Finite Group Databasebeta

← #28348271

Subgroups — #28348271

Order 13696.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293280C211yesmaximal, characteristic, complemented
2D4×D428C211yesmaximal, characteristic, complemented
3#20292975C211yesmaximal, characteristic, complemented
4#20292942C212yesmaximal, complemented
5#20292864C212yesmaximal, complemented
6#20292823C211yesmaximal, characteristic, complemented
7#20292762C212yesmaximal, complemented
8C4×#13133803C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#102×D107C212yesmaximal, complemented
11#474×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C22×D428—21no—
14C2×D4×D107C2212yes—
15C2×D856C2212yescomplemented
16C2×#13133803C2211yescharacteristic
17C2×C8×D107C2212yescomplemented
18#16812106C2211yescharacteristic
19#16812094—22no—
20#16812085—21no—
21#16812084C2212yes—
22#16812084C2212yescomplemented
23#16812083—21no—
24#16812079—22no—
25C4×D428C2211yescharacteristic
26C4×D428C2211yescharacteristic, complemented
27C4⋊C4×D107C2212yescomplemented
28#16812042C2211yescharacteristic, complemented
29#16812031C2212yescomplemented
30#16812030C2211yescharacteristic
31#16812029C2211yescharacteristic, complemented
32#16812029C2212yescomplemented
33#16812029C2212yescomplemented
34(C2×D107)⋊C8C2212yescomplemented
35C4×C107⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C107C2211yescharacteristic, complemented
37C4×D4×C107C2211yescharacteristic
38#107×C107C2212yescomplemented
39C4⋊C8×C107C2211yescharacteristic, complemented
40#102×C107C2212yes—
41C23×D107—22no—
42#13133819—44no—
43C2×D428—22no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—44no—
48C2×D428D411yescharacteristic
49C2×D428C2311yescharacteristic
50C2×D428C2312yes—
51C2×C22⋊D107—22no—
52D4×D107—24no—
53D4×D107—42no—
54C2×C4×D107—22no—
55C2×C4×D107—21no—
56C2×C4×D107C2312yes—
57D856—42no—
58#13133803—22no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—42no—
62C4⋊Dic107D411yescharacteristic
63C4⋊Dic107C2311yescharacteristic
64Dic107⋊C4—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4D412yes—
69C2×Q8×C107—21no—
70C2×D4×C107C2311yescharacteristic
71C22×C4×C107—21no—
72C2×C8×C107C2312yes—
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2311yescharacteristic
75C4⋊C4×C107C2312yes—
76C22⋊C4×C107—21no—
77C42×C107C2311yescharacteristic
78C23×C107—21no—
79C22×D107—21no—
80C22×D107—22no—
81C22×D107—44no—
82C22×D107—44no—
83C22×D107—82no—
84C22×D107C2×D412yes—
85Q8×C107—42no—
86D4×C107—21no—
87D4×C107#4112yescomplemented
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—42no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—44no—
95C2×Dic107—21no—
96C2×Dic107C2×D412yes—
97D428—21no—
98D428—22no—
99D428—21no—
100D428—22no—
101D428—44no—
102D428—44no—
103D428—42no—
104D428—81no—
105D428D814yescomplemented
106C4×D107—22no—
107C4×D107—22no—
108C4×D107—44no—
109C4×D107—42no—
110C8×C107—22no—
111C107⋊C8—21no—
112C107⋊C8—21no—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107C22×D411yescharacteristic
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—42no—
120C2×D107—41no—
121C2×D107—41no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—41no—
125C2×D107—88no—
126C2×D107C2×D814yes—
127C4×C107—21no—
128C4×C107—42no—
129C4×C107—41no—
130C4×C107C2×D812yes—
131C4×C107C2×#4111yescharacteristic
132C4×C107H(5,𝔽2)11yescharacteristic
133Dic107—22no—
134Dic107—41no—
135C2×C107—42no—
136C2×C107#54611yescharacteristic
137C2×C107C22×D811yescharacteristic
138C2×C107#57711yescharacteristic
139D107—24no—
140D107—81no—
141D107—82no—
142#3294—1071nomaximal, Sylow
143C107#329411yescharacteristic, complemented, Sylow
144#547—1071no—
145D42—1071no—
146#486—1071no—
147#480—1072no—
148#474D10711yescharacteristic, complemented
149#466—1072no—
150#459—1071no—
151#447—1072no—
152C4×D8—1071no—
153C8×D4—1071no—
154C2×#102—1072no—
155C2×#41—2141no—
156C22×D4—1072no—
157C22×D4—2141no—
158C2×D8—1071no—
159C2×D8—1072no—
160C22×C8—1072no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#127—1071no—
163#123—2142no—
164C22⋊Q8—2141no—
165#121—1072no—
166#121—1072no—
167#121—2141no—
168#120—2142no—
169C4×D4—1071no—
170C4×D4—1071no—
171C4×D4C2×D10711yescharacteristic
172C2×C4⋊C4—1072no—
173#107—1071no—
174#107C2×D10712yescomplemented
175C4⋊C8C2×D10711yescharacteristic, complemented
176#102—1071no—
177#102—1072no—
178#102—1072no—
179#102—1071no—
180#102—1072no—
181#102C2×D10712yes—
182C22⋊C8—1072no—
183C4×C8—1071no—
184C24—2142no—
185#41—4284no—
186C2×Q8—21no—
187C2×D4—1071no—
188C2×D4—1072no—
189C2×D4—1071no—
190C2×D4—4284no—
191C2×D4—4282no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4—2144no—
196C2×D4—2141no—
197C2×D4—2141no—
198C2×D4C22×D10711yescharacteristic
199C22×C4—21no—
200C22×C4—1072no—
201C22×C4—2142no—
202C22×C4—2141no—
203D8—4282no—
204D8—2142no—
205C2×C8—1071no—
206C2×C8—1071no—
207C2×C8—4282no—
208C2×C8C22×D10712yes—
209C4⋊C4—21no—
210C4⋊C4—1071no—
211C4⋊C4—1071no—
212C4⋊C4—2142no—
213C4⋊C4C22×D10712yes—
214C4⋊C4C22×D10711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1072no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4—2141no—
220C42C22×D10711yescharacteristic
221C23—21no—
222C23—8562no—
223C23—1072no—
224C23—4284no—
225C23—4284no—
226C23—2141no—
227C23—2142no—
228Q8—42no—
229D4—21no—
230D4—8561no—
231D4—1074no—
232D4—4284no—
233D4—4284no—
234D4—4284no—
235D4—4282no—
236D4—2141no—
237D4—2142no—
238D4—2141no—
239D4—2142no—
240D4#1313381912yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1072no—
245C2×C4—4284no—
246C2×C4—4282no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—22no—
254C8—2141no—
255C8—2141no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—8568no—
260C22—1074no—
261C22—4281no—
262C22—4282no—
263C22—4282no—
264C22—4281no—
265C22—4282no—
266C22—4281no—
267C22—2142no—
268C22—2142no—
269C22C2×D4×D10711yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—4281no—
274C4—2142no—
275C4D8×D10712yes—
276C4C2×#1313381911yescharacteristic
277C4#1681215511yescharacteristic
278C2—42no—
279C2—8562no—
280C2—8561no—
281C2—2144no—
282C2#2029326811yescharacteristic
283C2C2×D8×D10711yescharacteristic
284C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
hypercenter#474D107
hypocenterC107#3294
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20293280C2semidirect
2. D4×D428C2semidirect
3. #20292975C2semidirect
4. #20292942C2semidirect
5. #20292864C2semidirect
6. #20292823C2semidirect
7. #20292762C2semidirect
8. C4×#13133803C2semidirect
9. #20292692C2semidirect
10. #102×D107C2semidirect
11. #474×C107C2semidirect
15. C2×D856C22semidirect
17. C2×C8×D107C22semidirect
22. #16812084C22semidirect
26. C4×D428C22semidirect
27. C4⋊C4×D107C22semidirect
28. #16812042C22semidirect
29. #16812031C22semidirect
31. #16812029C22semidirect
32. #16812029C22semidirect
33. #16812029C22semidirect
34. (C2×D107)⋊C8C22semidirect
35. C4×C107⋊C8C22semidirect
36. C4⋊Q8×C107C22semidirect
38. #107×C107C22semidirect
39. C4⋊C8×C107C22semidirect
87. D4×C107#41semidirect
105. D428D8semidirect
143. C107#3294semidirect
148. #474D107semidirect
161. C4⋊Q8C2×D107semidirect
174. #107C2×D107semidirect
175. C4⋊C8C2×D107semidirect
240. D4#13133819semidirect

Maximal subgroups

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

subgroupindex
#202932802
D4×D4282
#202929752
#202929422
#202928642
#202928232
#202927622
C4×#131338032
#202926922
#102×D1072
#474×C1072
#3294107