Finite Group Databasebeta

← #497×D107 #28346897

Subgroups — #497×D107 #28346897

Order 13696.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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×Q16×D107C212yesmaximal, complemented
2C2×SD16×D107C212yesmaximal
3C4⋊Q8×D107C211yesmaximal, characteristic, complemented
4#124×D107C211yesmaximal, characteristic
5#20293022C211yesmaximal, characteristic, complemented
6#20293021C211yesmaximal, characteristic, complemented
7#20293020C212yesmaximal, complemented
8#20293019C211yesmaximal, characteristic, complemented
9#20293018C212yesmaximal, complemented
10#97×D107C211yesmaximal, characteristic, complemented
11#497×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2211yescharacteristic
13C2×Q8×D107C2212yes—
14C2×D4×D107C2211yescharacteristic
15C2×Dic428C2212yescomplemented
16C2×#13133808C2212yescomplemented
17C2×#13133808C2212yescomplemented
18Q16×D107—28no—
19C2×#13133807C2212yes—
20C2×#13133806C2212yes—
21C2×#13133805C2212yes—
22SD16×D107—28no—
23C2×C8×D107C2212yes—
24#16812109C2211yescharacteristic, complemented
25#16812108—22no—
26#16812107C2212yescomplemented
27#16812098C2211yescharacteristic
28#16812097C2211yescharacteristic
29#16812096C2211yescharacteristic
30#16812095—22no—
31C4⋊C4×D107—22no—
32C22⋊C4×D107—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic, complemented
35#16812019C2212yescomplemented
36C2×Q16×C107C2212yescomplemented
37C2×SD16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic, complemented
39#124×C107C2211yescharacteristic
40#97×C107C2211yescharacteristic, complemented
41C23×D107—21no—
42C2×Dic214—21no—
43C2×Dic214—22no—
44C2×Dic214C2311yescharacteristic
45C2×Dic214C2312yes—
46Q8×D107—24no—
47Q8×D107—24no—
48Q8×D107—24no—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—21no—
51D4×D107—24no—
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54C2×C4×D107D412yes—
55C2×C4×D107C2311yescharacteristic
56Dic428—24no—
57#13133808—24no—
58#13133808—24no—
59#13133807—24no—
60#13133806—24no—
61#13133805—24no—
62C2×C107⋊C8C2312yes—
63C8×D107D418yes—
64C4⋊Dic107—22no—
65Dic107⋊C4—24no—
66C22⋊Dic107—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107C2311yescharacteristic
72C2×Q8×C107C2312yes—
73C2×D4×C107C2311yescharacteristic
74Q16×C107—24no—
75SD16×C107—24no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—21no—
81C22×D107—21no—
82C22×D107—42no—
83C22×D107—44no—
84C22×D107C2×D411yescharacteristic
85Q8×C107—22no—
86Q8×C107—22no—
87Q8×C107—22no—
88D4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—42no—
94C2×Dic107—21no—
95C2×Dic107—22no—
96C2×Dic107C2×D412yes—
97C2×Dic107C2×D411yescharacteristic
98D428—22no—
99C4×D107—24no—
100C4×D107—42no—
101C4×D107—44no—
102C4×D107C2×D414yes—
103Dic214—22no—
104Dic214—22no—
105Dic214—22no—
106Dic214—44no—
107Dic214—42no—
108C8×C107C2×D414yes—
109C107⋊C8C2×D414yes—
110C22×C107—41no—
111C22×C107—42no—
112C22×C107C22×D411yescharacteristic
113C2×D107—41no—
114C2×D107—42no—
115C2×D107—84no—
116C2×D107#12712yes—
117C2×D107#13714yes—
118C4×C107—22no—
119C4×C107—41no—
120C4×C107—42no—
121C4×C107C22×D412yes—
122Dic107—22no—
123Dic107—42no—
124Dic107—41no—
125Dic107#12712yes—
126C2×C107—81no—
127C2×C107C2×#12711yescharacteristic
128C2×C107C2×#13712yes—
129D107—81no—
130D107#49714yesdirect product factor
131C2×#497—1071nomaximal, Sylow
132C107C2×#49711yescharacteristic, complemented, Sylow
133C22×Q16—1072no—
134C22×SD16—1072no—
135C2×C4⋊Q8—1071no—
136C2×#124—1071no—
137#497—1072no—
138#497—1072no—
139#497—1071no—
140#497—1071no—
141#497—1071no—
142#497D10711yescharacteristic, direct product factor
143C2×#97—1071no—
144C22×Q8—1071no—
145C22×Q8—1072no—
146C22×D4—1071no—
147C2×Q16—1072no—
148C2×Q16—1072no—
149C2×Q16—1072no—
150C2×Q16—2148no—
151C2×Q16C2×D10712yescomplemented
152C2×SD16—1072no—
153C2×SD16—1072no—
154C2×SD16—1072no—
155C2×SD16—2148no—
156C2×SD16C2×D10712yes—
157C22×C8—1072no—
158C4⋊Q8—1072no—
159C4⋊Q8—1071no—
160C4⋊Q8—2142no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#124—1071no—
163#124—1071no—
164#124—1071no—
165#124—2142no—
166#124C2×D10711yescharacteristic
167C2×C4⋊C4—2142no—
168C2×C22⋊C4—2142no—
169C2×C42—1071no—
170#97—1072no—
171#97—1071no—
172#97C2×D10711yescharacteristic, complemented
173C24—2141no—
174C2×Q8—1071no—
175C2×Q8—1072no—
176C2×Q8—2141no—
177C2×Q8—2142no—
178C2×Q8—2144no—
179C2×Q8—2144no—
180C2×Q8—2144no—
181C2×Q8C22×D10711yescharacteristic
182C2×Q8C22×D10712yes—
183C2×D4—1071no—
184C2×D4—2144no—
185C2×D4—2141no—
186C2×D4C22×D10711yescharacteristic
187C22×C4—1071no—
188C22×C4—1072no—
189C22×C4—2141no—
190C22×C4—2142no—
191Q16—24no—
192Q16—2144no—
193Q16—2144no—
194Q16—2144no—
195SD16—24no—
196SD16—2144no—
197SD16—2144no—
198SD16—2144no—
199C2×C8—1072no—
200C2×C8—1078no—
201C2×C8C22×D10712yes—
202C4⋊C4—22no—
203C4⋊C4—2144no—
204C4⋊C4—2142no—
205C22⋊C4—22no—
206C22⋊C4—2142no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C42—1072no—
210C42—1071no—
211C42C22×D10711yescharacteristic
212C23—21no—
213C23—1071no—
214C23—4282no—
215C23—4284no—
216C23—2141no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—4282no—
221Q8—4284no—
222Q8—2142no—
223Q8—2142no—
224Q8—2142no—
225D4—22no—
226D4—4282no—
227D4—2142no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—1071no—
231C2×C4—1074no—
232C2×C4—1072no—
233C2×C4—4282no—
234C2×C4—4284no—
235C2×C4—2144no—
236C2×C4—2141no—
237C2×C4—2142no—
238C2×C4D4×D10712yes—
239C2×C4C23×D10711yescharacteristic
240C8—1074no—
241C8D4×D10714yes—
242C22—42no—
243C22—41no—
244C22—8564no—
245C22—1074no—
246C22—1072no—
247C22—4282no—
248C22—4281no—
249C22C2×D4×D10711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—1072no—
254C4—4281no—
255C4—4282no—
256C4—2142no—
257C4C2×D4×D10712yes—
258C2—81no—
259C2—8561no—
260C2—1074no—
261C2#127×D10711yescharacteristic
262C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#497D107
hypocenterC107C2×#497
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D107C2semidirect
3. C4⋊Q8×D107C2semidirect
5. #20293022C2semidirect
6. #20293021C2semidirect
7. #20293020C2semidirect
8. #20293019C2semidirect
9. #20293018C2semidirect
10. #97×D107C2semidirect
11. #497×C107C2semidirect
15. C2×Dic428C22semidirect
16. C2×#13133808C22semidirect
17. C2×#13133808C22semidirect
24. #16812109C22semidirect
26. #16812107C22semidirect
34. #16812020C22semidirect
35. #16812019C22semidirect
36. C2×Q16×C107C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #97×C107C22semidirect
130. D107#497direct (N × H)
132. C107C2×#497semidirect
142. #497D107direct (N × H)
151. C2×Q16C2×D107semidirect
161. C4⋊Q8C2×D107semidirect
172. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
C2×SD16×D1072
C4⋊Q8×D1072
#124×D1072
#202930222
#202930212
#202930202
#202930192
#202930182
#97×D1072
#497×C1072
C2×#497107