Finite Group Databasebeta

← #28348109

Subgroups — #28348109

Order 13696.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812142C212yesmaximal, complemented
2D4×Dic214C211yesmaximal, characteristic, complemented
3#20293282C211yesmaximal, characteristic
4#20293007C211yesmaximal, characteristic, complemented
5#20292885C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292833C211yesmaximal, characteristic, complemented
8#20292791C212yesmaximal, complemented
9#20292785C212yesmaximal, complemented
10C4×#13133807C211yesmaximal, characteristic, complemented
11#436×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C22×Dic214C2212yes—
14#16812142—28no—
15C2×Dic428C2212yes—
16C2×Dic428C2212yes—
17C2×Dic428C2212yes—
18C2×#13133807C2212yes—
19C2×#13133807C2211yescharacteristic
20#16812109C2211yescharacteristic
21#16812109C2211yescharacteristic
22#16812108—21no—
23#16812091—22no—
24#16812088C2212yes—
25#16812088C2212yes—
26#16812086—22no—
27#16812080—21no—
28C4×Dic214C2211yescharacteristic
29C4×D428C2211yescharacteristic
30D4×Dic107—21no—
31C4⋊C4×D107—21no—
32#16812041C2211yescharacteristic
33#16812034C2212yes—
34#16812034C2212yes—
35#16812034C2211yescharacteristic
36#16812031C2211yescharacteristic
37C2×M16×C107C2212yes—
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40C22⋊C8×C107C2212yes—
41C4×C8×C107C2211yescharacteristic
42#13133818—24no—
43#13133818—42no—
44C2×Dic214—24no—
45C2×Dic214—22no—
46C2×Dic214—42no—
47C2×Dic214C2312yes—
48C2×Dic214C2312yes—
49C2×Dic214C2311yescharacteristic
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—22no—
52C22×Dic107—22no—
53C2×C4×D107—21no—
54C2×C4×D107—22no—
55Dic428—24no—
56Dic428—24no—
57Dic428—42no—
58#13133807—24no—
59#13133807D414yescomplemented
60C4⋊Dic107—21no—
61C4⋊Dic107—21no—
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107C2311yescharacteristic
65C4⋊Dic107C2311yescharacteristic
66Dic107⋊C4—22no—
67Dic107⋊C4—21no—
68Dic107⋊C4—22no—
69C22⋊Dic107—22no—
70C22⋊Dic107—22no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73C2×D4×C107D411yescharacteristic
74C22×C4×C107C2312yes—
75M16×C107—24no—
76C2×C8×C107C2312yes—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C4⋊C4×C107D411yescharacteristic
80C22⋊C4×C107D412yes—
81C42×C107C2311yescharacteristic
82C23×C107C2×D412yes—
83C22×D107—21no—
84D4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—44no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—44no—
98D428—21no—
99D428C2×D412yes—
100C4×D107—44no—
101C4×D107—42no—
102Dic214—21no—
103Dic214—22no—
104Dic214—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—44no—
108Dic214C2×D412yes—
109C8×C107—22no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C22×C107—22no—
113C22×C107—22no—
114C22×C107—22no—
115C22×C107C22×D411yescharacteristic
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—41no—
119C4×C107—22no—
120C4×C107—21no—
121C4×C107#13712yescomplemented
122C4×C107C22×D411yescharacteristic
123C4×C107H(5,𝔽2)11yescharacteristic
124Dic107—22no—
125Dic107—42no—
126Dic107—42no—
127Dic107—41no—
128Dic107—41no—
129C2×C107—42no—
130C2×C107D4211yescharacteristic
131C2×C107C2×#13711yescharacteristic
132C2×C107#57811yescharacteristic
133D107—42no—
134#3276—1071nomaximal, Sylow
135C107#327611yescharacteristic, complemented, Sylow
136C2×#137—1072no—
137D4×Q8—1071no—
138#547—1071no—
139#494—1071no—
140#470—1072no—
141#462—1071no—
142#461—1071no—
143#452—1072no—
144#451—1072no—
145C4×SD16—1071no—
146#436D10711yescharacteristic, complemented
147C2×#41—1072no—
148C22×Q8—1072no—
149#137—2148no—
150C2×Q16—1072no—
151C2×Q16—1072no—
152C2×Q16—1072no—
153C2×SD16—1071no—
154C2×SD16—1072no—
155C2×M16C2×D10712yes—
156C4⋊Q8—1071no—
157C4⋊Q8—1071no—
158C4⋊Q8—2141no—
159#123—2142no—
160C22⋊Q8—1072no—
161C22⋊Q8—1072no—
162C22⋊Q8—2142no—
163#121—2141no—
164C4×Q8—1071no—
165C4×D4—1071no—
166C4×D4—2141no—
167C4×D4C2×D10711yescharacteristic
168C2×C4⋊C4—2141no—
169#106—1071no—
170C4⋊C8C2×D10711yescharacteristic
171#103—1071no—
172#103—1072no—
173#103—1072no—
174#102—1071no—
175C22⋊C8C2×D10712yes—
176C4×C8C2×D10711yescharacteristic
177#41—4282no—
178#41—2144no—
179C2×Q8—1071no—
180C2×Q8—1072no—
181C2×Q8—1072no—
182C2×Q8—4282no—
183C2×Q8—2144no—
184C2×Q8—2142no—
185C2×D4—1071no—
186C2×D4—2142no—
187C2×D4D42811yescharacteristic
188C22×C4—2142no—
189C22×C4—2141no—
190C22×C4—2142no—
191C22×C4C22×D10712yes—
192Q16—4282no—
193Q16—2144no—
194Q16—2144no—
195SD16—1074no—
196SD16—2144no—
197M16—24no—
198C2×C8C22×D10711yescharacteristic
199C2×C8C22×D10712yes—
200C2×C8C22×D10711yescharacteristic
201C4⋊C4—1071no—
202C4⋊C4—1071no—
203C4⋊C4—2141no—
204C4⋊C4—2142no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2142no—
208C4⋊C4—2141no—
209C4⋊C4—2141no—
210C4⋊C4D42811yescharacteristic
211C22⋊C4—2142no—
212C22⋊C4—2142no—
213C22⋊C4—2141no—
214C22⋊C4D42812yes—
215C42—2141no—
216C42C22×D10711yescharacteristic
217C23—2141no—
218C23C2×D42812yes—
219Q8—1072no—
220Q8—4284no—
221Q8—2142no—
222Q8—2142no—
223Q8—2142no—
224Q8—2142no—
225Q8—2141no—
226D4—22no—
227D4—1072no—
228D4—4284no—
229D4—2141no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—4284no—
233C2×C4—4284no—
234C2×C4—4282no—
235C2×C4—2141no—
236C2×C4—2141no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2141no—
240C2×C4C2×D42811yescharacteristic
241C2×C4C2×D42812yes—
242C2×C4C2×D42811yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—22no—
245C8—21no—
246C8D4×D10712yes—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—4281no—
251C22—4281no—
252C22—2142no—
253C22C22×D42811yescharacteristic
254C4—21no—
255C4—22no—
256C4—4281no—
257C4—4281no—
258C4—4282no—
259C4—4282no—
260C4—2142no—
261C4#1681214212yescomplemented
262C4C2×D4×D10711yescharacteristic
263C4#1681215511yescharacteristic
264C2—42no—
265C2—4282no—
266C2D4×D42811yescharacteristic
267C2C2×#1681214211yescharacteristic
268C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#436×C107C2
socleC22×C107C22×D4
prime radical#20293282C2
hypercenter#436D107
hypocenterC107#3276
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. D4×Dic214C2semidirect
4. #20293007C2semidirect
5. #20292885C2semidirect
6. #20292838C2semidirect
7. #20292833C2semidirect
8. #20292791C2semidirect
9. #20292785C2semidirect
10. C4×#13133807C2semidirect
11. #436×C107C2semidirect
59. #13133807D4semidirect
121. C4×C107#137semidirect
135. C107#3276semidirect
146. #436D107semidirect
261. C4#16812142semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
D4×Dic2142
#202932822
#202930072
#202928852
#202928382
#202928332
#202927912
#202927852
C4×#131338072
#436×C1072
#3276107