Finite Group Databasebeta

← #28347056

Subgroups — #28347056

Order 13696.

408 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293177C211yesmaximal, characteristic, complemented
2#20293161C211yesmaximal, characteristic, complemented
3#20293036C211yesmaximal, characteristic, complemented
4#20293016C211yesmaximal, characteristic, complemented
5#20292992C211yesmaximal, characteristic, complemented
6#20292987C211yesmaximal, characteristic, complemented
7#20292880C212yesmaximal, complemented
8#20292873C212yesmaximal, complemented
9C4×#13133807C212yesmaximal, complemented
10C4×D856C212yesmaximal, complemented
11C4×M16×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×#13133807C2212yes—
14C2×D856C2212yes—
15#16812109C2211yescharacteristic
16#16812106C2211yescharacteristic
17#16812103—21no—
18#16812101C2211yescharacteristic
19#16812097—21no—
20#16812097C2211yescharacteristic
21#16812092—21no—
22#16812088C2211yescharacteristic
23#16812083—21no—
24#16812083C2212yes—
25#16812083C2211yescharacteristic
26C4×Dic214C2211yescharacteristic
27C4×D428—21no—
28C4×D428C2211yescharacteristic
29C4×D428C2212yes—
30C4×C22⋊D107—21no—
31#16812066—21no—
32#16812043C2212yes—
33#16812041C2212yes—
34#16812034C2212yes—
35#16812031C2212yes—
36#16812031C2212yes—
37#16812031C2212yes—
38C2×M16×C107C2212yes—
39C2×C42×C107C2211yescharacteristic
40#97×C107C2211yescharacteristic
41#97×C107C2211yescharacteristic
42C4×C8×C107C2212yes—
43#13133818—44no—
44C2×Dic214—21no—
45C2×Dic214C2311yescharacteristic
46C2×D428—21no—
47C2×D428—21no—
48C2×D428C2311yescharacteristic
49C2×D428C2312yes—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54#13133807—24no—
55D856—24no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—21no—
71C4×Dic107—21no—
72C22×C4×C107D411yescharacteristic
73C22×C4×C107D411yescharacteristic
74C22×C4×C107C2311yescharacteristic
75M16×C107—24no—
76C2×C8×C107C2312yes—
77C2×C8×C107C2312yes—
78C42×C107D411yescharacteristic
79C42×C107D411yescharacteristic
80C42×C107C2311yescharacteristic
81C42×C107C2311yescharacteristic
82C23×C107C2×D411yescharacteristic
83C22×D107—21no—
84C22×D107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—41no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—42no—
95C22⋊D107—44no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99C2×Dic107—21no—
100D428—22no—
101D428—22no—
102D428—22no—
103D428—42no—
104D428—42no—
105C4×D107—42no—
106C4×D107—44no—
107Dic214—22no—
108Dic214—42no—
109C8×C107—22no—
110C8×C107#4114yescomplemented
111C22×C107—21no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107C22×D411yescharacteristic
115C2×D107—41no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—41no—
119C2×D107—42no—
120C2×D107—41no—
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—21no—
124C4×C107—21no—
125C4×C107AGL(1,ℤ/8)12yes—
126C4×C107C2×#4112yes—
127Dic107—41no—
128Dic107—42no—
129Dic107—41no—
130Dic107—41no—
131C2×C107—41no—
132C2×C107#53211yescharacteristic
133C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C107#57511yescharacteristic
135D107—82no—
136D107—81no—
137#3171—1071nomaximal, Sylow
138C107#317111yescharacteristic, complemented, Sylow
139#532—1071no—
140#529—1071no—
141#501—1071no—
142#496—1071no—
143#490—1071no—
144#489—1071no—
145#469—1072no—
146#468—1072no—
147C4×SD16—1072no—
148C4×D8—1072no—
149C4×M16D10711yescharacteristic, complemented
150C2×#41—2141no—
151C2×SD16—1072no—
152C2×D8—1072no—
153C2×M16C2×D10712yes—
154C4⋊Q8—1071no—
155#127—1071no—
156#126—2141no—
157#125—1071no—
158#124—1071no—
159#124—2141no—
160#123—2141no—
161C22⋊Q8—1071no—
162#121—1071no—
163#121—1072no—
164#121—2141no—
165C4×Q8—1071no—
166C4×D4—1071no—
167C4×D4—1072no—
168C4×D4—2141no—
169C4×D4—2141no—
170#117—2141no—
171C2×C42C2×D10711yescharacteristic
172#107—1072no—
173#106—1072no—
174#103—1072no—
175#102—1072no—
176#102—1072no—
177#102—1072no—
178#97C2×D10711yescharacteristic
179#97C2×D10711yescharacteristic
180C4×C8C2×D10712yes—
181#41—4284no—
182C2×Q8—1071no—
183C2×Q8—2141no—
184C2×D4—1071no—
185C2×D4—1072no—
186C2×D4—2141no—
187C2×D4—2142no—
188C2×D4—2141no—
189C2×D4—2141no—
190C22×C4—2141no—
191C22×C4—2142no—
192C22×C4D42811yescharacteristic
193C22×C4D42811yescharacteristic
194C22×C4C22×D10711yescharacteristic
195SD16—2144no—
196D8—2144no—
197M16—24no—
198C2×C8C22×D10712yes—
199C2×C8C22×D10712yes—
200C4⋊C4—1071no—
201C4⋊C4—1071no—
202C4⋊C4—1072no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C22⋊C4—2141no—
209C22⋊C4—2141no—
210C22⋊C4—2141no—
211C22⋊C4—2142no—
212C22⋊C4—2141no—
213C22⋊C4—2142no—
214C22⋊C4—2141no—
215C42—2141no—
216C42D42811yescharacteristic
217C42D42811yescharacteristic
218C42C22×D10711yescharacteristic
219C42C22×D10711yescharacteristic
220C23—2141no—
221C23—2142no—
222C23C2×D42811yescharacteristic
223Q8—4282no—
224Q8—2142no—
225D4—4282no—
226D4—4282no—
227D4—4284no—
228D4—4282no—
229D4—2142no—
230D4—2142no—
231D4—2142no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—4284no—
236C2×C4—4282no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4—2142no—
241C2×C4C2×D42811yescharacteristic
242C2×C4C2×D42811yescharacteristic
243C2×C4C2×D42811yescharacteristic
244C2×C4C2×D42811yescharacteristic
245C2×C4C2×D42811yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—22no—
248C8#1313381814yescomplemented
249C22—21no—
250C22—21no—
251C22—21no—
252C22—4282no—
253C22—4281no—
254C22—4282no—
255C22—4281no—
256C22—4281no—
257C22—4282no—
258C22C22×D42811yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—4282no—
264C4—4281no—
265C4—4281no—
266C4—4281no—
267C4#1681213712yes—
268C4C2×#1313381812yes—
269C2—41no—
270C2—8562no—
271C2—8561no—
272C2#2029317711yescharacteristic
273C2C2×#1681213711yescharacteristic
274C2#2029342711yescharacteristic
subgroupG/H
centerC2×C4C2×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×M16×C107C2
socleC22×C107C22×D4
hypercenterC4×M16D107
hypocenterC107#3171
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. #20293177C2semidirect
2. #20293161C2semidirect
3. #20293036C2semidirect
4. #20293016C2semidirect
5. #20292992C2semidirect
6. #20292987C2semidirect
7. #20292880C2semidirect
8. #20292873C2semidirect
9. C4×#13133807C2semidirect
10. C4×D856C2semidirect
11. C4×M16×C107C2semidirect
110. C8×C107#41semidirect
138. C107#3171semidirect
149. C4×M16D107semidirect
248. C8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202931772
#202931612
#202930362
#202930162
#202929922
#202929872
#202928802
#202928732
C4×#131338072
C4×D8562
C4×M16×C1072
#3171107