Finite Group Databasebeta

← #28346844

Subgroups — #28346844

Order 13696.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293183C211yesmaximal, characteristic, complemented
2#124×D107C211yesmaximal, characteristic
3#20292988C212yesmaximal, complemented
4#20292985C212yesmaximal, complemented
5#20292984C211yesmaximal, characteristic, complemented
6#20292982C211yesmaximal, characteristic, complemented
7#20292981C211yesmaximal, characteristic, complemented
8#20292634C212yesmaximal, complemented
9#20292630C212yesmaximal
10C4×#13133801C211yesmaximal, characteristic, complemented
11#488×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2211yescharacteristic
13C2×D4×D107C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812108C2212yescomplemented
16#16812101C2211yescharacteristic, complemented
17#16812098C2211yescharacteristic
18#16812097C2211yescharacteristic
19#16812096C2211yescharacteristic
20#16812095—22no—
21#16812087—22no—
22#16812074—22no—
23#16812067C2212yes—
24C22⋊C4×D107—22no—
25C42×D107C2211yescharacteristic
26#16812034C2212yescomplemented
27#16812033C2212yescomplemented
28#16812032C2212yescomplemented
29#16812031C2212yes—
30#16812030C2212yes—
31#16812029C2212yes—
32#16812019C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34#125×C107C2211yescharacteristic, complemented
35#124×C107C2211yescharacteristic
36#103×C107C2212yescomplemented
37#102×C107C2212yes—
38C4×C8×C107C2211yescharacteristic, complemented
39C23×D107—21no—
40C2×Dic214—22no—
41C2×Dic214—21no—
42C2×Dic214C2311yescharacteristic
43Q8×D107—24no—
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—21no—
46D4×D107—24no—
47C2×C4×D107—21no—
48C2×C4×D107D412yes—
49C2×C4×D107C2311yescharacteristic
50#13133801—24no—
51C2×C107⋊C8C2312yes—
52C4⋊Dic107C2312yes—
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55C22⋊Dic107—22no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—22no—
59C4×Dic107—22no—
60C4×Dic107D412yes—
61C4×Dic107C2311yescharacteristic
62C2×Q8×C107C2311yescharacteristic
63C2×D4×C107C2311yescharacteristic
64C2×C8×C107C2312yes—
65C4⋊C4×C107—22no—
66C4⋊C4×C107C2312yes—
67C22⋊C4×C107—22no—
68C42×C107C2311yescharacteristic
69C23×C107—21no—
70C22×D107—21no—
71C22×D107—44no—
72C22×D107—42no—
73C22×D107C2×D411yescharacteristic
74Q8×C107—22no—
75D4×C107—22no—
76C2×C4×C107—21no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2411yescharacteristic
80C22⋊D107—42no—
81C2×Dic107—22no—
82C2×Dic107—21no—
83C2×Dic107C2×D412yes—
84C2×Dic107C2×D411yescharacteristic
85D428—22no—
86C4×D107—42no—
87C4×D107—42no—
88C4×D107#4114yes—
89Dic214—22no—
90Dic214—44no—
91Dic214—42no—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—41no—
95C22×C107—42no—
96C22×C107C22×D411yescharacteristic
97C2×D107—22no—
98C2×D107—41no—
99C2×D107—42no—
100C2×D107—84no—
101C2×D107#12412yes—
102C4×C107—22no—
103C4×C107—41no—
104C4×C107—42no—
105C4×C107C2×#4112yes—
106Dic107—22no—
107Dic107—41no—
108Dic107—42no—
109Dic107#12412yes—
110C2×C107—81no—
111C2×C107C2×#12411yescharacteristic
112C2×C107#57512yes—
113D107—22no—
114D107—81no—
115#3140—1071nomaximal, Sylow
116C107#314011yescharacteristic, complemented, Sylow
117#533—1071no—
118C2×#124—1071no—
119#489—1072no—
120#489—1072no—
121#488—1071no—
122#488—1071no—
123#488—1071no—
124#488D10711yescharacteristic, complemented
125#419—1072no—
126#418—1072no—
127C4×M16—1071no—
128C22×Q8—1071no—
129C22×D4—1071no—
130C2×M16—1072no—
131C4⋊Q8—1072no—
132#125—1071no—
133#125C2×D10711yescharacteristic, complemented
134#124—1071no—
135#124—1071no—
136#124—1071no—
137#124—2142no—
138#124C2×D10711yescharacteristic
139C22⋊Q8—2142no—
140C4×Q8—2142no—
141#117—1072no—
142C2×C22⋊C4—2142no—
143C2×C42—1071no—
144#103—1072no—
145#103—1072no—
146#103—1072no—
147#103C2×D10712yescomplemented
148#102—1072no—
149#102—1072no—
150#102—1072no—
151#102C2×D10712yes—
152#97—1072no—
153C4×C8—1071no—
154C4×C8C2×D10711yescharacteristic, complemented
155C24—2141no—
156C2×Q8—1071no—
157C2×Q8—2142no—
158C2×Q8—2141no—
159C2×Q8—2144no—
160C2×Q8C22×D10711yescharacteristic
161C2×D4—1071no—
162C2×D4—2144no—
163C2×D4—2141no—
164C2×D4C22×D10711yescharacteristic
165C22×C4—1071no—
166C22×C4—1072no—
167C22×C4—2141no—
168M16—2144no—
169C2×C8—1072no—
170C2×C8C22×D10712yes—
171C4⋊C4—22no—
172C4⋊C4—1072no—
173C4⋊C4—2142no—
174C4⋊C4—2142no—
175C4⋊C4C22×D10712yes—
176C22⋊C4—22no—
177C22⋊C4—2142no—
178C22⋊C4—2142no—
179C22⋊C4—2142no—
180C22⋊C4—2142no—
181C42—1072no—
182C42—1071no—
183C42—2142no—
184C42C22×D10711yescharacteristic
185C23—21no—
186C23—1071no—
187C23—4282no—
188C23—4284no—
189C23—2141no—
190Q8—22no—
191Q8—4282no—
192Q8—4284no—
193Q8—2142no—
194D4—22no—
195D4—4282no—
196D4—2142no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—1071no—
200C2×C4—1074no—
201C2×C4—1072no—
202C2×C4—4282no—
203C2×C4—4282no—
204C2×C4—2142no—
205C2×C4—2141no—
206C2×C4D4×D10712yes—
207C2×C4C23×D10711yescharacteristic
208C8—22no—
209C8—2142no—
210C22—41no—
211C22—42no—
212C22—8564no—
213C22—1072no—
214C22—4281no—
215C22—4282no—
216C22—2142no—
217C22C2×D4×D10711yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—1072no—
222C4—4281no—
223C4—4282no—
224C4—2142no—
225C4#41×D10712yes—
226C2—81no—
227C2—8561no—
228C2—2142no—
229C2#124×D10711yescharacteristic
230C2#2029342412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#488×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#488D107
hypocenterC107#3140
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20293183C2semidirect
3. #20292988C2semidirect
4. #20292985C2semidirect
5. #20292984C2semidirect
6. #20292982C2semidirect
7. #20292981C2semidirect
8. #20292634C2semidirect
10. C4×#13133801C2semidirect
11. #488×C107C2semidirect
15. #16812108C22semidirect
16. #16812101C22semidirect
26. #16812034C22semidirect
27. #16812033C22semidirect
28. #16812032C22semidirect
32. #16812019C22semidirect
33. C4×C107⋊C8C22semidirect
34. #125×C107C22semidirect
36. #103×C107C22semidirect
38. C4×C8×C107C22semidirect
116. C107#3140semidirect
124. #488D107semidirect
133. #125C2×D107semidirect
147. #103C2×D107semidirect
154. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931832
#124×D1072
#202929882
#202929852
#202929842
#202929822
#202929812
#202926342
#202926302
C4×#131338012
#488×C1072
#3140107