Finite Group Databasebeta

← #28349596

Subgroups — #28349596

Order 13696.

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

normal subgroups only

This group has 364 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
1C2×#16812132C211yesmaximal, characteristic, complemented
2#135×D107C211yesmaximal, characteristic, complemented
3#20293069C212yesmaximal, complemented
4#20293068C211yesmaximal, characteristic, complemented
5#20293065C211yesmaximal, characteristic, complemented
6#20293064C212yesmaximal, complemented
7#20293062C211yesmaximal, characteristic, complemented
8#20293060C212yesmaximal, complemented
9#20293058C212yesmaximal, complemented
10M32×D107C211yesmaximal, characteristic, complemented
11#510×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13#41×D107—21no—
14#16812132—21no—
15#16812132—21no—
16#16812132C2212yes—
17#16812132C2211yescharacteristic, complemented
18#16812132C2212yescomplemented
19#16812131C2211yescharacteristic
20#16812130—21no—
21#16812129—21no—
22#16812128C2211yescharacteristic, complemented
23Q16×D107C2211yescharacteristic, complemented
24C2×#13133805—21no—
25SD16×D107—21no—
26C2×D856C2211yescharacteristic, complemented
27D8×D107C2211yescharacteristic, complemented
28C2×C8×D107C2211yescharacteristic
29#16812055C2212yescomplemented
30#16812054C2212yescomplemented
31#16812052C2212yescomplemented
32#16812052C2212yescomplemented
33D1712C2212yescomplemented
34#16812050C2212yescomplemented
35#16812049C2211yescharacteristic, complemented
36#16812048C2212yescomplemented
37C16×D107C2212yescomplemented
38#135×C107C2211yescharacteristic
39C2×Q16×C107C2211yescharacteristic, complemented
40Q32×C107C2212yescomplemented
41SD32×C107C2212yescomplemented
42M32×C107C2211yescharacteristic, complemented
43#13133819—21no—
44#13133819—22no—
45#13133819—22no—
46#13133819—41no—
47#13133819—41no—
48#13133818—21no—
49#13133818—41no—
50#13133817—21no—
51#13133817—41no—
52Q8×D107—21no—
53C2×D428—21no—
54C2×D428—41no—
55D4×D107—21no—
56D4×D107—41no—
57C2×C4×D107—41no—
58C2×C4×D107—41no—
59C2×C4×D107D411yescharacteristic
60Dic428C2311yescharacteristic
61#13133808—21no—
62#13133807—21no—
63#13133806—21no—
64#13133805—21no—
65#13133805—22no—
66#13133805—22no—
67D856—21no—
68D856C2312yes—
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×C107C2311yescharacteristic
81Q16×C107C2312yes—
82SD16×C107—21no—
83D8×C107C2311yescharacteristic
84C2×C8×C107C2311yescharacteristic
85C16×C107C2312yes—
86C22×D107—41no—
87C22×D107—41no—
88C22×D107D811yescharacteristic
89Q8×C107—21no—
90Q8×C107—22no—
91Q8×C107—41no—
92D4×C107—21no—
93D4×C107—41no—
94C2×C4×C107—41no—
95C2×C4×C107—41no—
96C2×C4×C107C2×D411yescharacteristic
97C22⋊D107—41no—
98C22⋊D107—41no—
99C22⋊D107—41no—
100C2×Dic107—41no—
101C2×Dic107D811yescharacteristic
102D428—21no—
103D428—22no—
104D428—41no—
105D428—42no—
106D428—42no—
107D428—41no—
108C4×D107—41no—
109C4×D107—41no—
110C4×D107—41no—
111C4×D107—42no—
112C4×D107—42no—
113C4×D107D811yescharacteristic
114C4×D107D811yescharacteristic
115C4×D107C2×D411yescharacteristic
116C4×D107C2×D411yescharacteristic
117Dic214—21no—
118Dic214—41no—
119C8×C107C2×D411yescharacteristic
120C8×C107C2411yescharacteristic
121C107⋊C8C2×D411yescharacteristic
122C107⋊C8C2×D411yescharacteristic
123C22×C107—41no—
124C22×C107C2×D811yescharacteristic
125C2×D107—41no—
126C2×D107—42no—
127C2×D107—41no—
128C2×D107—82no—
129C2×D107—82no—
130C2×D107C2×D811yescharacteristic
131C2×D107C2×D811yescharacteristic
132C4×C107—41no—
133C4×C107—42no—
134C4×C107C2×D811yescharacteristic
135C4×C107C22×D411yescharacteristic
136Dic107—41no—
137Dic107C2×D811yescharacteristic
138Dic107C2×D811yescharacteristic
139C2×C107—21no—
140C2×C107—81no—
141C2×C107C22×D811yescharacteristic
142D107—21no—
143D107—21no—
144D107—82no—
145D107—81no—
146#3414—1071nomaximal, Sylow
147C107#341411yescharacteristic, complemented, Sylow
148C2×#135—1071no—
149C2×#135—1071no—
150#510—1071no—
151#510—1072no—
152#510D10711yescharacteristic, complemented
153#509—1071no—
154#509—1072no—
155#509—1071no—
156#508—1072no—
157#508—1072no—
158C2×M32—1071no—
159C2×#41—2141no—
160C2×#41—2141no—
161#135—1072no—
162#135—1071no—
163#135—1072no—
164#135—1071no—
165#135—1071no—
166#135—2141no—
167#135—2141no—
168#135—2141no—
169#135—2141no—
170#135C2×D10711yescharacteristic
171C2×Q16—1071no—
172C2×Q16C2×D10711yescharacteristic, complemented
173C2×SD16—2141no—
174C2×SD16—2141no—
175C2×D8—1071no—
176C2×D8—1071no—
177C22×C8—1071no—
178Q32—1072no—
179Q32C2×D10712yescomplemented
180SD32—1072no—
181SD32—1072no—
182SD32—1072no—
183SD32C2×D10712yescomplemented
184D16—1072no—
185D16—1072no—
186M32—1071no—
187M32—1072no—
188M32C2×D10711yescharacteristic, complemented
189C2×C16—1072no—
190#41—21no—
191#41—4281no—
192#41—4281no—
193#41—4281no—
194#41—4281no—
195#41—2141no—
196#41—2142no—
197#41—2141no—
198#41—2142no—
199#41—2141no—
200C2×Q8—21no—
201C2×Q8—2141no—
202C2×D4—4281no—
203C2×D4—4281no—
204C2×D4—2141no—
205C2×D4—2141no—
206C22×C4—1071no—
207C22×C4—4281no—
208C22×C4—4281no—
209Q16—21no—
210Q16—1071no—
211Q16—2141no—
212Q16C22×D10711yescharacteristic
213Q16C22×D10712yes—
214SD16—21no—
215SD16—2141no—
216SD16—2142no—
217SD16—2141no—
218SD16—2142no—
219SD16—2141no—
220D8—1072no—
221D8—1071no—
222D8—2141no—
223D8—2141no—
224D8C22×D10711yescharacteristic
225C2×C8—1071no—
226C2×C8—1071no—
227C2×C8—1071no—
228C2×C8—1071no—
229C2×C8—1071no—
230C2×C8C22×D10711yescharacteristic
231C16—1072no—
232C16C22×D10712yes—
233C23—1071no—
234C23—4281no—
235C23—4281no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—4281no—
240Q8—2141no—
241D4—21no—
242D4—41no—
243D4—4281no—
244D4—4282no—
245D4—4282no—
246D4—4281no—
247D4—4281no—
248D4—4281no—
249D4—4281no—
250D4—2141no—
251D4—2142no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1071no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—1071no—
258C2×C4—1071no—
259C2×C4—4281no—
260C2×C4—4282no—
261C2×C4—4282no—
262C2×C4—4281no—
263C2×C4—4281no—
264C2×C4—4281no—
265C2×C4D4×D10711yescharacteristic
266C8—1071no—
267C8—1071no—
268C8D4×D10711yescharacteristic
269C8C23×D10711yescharacteristic
270C22—41no—
271C22—8562no—
272C22—8562no—
273C22—1071no—
274C22—1071no—
275C22—4281no—
276C22—4282no—
277C22—4281no—
278C22D8×D10711yescharacteristic
279C4—41no—
280C4—42no—
281C4—1071no—
282C4—1071no—
283C4—4281no—
284C4D8×D10711yescharacteristic
285C4C2×D4×D10711yescharacteristic
286C2—21no—
287C2—81no—
288C2—8562no—
289C2—8561no—
290C2—2141no—
291C2—2141no—
292C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#510×C107C2
socleC2×C107C22×D8
hypercenter#510D107
hypocenterC107#3414
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
1. C2×#16812132C2semidirect
2. #135×D107C2semidirect
3. #20293069C2semidirect
4. #20293068C2semidirect
5. #20293065C2semidirect
6. #20293064C2semidirect
7. #20293062C2semidirect
8. #20293060C2semidirect
9. #20293058C2semidirect
10. M32×D107C2semidirect
11. #510×C107C2semidirect
17. #16812132C22semidirect
18. #16812132C22semidirect
22. #16812128C22semidirect
23. Q16×D107C22semidirect
26. C2×D856C22semidirect
27. D8×D107C22semidirect
29. #16812055C22semidirect
30. #16812054C22semidirect
31. #16812052C22semidirect
32. #16812052C22semidirect
33. D1712C22semidirect
34. #16812050C22semidirect
35. #16812049C22semidirect
36. #16812048C22semidirect
37. C16×D107C22semidirect
39. C2×Q16×C107C22semidirect
40. Q32×C107C22semidirect
41. SD32×C107C22semidirect
42. M32×C107C22semidirect
147. C107#3414semidirect
152. #510D107semidirect
172. C2×Q16C2×D107semidirect
179. Q32C2×D107semidirect
183. SD32C2×D107semidirect
188. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
#135×D1072
#202930692
#202930682
#202930652
#202930642
#202930622
#202930602
#202930582
M32×D1072
#510×C1072
#3414107