Finite Group Databasebeta

← #28348178

Subgroups — #28348178

Order 13696.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812130C212yesmaximal, complemented
2#20293281C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic, complemented
4#20293006C211yesmaximal, characteristic, complemented
5#20292867C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292792C212yesmaximal, complemented
8#20292786C212yesmaximal, complemented
9C4×#13133803C211yesmaximal, characteristic, complemented
10#20292692C211yesmaximal, characteristic, complemented
11#461×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133817C2212yes—
14#16812130—28no—
15C2×#13133808C2212yes—
16C2×#13133808C2212yes—
17C2×#13133807C2212yes—
18C2×#13133803C2211yescharacteristic
19C2×C8×D107C2212yes—
20#16812109C2211yescharacteristic
21#16812094—22no—
22#16812093—22no—
23#16812089C2212yes—
24#16812085C2212yes—
25#16812084—21no—
26#16812083—21no—
27C4×D428C2211yescharacteristic
28C4×D428C2211yescharacteristic
29C2×C4⋊Dic107—21no—
30C4⋊C4×D107—21no—
31#16812042C2211yescharacteristic
32#16812034C2212yes—
33#16812030C2211yescharacteristic
34#16812029C2211yescharacteristic
35(C2×D107)⋊C8C2212yes—
36C4×C107⋊C8C2211yescharacteristic
37C2×SD16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic
41#103×C107C2212yes—
42#13133819—24no—
43#13133819—42no—
44#13133817—24no—
45#13133817—42no—
46C2×Dic214C2312yes—
47C2×D428—22no—
48C2×D428D411yescharacteristic
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—22no—
51C22×Dic107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54C2×C4×D107C2312yes—
55#13133808—24no—
56#13133808—42no—
57#13133807—24no—
58#13133803D414yescomplemented
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—24no—
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107—22no—
65C4⋊Dic107D411yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4D412yes—
72C2×Q8×C107C2312yes—
73C2×D4×C107C2311yescharacteristic
74C22×C4×C107—21no—
75SD16×C107—24no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C22×D107—21no—
84C22×D107C2×D412yes—
85Q8×C107—22no—
86Q8×C107—22no—
87D4×C107—21no—
88D4×C107C2×D412yes—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—42no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—44no—
96C2×Dic107—21no—
97C2×Dic107—22no—
98C2×Dic107—44no—
99C2×Dic107C2×D412yes—
100D428—22no—
101D428—21no—
102D428—44no—
103D428C2×D412yes—
104C4×D107—22no—
105C4×D107—22no—
106C4×D107—44no—
107C4×D107—42no—
108Dic214—22no—
109Dic214—22no—
110C8×C107—22no—
111C107⋊C8—21no—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107C22×D411yescharacteristic
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—22no—
121C2×D107—41no—
122C2×D107—41no—
123C4×C107—21no—
124C4×C107—42no—
125C4×C107—41no—
126C4×C107#13512yescomplemented
127C4×C107C22×D411yescharacteristic
128C4×C107H(5,𝔽2)11yescharacteristic
129Dic107—22no—
130Dic107—42no—
131Dic107—41no—
132C2×C107—42no—
133C2×C107D4211yescharacteristic
134C2×C107C2×#13511yescharacteristic
135C2×C107#57811yescharacteristic
136D107—42no—
137D107—42no—
138#3283—1071nomaximal, Sylow
139C107#328311yescharacteristic, complemented, Sylow
140C2×#135—1072no—
141#547—1071no—
142#547—1071no—
143#494—1071no—
144#467—1072no—
145#461—1071no—
146#461D10711yescharacteristic, complemented
147#452—1072no—
148#452—1072no—
149C4×D8—1071no—
150C8×D4—1071no—
151C2×#41—1072no—
152C2×#41—1072no—
153#135—2148no—
154C2×Q16—1072no—
155C2×Q16—1072no—
156C2×SD16—1072no—
157C2×SD16C2×D10712yes—
158C2×D8—1071no—
159C22×C8—1072no—
160C4⋊Q8—1071no—
161C4⋊Q8C2×D10711yescharacteristic
162#123—2142no—
163#123—2142no—
164C22⋊Q8—1072no—
165C22⋊Q8—1072no—
166#121—2141no—
167#121—2141no—
168C4×D4—1071no—
169C4×D4—1071no—
170C4×D4C2×D10711yescharacteristic
171C2×C4⋊C4—2141no—
172C2×C4⋊C4—2141no—
173#107—1071no—
174C4⋊C8C2×D10711yescharacteristic
175#103—1072no—
176#103C2×D10712yes—
177#102—1071no—
178#102—1071no—
179C22⋊C8—1072no—
180C4×C8—1071no—
181#41—4282no—
182#41—4282no—
183#41—2144no—
184#41—2144no—
185C2×Q8—1072no—
186C2×Q8C22×D10712yes—
187C2×D4—1071no—
188C2×D4—1071no—
189C2×D4—2142no—
190C2×D4—2142no—
191C2×D4C22×D10711yescharacteristic
192C22×C4—21no—
193C22×C4—1072no—
194C22×C4—2142no—
195C22×C4—2142no—
196C22×C4—2141no—
197Q16—4282no—
198Q16—2144no—
199SD16—24no—
200SD16—2144no—
201D8—1074no—
202C2×C8—1071no—
203C2×C8—1071no—
204C2×C8—2144no—
205C2×C8C22×D10712yes—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—1071no—
209C4⋊C4—1071no—
210C4⋊C4—2142no—
211C4⋊C4—2142no—
212C4⋊C4—2141no—
213C4⋊C4—2141no—
214C4⋊C4C22×D10711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1072no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4—2141no—
220C42C22×D10711yescharacteristic
221C23—21no—
222C23—1072no—
223C23—2141no—
224Q8—22no—
225Q8—22no—
226Q8—2142no—
227Q8—2142no—
228D4—21no—
229D4—1072no—
230D4—4284no—
231D4—4284no—
232D4—2142no—
233D4—2141no—
234D4D4×D10712yes—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—1072no—
239C2×C4—4284no—
240C2×C4—4284no—
241C2×C4—4282no—
242C2×C4—2141no—
243C2×C4—2142no—
244C2×C4—2142no—
245C2×C4—2142no—
246C2×C4D4×D10711yescharacteristic
247C2×C4D4×D10711yescharacteristic
248C2×C4C23×D10711yescharacteristic
249C8—22no—
250C8—1072no—
251C8—2141no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—4281no—
256C22—4281no—
257C22—2142no—
258C22—2142no—
259C22—2142no—
260C22—2142no—
261C22C2×D4×D10711yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—4281no—
266C4—4282no—
267C4—2142no—
268C4#1681213012yescomplemented
269C4C2×D4×D10711yescharacteristic
270C4#1681215511yescharacteristic
271C2—42no—
272C2—4282no—
273C2—4282no—
274C2#2029326111yescharacteristic
275C2C2×#1681213011yescharacteristic
276C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
hypercenter#461D107
hypocenterC107#3283
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16812130C2semidirect
2. #20293281C2semidirect
3. #20293280C2semidirect
4. #20293006C2semidirect
5. #20292867C2semidirect
6. #20292833C2semidirect
7. #20292792C2semidirect
8. #20292786C2semidirect
9. C4×#13133803C2semidirect
10. #20292692C2semidirect
11. #461×C107C2semidirect
58. #13133803D4semidirect
126. C4×C107#135semidirect
139. C107#3283semidirect
146. #461D107semidirect
268. C4#16812130semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202932812
#202932802
#202930062
#202928672
#202928332
#202927922
#202927862
C4×#131338032
#202926922
#461×C1072
#3283107