Finite Group Databasebeta

← #28346903

Subgroups — #28346903

Order 13696.

564 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#16812128C212yesmaximal, complemented
3#20293176C211yesmaximal, characteristic, complemented
4#20293175C211yesmaximal, characteristic, complemented
5#20293022C211yesmaximal, characteristic, complemented
6#20293021C211yesmaximal, characteristic, complemented
7#20293018C212yesmaximal, complemented
8#20293015C212yesmaximal, complemented
9#20293014C211yesmaximal, characteristic, complemented
10#97×D107C211yesmaximal, characteristic, complemented
11#496×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×#13133817C2212yes—
15#16812130—28no—
16#16812128—28no—
17C2×Dic428C2212yes—
18C2×#13133808C2212yes—
19C2×#13133807C2212yes—
20C2×#13133806C2212yes—
21C2×#13133806C2212yes—
22C2×#13133803C2212yes—
23C2×C8×D107C2212yes—
24#16812109C2211yescharacteristic
25#16812107C2211yescharacteristic
26#16812105C2211yescharacteristic
27#16812097C2211yescharacteristic
28#16812096C2212yes—
29#16812082—22no—
30#16812080—22no—
31D4×Dic107—22no—
32(C22⋊D107)⋊C4—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic
35#16812019C2212yes—
36C2×SD16×C107C2212yes—
37C2×D8×C107C2212yes—
38#127×C107C2211yescharacteristic
39#124×C107C2211yescharacteristic
40#97×C107C2211yescharacteristic
41#13133819—24no—
42#13133817—24no—
43#13133817—24no—
44#13133817—24no—
45C2×Dic214C2312yes—
46C2×Dic214C2311yescharacteristic
47C2×D428—21no—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—22no—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—22no—
52C22×Dic107—21no—
53C22×Dic107—22no—
54C2×C4×D107—21no—
55C2×C4×D107D412yes—
56C2×C4×D107C2311yescharacteristic
57Dic428—24no—
58#13133808—24no—
59#13133807—24no—
60#13133806—24no—
61#13133806—24no—
62#13133803—24no—
63C2×C107⋊C8C2312yes—
64C8×D107D418yescomplemented
65C4⋊Dic107—22no—
66Dic107⋊C4—22no—
67C22⋊Dic107—24no—
68(C2×D107)⋊C4—22no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107C2311yescharacteristic
72C2×D4×C107—22no—
73C2×D4×C107C2312yes—
74C2×D4×C107C2311yescharacteristic
75SD16×C107—24no—
76D8×C107—24no—
77C2×C8×C107C2312yes—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—21no—
81C23×C107—22no—
82C22×D107—21no—
83C22×D107C2×D411yescharacteristic
84Q8×C107—22no—
85D4×C107—22no—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—44no—
89C2×C4×C107—21no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—44no—
93C22⋊D107—44no—
94C22⋊D107—42no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—42no—
98C2×Dic107—44no—
99C2×Dic107C2×D412yes—
100C2×Dic107C2×D411yescharacteristic
101D428—22no—
102D428—42no—
103C4×D107—42no—
104C4×D107—42no—
105C4×D107C2×D414yes—
106Dic214—22no—
107Dic214—22no—
108Dic214—22no—
109C8×C107C2×D414yes—
110C107⋊C8C2×D414yes—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107—44no—
115C22×C107C22×D411yescharacteristic
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—42no—
119C2×D107#12712yes—
120C4×C107—22no—
121C4×C107—41no—
122C4×C107C22×D412yes—
123Dic107—22no—
124Dic107—41no—
125Dic107—42no—
126Dic107#12712yes—
127C2×C107—81no—
128C2×C107—82no—
129C2×C107C2×#12711yescharacteristic
130C2×C107#57512yes—
131D107—22no—
132D107—81no—
133#3150—1071nomaximal, Sylow
134C107#315011yescharacteristic, complemented, Sylow
135C2×#135—1072no—
136C2×#135—1072no—
137#532—1071no—
138#532—1071no—
139#497—1072no—
140#497—1071no—
141#497—1071no—
142#496—1072no—
143#496—1071no—
144#496D10711yescharacteristic, complemented
145C2×#97—1071no—
146C2×#41—1071no—
147C2×#41—1072no—
148C2×#41—1071no—
149#135—2148no—
150#135—2148no—
151C2×Q16—1072no—
152C2×Q16—1072no—
153C2×SD16—1072no—
154C2×SD16—1072no—
155C2×SD16—1072no—
156C2×SD16C2×D10712yes—
157C2×D8—1072no—
158C2×D8C2×D10712yes—
159C22×C8—1072no—
160C4⋊Q8—1071no—
161C4⋊Q8—1071no—
162#127—1071no—
163#127C2×D10711yescharacteristic
164#124—1072no—
165#124—1071no—
166#124C2×D10711yescharacteristic
167#121—2142no—
168#121—2142no—
169C4×D4—2142no—
170C4×D4—2142no—
171C2×C42—1071no—
172#97—1072no—
173#97—1071no—
174#97C2×D10711yescharacteristic
175#41—2144no—
176#41—2144no—
177#41—2144no—
178#41—2144no—
179C2×Q8—1072no—
180C2×Q8—1071no—
181C2×Q8C22×D10711yescharacteristic
182C2×D4—22no—
183C2×D4—1071no—
184C2×D4—2141no—
185C2×D4—2142no—
186C2×D4—2141no—
187C2×D4—2142no—
188C2×D4C22×D10712yes—
189C2×D4C22×D10711yescharacteristic
190C22×C4—1071no—
191C22×C4—1072no—
192C22×C4—2141no—
193C22×C4—2141no—
194C22×C4—2142no—
195Q16—2144no—
196Q16—2144no—
197SD16—24no—
198SD16—2144no—
199SD16—2144no—
200SD16—2144no—
201D8—24no—
202D8—2144no—
203C2×C8—1072no—
204C2×C8—1078no—
205C2×C8C22×D10712yes—
206C4⋊C4—2142no—
207C4⋊C4—2142no—
208C22⋊C4—22no—
209C22⋊C4—2144no—
210C22⋊C4—2142no—
211C42—1072no—
212C42—1071no—
213C42C22×D10711yescharacteristic
214C23—21no—
215C23—22no—
216C23—1071no—
217C23—2141no—
218Q8—22no—
219Q8—2142no—
220Q8—2142no—
221Q8—2142no—
222D4—22no—
223D4—22no—
224D4—22no—
225D4—44no—
226D4—4282no—
227D4—4282no—
228D4—4284no—
229D4—4284no—
230D4—2142no—
231C2×C4—21no—
232C2×C4—1071no—
233C2×C4—1074no—
234C2×C4—1072no—
235C2×C4—4282no—
236C2×C4—4282no—
237C2×C4—4282no—
238C2×C4—4284no—
239C2×C4—2142no—
240C2×C4—2141no—
241C2×C4D4×D10712yes—
242C2×C4C23×D10711yescharacteristic
243C8—1074no—
244C8D4×D10714yes—
245C22—41no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—1072no—
250C22—4281no—
251C22—4282no—
252C22—2142no—
253C22C2×D4×D10711yescharacteristic
254C4—22no—
255C4—41no—
256C4—1072no—
257C4—4282no—
258C4—4281no—
259C4—2142no—
260C4C2×D4×D10712yes—
261C2—82no—
262C2—81no—
263C2—8561no—
264C2—2142no—
265C2#127×D10711yescharacteristic
266C2#2029342312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3150
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. C2×#16812130C2semidirect
2. C2×#16812128C2semidirect
3. #20293176C2semidirect
4. #20293175C2semidirect
5. #20293022C2semidirect
6. #20293021C2semidirect
7. #20293018C2semidirect
8. #20293015C2semidirect
9. #20293014C2semidirect
10. #97×D107C2semidirect
11. #496×C107C2semidirect
64. C8×D107D4semidirect
134. C107#3150semidirect
144. #496D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
C2×#168121282
#202931762
#202931752
#202930222
#202930212
#202930182
#202930152
#202930142
#97×D1072
#496×C1072
#3150107