Finite Group Databasebeta

← #28345982

Subgroups — #28345982

Order 13696.

490 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812140C211yesmaximal, characteristic, complemented
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293196C211yesmaximal, characteristic
4#20292849C214yesmaximal, complemented
5#20292827C214yesmaximal, complemented
6#20292654C211yesmaximal, characteristic, complemented
7#20292631C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#519×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812140—24no—
13C2×#13133808C2212yescomplemented
14C2×#13133805—24no—
15C2×#13133805C2214yes—
16C2×#13133805C2212yescomplemented
17C2×#13133802C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic
19#16812106C2212yes—
20#16812097C2212yes—
21#16812084—22no—
22#16812079—22no—
23C4⋊(C107⋊C8)C2214yescomplemented
24#16812032C2212yescomplemented
25#16812029C2214yes—
26#16812029C2212yescomplemented
27C22×Q8×C107C2211yescharacteristic
28C4×Q8×C107—22no—
29C4×Q8×C107C2214yescomplemented
30#117×C107—22no—
31#117×C107C2211yescharacteristic
32#117×C107C2211yescharacteristic
33C23×D107—21no—
34#13133818—22no—
35#13133818—22no—
36C2×Dic214C2311yescharacteristic
37C2×D428—22no—
38C2×D428—22no—
39C2×D428—21no—
40C2×D428—24no—
41C2×D428C2312yes—
42C2×D428C2311yescharacteristic
43C2×C22⋊D107—21no—
44C2×C4×D107—21no—
45#13133808—24no—
46#13133805—28no—
47#13133805—24no—
48#13133802—22no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52(C2×D107)⋊C4—24no—
53C2×Q8×C107—21no—
54C2×Q8×C107—22no—
55C2×Q8×C107D414yescomplemented
56C2×Q8×C107C2312yes—
57C22×C4×C107—21no—
58C22×C4×C107C2311yescharacteristic
59C4⋊C4×C107—24no—
60C4⋊C4×C107D412yes—
61C4⋊C4×C107C2312yes—
62C22⋊C4×C107—21no—
63C22⋊C4×C107D412yes—
64C42×C107—24no—
65C42×C107—21no—
66C42×C107C2312yes—
67C23×C107C2×D411yescharacteristic
68C22×D107—22no—
69C22×D107—21no—
70C22×D107—42no—
71C22×D107—82no—
72Q8×C107—24no—
73Q8×C107—22no—
74Q8×C107C2×D414yes—
75C2×C4×C107—24no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D414yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107#4112yes—
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—42no—
84C2×Dic107—21no—
85D428—22no—
86D428—21no—
87D428—22no—
88D428—21no—
89D428—42no—
90D428—84no—
91C4×D107—42no—
92Dic214—21no—
93Dic214—21no—
94C107⋊C8—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C2×D107—41no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—44no—
103C2×D107—84no—
104C4×C107—24no—
105C4×C107—24no—
106C4×C107—42no—
107C4×C107#12112yes—
108C4×C107C22×D411yescharacteristic
109C4×C107C2×#4111yescharacteristic
110Dic107—41no—
111C2×C107—22no—
112C2×C107C2×#12111yescharacteristic
113C2×C107#57712yes—
114D107—81no—
115D107—82no—
116#3043—1071nomaximal, Sylow
117C107#304311yescharacteristic, complemented, Sylow
118C2×#137—1071no—
119C22×SD16—1071no—
120#535—1071no—
121#519D10711yescharacteristic, complemented
122#464—1074no—
123#460—1074no—
124#424—1071no—
125#418—1071no—
126#416—1071no—
127C2×#41—1071no—
128C22×Q8C2×D10711yescharacteristic
129C22×D4—1071no—
130#137—2144no—
131C2×Q16—1072no—
132C2×SD16—1074no—
133C2×SD16—1072no—
134C2×SD16—2144no—
135C2×M16—1071no—
136C22×C8—1071no—
137#127—1072no—
138#124—1072no—
139#121—2142no—
140#120—2142no—
141C4×Q8—22no—
142C4×Q8C2×D10714yescomplemented
143#117—22no—
144#117C2×D10711yescharacteristic
145#117C2×D10711yescharacteristic
146C4⋊C8—1074no—
147#103—1072no—
148#102—1072no—
149#102—1074no—
150C24—2141no—
151#41—2142no—
152#41—2142no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—1071no—
156C2×Q8D42814yescomplemented
157C2×Q8C22×D10712yes—
158C2×D4—1072no—
159C2×D4—1071no—
160C2×D4—2141no—
161C2×D4—2144no—
162C2×D4—2141no—
163C2×D4—2142no—
164C2×D4—2142no—
165C22×C4—21no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168Q16—2144no—
169SD16—2148no—
170SD16—2144no—
171M16—2142no—
172C2×C8—1072no—
173C2×C8—1072no—
174C2×C8—2142no—
175C4⋊C4—24no—
176C4⋊C4C22⋊D10712yes—
177C4⋊C4C22×D10712yes—
178C22⋊C4—21no—
179C22⋊C4—2144no—
180C22⋊C4C22⋊D10712yes—
181C42—24no—
182C42—21no—
183C42C22×D10712yes—
184C23—8562no—
185C23—4282no—
186C23—2142no—
187C23—2141no—
188C23C2×C22⋊D10711yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—2141no—
192Q8—2141no—
193Q8C2×D42814yes—
194D4—8564no—
195D4—4282no—
196D4—4282no—
197D4—2142no—
198D4—2141no—
199D4—2142no—
200D4—2141no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—4282no—
205C2×C4—2141no—
206C2×C4C2×C22⋊D10714yes—
207C2×C4C2×C22⋊D10711yescharacteristic
208C2×C4C2×D42812yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—2142no—
212C8—2142no—
213C22—22no—
214C22—8564no—
215C22—4282no—
216C22—4284no—
217C22—4281no—
218C22—4282no—
219C22#1681208312yes—
220C22C22×C22⋊D10711yescharacteristic
221C4—24no—
222C4—24no—
223C4—42no—
224C4—4281no—
225C4#1681208312yes—
226C4C22×D42811yescharacteristic
227C4C2×#1313381811yescharacteristic
228C2—22no—
229C2—8562no—
230C2—8561no—
231C2C2×#1681208311yescharacteristic
232C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#519×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#519D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C22×#13133805C2semidirect
4. #20292849C2semidirect
5. #20292827C2semidirect
6. #20292654C2semidirect
7. #20292631C2semidirect
8. #20292621C2semidirect
9. #519×C107C2semidirect
13. C2×#13133808C22semidirect
16. C2×#13133805C22semidirect
23. C4⋊(C107⋊C8)C22semidirect
24. #16812032C22semidirect
26. #16812029C22semidirect
29. C4×Q8×C107C22semidirect
55. C2×Q8×C107D4semidirect
117. C107#3043semidirect
121. #519D107semidirect
142. C4×Q8C2×D107semidirect
156. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C22×#131338052
#202931962
#202928492
#202928272
#202926542
#202926312
#202926212
#519×C1072
#3043107