Finite Group Databasebeta

← #28345928

Subgroups — #28345928

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×#16812138C212yesmaximal, complemented
2#20293175C211yesmaximal, characteristic
3C4×Q8×D107C211yesmaximal, characteristic, complemented
4#20292848C212yesmaximal, complemented
5#20292847C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292837C211yesmaximal, characteristic, complemented
8#20292824C211yesmaximal, characteristic, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292625C212yesmaximal, complemented
11#460×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×Q8×D107C2211yescharacteristic
14#16812138—28no—
15C2×Dic428C2212yes—
16C2×#13133808C2212yes—
17C2×#13133806C2212yes—
18C2×#13133801C2212yes—
19#16812109C2211yescharacteristic
20#16812096C2212yes—
21#16812080—21no—
22#16812080—21no—
23Q8×Dic107C2211yescharacteristic
24C4×Dic214—21no—
25C4×Dic214C2211yescharacteristic
26#16812074—21no—
27#16812074C2211yescharacteristic
28D4×Dic107—21no—
29D4×Dic107—21no—
30C4⋊C4×D107—21no—
31C4⋊C4×D107C2211yescharacteristic
32C42×D107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35Dic107⋊C8C2212yes—
36#16812034C2212yes—
37#16812032C2212yes—
38#16812030C2212yes—
39C2×SD16×C107C2212yes—
40#127×C107C2211yescharacteristic
41C4×Q8×C107C2211yescharacteristic
42C4⋊C8×C107C2211yescharacteristic
43#102×C107C2212yes—
44#13133817—24no—
45#13133817—24no—
46C2×Dic214—21no—
47C2×Dic214C2312yes—
48C2×Dic214C2311yescharacteristic
49Q8×D107—22no—
50Q8×D107D414yescomplemented
51C2×C22⋊D107—22no—
52C22×Dic107—22no—
53C2×C4×D107—21no—
54C2×C4×D107—21no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58Dic428—24no—
59#13133808—24no—
60#13133806—24no—
61#13133801—24no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107—21no—
64C4⋊Dic107—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—21no—
68Dic107⋊C4—21no—
69Dic107⋊C4—21no—
70C22⋊Dic107—22no—
71C22⋊Dic107—22no—
72C4×Dic107—21no—
73C4×Dic107—21no—
74C4×Dic107—21no—
75C4×Dic107D411yescharacteristic
76C4×Dic107D411yescharacteristic
77C4×Dic107C2311yescharacteristic
78C2×Q8×C107C2311yescharacteristic
79C2×D4×C107—21no—
80C2×D4×C107—21no—
81C2×D4×C107C2312yes—
82SD16×C107—24no—
83C2×C8×C107C2312yes—
84C4⋊C4×C107—21no—
85C4⋊C4×C107C2311yescharacteristic
86C42×C107—21no—
87C42×C107C2311yescharacteristic
88C23×C107—22no—
89C22×D107C2×D411yescharacteristic
90Q8×C107—21no—
91Q8×C107C2×D412yes—
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—42no—
95D4×C107—42no—
96C2×C4×C107—21no—
97C2×C4×C107—21no—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—44no—
102C2×Dic107—21no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—44no—
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108C2×Dic107C2×D411yescharacteristic
109C4×D107—24no—
110C4×D107—22no—
111C4×D107—41no—
112C4×D107—42no—
113C4×D107C2×D412yes—
114C4×D107#4112yes—
115Dic214—21no—
116Dic214—24no—
117Dic214—22no—
118Dic214—22no—
119Dic214C2×D412yes—
120C8×C107—22no—
121C107⋊C8—22no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107C22×D411yescharacteristic
126C2×D107—21no—
127C2×D107—21no—
128C2×D107#12112yes—
129C4×C107—22no—
130C4×C107—21no—
131C4×C107—41no—
132C4×C107#13712yescomplemented
133C4×C107C22×D411yescharacteristic
134C4×C107C2×#4111yescharacteristic
135Dic107—22no—
136Dic107—21no—
137Dic107—21no—
138Dic107—42no—
139Dic107—41no—
140Dic107#12112yes—
141C2×C107—82no—
142C2×C107C2×#12111yescharacteristic
143C2×C107C2×#13711yescharacteristic
144C2×C107#57511yescharacteristic
145D107—22no—
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#462—1071no—
155#460—1071no—
156#460D10711yescharacteristic, complemented
157#423—1071no—
158#417—1072no—
159C2×#41—1072no—
160C22×Q8—1071no—
161#137—2148no—
162C2×Q16—1072no—
163C2×Q16—1072no—
164C2×SD16—1072no—
165C2×SD16C2×D10712yes—
166C2×M16—1072no—
167C4⋊Q8—1071no—
168#127C2×D10711yescharacteristic
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#103—1072no—
189#102—1072no—
190#102C2×D10712yes—
191#41—2144no—
192#41—2144no—
193C2×Q8—1074no—
194C2×Q8—1071no—
195C2×Q8—1072no—
196C2×Q8—2141no—
197C2×Q8—2142no—
198C2×Q8C22×D10711yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2142no—
202C2×D4C22×D10712yes—
203C22×C4—1071no—
204C22×C4—1071no—
205C22×C4—1071no—
206C22×C4—2141no—
207C22×C4—2141no—
208C22×C4—2142no—
209Q16—2144no—
210Q16—2144no—
211SD16—24no—
212SD16—2144no—
213M16—2144no—
214C2×C8—1072no—
215C2×C8C22×D10712yes—
216C4⋊C4—21no—
217C4⋊C4—1071no—
218C4⋊C4—2141no—
219C4⋊C4—2141no—
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—22no—
236C23—1071no—
237Q8—21no—
238Q8—1072no—
239Q8—2144no—
240Q8—2142no—
241Q8—2142no—
242Q8—2141no—
243Q8D4×D10712yes—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—42no—
248D4—4284no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—1071no—
252C2×C4—1072no—
253C2×C4—1072no—
254C2×C4—1071no—
255C2×C4—1071no—
256C2×C4—4281no—
257C2×C4—4282no—
258C2×C4—4284no—
259C2×C4—2144no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4—2141no—
263C2×C4—2142no—
264C2×C4D4×D10711yescharacteristic
265C2×C4D4×D10711yescharacteristic
266C2×C4C23×D10711yescharacteristic
267C8—22no—
268C8—2142no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—1072no—
273C22—2141no—
274C22—2141no—
275C22C2×D4×D10711yescharacteristic
276C4—21no—
277C4—22no—
278C4—41no—
279C4—1072no—
280C4—4281no—
281C4—4282no—
282C4—2141no—
283C4—2141no—
284C4—2142no—
285C4#1681213812yescomplemented
286C4C2×D4×D10711yescharacteristic
287C4#41×D10711yescharacteristic
288C2—82no—
289C2—2142no—
290C2#121×D10711yescharacteristic
291C2C2×#1681213811yescharacteristic
292C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293175C2
hypercenter#460D107
hypocenterC107#3037
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
3. C4×Q8×D107C2semidirect
4. #20292848C2semidirect
5. #20292847C2semidirect
6. #20292838C2semidirect
7. #20292837C2semidirect
8. #20292824C2semidirect
9. #20292649C2semidirect
10. #20292625C2semidirect
11. #460×C107C2semidirect
50. Q8×D107D4semidirect
132. C4×C107#137semidirect
147. C107#3037semidirect
156. #460D107semidirect
285. C4#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
#202931752
C4×Q8×D1072
#202928482
#202928472
#202928382
#202928372
#202928242
#202926492
#202926252
#460×C1072
#3037107