Finite Group Databasebeta

← #28346259

Subgroups — #28346259

Order 13696.

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

normal subgroups only

This group has 440 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
1#20293182C211yesmaximal, characteristic, complemented
2C4×D4×D107C211yesmaximal, characteristic
3#20292926C212yesmaximal, complemented
4#20292925C212yesmaximal, complemented
5#20292914C211yesmaximal, characteristic, complemented
6#20292908C211yesmaximal, characteristic, complemented
7#20292907C211yesmaximal, characteristic, complemented
8#20292666C212yesmaximal, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292630C212yesmaximal
11#476×C107C211yesmaximal, characteristic, complemented
12C2×D4×D107C2211yescharacteristic
13C22×C4×D107—21no—
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic, complemented
16#16812100C2212yescomplemented
17#16812089—21no—
18#16812085—21no—
19Q8×Dic107—21no—
20#16812074—21no—
21C4×D428C2211yescharacteristic
22D4×Dic107C2211yescharacteristic
23#16812071C2211yescharacteristic
24C4×C22⋊D107—21no—
25(C22⋊D107)⋊C4—21no—
26#16812067C2212yes—
27C4⋊C4×D107C2211yescharacteristic
28C22⋊C4×D107—21no—
29C42×D107C2211yescharacteristic
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812040C2212yescomplemented
33C4⋊(C107⋊C8)C2211yescharacteristic, complemented
34Dic107⋊C8C2212yescomplemented
35#16812031C2212yes—
36#16812030C2212yes—
37#16812029C2212yes—
38C4⋊Q8×C107C2211yescharacteristic, complemented
39C4×D4×C107C2211yescharacteristic
40#106×C107C2212yescomplemented
41C4⋊C8×C107C2211yescharacteristic, complemented
42#102×C107C2212yes—
43C23×D107—21no—
44C2×Dic214—21no—
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47D4×D107—22no—
48D4×D107Q814yescomplemented
49C22×Dic107—21no—
50C2×C4×D107—21no—
51C2×C4×D107—21no—
52C2×C4×D107—21no—
53C2×C4×D107—44no—
54C2×C4×D107D411yescharacteristic
55C2×C4×D107D411yescharacteristic
56C2×C4×D107C2311yescharacteristic
57#13133801—24no—
58C2×C107⋊C8C2312yes—
59C4⋊Dic107—21no—
60C4⋊Dic107—22no—
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2312yes—
63Dic107⋊C4—21no—
64Dic107⋊C4—22no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69C4×Dic107—22no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C2×Q8×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C22×D107—21no—
84C22×D107—24no—
85C22×D107—42no—
86C22×D107—42no—
87C22×D107C2×D411yescharacteristic
88Q8×C107—42no—
89D4×C107—21no—
90D4×C107C2×Q812yes—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—42no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—24no—
98C2×Dic107—22no—
99C2×Dic107—21no—
100C2×Dic107—42no—
101C2×Dic107C2×D411yescharacteristic
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104D428—21no—
105D428C2×Q812yes—
106C4×D107—22no—
107C4×D107—41no—
108C4×D107—42no—
109C4×D107—42no—
110C4×D107—42no—
111C4×D107C2×Q812yes—
112C4×D107#4112yes—
113Dic214—42no—
114C8×C107—22no—
115C107⋊C8—22no—
116C22×C107—22no—
117C22×C107—41no—
118C22×C107—41no—
119C22×C107C22×D411yescharacteristic
120C2×D107—21no—
121C2×D107—21no—
122C2×D107—22no—
123C2×D107—48no—
124C2×D107—41no—
125C2×D107—41no—
126C2×D107C22⋊Q812yes—
127C4×C107—21no—
128C4×C107—41no—
129C4×C107—42no—
130C4×C107AGL(1,ℤ/8)12yes—
131C4×C107C22×Q811yescharacteristic
132C4×C107C2×#4111yescharacteristic
133Dic107—21no—
134Dic107—21no—
135Dic107—42no—
136Dic107—41no—
137Dic107C22⋊Q812yes—
138C2×C107—42no—
139C2×C107C2×C22⋊Q811yescharacteristic
140C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C107#57511yescharacteristic
142D107—22no—
143D107—42no—
144#3077—1071nomaximal, Sylow
145C107#307711yescharacteristic, complemented, Sylow
146#533—1071no—
147C2×C4×D4—1071no—
148#478—1072no—
149#478—1072no—
150#476—1071no—
151#476D10711yescharacteristic, complemented
152#474—1071no—
153#474—1071no—
154#428—1072no—
155#423—1071no—
156#418—1072no—
157C22×D4—1071no—
158C23×C4—2141no—
159C2×M16—1072no—
160C4⋊Q8—1071no—
161C4⋊Q8C2×D10711yescharacteristic, complemented
162#125—1072no—
163C22⋊Q8—2141no—
164C22⋊Q8—2141no—
165C4×Q8—2141no—
166C4×Q8—2141no—
167C4×D4—1071no—
168C4×D4—1071no—
169C4×D4—1071no—
170C4×D4—2141no—
171C4×D4—2141no—
172C4×D4C2×D10711yescharacteristic
173#117—1072no—
174C2×C4⋊C4—1071no—
175C2×C22⋊C4—2141no—
176C2×C42—1071no—
177#107—1072no—
178#107—1072no—
179#106—1072no—
180#106C2×D10712yescomplemented
181C4⋊C8—1072no—
182C4⋊C8—1071no—
183C4⋊C8C2×D10711yescharacteristic, complemented
184#102—1072no—
185#102—1072no—
186#102—1072no—
187#102C2×D10712yes—
188C24—2141no—
189C2×Q8—21no—
190C2×Q8—2141no—
191C2×D4—1074no—
192C2×D4—1071no—
193C2×D4—2142no—
194C2×D4—2141no—
195C2×D4C22×D10711yescharacteristic
196C22×C4—21no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—4284no—
201C22×C4—2141no—
202C22×C4—2141no—
203C22×C4—2141no—
204C22×C4—2141no—
205M16—2144no—
206C2×C8—1072no—
207C2×C8C22×D10712yes—
208C4⋊C4—21no—
209C4⋊C4—1072no—
210C4⋊C4—1071no—
211C4⋊C4—2141no—
212C4⋊C4—2142no—
213C4⋊C4—2142no—
214C4⋊C4—2141no—
215C4⋊C4C22×D10712yes—
216C4⋊C4C22×D10711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—2141no—
219C22⋊C4—2142no—
220C22⋊C4—2141no—
221C22⋊C4—2141no—
222C42—1071no—
223C42—1071no—
224C42—1071no—
225C42—2142no—
226C42C22×D10711yescharacteristic
227C23—21no—
228C23—1071no—
229C23—4282no—
230C23—4282no—
231C23—2144no—
232C23—2141no—
233Q8—42no—
234Q8—4282no—
235D4—21no—
236D4—1072no—
237D4—2144no—
238D4—2141no—
239D4Q8×D10712yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—1071no—
244C2×C4—1072no—
245C2×C4—1072no—
246C2×C4—1071no—
247C2×C4—1071no—
248C2×C4—4281no—
249C2×C4—4282no—
250C2×C4—4282no—
251C2×C4—4282no—
252C2×C4—4282no—
253C2×C4—2142no—
254C2×C4—2141no—
255C2×C4—2142no—
256C2×C4D4×D10711yescharacteristic
257C2×C4D4×D10711yescharacteristic
258C2×C4C23×D10711yescharacteristic
259C8—22no—
260C8—2142no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—1072no—
265C22—4288no—
266C22—4281no—
267C22—4281no—
268C22—2142no—
269C22—2141no—
270C22—2141no—
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—1072no—
276C4—4281no—
277C4—4282no—
278C4—2141no—
279C4—2141no—
280C4#1681213612yes—
281C4C2×Q8×D10711yescharacteristic
282C4#41×D10711yescharacteristic
283C2—42no—
284C2—4282no—
285C2—2142no—
286C2C22⋊Q8×D10711yescharacteristic
287C2C2×#1681213611yescharacteristic
288C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#476×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#476D107
hypocenterC107#3077
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20292926C2semidirect
4. #20292925C2semidirect
5. #20292914C2semidirect
6. #20292908C2semidirect
7. #20292907C2semidirect
8. #20292666C2semidirect
9. #20292649C2semidirect
11. #476×C107C2semidirect
15. #16812109C22semidirect
16. #16812100C22semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. C4⋊(C107⋊C8)C22semidirect
34. Dic107⋊C8C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #106×C107C22semidirect
41. C4⋊C8×C107C22semidirect
48. D4×D107Q8semidirect
145. C107#3077semidirect
151. #476D107semidirect
161. C4⋊Q8C2×D107semidirect
180. #106C2×D107semidirect
183. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
C4×D4×D1072
#202929262
#202929252
#202929142
#202929082
#202929072
#202926662
#202926492
#202926302
#476×C1072
#3077107