Finite Group Databasebeta

← #28347053

Subgroups — #28347053

Order 13696.

408 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 119 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#20293175C211yesmaximal, characteristic, complemented
2#20293156C211yesmaximal, characteristic, complemented
3#20293036C211yesmaximal, characteristic, complemented
4#20292991C211yesmaximal, characteristic, complemented
5#20292986C211yesmaximal, characteristic, complemented
6#20292877C212yesmaximal, complemented
7#20292870C212yesmaximal, complemented
8SD16×Dic107C212yesmaximal, complemented
9D8×Dic107C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#496×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—21no—
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812102—21no—
16#16812101C2211yescharacteristic
17#16812096—21no—
18#16812091—21no—
19#16812085C2211yescharacteristic
20#16812080—21no—
21#16812080C2212yes—
22#16812080C2211yescharacteristic
23Q8×Dic107C2211yescharacteristic
24D4×Dic107—21no—
25D4×Dic107C2211yescharacteristic
26D4×Dic107C2212yes—
27(C22⋊D107)⋊C4—21no—
28#16812065—21no—
29C42×D107C2211yescharacteristic
30#16812043C2212yes—
31#16812041C2212yes—
32#16812033C2212yes—
33#16812030C2212yes—
34#16812030C2212yes—
35#16812030C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C2×SD16×C107C2212yes—
39C2×D8×C107C2212yes—
40#127×C107C2211yescharacteristic
41#124×C107C2211yescharacteristic
42#97×C107C2211yescharacteristic
43#13133817—44no—
44C2×Dic214—21no—
45C2×C22⋊D107—21no—
46C2×C22⋊D107—22no—
47C22×Dic107—21no—
48C22×Dic107—22no—
49C2×C4×D107D411yescharacteristic
50C2×C4×D107D411yescharacteristic
51C2×C4×D107C2311yescharacteristic
52#13133801—24no—
53C2×C107⋊C8C2312yes—
54C4⋊Dic107—21no—
55C4⋊Dic107—21no—
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62C22⋊Dic107—21no—
63C22⋊Dic107—22no—
64C22⋊Dic107—22no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67C4×Dic107—21no—
68C4×Dic107D411yescharacteristic
69C4×Dic107D411yescharacteristic
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107C2311yescharacteristic
72C2×D4×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C2×D4×C107C2312yes—
76SD16×C107—24no—
77D8×C107—24no—
78C2×C8×C107C2312yes—
79C22⋊C4×C107—21no—
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C23×C107—22no—
84C22×D107C2×D411yescharacteristic
85Q8×C107—22no—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—42no—
90D4×C107—42no—
91C2×C4×C107—21no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—42no—
96C22⋊D107—44no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—22no—
100C2×Dic107—42no—
101C2×Dic107—44no—
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105C4×D107—22no—
106C4×D107—22no—
107C4×D107—41no—
108Dic214—42no—
109C8×C107#4114yescomplemented
110C107⋊C8—22no—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—41no—
124C4×C107C2×#4112yes—
125Dic107—21no—
126Dic107—21no—
127Dic107—42no—
128Dic107—41no—
129Dic107—41no—
130Dic107AGL(1,ℤ/8)12yes—
131C2×C107—82no—
132C2×C107—81no—
133C2×C107#53211yescharacteristic
134C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C107#57511yescharacteristic
136D107—41no—
137#3171—1071nomaximal, Sylow
138C107#317111yescharacteristic, complemented, Sylow
139#532—1071no—
140#529—1071no—
141#501—1071no—
142#496D10711yescharacteristic, complemented
143#490—1071no—
144#489—1071no—
145#469—1072no—
146#468—1072no—
147C4×SD16—1072no—
148C4×D8—1072no—
149C4×M16—1071no—
150C2×#41—2141no—
151C2×SD16C2×D10712yes—
152C2×D8C2×D10712yes—
153C2×M16—1072no—
154C4⋊Q8—1071no—
155#127C2×D10711yescharacteristic
156#126—2141no—
157#125—1071no—
158#124—2141no—
159#124C2×D10711yescharacteristic
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×C42—1071no—
172#107—1072no—
173#106—1072no—
174#103—1072no—
175#102—1072no—
176#102—1072no—
177#102—1072no—
178#97—1071no—
179#97C2×D10711yescharacteristic
180C4×C8—1072no—
181#41—4284no—
182C2×Q8—2141no—
183C2×Q8C22×D10711yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—2141no—
187C2×D4—2142no—
188C2×D4C22×D10711yescharacteristic
189C2×D4C22×D10712yes—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—2141no—
194C22×C4—2142no—
195SD16—24no—
196D8—24no—
197M16—2144no—
198C2×C8—1072no—
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—21no—
209C22⋊C4—21no—
210C22⋊C4—2141no—
211C22⋊C4—2141no—
212C22⋊C4—2142no—
213C22⋊C4—2141no—
214C22⋊C4—2142no—
215C42—1071no—
216C42—1071no—
217C42—1071no—
218C42—2141no—
219C42C22×D10711yescharacteristic
220C23—21no—
221C23—22no—
222C23—1071no—
223Q8—22no—
224Q8—4282no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—42no—
230D4—4284no—
231D4—4282no—
232C2×C4—21no—
233C2×C4—1071no—
234C2×C4—1071no—
235C2×C4—1071no—
236C2×C4—4281no—
237C2×C4—4284no—
238C2×C4—4282no—
239C2×C4—2142no—
240C2×C4—2141no—
241C2×C4—2141no—
242C2×C4—2142no—
243C2×C4—2142no—
244C2×C4D4×D10711yescharacteristic
245C2×C4D4×D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—2142no—
248C8#1313381714yescomplemented
249C22—42no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—2141no—
256C22—2141no—
257C22—2141no—
258C22C2×D4×D10711yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—1072no—
263C4—4282no—
264C4—4281no—
265C4—4281no—
266C4—2141no—
267C4—2141no—
268C4C2×#1313381712yes—
269C2—82no—
270C2—81no—
271C2—4281no—
272C2#2029317511yescharacteristic
273C2AGL(1,ℤ/8)×D10711yescharacteristic
274C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3171
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20293175C2semidirect
2. #20293156C2semidirect
3. #20293036C2semidirect
4. #20292991C2semidirect
5. #20292986C2semidirect
6. #20292877C2semidirect
7. #20292870C2semidirect
8. SD16×Dic107C2semidirect
9. D8×Dic107C2semidirect
10. #20292579C2semidirect
11. #496×C107C2semidirect
109. C8×C107#41semidirect
138. C107#3171semidirect
142. #496D107semidirect
248. C8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202931752
#202931562
#202930362
#202929912
#202929862
#202928772
#202928702
SD16×Dic1072
D8×Dic1072
#202925792
#496×C1072
#3171107