Finite Group Databasebeta

← #28345981

Subgroups — #28345981

Order 13696.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812142C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3#20293194C211yesmaximal, characteristic
4#20293104C211yesmaximal, characteristic, complemented
5#20292848C214yesmaximal, complemented
6#20292826C214yesmaximal, complemented
7#20292655C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812142—24no—
13C2×Dic428C2212yescomplemented
14C2×#13133807C2212yescomplemented
15C2×#13133806—24no—
16C2×#13133806C2214yes—
17C22×C107⋊C8C2211yescharacteristic
18#16812105C2212yes—
19#16812096C2212yes—
20#16812080—22no—
21#16812078—22no—
22C4×Dic214—22no—
23#16812074C2214yescomplemented
24#16812068C2211yescharacteristic
25#16812065—22no—
26Dic107⋊C8C2214yescomplemented
27#16812032C2212yescomplemented
28#16812029C2212yescomplemented
29C22×D4×C107C2211yescharacteristic
30C2×M16×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32#102×C107C2214yes—
33#13133818—22no—
34#13133818—22no—
35C2×Dic214—21no—
36C2×Dic214—22no—
37C2×Dic214D414yescomplemented
38C2×Dic214C2311yescharacteristic
39C2×Dic214C2312yes—
40C2×D428C2311yescharacteristic
41C2×C22⋊D107—21no—
42C2×C22⋊D107—24no—
43C22×Dic107—21no—
44C2×C4×D107—21no—
45Dic428—24no—
46#13133807—24no—
47#13133806—28no—
48C2×C107⋊C8—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107D412yes—
51Dic107⋊C4—24no—
52C22⋊Dic107—24no—
53C22⋊Dic107D412yes—
54C4×Dic107—24no—
55C4×Dic107C2312yes—
56C24×C107—21no—
57C2×D4×C107—22no—
58C2×D4×C107—21no—
59C2×D4×C107—22no—
60C2×D4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107—22no—
68C23×C107—42no—
69C23×C107—82no—
70C23×C107C2×D411yescharacteristic
71C22×D107—21no—
72D4×C107—22no—
73D4×C107—22no—
74D4×C107—42no—
75C2×C4×C107—22no—
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D412yes—
78C2×C4×C107#4112yes—
79C2×C4×C107C2411yescharacteristic
80C22⋊D107—42no—
81C22⋊D107—84no—
82C2×Dic107—24no—
83C2×Dic107—22no—
84C2×Dic107—21no—
85C2×Dic107C2×D414yes—
86D428—21no—
87D428—21no—
88C4×D107—42no—
89Dic214—24no—
90Dic214—22no—
91Dic214—21no—
92Dic214—21no—
93Dic214C2×D414yes—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—42no—
98C22×C107—44no—
99C22×C107—84no—
100C22×C107#12112yes—
101C22×C107C22×D411yescharacteristic
102C2×D107—41no—
103C2×D107—42no—
104C4×C107—42no—
105C4×C107#12112yes—
106C4×C107C22×D411yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—24no—
109Dic107—24no—
110Dic107—41no—
111C2×C107—22no—
112C2×C107—82no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—81no—
116#3043—1071nomaximal, Sylow
117C107#304311yescharacteristic, complemented, Sylow
118C2×#137—1071no—
119C22×SD16—1071no—
120#535—1071no—
121#519—1071no—
122#464—1074no—
123#460—1074no—
124#424—1071no—
125#418D10711yescharacteristic, complemented
126#416—1071no—
127C2×#41—1071no—
128C22×Q8—1071no—
129C22×D4C2×D10711yescharacteristic
130#137—2144no—
131C2×Q16—1072no—
132C2×SD16—1074no—
133C2×SD16—1072no—
134C2×SD16—2144no—
135C2×M16C2×D10711yescharacteristic
136C22×C8—1071no—
137#127—1072no—
138#124—1072no—
139#121—2142no—
140#120—2142no—
141C4×Q8—1074no—
142C4×Q8—2142no—
143#117—1071no—
144#117—2142no—
145#117C2×D10711yescharacteristic
146C4⋊C8—1074no—
147#103—1072no—
148#102—1072no—
149#102C2×D10714yes—
150C24—21no—
151#41—2142no—
152#41—2142no—
153C2×Q8—1074no—
154C2×Q8—1072no—
155C2×Q8—1071no—
156C2×Q8—2141no—
157C2×Q8—2142no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—1071no—
162C2×D4—2144no—
163C2×D4—2141no—
164C2×D4C22×D10712yes—
165C22×C4—2141no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168Q16—2144no—
169SD16—2144no—
170SD16—2148no—
171M16—22no—
172C2×C8—1072no—
173C2×C8—2142no—
174C2×C8C22×D10712yes—
175C4⋊C4—1072no—
176C4⋊C4—2144no—
177C4⋊C4C22×D10712yes—
178C22⋊C4—21no—
179C22⋊C4—1072no—
180C22⋊C4—2144no—
181C42—21no—
182C42—1072no—
183C42—2144no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—2141no—
188C23D4×D10711yescharacteristic
189Q8—1074no—
190Q8—2141no—
191Q8—2141no—
192Q8—2144no—
193Q8—2142no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—8564no—
198D4—4282no—
199D4—2141no—
200D4—2141no—
201C2×C4—22no—
202C2×C4—1074no—
203C2×C4—4282no—
204C2×C4—2141no—
205C2×C4—2144no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4D4×D10712yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—44no—
215C22—42no—
216C22—84no—
217C22—4282no—
218C22—4281no—
219C22#1681208212yes—
220C22C2×D4×D10711yescharacteristic
221C4—42no—
222C4—4281no—
223C4—2144no—
224C4—2144no—
225C4#1681208212yes—
226C4C2×D4×D10711yescharacteristic
227C4C2×#1313381811yescharacteristic
228C2—22no—
229C2—82no—
230C2—8561no—
231C2C2×#1681208211yescharacteristic
232C2#2029343712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radical#20293194C2
hypercenter#418D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C22×#13133806C2semidirect
4. #20293104C2semidirect
5. #20292848C2semidirect
6. #20292826C2semidirect
7. #20292655C2semidirect
8. #20292621C2semidirect
9. #418×C107C2semidirect
13. C2×Dic428C22semidirect
14. C2×#13133807C22semidirect
23. #16812074C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812032C22semidirect
28. #16812029C22semidirect
37. C2×Dic214D4semidirect
117. C107#3043semidirect
125. #418D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C22×#131338062
#202931942
#202931042
#202928482
#202928262
#202926552
#202926212
#418×C1072
#3043107