Finite Group Databasebeta

← #28346634

Subgroups — #28346634

Order 13696.

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

normal subgroups only

This group has 386 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
1#20293161C211yesmaximal, characteristic
2C2×C4×Dic214C211yesmaximal, characteristic, complemented
3#20292963C211yesmaximal, characteristic, complemented
4#20292946C211yesmaximal, characteristic
5#20292935C212yesmaximal, complemented
6#20292850C212yesmaximal, complemented
7#20292785C211yesmaximal, characteristic, complemented
8#20292768C211yesmaximal, characteristic
9#20292735C212yesmaximal, complemented
10#20292729C212yesmaximal, complemented
11#432×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C2×Dic428C2212yes—
14C2×#13133807C2212yes—
15#16812103—21no—
16#16812101C2211yescharacteristic
17#16812097C2211yescharacteristic
18#16812092—21no—
19#16812088C2211yescharacteristic
20#16812083C2211yescharacteristic
21C4×Dic214—21no—
22C4×Dic214—21no—
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2212yes—
25C4×Dic214C2212yes—
26C4×D428C2211yescharacteristic
27C4×C22⋊D107—21no—
28#16812066—21no—
29C2×C4⋊Dic107C2211yescharacteristic
30C2×Dic107⋊C4—21no—
31C2×C4×Dic107—21no—
32#16812043C2212yes—
33#16812041C2212yes—
34#16812034C2212yes—
35#16812034C2212yes—
36#16812034C2212yes—
37#16812031C2212yes—
38C2×C42×C107C2211yescharacteristic
39C4⋊C8×C107C2212yes—
40C22⋊C8×C107C2211yescharacteristic
41C22⋊C8×C107C2211yescharacteristic
42#97×C107C2212yes—
43C2×Dic214—21no—
44C2×Dic214—24no—
45C2×Dic214C2311yescharacteristic
46C2×Dic214C2312yes—
47C2×D428C2311yescharacteristic
48C2×C22⋊D107—21no—
49C22×Dic107—21no—
50C22×Dic107—21no—
51C2×C4×D107—21no—
52Dic428—24no—
53#13133807—24no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64C22⋊Dic107—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69C4×Dic107—22no—
70C4×Dic107—22no—
71C4×Dic107—21no—
72C22×C4×C107D411yescharacteristic
73C22×C4×C107D411yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C2×C8×C107C2312yes—
77C42×C107D411yescharacteristic
78C42×C107D411yescharacteristic
79C42×C107C2311yescharacteristic
80C42×C107C2311yescharacteristic
81C23×C107C2×D411yescharacteristic
82C22×D107—21no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—41no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—42no—
93C2×Dic107—24no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—42no—
100D428—22no—
101C4×D107—42no—
102Dic214—24no—
103Dic214—22no—
104Dic214—22no—
105Dic214#4114yes—
106C8×C107—22no—
107C8×C107—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107#13712yes—
111C22×C107C22×D411yescharacteristic
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—21no—
116C4×C107—21no—
117C4×C107—21no—
118C4×C107—21no—
119C4×C107—21no—
120C4×C107C2×#4112yes—
121Dic107—24no—
122Dic107—41no—
123Dic107—41no—
124Dic107—41no—
125Dic107—42no—
126C2×C107—22no—
127C2×C107#52511yescharacteristic
128C2×C107C2×#13711yescharacteristic
129C2×C107#57511yescharacteristic
130D107—81no—
131#3117—1071nomaximal, Sylow
132C107#311711yescharacteristic, complemented, Sylow
133#529—1071no—
134C2×C4×Q8—1071no—
135#484—1071no—
136#481—1071no—
137#479—1072no—
138#464—1072no—
139#451—1071no—
140#448—1071no—
141#441—1072no—
142#440—1072no—
143#432D10711yescharacteristic, complemented
144C22×Q8—1071no—
145C2×Q16—1072no—
146C2×SD16—1072no—
147#126—2141no—
148#125—1071no—
149#124—1071no—
150#123—2141no—
151C22⋊Q8—1071no—
152#121—1071no—
153C4×Q8—1072no—
154C4×Q8—1072no—
155C4×Q8—1071no—
156C4×Q8—2141no—
157C4×Q8—2141no—
158C4×D4—1071no—
159C4×D4—2141no—
160#117—2141no—
161C2×C4⋊C4—1071no—
162C2×C4⋊C4—2141no—
163C2×C42—2141no—
164C2×C42C2×D10711yescharacteristic
165#107—1072no—
166#106—1072no—
167C4⋊C8C2×D10712yes—
168#103—1072no—
169#103—1072no—
170#103—1072no—
171#102—1072no—
172C22⋊C8C2×D10711yescharacteristic
173C22⋊C8C2×D10711yescharacteristic
174#97C2×D10712yes—
175C2×Q8—1072no—
176C2×Q8—1071no—
177C2×Q8—2144no—
178C2×Q8—2141no—
179C2×D4—1071no—
180C2×D4—2141no—
181C22×C4—2141no—
182C22×C4—2141no—
183C22×C4—2141no—
184C22×C4D42811yescharacteristic
185C22×C4D42811yescharacteristic
186C22×C4C22×D10711yescharacteristic
187Q16—2144no—
188SD16—2144no—
189C2×C8C22×D10712yes—
190C2×C8C22×D10712yes—
191C4⋊C4—1072no—
192C4⋊C4—1071no—
193C4⋊C4—1071no—
194C4⋊C4—2141no—
195C4⋊C4—2142no—
196C4⋊C4—2142no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4—2141no—
201C22⋊C4—2141no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C22⋊C4—2141no—
206C42—2142no—
207C42—2142no—
208C42—2141no—
209C42D42811yescharacteristic
210C42D42811yescharacteristic
211C42C22×D10711yescharacteristic
212C42C22×D10711yescharacteristic
213C23—2141no—
214C23C2×D42811yescharacteristic
215Q8—1074no—
216Q8—2144no—
217Q8—2142no—
218Q8—2142no—
219D4—4282no—
220D4—2142no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—4282no—
225C2×C4—4282no—
226C2×C4—2142no—
227C2×C4—2141no—
228C2×C4—2141no—
229C2×C4—2141no—
230C2×C4—2144no—
231C2×C4—2142no—
232C2×C4C2×D42811yescharacteristic
233C2×C4C2×D42811yescharacteristic
234C2×C4C2×D42811yescharacteristic
235C2×C4C2×D42811yescharacteristic
236C2×C4C2×D42811yescharacteristic
237C2×C4C23×D10711yescharacteristic
238C8—22no—
239C8—22no—
240C22—21no—
241C22—21no—
242C22—4281no—
243C22—4281no—
244C22—4281no—
245C22#1681214212yes—
246C22C22×D42811yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—4281no—
253C4—4281no—
254C4—4281no—
255C4—4282no—
256C4—2144no—
257C4#41×D10712yes—
258C2—22no—
259C2—8561no—
260C2#2029313911yescharacteristic
261C2C2×#1681214211yescharacteristic
262C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#432×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenter#432D107
hypocenterC107#3117
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic214C2semidirect
3. #20292963C2semidirect
5. #20292935C2semidirect
6. #20292850C2semidirect
7. #20292785C2semidirect
9. #20292735C2semidirect
10. #20292729C2semidirect
11. #432×C107C2semidirect
132. C107#3117semidirect
143. #432D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
C2×C4×Dic2142
#202929632
#202929462
#202929352
#202928502
#202927852
#202927682
#202927352
#202927292
#432×C1072
#3117107