Finite Group Databasebeta

← #28346383

Subgroups — #28346383

Order 13696.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293137C211yesmaximal, characteristic
2C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
3#20292963C211yesmaximal, characteristic, complemented
4#20292962C211yesmaximal, characteristic, complemented
5#20292959C212yesmaximal, complemented
6#20292943C211yesmaximal, characteristic, complemented
7#20292939C212yesmaximal, complemented
8#20292666C212yesmaximal, complemented
9#20292625C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#481×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2211yescharacteristic
13C22×C4×D107—21no—
14C22×C4×D107C2211yescharacteristic
15C2×#13133801C2212yes—
16#16812091—21no—
17#16812089C2211yescharacteristic
18#16812088C2211yescharacteristic
19#16812080C2211yescharacteristic
20#16812077—21no—
21D4×Dic107—21no—
22(C22⋊D107)⋊C4—21no—
23#16812067C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4—21no—
26C4⋊C4×D107—21no—
27C4⋊C4×D107—21no—
28C4⋊C4×D107C2212yes—
29C4⋊C4×D107C2212yes—
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812040C2212yescomplemented
33#16812034C2212yescomplemented
34#16812032C2212yescomplemented
35#16812030C2212yescomplemented
36C22⋊(C107⋊C8)C2211yescharacteristic
37(C2×D107)⋊C8C2212yes—
38#121×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2211yescharacteristic
40#106×C107C2212yescomplemented
41#102×C107C2212yescomplemented
42C22⋊C8×C107C2211yescharacteristic
43C23×D107D411yescharacteristic
44#13133817—24no—
45C2×Dic214C2311yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C22×Dic107—21no—
50C22×Dic107D411yescharacteristic
51C2×C4×D107—24no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—44no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58C2×C4×D107C2311yescharacteristic
59#13133801—24no—
60C2×C107⋊C8C2312yes—
61C4⋊Dic107—21no—
62C4⋊Dic107—21no—
63C4⋊Dic107C2312yes—
64C4⋊Dic107C2311yescharacteristic
65Dic107⋊C4—21no—
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68C22⋊Dic107—21no—
69C22⋊Dic107—21no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73C2×D4×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77C2×C8×C107C2312yes—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2312yes—
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C23×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107C2×D411yescharacteristic
88C22×D107C2×D411yescharacteristic
89D4×C107—22no—
90D4×C107—42no—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—42no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—42no—
98C22⋊D107—42no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—42no—
104C2×Dic107—42no—
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107C4×D107—22no—
108C4×D107—22no—
109C4×D107—22no—
110C4×D107—44no—
111C4×D107—44no—
112C4×D107#4114yes—
113Dic214—22no—
114C8×C107—22no—
115C107⋊C8—22no—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107—41no—
119C22×C107—41no—
120C22×C107—41no—
121C22×C107#13712yes—
122C22×C107C22×D411yescharacteristic
123C2×D107—28no—
124C2×D107—21no—
125C2×D107—21no—
126C2×D107—21no—
127C2×D107—21no—
128C2×D107#12312yes—
129C2×D107AGL(1,ℤ/8)12yes—
130C4×C107—21no—
131C4×C107—42no—
132C4×C107—41no—
133C4×C107C2×#4112yes—
134Dic107—21no—
135Dic107—41no—
136Dic107—42no—
137Dic107—41no—
138Dic107#12312yes—
139C2×C107—22no—
140C2×C107—81no—
141C2×C107C2×#12311yescharacteristic
142C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C107C2×#13711yescharacteristic
144D107—22no—
145D107—22no—
146#3092—1071nomaximal, Sylow
147C107#309211yescharacteristic, complemented, Sylow
148#525—1071no—
149C22×C4⋊C4—1071no—
150#484—1071no—
151#484—1071no—
152#483—1072no—
153#481—1071no—
154#481D10711yescharacteristic, complemented
155#480—1072no—
156#428—1072no—
157#417—1072no—
158#407—1071no—
159C2×#41—1071no—
160C23×C4—1071no—
161C23×C4—2141no—
162C2×M16—1072no—
163#123—2141no—
164C22⋊Q8—1071no—
165C22⋊Q8—1071no—
166#121—1071no—
167#121C2×D10711yescharacteristic
168#120—2141no—
169C4×D4—2141no—
170C4×D4—2141no—
171#117—1071no—
172C2×C4⋊C4—1071no—
173C2×C4⋊C4—1072no—
174C2×C4⋊C4—1072no—
175C2×C4⋊C4—2141no—
176C2×C4⋊C4—2141no—
177C2×C4⋊C4—2141no—
178C2×C4⋊C4C2×D10711yescharacteristic
179#107—1072no—
180#107—1072no—
181#106—1072no—
182#106C2×D10712yescomplemented
183#103—1072no—
184#103—1072no—
185#102—1072no—
186#102C2×D10712yescomplemented
187C22⋊C8—1071no—
188C22⋊C8—1072no—
189C22⋊C8C2×D10711yescharacteristic
190C24—1071no—
191#41—2144no—
192C2×Q8—1071no—
193C2×D4—21no—
194C2×D4—2141no—
195C2×D4—2141no—
196C2×D4C22×D10711yescharacteristic
197C22×C4—21no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—1071no—
202C22×C4—1071no—
203C22×C4—4284no—
204C22×C4—2144no—
205C22×C4—2142no—
206C22×C4—2142no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4C22×D10711yescharacteristic
210M16—2144no—
211C2×C8—1072no—
212C2×C8C22×D10712yes—
213C4⋊C4—21no—
214C4⋊C4—1072no—
215C4⋊C4—1071no—
216C4⋊C4—2142no—
217C4⋊C4—2142no—
218C4⋊C4—2141no—
219C4⋊C4—2141no—
220C4⋊C4—2141no—
221C4⋊C4C22×D10712yes—
222C4⋊C4C22×D10711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2141no—
225C22⋊C4—2141no—
226C22⋊C4—2141no—
227C22⋊C4—2141no—
228C42—2141no—
229C23—21no—
230C23—1071no—
231C23—1071no—
232C23—2142no—
233C23—2142no—
234C23—2142no—
235C23D4×D10711yescharacteristic
236Q8—2142no—
237D4—22no—
238D4—42no—
239D4—4282no—
240D4—4282no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—1071no—
246C2×C4—1074no—
247C2×C4—1071no—
248C2×C4—4284no—
249C2×C4—4284no—
250C2×C4—4282no—
251C2×C4—4282no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2141no—
256C2×C4—2141no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4D4×D10711yescharacteristic
260C2×C4C23×D10711yescharacteristic
261C8—22no—
262C8—2142no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—1072no—
269C22—1072no—
270C22—2148no—
271C22—2141no—
272C22—2141no—
273C22—2141no—
274C22—2141no—
275C22#1681213812yes—
276C22C2×D4×D10711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—1072no—
281C4—4281no—
282C4—4282no—
283C4—4281no—
284C4—2141no—
285C4#41×D10712yes—
286C2—22no—
287C2—81no—
288C2—2142no—
289C2—2142no—
290C2#123×D10711yescharacteristic
291C2AGL(1,ℤ/8)×D10711yescharacteristic
292C2C2×#1681213811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#481×C107C2
socleC22×C107C22×D4
prime radical#20293137C2
hypercenter#481D107
hypocenterC107#3092
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D107C2semidirect
3. #20292963C2semidirect
4. #20292962C2semidirect
5. #20292959C2semidirect
6. #20292943C2semidirect
7. #20292939C2semidirect
8. #20292666C2semidirect
9. #20292625C2semidirect
10. #20292589C2semidirect
11. #481×C107C2semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812034C22semidirect
34. #16812032C22semidirect
35. #16812030C22semidirect
40. #106×C107C22semidirect
41. #102×C107C22semidirect
147. C107#3092semidirect
154. #481D107semidirect
182. #106C2×D107semidirect
186. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931372
C2×C4⋊C4×D1072
#202929632
#202929622
#202929592
#202929432
#202929392
#202926662
#202926252
#202925892
#481×C1072
#3092107