Finite Group Databasebeta

← #28348268

Subgroups — #28348268

Order 13696.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D428C211yesmaximal, characteristic, complemented
2#20292975C211yesmaximal, characteristic, complemented
3#20292940C212yesmaximal, complemented
4#20292904C211yesmaximal, characteristic, complemented
5#20292861C212yesmaximal, complemented
6#20292822C211yesmaximal, characteristic, complemented
7#20292760C212yesmaximal, complemented
8C4×#13133803C211yesmaximal, characteristic, complemented
9D4×C107⋊C8C211yesmaximal, characteristic, complemented
10C2×#16812029C212yesmaximal, complemented
11#547×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14C2×#13133803C2211yescharacteristic
15C2×#13133803C2212yescomplemented
16C22×C107⋊C8C2212yescomplemented
17#16812106C2211yescharacteristic
18#16812084—21no—
19#16812083C2212yes—
20#16812079—22no—
21C4×D428C2211yescharacteristic
22#16812042C2212yescomplemented
23#16812042C2211yescharacteristic, complemented
24C4⋊(C107⋊C8)C2211yescharacteristic, complemented
25#16812030C2211yescharacteristic, complemented
26#16812029C2211yescharacteristic
27#16812029C2212yes—
28#16812029C2212yescomplemented
29#16812029C2212yescomplemented
30#16812029C2212yescomplemented
31C22⋊(C107⋊C8)C2212yescomplemented
32C4×C107⋊C8C2211yescharacteristic, complemented
33C2×#41×C107—21no—
34C4⋊Q8×C107C2211yescharacteristic, complemented
35#123×C107—22no—
36C22⋊Q8×C107—21no—
37#121×C107C2212yescomplemented
38C4×D4×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic, complemented
40C2×C4⋊C4×C107C2212yescomplemented
41C23×D107—22no—
42C2×D428—24no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428—21no—
46C2×D428—42no—
47C2×D428C2312yes—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—22no—
50D4×D107—44no—
51C2×C4×D107—21no—
52#13133803—22no—
53#13133803—42no—
54C2×C107⋊C8—42no—
55C2×C107⋊C8C2312yes—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—22no—
61#41×C107—44no—
62C2×Q8×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107D411yescharacteristic
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—22no—
67C22×C4×C107—21no—
68C22×C4×C107C2312yes—
69C4⋊C4×C107—22no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107D411yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C4⋊C4×C107C2312yes—
74C22⋊C4×C107—21no—
75C22⋊C4×C107—22no—
76C22⋊C4×C107D412yes—
77C42×C107C2311yescharacteristic
78C23×C107—21no—
79C23×C107C2×D412yes—
80C22×D107—21no—
81C22×D107—22no—
82C22×D107—44no—
83C22×D107—44no—
84C22×D107—82no—
85Q8×C107—42no—
86D4×C107—21no—
87D4×C107—21no—
88D4×C107—44no—
89D4×C107D814yescomplemented
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—44no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—44no—
101C2×Dic107—21no—
102D428—22no—
103D428—21no—
104D428—22no—
105D428—44no—
106D428—42no—
107D428—81no—
108D428#4112yescomplemented
109C4×D107—42no—
110C107⋊C8—21no—
111C107⋊C8—21no—
112C107⋊C8—22no—
113C22×C107—22no—
114C22×C107—22no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107—41no—
118C22×C107C2×D814yes—
119C22×C107C22×D411yescharacteristic
120C2×D107—22no—
121C2×D107—41no—
122C2×D107—41no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—88no—
127C4×C107—22no—
128C4×C107—21no—
129C4×C107—41no—
130C4×C107—42no—
131C4×C107C2×D812yes—
132C4×C107C2×#4111yescharacteristic
133C4×C107H(5,𝔽2)11yescharacteristic
134Dic107—41no—
135C2×C107—24no—
136C2×C107—81no—
137C2×C107#54611yescharacteristic
138C2×C107C22×D811yescharacteristic
139C2×C107#57711yescharacteristic
140D107—42no—
141D107—82no—
142#3294—1071nomaximal, Sylow
143C107#329411yescharacteristic, complemented, Sylow
144#547D10711yescharacteristic, complemented
145D42—1071no—
146#486—1071no—
147#480—1072no—
148#474—1071no—
149#466—1072no—
150#459—1071no—
151#447—1072no—
152C4×D8—1071no—
153C8×D4—1071no—
154C2×#102—1072no—
155C2×#41—21no—
156C22×D4—1072no—
157C22×D4—2141no—
158C2×D8—1072no—
159C2×D8—1071no—
160C22×C8—1072no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#127—1071no—
163#123—22no—
164C22⋊Q8—21no—
165#121—1072no—
166#121—2141no—
167#121C2×D10712yescomplemented
168#120—2142no—
169C4×D4—1071no—
170C4×D4C2×D10711yescharacteristic
171C4×D4C2×D10711yescharacteristic, complemented
172C2×C4⋊C4C2×D10712yescomplemented
173#107—1071no—
174#107—1072no—
175C4⋊C8—1071no—
176#102—1071no—
177#102—1071no—
178#102—1072no—
179#102—1072no—
180#102—1072no—
181#102—1072no—
182C22⋊C8—1072no—
183C4×C8—1071no—
184C24—2142no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—1071no—
189C2×D4—1072no—
190C2×D4—4282no—
191C2×D4—4284no—
192C2×D4—2144no—
193C2×D4—2142no—
194C2×D4—2141no—
195C2×D4—2142no—
196C2×D4—2141no—
197C2×D4C22⋊D10711yescharacteristic
198C2×D4C22×D10711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2141no—
202C22×C4C22×D10712yes—
203D8—4282no—
204D8—2142no—
205C2×C8—1071no—
206C2×C8—1071no—
207C2×C8—1072no—
208C2×C8—4282no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1071no—
212C4⋊C4C22⋊D10711yescharacteristic
213C4⋊C4C22×D10711yescharacteristic
214C4⋊C4C22×D10712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2141no—
218C22⋊C4—2142no—
219C22⋊C4C22⋊D10712yes—
220C42C22×D10711yescharacteristic
221C23—21no—
222C23—8562no—
223C23—4284no—
224C23—4284no—
225C23—2142no—
226C23—2141no—
227C23C2×C22⋊D10712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—8561no—
233D4—1072no—
234D4—4284no—
235D4—4284no—
236D4—4282no—
237D4—2142no—
238D4—2142no—
239D4—2141no—
240D4#1313380314yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—4282no—
248C2×C4—2141no—
249C2×C4C2×C22⋊D10711yescharacteristic
250C2×C4C2×C22⋊D10712yes—
251C2×C4C2×C22⋊D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—2141no—
254C8—2141no—
255C8—2142no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—8568no—
262C22—4282no—
263C22—4282no—
264C22—4282no—
265C22—4281no—
266C22—4281no—
267C22—2142no—
268C22C2×#1313380314yes—
269C22C22×C22⋊D10711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—4281no—
275C4C2×#1313380312yes—
276C4#41×D10711yescharacteristic
277C4#1681215511yescharacteristic
278C2—24no—
279C2—81no—
280C2—8562no—
281C2—4282no—
282C2#2029326511yescharacteristic
283C2C22×#1313380311yescharacteristic
284C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3294
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D428C2semidirect
2. #20292975C2semidirect
3. #20292940C2semidirect
4. #20292904C2semidirect
5. #20292861C2semidirect
6. #20292822C2semidirect
7. #20292760C2semidirect
8. C4×#13133803C2semidirect
9. D4×C107⋊C8C2semidirect
10. C2×#16812029C2semidirect
11. #547×C107C2semidirect
15. C2×#13133803C22semidirect
16. C22×C107⋊C8C22semidirect
22. #16812042C22semidirect
23. #16812042C22semidirect
24. C4⋊(C107⋊C8)C22semidirect
25. #16812030C22semidirect
28. #16812029C22semidirect
29. #16812029C22semidirect
30. #16812029C22semidirect
31. C22⋊(C107⋊C8)C22semidirect
32. C4×C107⋊C8C22semidirect
34. C4⋊Q8×C107C22semidirect
37. #121×C107C22semidirect
39. C4×D4×C107C22semidirect
40. C2×C4⋊C4×C107C22semidirect
89. D4×C107D8semidirect
108. D428#41semidirect
143. C107#3294semidirect
144. #547D107semidirect
161. C4⋊Q8C2×D107semidirect
167. #121C2×D107semidirect
171. C4×D4C2×D107semidirect
172. C2×C4⋊C4C2×D107semidirect
240. D4#13133803semidirect

Maximal subgroups

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

subgroupindex
D4×D4282
#202929752
#202929402
#202929042
#202928612
#202928222
#202927602
C4×#131338032
D4×C107⋊C82
C2×#168120292
#547×C1072
#3294107