Finite Group Databasebeta

← #28348276

Subgroups — #28348276

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#20293282C211yesmaximal, characteristic, complemented
2D4×D428C211yesmaximal, characteristic, complemented
3#20292974C211yesmaximal, characteristic, complemented
4#20292942C212yesmaximal, complemented
5#20292907C211yesmaximal, characteristic, complemented
6#20292862C212yesmaximal, complemented
7#20292822C211yesmaximal, characteristic, complemented
8#20292759C212yesmaximal, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#102×D107C212yesmaximal, complemented
11C4×D8×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C22×D428—21no—
14C2×D4×D107C2212yes—
15C2×#13133803C2212yescomplemented
16C2×C8×D107C2212yescomplemented
17#16812109C2211yescharacteristic, complemented
18#16812106C2211yescharacteristic
19#16812091—22no—
20#16812088—21no—
21#16812084C2212yes—
22#16812083—21no—
23#16812080C2212yescomplemented
24#16812079—22no—
25C4×D428C2211yescharacteristic
26C4⋊C4×D107C2212yescomplemented
27#16812042C2212yescomplemented
28C4⋊(C107⋊C8)C2211yescharacteristic, complemented
29#16812031C2212yescomplemented
30#16812031C2212yescomplemented
31#16812030C2212yes—
32#16812029C2212yescomplemented
33(C2×D107)⋊C8C2212yescomplemented
34C2×D8×C107C2211yescharacteristic
35C4×D4×C107C2211yescharacteristic
36C4×D4×C107C2211yescharacteristic, complemented
37#107×C107C2211yescharacteristic, complemented
38#102×C107C2211yescharacteristic
39#102×C107C2211yescharacteristic, complemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42#13133818—44no—
43C2×Dic214—21no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—44no—
48C2×D428D411yescharacteristic
49C2×D428C2312yes—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52D4×D107—24no—
53D4×D107—42no—
54C2×C4×D107—22no—
55C2×C4×D107C2312yes—
56#13133803—42no—
57C2×C107⋊C8C2312yes—
58C8×D107—42no—
59C4⋊Dic107—21no—
60C4⋊Dic107D411yescharacteristic
61C4⋊Dic107C2312yes—
62Dic107⋊C4—22no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4D412yes—
66C2×D4×C107C2311yescharacteristic
67C2×D4×C107C2311yescharacteristic
68C22×C4×C107—21no—
69C22×C4×C107—21no—
70D8×C107—22no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107C2311yescharacteristic
74C4⋊C4×C107C2311yescharacteristic
75C22⋊C4×C107—21no—
76C22⋊C4×C107—21no—
77C42×C107C2311yescharacteristic
78C23×C107—21no—
79C23×C107—21no—
80C22×D107—22no—
81C22×D107—44no—
82C22×D107—44no—
83C22×D107—82no—
84C22×D107C2×D412yes—
85D4×C107—21no—
86D4×C107—21no—
87D4×C107—21no—
88D4×C107#4112yescomplemented
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—42no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—44no—
97C22⋊D107—44no—
98C2×Dic107—22no—
99C2×Dic107C2×D412yes—
100D428—22no—
101D428—22no—
102D428—44no—
103D428—42no—
104D428—81no—
105D428D814yescomplemented
106C4×D107—22no—
107C4×D107—22no—
108C4×D107—44no—
109Dic214—42no—
110C8×C107—21no—
111C8×C107—21no—
112C107⋊C8—22no—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107—41no—
118C22×C107—41no—
119C22×C107C22×D411yescharacteristic
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107—88no—
126C2×D107C2×D814yes—
127C4×C107—21no—
128C4×C107—41no—
129C4×C107—41no—
130C4×C107C2×D812yes—
131C4×C107C2×#4111yescharacteristic
132C4×C107H(5,𝔽2)11yescharacteristic
133Dic107—22no—
134Dic107—42no—
135C2×C107—42no—
136C2×C107—81no—
137C2×C107#54611yescharacteristic
138C2×C107C22×D811yescharacteristic
139C2×C107#57711yescharacteristic
140D107—24no—
141D107—82no—
142#3294—1071nomaximal, Sylow
143C107#329411yescharacteristic, complemented, Sylow
144#547—1071no—
145D42—1071no—
146#486—1071no—
147#480—1072no—
148#474—1071no—
149#466—1072no—
150#459—1071no—
151#447—1072no—
152C4×D8D10711yescharacteristic, complemented
153C8×D4—1071no—
154C2×#102—1072no—
155C2×#41—2141no—
156C22×D4—1072no—
157C22×D4—2141no—
158C2×D8—1072no—
159C2×D8C2×D10711yescharacteristic
160C22×C8—1072no—
161C4⋊Q8—1071no—
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×D4C2×D10711yescharacteristic
171C4×D4C2×D10711yescharacteristic, complemented
172C2×C4⋊C4—1072no—
173#107—1072no—
174#107C2×D10711yescharacteristic, complemented
175C4⋊C8—1071no—
176#102—1072no—
177#102—1072no—
178#102—1072no—
179#102—1072no—
180#102C2×D10711yescharacteristic
181#102C2×D10711yescharacteristic, complemented
182C22⋊C8—1072no—
183C4×C8C2×D10711yescharacteristic, complemented
184C24—2142no—
185#41—4284no—
186C2×Q8—2141no—
187C2×D4—1071no—
188C2×D4—1072no—
189C2×D4—4284no—
190C2×D4—4282no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2144no—
195C2×D4—2141no—
196C2×D4—2141no—
197C2×D4C22×D10711yescharacteristic
198C2×D4C22×D10711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1072no—
202C22×C4—2142no—
203D8—22no—
204D8—4282no—
205C2×C8—1072no—
206C2×C8—4282no—
207C2×C8C22×D10711yescharacteristic
208C2×C8C22×D10711yescharacteristic
209C4⋊C4—1071no—
210C4⋊C4—1072no—
211C4⋊C4—2142no—
212C4⋊C4—2141no—
213C4⋊C4C22×D10711yescharacteristic
214C4⋊C4C22×D10711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1072no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C42C22×D10711yescharacteristic
221C23—21no—
222C23—21no—
223C23—8562no—
224C23—1072no—
225C23—4284no—
226C23—4284no—
227C23—2142no—
228Q8—4282no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—8561no—
233D4—1074no—
234D4—4284no—
235D4—4284no—
236D4—4284no—
237D4—4282no—
238D4—2142no—
239D4—2142no—
240D4#1313381812yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—1072no—
246C2×C4—4284no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—21no—
254C8—21no—
255C8—2142no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—8568no—
263C22—1074no—
264C22—4282no—
265C22—4282no—
266C22—4282no—
267C22—2142no—
268C22—2142no—
269C22C2×D4×D10711yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—4282no—
274C4—2142no—
275C4D8×D10712yes—
276C4C2×#1313381811yescharacteristic
277C4#1681215511yescharacteristic
278C2—42no—
279C2—81no—
280C2—8562no—
281C2—2144no—
282C2#2029327311yescharacteristic
283C2C2×D8×D10711yescharacteristic
284C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3294
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20293282C2semidirect
2. D4×D428C2semidirect
3. #20292974C2semidirect
4. #20292942C2semidirect
5. #20292907C2semidirect
6. #20292862C2semidirect
7. #20292822C2semidirect
8. #20292759C2semidirect
9. C8×D428C2semidirect
10. #102×D107C2semidirect
11. C4×D8×C107C2semidirect
15. C2×#13133803C22semidirect
16. C2×C8×D107C22semidirect
17. #16812109C22semidirect
23. #16812080C22semidirect
26. C4⋊C4×D107C22semidirect
27. #16812042C22semidirect
28. C4⋊(C107⋊C8)C22semidirect
29. #16812031C22semidirect
30. #16812031C22semidirect
32. #16812029C22semidirect
33. (C2×D107)⋊C8C22semidirect
36. C4×D4×C107C22semidirect
37. #107×C107C22semidirect
39. #102×C107C22semidirect
40. C4×C8×C107C22semidirect
88. D4×C107#41semidirect
105. D428D8semidirect
143. C107#3294semidirect
152. C4×D8D107semidirect
171. C4×D4C2×D107semidirect
174. #107C2×D107semidirect
181. #102C2×D107semidirect
183. C4×C8C2×D107semidirect
240. D4#13133818semidirect

Maximal subgroups

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

subgroupindex
#202932822
D4×D4282
#202929742
#202929422
#202929072
#202928622
#202928222
#202927592
C8×D4282
#102×D1072
C4×D8×C1072
#3294107