—
219D4—4282no—
220D4—4282no—
221D4—4281no—
222D4—2142no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1072no—
226C2×C4—4282no—
227C2×C4—4282no—
228C2×C4—4282no—
229C2×C4—4281no—
230C2×C4D4×D10711yescharacteristic
231C8—1072no—
232C8D4×D10711yescharacteristic
233C8C23×D10711yescharacteristic
234C22—41no—
235C22—41no—
236C22—1072no—
237C22—4282no—
238C22—17121no—
239C22—17122no—
240C22D8×D10711yescharacteristic
241C4—41no—
242C4—41no—
243C4—1072no—
244C4—4282no—
245C4D8×D10711yescharacteristic
246C4C2×D4×D10711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—8562no—
251C2—2142no—
252C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#508×C107C2
socleC2×C107C22×D8
hypercenter#508D107
hypocenterC107#3416
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #20293441C2semidirect
2. #20293429C2semidirect
3. #20293069C2semidirect
4. #20293067C2semidirect
5. #20293062C2semidirect
6. #20293061C2semidirect
7. #20293058C2semidirect
8. C2×#16812054C2semidirect
9. SD32×D107C2semidirect
10. #20293043C2semidirect
11. #508×C107C2semidirect
19. #16812128C22semidirect
20. C2×Dic428C22semidirect
21. Q16×D107C22semidirect
25. #16812055C22semidirect
26. #16812054C22semidirect
27. #16812054C22semidirect
28. #16812053C22semidirect
29. #16812052C22semidirect
30. #16812050C22semidirect
31. #16812049C22semidirect
32. #16812048C22semidirect
33. C16×D107C22semidirect
35. #135×C107C22semidirect
36. Q32×C107C22semidirect
37. SD32×C107C22semidirect
38. SD32×C107C22semidirect
39. D16×C107C22semidirect
40. C2×C16×C107C22semidirect
127. C107#3416semidirect
135. #508D107semidirect
148. #135C2×D107semidirect
155. Q32C2×D107semidirect
160. SD32C2×D107semidirect
161. SD32C2×D107semidirect
163. D16C2×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
#202934412
#202934292
#202930692
#202930672
#202930622
#202930612
#202930582
C2×#168120542
SD32×D1072
#202930432
#508×C1072
#3416107