Finite Group Databasebeta

← #28345973

Subgroups — #28345973

Order 13696.

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

normal subgroups only

This group has 494 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×#16812135C211yesmaximal, characteristic
2C22×#13133806C211yesmaximal, characteristic, complemented
3#20293203C211yesmaximal, characteristic, complemented
4#20292842C214yesmaximal, complemented
5#20292832C214yesmaximal, complemented
6#20292654C211yesmaximal, characteristic, complemented
7#20292635C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#518×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812135—24no—
13C2×#13133806—24no—
14C2×#13133806C2214yes—
15C2×#13133806C2212yes—
16C2×#13133803C2212yes—
17C2×#13133802C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic
19#16812109C2212yescomplemented
20#16812097C2212yescomplemented
21#16812093—22no—
22#16812089—22no—
23C4⋊(C107⋊C8)C2214yescomplemented
24#16812032C2212yescomplemented
25#16812032C2214yescomplemented
26#16812029C2212yescomplemented
27C22×D4×C107C2211yescharacteristic
28C4×D4×C107—22no—
29C4×D4×C107C2214yes—
30#117×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32C2×C22⋊C4×C107—22no—
33#13133818—22no—
34#13133818—22no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—21no—
38C2×Dic214C2312yes—
39C2×Dic214C2311yescharacteristic
40C2×D428C2311yescharacteristic
41C2×C22⋊D107—21no—
42C22×Dic107—21no—
43C2×C4×D107—21no—
44#13133806—28no—
45#13133806—24no—
46#13133803—24no—
47#13133802—22no—
48C2×C107⋊C8—22no—
49C2×C107⋊C8C2312yes—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107—24no—
52(C2×D107)⋊C4—24no—
53C24×C107—21no—
54C2×D4×C107—21no—
55C2×D4×C107—22no—
56C2×D4×C107D414yes—
57C2×D4×C107C2312yes—
58C22×C4×C107—24no—
59C22×C4×C107C2311yescharacteristic
60C4⋊C4×C107D412yes—
61C4⋊C4×C107C2312yes—
62C22⋊C4×C107—21no—
63C22⋊C4×C107—24no—
64C22⋊C4×C107D412yes—
65C42×C107—21no—
66C42×C107C2312yes—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—82no—
70C23×C107C2×D411yescharacteristic
71C22×D107—21no—
72D4×C107—24no—
73D4×C107—22no—
74D4×C107C2×D414yes—
75C2×C4×C107—22no—
76C2×C4×C107—84no—
77C2×C4×C107C2×D414yes—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D412yes—
80C2×C4×C107#4112yes—
81C2×C4×C107C2411yescharacteristic
82C22⋊D107—42no—
83C2×Dic107—22no—
84C2×Dic107—21no—
85C2×Dic107—42no—
86D428—21no—
87D428—21no—
88C4×D107—42no—
89Dic214—22no—
90Dic214—21no—
91Dic214—22no—
92Dic214—21no—
93Dic214—42no—
94C107⋊C8—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—24no—
98C22×C107—44no—
99C22×C107—84no—
100C22×C107#12112yes—
101C22×C107C22×D411yescharacteristic
102C2×D107—41no—
103C2×D107—42no—
104C4×C107—24no—
105C4×C107—42no—
106C4×C107#12112yes—
107C4×C107C22×D411yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—42no—
110Dic107—41no—
111C2×C107—22no—
112C2×C107—44no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—81no—
116#3042—1071nomaximal, Sylow
117C107#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1071no—
119C22×SD16—1071no—
120#536—1071no—
121#518D10711yescharacteristic, complemented
122#463—1074no—
123#461—1074no—
124#424—1071no—
125#419—1071no—
126#416—1071no—
127C2×#41—1071no—
128C22×Q8—1071no—
129C22×D4C2×D10711yescharacteristic
130AGL(1,ℤ/8)—2144no—
131C2×SD16—1074no—
132C2×SD16—1072no—
133C2×SD16—2144no—
134C2×D8—1072no—
135C2×M16—1071no—
136C22×C8—1071no—
137C4⋊Q8—1072no—
138#124—1072no—
139#123—2142no—
140C22⋊Q8—2142no—
141C4×D4—22no—
142C4×D4C2×D10714yes—
143#117C2×D10711yescharacteristic
144#117C2×D10711yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—1074no—
147#103—1072no—
148#103—1074no—
149#102—1072no—
150C24—21no—
151#41—2142no—
152#41—2142no—
153C2×Q8—1072no—
154C2×Q8—1071no—
155C2×Q8—2142no—
156C2×Q8—2142no—
157C2×Q8—2141no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1071no—
161C2×D4—2141no—
162C2×D4D42814yes—
163C2×D4C22×D10712yes—
164C22×C4—24no—
165C22×C4—2141no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168SD16—2148no—
169SD16—2144no—
170D8—2144no—
171M16—2142no—
172C2×C8—1072no—
173C2×C8—1072no—
174C2×C8—2142no—
175C4⋊C4—2144no—
176C4⋊C4C22⋊D10712yes—
177C4⋊C4C22×D10712yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—2144no—
181C22⋊C4C22⋊D10712yes—
182C42—21no—
183C42C22×D10712yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2141no—
188C23C2×C22⋊D10711yescharacteristic
189Q8—4282no—
190Q8—2142no—
191Q8—2141no—
192Q8—2142no—
193Q8—2141no—
194D4—24no—
195D4—22no—
196D4—4282no—
197D4—2141no—
198D4—2141no—
199D4C2×D42814yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—4282no—
203C2×C4—4282no—
204C2×C4—2142no—
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—24no—
215C22—44no—
216C22—84no—
217C22—4282no—
218C22—4281no—
219C22#1681208312yes—
220C22C22×C22⋊D10711yescharacteristic
221C4—24no—
222C4—42no—
223C4—4281no—
224C4—4282no—
225C4#1681208312yes—
226C4C22×D42811yescharacteristic
227C4C2×#1313381811yescharacteristic
228C2—22no—
229C2—44no—
230C2—8561no—
231C2C2×#1681208311yescharacteristic
232C2#2029343812yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#518×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812135C2
hypercenter#518D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#13133806C2semidirect
3. #20293203C2semidirect
4. #20292842C2semidirect
5. #20292832C2semidirect
6. #20292654C2semidirect
7. #20292635C2semidirect
8. #20292621C2semidirect
9. #518×C107C2semidirect
19. #16812109C22semidirect
20. #16812097C22semidirect
23. C4⋊(C107⋊C8)C22semidirect
24. #16812032C22semidirect
25. #16812032C22semidirect
26. #16812029C22semidirect
117. C107#3042semidirect
121. #518D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C22×#131338062
#202932032
#202928422
#202928322
#202926542
#202926352
#202926212
#518×C1072
#3042107