Finite Group Databasebeta

← #28345945

Subgroups — #28345945

Order 13696.

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

normal subgroups only

This group has 480 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×#16812141C212yesmaximal
2C4⋊Q8×D107C211yesmaximal, characteristic, complemented
3C4×#13133819C211yesmaximal, characteristic
4#20292837C211yesmaximal, characteristic, complemented
5#20292835C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292831C212yesmaximal, complemented
8#20292829C211yesmaximal, characteristic, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292634C212yesmaximal, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×Q8×D107C2212yes—
14#16812141—28no—
15C2×#13133808C2212yes—
16C2×#13133807C2212yes—
17C2×#13133805C2212yes—
18C2×#13133801C2212yes—
19#16812109C2211yescharacteristic
20#16812108—21no—
21#16812108—21no—
22#16812107C2212yes—
23Q8×Dic107C2211yescharacteristic
24C4×D428—21no—
25C4×D428C2211yescharacteristic
26#16812071—21no—
27#16812071C2211yescharacteristic
28#16812067—21no—
29#16812067C2211yescharacteristic
30C4⋊C4×D107—21no—
31C4⋊C4×D107—21no—
32C42×D107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35Dic107⋊C8C2212yes—
36#16812034C2212yes—
37#16812033C2212yes—
38#16812032C2212yes—
39C2×Q16×C107C2212yes—
40C4⋊Q8×C107C2211yescharacteristic
41C4×Q8×C107C2211yescharacteristic
42C4⋊C8×C107C2211yescharacteristic
43#103×C107C2212yes—
44#13133819—22no—
45#13133819D414yes—
46C2×Dic214—22no—
47C2×Dic214C2312yes—
48Q8×D107—24no—
49Q8×D107—24no—
50C2×D428—21no—
51C2×D428C2311yescharacteristic
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54C2×C4×D107—21no—
55C2×C4×D107—21no—
56C2×C4×D107D411yescharacteristic
57C2×C4×D107D411yescharacteristic
58C2×C4×D107C2311yescharacteristic
59#13133808—24no—
60#13133807—24no—
61#13133805—24no—
62#13133801—24no—
63C2×C107⋊C8C2312yes—
64C4⋊Dic107—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107—21no—
67C4⋊Dic107C2311yescharacteristic
68Dic107⋊C4—22no—
69Dic107⋊C4—22no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—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—21no—
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—22no—
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105D428—24no—
106D428—21no—
107D428C2×D412yes—
108C4×D107—22no—
109C4×D107—24no—
110C4×D107—22no—
111C4×D107—44no—
112C4×D107—42no—
113C4×D107—42no—
114C4×D107C2×D412yes—
115C4×D107#4112yes—
116Dic214—22no—
117Dic214—22no—
118Dic214—44no—
119C8×C107—22no—
120C107⋊C8—22no—
121C22×C107C22×D411yescharacteristic
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107#12112yes—
128C4×C107—22no—
129C4×C107—21no—
130C4×C107—42no—
131C4×C107—41no—
132C4×C107#13712yescomplemented
133C4×C107C22×D411yescharacteristic
134C4×C107C2×#4111yescharacteristic
135Dic107—21no—
136Dic107—42no—
137Dic107—41no—
138Dic107#12112yes—
139Dic107#13712yescomplemented
140C2×C107C2×#12111yescharacteristic
141C2×C107C2×#13711yescharacteristic
142C2×C107C2×#13711yescharacteristic
143D107—22no—
144D107—42no—
145#3039—1071nomaximal, Sylow
146C107#303911yescharacteristic, complemented, Sylow
147C2×#137—1072no—
148C2×C4⋊Q8—1071no—
149C4×#41—1071no—
150#462—1072no—
151#462—1071no—
152#462D10711yescharacteristic, complemented
153#461—1071no—
154#461—1071no—
155#461—1072no—
156#423—1071no—
157#419—1072no—
158C2×#41—1071no—
159C22×Q8—1072no—
160#137—2148no—
161C2×Q16—1072no—
162C2×Q16C2×D10712yes—
163C2×SD16—1072no—
164C2×SD16—1072no—
165C2×M16—1072no—
166C4⋊Q8—1072no—
167C4⋊Q8—1071no—
168C4⋊Q8—2141no—
169C4⋊Q8—2141no—
170C4⋊Q8C2×D10711yescharacteristic
171C4×Q8—1071no—
172C4×Q8C2×D10711yescharacteristic
173C4×D4—1071no—
174C4×D4—1071no—
175C4×D4—2141no—
176C4×D4—2141no—
177#117—1071no—
178#117—2141no—
179C2×C4⋊C4—2141no—
180C2×C4⋊C4—2141no—
181C2×C42—1071no—
182C2×C42—2141no—
183C4⋊C8—1072no—
184C4⋊C8—1071no—
185C4⋊C8C2×D10711yescharacteristic
186#103—1072no—
187#103—1072no—
188#103—1072no—
189#103C2×D10712yes—
190#41—1074no—
191#41—2142no—
192C2×Q8—1072no—
193C2×Q8—2142no—
194C2×Q8—2144no—
195C2×Q8—2144no—
196C2×Q8C22×D10712yes—
197C2×Q8C22×D10711yescharacteristic
198C2×D4—1071no—
199C2×D4—2141no—
200C22×C4—1071no—
201C22×C4—1071no—
202C22×C4—1071no—
203C22×C4—2141no—
204C22×C4—2141no—
205C22×C4—2142no—
206C22×C4—2141no—
207Q16—24no—
208Q16—2144no—
209SD16—2144no—
210SD16—2144no—
211M16—2144no—
212C2×C8—1072no—
213C2×C8C22×D10712yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—1071no—
218C4⋊C4—2142no—
219C4⋊C4—2142no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4—2141no—
223C4⋊C4C22×D10711yescharacteristic
224C22⋊C4—2141no—
225C22⋊C4—2141no—
226C22⋊C4—2141no—
227C42—21no—
228C42—1071no—
229C42—1071no—
230C42—1071no—
231C42—2141no—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—1071no—
235C23—2141no—
236Q8—21no—
237Q8—22no—
238Q8—22no—
239Q8—4284no—
240Q8—2142no—
241Q8—2142no—
242Q8D4×D10712yes—
243D4—1072no—
244D4—2144no—
245D4—2141no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—1071no—
250C2×C4—1071no—
251C2×C4—1072no—
252C2×C4—1072no—
253C2×C4—1071no—
254C2×C4—4282no—
255C2×C4—4284no—
256C2×C4—4282no—
257C2×C4—2142no—
258C2×C4—2144no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4D4×D10711yescharacteristic
263C2×C4D4×D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—22no—
266C8—2142no—
267C22—1072no—
268C22—4281no—
269C22—4281no—
270C22—2142no—
271C22—2141no—
272C22—2141no—
273C22C2×D4×D10711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—42no—
278C4—1072no—
279C4—1072no—
280C4—4281no—
281C4—4282no—
282C4—2141no—
283C4#1681214112yescomplemented
284C4C2×D4×D10711yescharacteristic
285C4#41×D10711yescharacteristic
286C2—4282no—
287C2—2142no—
288C2#121×D10711yescharacteristic
289C2#137×D10711yescharacteristic
290C2C2×#1681214111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133819C22
hypercenter#462D107
hypocenterC107#3039
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
2. C4⋊Q8×D107C2semidirect
4. #20292837C2semidirect
5. #20292835C2semidirect
6. #20292833C2semidirect
7. #20292831C2semidirect
8. #20292829C2semidirect
9. #20292649C2semidirect
10. #20292634C2semidirect
11. #462×C107C2semidirect
132. C4×C107#137semidirect
139. Dic107#137semidirect
146. C107#3039semidirect
152. #462D107semidirect
283. C4#16812141semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C4⋊Q8×D1072
C4×#131338192
#202928372
#202928352
#202928332
#202928312
#202928292
#202926492
#202926342
#462×C1072
#3039107