Finite Group Databasebeta

← #28345934

Subgroups — #28345934

Order 13696.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16812141C212yesmaximal, complemented
2#20293178C211yesmaximal, characteristic
3C4×Q8×D107C211yesmaximal, characteristic, complemented
4#20292852C212yesmaximal, complemented
5#20292851C212yesmaximal, complemented
6#20292834C211yesmaximal, characteristic, complemented
7#20292828C211yesmaximal, characteristic, complemented
8#20292827C211yesmaximal, characteristic, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292627C212yesmaximal, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×Q8×D107C2211yescharacteristic
14#16812141—28no—
15C2×#13133808C2212yes—
16C2×#13133807C2212yes—
17C2×#13133805C2212yes—
18C2×#13133801C2212yes—
19#16812106C2211yescharacteristic
20#16812098C2212yes—
21#16812084—21no—
22#16812084—21no—
23Q8×Dic107C2211yescharacteristic
24C4×Dic214—21no—
25C4×Dic214C2211yescharacteristic
26#16812074—21no—
27#16812074C2211yescharacteristic
28#16812071—21no—
29#16812071—21no—
30C4⋊C4×D107—21no—
31C4⋊C4×D107C2211yescharacteristic
32C42×D107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35Dic107⋊C8C2212yes—
36#16812033C2212yes—
37#16812031C2212yes—
38#16812029C2212yes—
39C2×Q16×C107C2212yes—
40C4⋊Q8×C107C2211yescharacteristic
41C4×Q8×C107C2211yescharacteristic
42C4⋊C8×C107C2211yescharacteristic
43#103×C107C2212yes—
44#13133819—24no—
45#13133819—24no—
46C2×Dic214—21no—
47C2×Dic214C2311yescharacteristic
48Q8×D107—22no—
49Q8×D107D414yescomplemented
50C2×D428—22no—
51C2×D428—21no—
52C2×D428—21no—
53C2×D428C2312yes—
54C2×C4×D107—21no—
55C2×C4×D107—21no—
56C2×C4×D107—22no—
57C2×C4×D107D411yescharacteristic
58C2×C4×D107D411yescharacteristic
59C2×C4×D107C2311yescharacteristic
60#13133808—24no—
61#13133807—24no—
62#13133805—24no—
63#13133801—24no—
64C2×C107⋊C8C2312yes—
65C4⋊Dic107—21no—
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—21no—
68Dic107⋊C4—21no—
69Dic107⋊C4—21no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C4×Dic107—21no—
73C4×Dic107—21no—
74C4×Dic107—21no—
75C4×Dic107D411yescharacteristic
76C4×Dic107D411yescharacteristic
77C4×Dic107C2311yescharacteristic
78C2×Q8×C107C2312yes—
79C2×Q8×C107C2311yescharacteristic
80Q16×C107—24no—
81C2×C8×C107C2312yes—
82C4⋊C4×C107—21no—
83C4⋊C4×C107—21no—
84C4⋊C4×C107—21no—
85C4⋊C4×C107C2311yescharacteristic
86C42×C107—21no—
87C42×C107C2311yescharacteristic
88C22×D107—22no—
89C22×D107C2×D411yescharacteristic
90Q8×C107—21no—
91Q8×C107—22no—
92Q8×C107—22no—
93Q8×C107C2×D412yes—
94C2×C4×C107—21no—
95C2×C4×C107—21no—
96C2×C4×C107—22no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105D428—22no—
106D428—22no—
107D428—42no—
108D428—44no—
109D428—42no—
110C4×D107—24no—
111C4×D107—22no—
112C4×D107—41no—
113C4×D107—42no—
114C4×D107—44no—
115C4×D107C2×D412yes—
116C4×D107#4112yes—
117Dic214—21no—
118Dic214—24no—
119Dic214C2×D412yes—
120C8×C107—22no—
121C107⋊C8—22no—
122C22×C107C22×D411yescharacteristic
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C2×D107#12112yes—
129C4×C107—22no—
130C4×C107—21no—
131C4×C107—42no—
132C4×C107—41no—
133C4×C107#13712yescomplemented
134C4×C107C22×D411yescharacteristic
135C4×C107C2×#4111yescharacteristic
136Dic107—22no—
137Dic107—21no—
138Dic107—21no—
139Dic107—41no—
140Dic107#12112yes—
141C2×C107C2×#12111yescharacteristic
142C2×C107C2×#13711yescharacteristic
143C2×C107#57511yescharacteristic
144D107—22no—
145D107—82no—
146#3037—1071nomaximal, Sylow
147C107#303711yescharacteristic, complemented, Sylow
148C2×#137—1072no—
149#532—1071no—
150C2×C4×Q8—1071no—
151#464—1072no—
152#464—1072no—
153#462—1071no—
154#462D10711yescharacteristic, complemented
155#460—1071no—
156#460—1071no—
157#423—1071no—
158#417—1072no—
159C2×#41—1072no—
160C22×Q8—1071no—
161#137—2148no—
162C2×Q16—1072no—
163C2×Q16C2×D10712yes—
164C2×SD16—1072no—
165C2×SD16—1072no—
166C2×M16—1072no—
167C4⋊Q8C2×D10711yescharacteristic
168#127—1071no—
169#124—1072no—
170#121—2141no—
171#121—2141no—
172C4×Q8—1071no—
173C4×Q8—1071no—
174C4×Q8—1071no—
175C4×Q8—2141no—
176C4×Q8—2141no—
177C4×Q8C2×D10711yescharacteristic
178C4×D4—2141no—
179C4×D4—2141no—
180C2×C4⋊C4—1071no—
181C2×C4⋊C4—2141no—
182C2×C42—1071no—
183C2×C42—2141no—
184C4⋊C8—1072no—
185C4⋊C8—1071no—
186C4⋊C8C2×D10711yescharacteristic
187#103—1072no—
188#103C2×D10712yes—
189#102—1072no—
190#102—1072no—
191#41—2144no—
192#41—2144no—
193C2×Q8—1074no—
194C2×Q8—1071no—
195C2×Q8—2141no—
196C2×Q8—2142no—
197C2×Q8C22×D10712yes—
198C2×Q8C22×D10711yescharacteristic
199C2×D4—1072no—
200C2×D4—2142no—
201C2×D4—2141no—
202C2×D4—2141no—
203C22×C4—1071no—
204C22×C4—1071no—
205C22×C4—1071no—
206C22×C4—2142no—
207C22×C4—2141no—
208C22×C4—2141no—
209Q16—24no—
210Q16—2144no—
211SD16—2144no—
212SD16—2144no—
213M16—2144no—
214C2×C8—1072no—
215C2×C8C22×D10712yes—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—1071no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4—2141no—
223C4⋊C4—2141no—
224C4⋊C4C22×D10711yescharacteristic
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C42—21no—
228C42—1071no—
229C42—1071no—
230C42—1071no—
231C42—2141no—
232C42—2141no—
233C42—2141no—
234C42C22×D10711yescharacteristic
235C23—1071no—
236C23—2142no—
237Q8—21no—
238Q8—22no—
239Q8—22no—
240Q8—1072no—
241Q8—2144no—
242Q8—2141no—
243Q8D4×D10712yes—
244D4—4282no—
245D4—4284no—
246D4—4282no—
247D4—2142no—
248D4—2142no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—1071no—
253C2×C4—1072no—
254C2×C4—1072no—
255C2×C4—1071no—
256C2×C4—1071no—
257C2×C4—4281no—
258C2×C4—4284no—
259C2×C4—4282no—
260C2×C4—2144no—
261C2×C4—2141no—
262C2×C4—2141no—
263C2×C4—2142no—
264C2×C4D4×D10711yescharacteristic
265C2×C4D4×D10711yescharacteristic
266C2×C4C23×D10711yescharacteristic
267C8—22no—
268C8—2142no—
269C22—1072no—
270C22—4282no—
271C22—4282no—
272C22—4282no—
273C22—2141no—
274C22—2141no—
275C22C2×D4×D10711yescharacteristic
276C4—21no—
277C4—22no—
278C4—41no—
279C4—42no—
280C4—1072no—
281C4—4281no—
282C4—2141no—
283C4—2141no—
284C4—2142no—
285C4#1681214112yescomplemented
286C4C2×D4×D10711yescharacteristic
287C4#41×D10711yescharacteristic
288C2—8562no—
289C2—2142no—
290C2#121×D10711yescharacteristic
291C2C2×#1681214111yescharacteristic
292C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radical#20293178C2
hypercenter#462D107
hypocenterC107#3037
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
3. C4×Q8×D107C2semidirect
4. #20292852C2semidirect
5. #20292851C2semidirect
6. #20292834C2semidirect
7. #20292828C2semidirect
8. #20292827C2semidirect
9. #20292649C2semidirect
10. #20292627C2semidirect
11. #462×C107C2semidirect
49. Q8×D107D4semidirect
133. C4×C107#137semidirect
147. C107#3037semidirect
154. #462D107semidirect
285. C4#16812141semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
#202931782
C4×Q8×D1072
#202928522
#202928512
#202928342
#202928282
#202928272
#202926492
#202926272
#462×C1072
#3037107