Finite Group Databasebeta

← #28349601

Subgroups — #28349601

Order 13696.

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

normal subgroups only

This group has 370 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#20293429C211yesmaximal, characteristic, complemented
2#20293428C211yesmaximal, characteristic, complemented
3#20293065C211yesmaximal, characteristic, complemented
4#20293064C212yesmaximal, complemented
5#20293061C212yesmaximal, complemented
6#20293058C212yesmaximal, complemented
7#20293057C211yesmaximal, characteristic, complemented
8C2×#16812050C211yesmaximal, characteristic, complemented
9D16×D107C212yesmaximal, complemented
10#20293045C211yesmaximal, characteristic, complemented
11#509×C107C211yesmaximal, characteristic, complemented
12#16812155—21no—
13#16812154—21no—
14#16812136—22no—
15#16812135—21no—
16#16812134—21no—
17#16812133—22no—
18#16812132C2212yescomplemented
19#16812131C2211yescharacteristic, complemented
20#16812128C2212yescomplemented
21C2×D856C2211yescharacteristic, complemented
22D8×D107C2212yescomplemented
23D8×D107C2212yescomplemented
24#16812115C2211yescharacteristic
25#16812055C2211yescharacteristic, complemented
26#16812054C2212yescomplemented
27#16812053C2211yescharacteristic, complemented
28#16812052C2211yescharacteristic, complemented
29D1712C2212yescomplemented
30#16812050C2211yescharacteristic, complemented
31#16812050C2212yescomplemented
32#16812050C2212yescomplemented
33#16812048C2212yescomplemented
34C2×C107⋊C16C2211yescharacteristic, complemented
35C16×D107C2212yescomplemented
36#135×C107C2211yescharacteristic, complemented
37C2×D8×C107C2211yescharacteristic, complemented
38SD32×C107C2212yescomplemented
39D16×C107C2212yescomplemented
40M32×C107C2211yescharacteristic, complemented
41#13133819—22no—
42#13133818—21no—
43#13133818—41no—
44#13133818D411yescharacteristic
45#13133817—22no—
46#13133817—41no—
47C2×D428—21no—
48C2×D428—41no—
49C2×C22⋊D107—42no—
50D4×D107—22no—
51D4×D107—22no—
52D4×D107—42no—
53D4×D107—41no—
54Dic428C2311yescharacteristic
55#13133807—21no—
56#13133806—22no—
57#13133805—22no—
58D856—21no—
59D856C2312yes—
60D856C2311yescharacteristic
61#13133803—22no—
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×C107C2312yes—
74D8×C107C2311yescharacteristic
75C2×C8×C107C2311yescharacteristic
76C16×C107C2312yes—
77C23×C107—41no—
78C22×D107—41no—
79C22×D107—42no—
80C22×D107—42no—
81Q8×C107—21no—
82D4×C107—22no—
83D4×C107—21no—
84D4×C107—41no—
85D4×C107—41no—
86C2×C4×C107—41no—
87C2×C4×C107C2×D411yescharacteristic
88C22⋊D107—41no—
89C22⋊D107—42no—
90C22⋊D107—42no—
91C22⋊D107—42no—
92C22⋊D107D812yes—
93C2×Dic107—42no—
94D428—21no—
95D428—22no—
96D428—42no—
97D428—41no—
98D428D811yescharacteristic
99C4×D107—41no—
100C4×D107—42no—
101C4×D107C2×D412yes—
102Dic214—21no—
103Dic214D811yescharacteristic
104C8×C107C2×D411yescharacteristic
105C8×C107C2411yescharacteristic
106C107⋊C8C2×D412yes—
107C22×C107—42no—
108C22×C107—41no—
109C22×C107—161no—
110C22×C107C2×D811yescharacteristic
111C2×D107—42no—
112C2×D107—41no—
113C2×D107—162no—
114C2×D107—162no—
115C2×D107—161no—
116C2×D107C2×D812yes—
117C4×C107—41no—
118C4×C107C2×D811yescharacteristic
119C4×C107C22×D411yescharacteristic
120Dic107—41no—
121Dic107C2×D812yes—
122C2×C107—21no—
123C2×C107—81no—
124C2×C107—82no—
125C2×C107C22×D811yescharacteristic
126D107—22no—
127D107—82no—
128D107—81no—
129#3415—1071nomaximal, Sylow
130C107#341511yescharacteristic, complemented, Sylow
131#576—1071no—
132#576—1071no—
133#509—1072no—
134#509—1072no—
135#509—1071no—
136#509D10711yescharacteristic, complemented
137#508—1071no—
138#508—1072no—
139C2×D16—1071no—
140C2×D16—1072no—
141#504—1071no—
142H(5,𝔽2)—2141no—
143H(5,𝔽2)—2141no—
144AGL(1,ℤ/8)—2142no—
145AGL(1,ℤ/8)—2142no—
146AGL(1,ℤ/8)—2141no—
147AGL(1,ℤ/8)—2141no—
148#135—1072no—
149#135—1072no—
150#135—1071no—
151#135C2×D10711yescharacteristic, complemented
152C2×D8—1072no—
153C2×D8—1072no—
154C2×D8—1071no—
155C2×D8C2×D10711yescharacteristic, complemented
156#131—1071no—
157Q32—1071no—
158SD32—1071no—
159SD32—1071no—
160SD32—1072no—
161SD32C2×D10712yescomplemented
162D16—1072no—
163D16—1071no—
164D16—1072no—
165D16—1072no—
166D16C2×D10712yescomplemented
167M32—1072no—
168M32C2×D10711yescharacteristic, complemented
169C2×C16—1071no—
170C2×C16—1072no—
171#41—21no—
172#41—1071no—
173#41—4281no—
174#41—4281no—
175#41—2142no—
176#41—2142no—
177#41—2141no—
178C2×D4—21no—
179C2×D4—4281no—
180C2×D4—4282no—
181C2×D4—4281no—
182C2×D4—4282no—
183C2×D4—2142no—
184C2×D4—2142no—
185C2×D4—2141no—
186Q16—1071no—
187Q16C22×D10711yescharacteristic
188SD16—21no—
189SD16—2142no—
190SD16—2142no—
191SD16—2141no—
192D8—21no—
193D8—1072no—
194D8—1071no—
195D8—2142no—
196D8—2142no—
197D8—2141no—
198D8C22×D10712yes—
199D8C22×D10711yescharacteristic
200M16—1071no—
201M16—1072no—
202C2×C8—1072no—
203C2×C8C22×D10711yescharacteristic
204C16—1071no—
205C16—1071no—
206C16C22×D10712yes—
207C23—41no—
208C23—4282no—
209C23—4282no—
210C23—4281no—
211Q8—21no—
212Q8—1071no—
213Q8—2141no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—1072no—
219D4—1071no—
220D4—4282no—
221D4—4282no—
222D4—4282no—
223D4—4282no—
224D4—4281no—
225D4—4281no—
226D4—2141no—
227D4—2142no—
228C2×C4—41no—
229C2×C4—1072no—
230C2×C4—4282no—
231C2×C4—4282no—
232C2×C4—4281no—
233C2×C4D4×D10711yescharacteristic
234C8—1072no—
235C8D4×D10711yescharacteristic
236C8C23×D10711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—1072no—
241C22—4282no—
242C22—4281no—
243C22—17121no—
244C22—17122no—
245C22—17122no—
246C22D8×D10711yescharacteristic
247C4—41no—
248C4—1072no—
249C4—4281no—
250C4D8×D10711yescharacteristic
251C4C2×D4×D10711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—8562no—
256C2—8561no—
257C2—2142no—
258C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#509×C107C2
socleC2×C107C22×D8
hypercenter#509D107
hypocenterC107#3415
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. #20293429C2semidirect
2. #20293428C2semidirect
3. #20293065C2semidirect
4. #20293064C2semidirect
5. #20293061C2semidirect
6. #20293058C2semidirect
7. #20293057C2semidirect
8. C2×#16812050C2semidirect
9. D16×D107C2semidirect
10. #20293045C2semidirect
11. #509×C107C2semidirect
18. #16812132C22semidirect
19. #16812131C22semidirect
20. #16812128C22semidirect
21. C2×D856C22semidirect
22. D8×D107C22semidirect
23. D8×D107C22semidirect
25. #16812055C22semidirect
26. #16812054C22semidirect
27. #16812053C22semidirect
28. #16812052C22semidirect
29. D1712C22semidirect
30. #16812050C22semidirect
31. #16812050C22semidirect
32. #16812050C22semidirect
33. #16812048C22semidirect
34. C2×C107⋊C16C22semidirect
35. C16×D107C22semidirect
36. #135×C107C22semidirect
37. C2×D8×C107C22semidirect
38. SD32×C107C22semidirect
39. D16×C107C22semidirect
40. M32×C107C22semidirect
130. C107#3415semidirect
136. #509D107semidirect
151. #135C2×D107semidirect
155. C2×D8C2×D107semidirect
161. SD32C2×D107semidirect
166. D16C2×D107semidirect
168. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934292
#202934282
#202930652
#202930642
#202930612
#202930582
#202930572
C2×#168120502
D16×D1072
#202930452
#509×C1072
#3415107