Finite Group Databasebeta

← #28346714

Subgroups — #28346714

Order 13696.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#20293239C211yesmaximal, characteristic, complemented
2#20293101C211yesmaximal, characteristic
3#20292969C212yesmaximal, complemented
4#20292928C212yesmaximal, complemented
5#20292833C212yesmaximal, complemented
6#20292780C212yesmaximal
7#20292729C212yesmaximal, complemented
8C4×#13133807C212yesmaximal, complemented
9#433×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133807C2212yes—
12C2×#13133807C2212yes—
13#16812109C2211yescharacteristic, complemented
14#16812102—22no—
15#16812101C2211yescharacteristic, complemented
16#16812099—22no—
17#16812088C2212yes—
18C4×Dic214C2212yescomplemented
19C4×D428C2212yes—
20C4×D428C2212yes—
21#16812071—22no—
22#16812068C2211yescharacteristic
23C22⋊C4×D107—22no—
24#16812043C2212yescomplemented
25#16812041C2212yescomplemented
26#16812034C2212yescomplemented
27#16812034C2212yescomplemented
28#16812031C2212yes—
29#16812031C2212yes—
30#117×C107C2211yescharacteristic
31C4⋊C8×C107C2212yescomplemented
32C22⋊C8×C107C2212yes—
33#97×C107C2211yescharacteristic, complemented
34C4×C8×C107C2211yescharacteristic, complemented
35C23×D107—21no—
36C2×Dic214C2312yes—
37C2×D428—21no—
38C2×D428—24no—
39C2×D428C2312yes—
40C2×C4×D107—22no—
41C2×C4×D107—22no—
42#13133807—24no—
43#13133807—24no—
44C4⋊Dic107—22no—
45C4⋊Dic107C2312yes—
46C4⋊Dic107C2312yes—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49Dic107⋊C4—22no—
50C22⋊Dic107—21no—
51C22⋊Dic107—22no—
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—21no—
55C4×Dic107—22no—
56C22×C4×C107C2311yescharacteristic
57C2×C8×C107C2312yes—
58C2×C8×C107C2312yes—
59C4⋊C4×C107D412yes—
60C22⋊C4×C107D412yes—
61C42×C107C2311yescharacteristic
62C42×C107C2311yescharacteristic
63C23×C107C2×D411yescharacteristic
64C22×D107—22no—
65C22×D107—24no—
66C22×D107—82no—
67C2×C4×C107—22no—
68C2×C4×C107C2×D412yes—
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2411yescharacteristic
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75D428—24no—
76D428—22no—
77D428#4114yes—
78C4×D107—82no—
79C4×D107—82no—
80Dic214—22no—
81Dic214—22no—
82C8×C107—22no—
83C8×C107—22no—
84C22×C107—21no—
85C22×C107—22no—
86C22×C107C22×D411yescharacteristic
87C2×D107—24no—
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—84no—
91C4×C107—21no—
92C4×C107—22no—
93C4×C107—22no—
94C4×C107C2×#4112yes—
95Dic107—42no—
96Dic107—42no—
97Dic107—42no—
98C2×C107—41no—
99C2×C107#52511yescharacteristic
100C2×C107#57712yes—
101D107—44no—
102#3123—1071nomaximal, Sylow
103C107#312311yescharacteristic, complemented, Sylow
104#541—1071no—
105#518—1071no—
106#485—1072no—
107#478—1072no—
108#461—1072no—
109#450—1072no—
110#440—1072no—
111C4×SD16—1072no—
112#433D10711yescharacteristic, complemented
113C22×D4—1071no—
114C2×SD16—1072no—
115C2×SD16—1072no—
116C4⋊Q8—1071no—
117#126—2142no—
118#125—1071no—
119#125—2142no—
120C22⋊Q8—1072no—
121C4×Q8—1072no—
122C4×D4—1072no—
123C4×D4—1072no—
124C4×D4—2142no—
125#117—1071no—
126#117C2×D10711yescharacteristic
127C2×C22⋊C4—2142no—
128#107—1072no—
129#106—1072no—
130C4⋊C8C2×D10712yescomplemented
131#103—1072no—
132#103—1072no—
133#102—1072no—
134#102—1072no—
135C22⋊C8C2×D10712yes—
136#97C2×D10711yescharacteristic, complemented
137C4×C8C2×D10711yescharacteristic, complemented
138C24—2141no—
139C2×Q8—1072no—
140C2×D4—1072no—
141C2×D4—2144no—
142C2×D4—2141no—
143C22×C4—2142no—
144C22×C4—2142no—
145C22×C4C22×D10711yescharacteristic
146SD16—2144no—
147SD16—2144no—
148C2×C8C22×D10712yes—
149C2×C8C22×D10712yes—
150C4⋊C4—1072no—
151C4⋊C4—1072no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4—2142no—
155C4⋊C4—2142no—
156C4⋊C4D42812yes—
157C22⋊C4—2141no—
158C22⋊C4—2142no—
159C22⋊C4—2142no—
160C22⋊C4—2142no—
161C22⋊C4D42812yes—
162C42—2141no—
163C42—2142no—
164C42C22×D10711yescharacteristic
165C42C22×D10711yescharacteristic
166C23—8562no—
167C23—2144no—
168C23—2142no—
169C23C2×D42811yescharacteristic
170Q8—2142no—
171Q8—2142no—
172D4—1074no—
173D4—2144no—
174D4—2142no—
175C2×C4—22no—
176C2×C4—8562no—
177C2×C4—8562no—
178C2×C4—2142no—
179C2×C4—2142no—
180C2×C4—2142no—
181C2×C4C2×D42812yes—
182C2×C4C2×D42811yescharacteristic
183C2×C4C2×D42812yes—
184C2×C4C23×D10711yescharacteristic
185C8—22no—
186C8—22no—
187C22—21no—
188C22—22no—
189C22—8564no—
190C22—4282no—
191C22—4282no—
192C22—2144no—
193C22C22×D42811yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—4282no—
198C4—4282no—
199C4—4282no—
200C4#41×D10712yes—
201C2—41no—
202C2—4284no—
203C2#2029313911yescharacteristic
204C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#433×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#433D107
hypocenterC107#3123
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #20293239C2semidirect
3. #20292969C2semidirect
4. #20292928C2semidirect
5. #20292833C2semidirect
7. #20292729C2semidirect
8. C4×#13133807C2semidirect
9. #433×C107C2semidirect
13. #16812109C22semidirect
15. #16812101C22semidirect
18. C4×Dic214C22semidirect
24. #16812043C22semidirect
25. #16812041C22semidirect
26. #16812034C22semidirect
27. #16812034C22semidirect
31. C4⋊C8×C107C22semidirect
33. #97×C107C22semidirect
34. C4×C8×C107C22semidirect
103. C107#3123semidirect
112. #433D107semidirect
130. C4⋊C8C2×D107semidirect
136. #97C2×D107semidirect
137. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932392
#202931012
#202929692
#202929282
#202928332
#202927802
#202927292
C4×#131338072
#433×C1072
#3123107