Finite Group Databasebeta

← #28346859

Subgroups — #28346859

Order 13696.

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

normal subgroups only

This group has 392 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#20293173C211yesmaximal, characteristic
2C2×#16812100C211yesmaximal, characteristic, complemented
3#20292994C212yesmaximal, complemented
4#20292990C212yesmaximal, complemented
5#20292983C212yesmaximal, complemented
6#20292980C212yesmaximal, complemented
7#20292629C211yesmaximal, characteristic, complemented
8#20292628C211yesmaximal, characteristic, complemented
9#20292626C211yesmaximal, characteristic, complemented
10M16×Dic107C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14#16812107C2211yescharacteristic
15#16812105C2211yescharacteristic
16#16812100—21no—
17#16812100—21no—
18#16812100C2212yes—
19#16812100C2212yes—
20#16812098C2211yescharacteristic
21#16812096C2211yescharacteristic
22#16812081—21no—
23#16812081—21no—
24C4×C22⋊D107—21no—
25C4×C22⋊D107—21no—
26C2×C4⋊Dic107—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×Dic107⋊C4—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812034C2212yes—
31#16812033C2212yes—
32#16812032C2212yes—
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812019C2212yes—
37C8×Dic107C2212yes—
38C2×#41×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yes—
42#102×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45C2×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C2×C22⋊D107—21no—
49C2×C22⋊D107—21no—
50C22×Dic107—21no—
51C22×Dic107D411yescharacteristic
52C22×Dic107D411yescharacteristic
53C2×C4×D107—21no—
54#13133802—22no—
55C2×C107⋊C8C2312yes—
56C4⋊Dic107—21no—
57C4⋊Dic107—22no—
58C4⋊Dic107—22no—
59C4⋊Dic107C2312yes—
60Dic107⋊C4—22no—
61Dic107⋊C4—22no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64C22⋊Dic107—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68C4×Dic107D411yescharacteristic
69C4×Dic107D411yescharacteristic
70C4×Dic107C2311yescharacteristic
71C4×Dic107C2311yescharacteristic
72#41×C107—22no—
73#41×C107—22no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107—21no—
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78C22×C4×C107—21no—
79C22×C4×C107C2311yescharacteristic
80M16×C107—22no—
81C2×C8×C107C2312yes—
82C4⋊C4×C107—21no—
83C4⋊C4×C107C2312yes—
84C23×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—21no—
87Q8×C107—21no—
88Q8×C107—21no—
89D4×C107—21no—
90D4×C107—21no—
91D4×C107—42no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—42no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107#4112yes—
98C2×C4×C107#4112yes—
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—42no—
101C22⋊D107—42no—
102C22⋊D107—42no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—41no—
106C2×Dic107—41no—
107C2×Dic107—42no—
108C2×Dic107C2×D411yescharacteristic
109C2×Dic107C2×D411yescharacteristic
110C2×Dic107C2×D411yescharacteristic
111C2×Dic107C2×D411yescharacteristic
112D428—21no—
113D428—21no—
114C4×D107—42no—
115Dic214—21no—
116Dic214—21no—
117C8×C107—22no—
118C107⋊C8—22no—
119C22×C107—21no—
120C22×C107—21no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107#12412yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—41no—
129C4×C107—41no—
130C4×C107—42no—
131C4×C107#12412yes—
132C4×C107C2×#4111yescharacteristic
133C4×C107C2×#4111yescharacteristic
134Dic107—21no—
135Dic107—21no—
136Dic107—21no—
137Dic107—21no—
138Dic107—41no—
139Dic107—42no—
140C2×C107—22no—
141C2×C107—81no—
142C2×C107C2×#12411yescharacteristic
143C2×C107#57511yescharacteristic
144C2×C107#57511yescharacteristic
145D107—81no—
146#3142—1071nomaximal, Sylow
147C107#314211yescharacteristic, complemented, Sylow
148#532—1071no—
149C2×#125—1071no—
150#491—1072no—
151#490—1072no—
152#488—1072no—
153#488—1072no—
154#417—1071no—
155#417—1071no—
156#417—1071no—
157#417D10711yescharacteristic, complemented
158C4×M16—1071no—
159C2×#41—1071no—
160C2×#41C2×D10711yescharacteristic
161C2×M16—1071no—
162C2×M16C2×D10711yescharacteristic
163C4⋊Q8—1071no—
164#127—1071no—
165#125—1072no—
166#125—1072no—
167#125—2141no—
168#125—2141no—
169#124—1071no—
170#124—1071no—
171#121—2141no—
172#121—2141no—
173C4×D4—2141no—
174C4×D4—2141no—
175C2×C4⋊C4—1071no—
176C2×C4⋊C4—2141no—
177C2×C4⋊C4—2141no—
178C2×C4⋊C4C2×D10711yescharacteristic
179C2×C42—1071no—
180#103—1072no—
181#103—1072no—
182#103—1072no—
183#103C2×D10712yes—
184#102—1072no—
185#102—1072no—
186#102—1072no—
187#102C2×D10712yes—
188#97—1072no—
189C4×C8—1072no—
190#41—22no—
191#41—22no—
192#41—2142no—
193#41—2142no—
194C2×Q8—1071no—
195C2×Q8C22×D10711yescharacteristic
196C2×D4—21no—
197C2×D4—1071no—
198C2×D4—2141no—
199C2×D4—2141no—
200C2×D4—2141no—
201C2×D4C22×D10711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1071no—
205C22×C4—1071no—
206C22×C4—2141no—
207C22×C4—2141no—
208C22×C4C22×D10711yescharacteristic
209M16—22no—
210M16—2142no—
211C2×C8—1072no—
212C2×C8C22×D10712yes—
213C4⋊C4—21no—
214C4⋊C4—1072no—
215C4⋊C4—2142no—
216C4⋊C4—2142no—
217C4⋊C4—2142no—
218C4⋊C4—2142no—
219C4⋊C4—2141no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4C22×D10712yes—
223C22⋊C4—2141no—
224C22⋊C4—2141no—
225C22⋊C4—2141no—
226C22⋊C4—2141no—
227C42—1071no—
228C42—1071no—
229C42—1071no—
230C42—1071no—
231C23—21no—
232C23—2141no—
233C23D4×D10711yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—2141no—
237Q8—2141no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—4282no—
242D4—4282no—
243D4—4282no—
244D4—2141no—
245D4—2141no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—1071no—
251C2×C4—1071no—
252C2×C4—1071no—
253C2×C4—1071no—
254C2×C4—4282no—
255C2×C4—4282no—
256C2×C4—4281no—
257C2×C4—4281no—
258C2×C4—2141no—
259C2×C4—2142no—
260C2×C4D4×D10711yescharacteristic
261C2×C4#1313381712yes—
262C2×C4#1313381812yes—
263C2×C4C23×D10711yescharacteristic
264C8—22no—
265C8—2142no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—4281no—
272C22—4281no—
273C22—4281no—
274C22#1681209512yes—
275C22C2×D4×D10711yescharacteristic
276C4—41no—
277C4—42no—
278C4—4281no—
279C4—4282no—
280C4—2141no—
281C4—2141no—
282C4—2141no—
283C4—2141no—
284C4#1681209512yes—
285C4C2×#1313381711yescharacteristic
286C4C2×#1313381811yescharacteristic
287C2—22no—
288C2—81no—
289C2—8561no—
290C2C2×#1681209511yescharacteristic
291C2#2029342311yescharacteristic
292C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radical#20293173C2
hypercenter#417D107
hypocenterC107#3142
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16812100C2semidirect
3. #20292994C2semidirect
4. #20292990C2semidirect
5. #20292983C2semidirect
6. #20292980C2semidirect
7. #20292629C2semidirect
8. #20292628C2semidirect
9. #20292626C2semidirect
10. M16×Dic107C2semidirect
11. #417×C107C2semidirect
147. C107#3142semidirect
157. #417D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
C2×#168121002
#202929942
#202929902
#202929832
#202929802
#202926292
#202926282
#202926262
M16×Dic1072
#417×C1072
#3142107