Finite Group Databasebeta

← #28346853

Subgroups — #28346853

Order 13696.

424 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 121 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#20293176C211yesmaximal, characteristic
2C4⋊Q8×D107C211yesmaximal, characteristic, complemented
3#20292989C211yesmaximal, characteristic, complemented
4#20292987C211yesmaximal, characteristic, complemented
5#20292986C211yesmaximal, characteristic, complemented
6#20292976C212yesmaximal, complemented
7#20292972C212yesmaximal, complemented
8#20292627C212yesmaximal, complemented
9#20292625C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#489×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×Q8×D107—21no—
15C2×#13133801C2212yes—
16#16812109C2211yescharacteristic
17#16812108—21no—
18#16812108C2212yes—
19#16812107—21no—
20#16812107C2211yescharacteristic
21#16812105C2211yescharacteristic
22#16812097C2211yescharacteristic
23#16812082—21no—
24#16812082—21no—
25(C22⋊D107)⋊C4—21no—
26(C22⋊D107)⋊C4—21no—
27C4⋊C4×D107—21no—
28C4⋊C4×D107C2212yes—
29C42×D107C2211yescharacteristic
30#16812034C2212yes—
31#16812033C2212yes—
32#16812032C2212yes—
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#124×C107C2211yescharacteristic
40#103×C107C2212yes—
41#102×C107C2212yes—
42#97×C107C2211yescharacteristic
43#13133819—24no—
44#13133817—24no—
45C2×Dic214—21no—
46C2×Dic214—22no—
47C2×Dic214C2311yescharacteristic
48Q8×D107—44no—
49C2×D428—21no—
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C2×C22⋊D107—21no—
53C2×C22⋊D107—21no—
54C22×Dic107—21no—
55C2×C4×D107—22no—
56C2×C4×D107—21no—
57C2×C4×D107D411yescharacteristic
58C2×C4×D107D411yescharacteristic
59C2×C4×D107C2311yescharacteristic
60#13133801—24no—
61C2×C107⋊C8C2312yes—
62C4⋊Dic107—21no—
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65Dic107⋊C4—22no—
66Dic107⋊C4—21no—
67Dic107⋊C4—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C2×Q8×C107—21no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107C2311yescharacteristic
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C22⋊C4×C107—21no—
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C22×D107—21no—
84C22×D107C2×D411yescharacteristic
85Q8×C107—22no—
86Q8×C107—42no—
87D4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—42no—
94C22⋊D107—42no—
95C22⋊D107—42no—
96C2×Dic107—21no—
97C2×Dic107—22no—
98C2×Dic107—42no—
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102D428—22no—
103D428—42no—
104C4×D107—22no—
105C4×D107—22no—
106C4×D107—44no—
107C4×D107—42no—
108C4×D107#4114yes—
109Dic214—22no—
110Dic214—42no—
111Dic214—44no—
112C8×C107—22no—
113C107⋊C8—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—41no—
121C2×D107—41no—
122C2×D107—41no—
123C2×D107#12412yes—
124C4×C107—21no—
125C4×C107—21no—
126C4×C107—41no—
127C4×C107—42no—
128C4×C107C2×#4112yes—
129Dic107—41no—
130Dic107—42no—
131Dic107#12412yes—
132Dic107AGL(1,ℤ/8)12yes—
133Dic107#13712yes—
134C2×C107—81no—
135C2×C107C2×#12411yescharacteristic
136C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C107C2×#13711yescharacteristic
138D107—22no—
139D107—81no—
140#3141—1071nomaximal, Sylow
141C107#314111yescharacteristic, complemented, Sylow
142#532—1071no—
143C2×C4⋊Q8—1071no—
144#489—1071no—
145#489—1071no—
146#489—1071no—
147#489D10711yescharacteristic, complemented
148#487—1072no—
149#486—1072no—
150#417—1072no—
151#417—1072no—
152C4×M16—1071no—
153C2×#41—1071no—
154C2×#41—1071no—
155C22×Q8—2141no—
156C2×M16—1072no—
157C4⋊Q8—1072no—
158C4⋊Q8—1071no—
159C4⋊Q8—1071no—
160C4⋊Q8—2141no—
161C4⋊Q8—2141no—
162C4⋊Q8C2×D10711yescharacteristic
163#127—1071no—
164#124—1071no—
165#124C2×D10711yescharacteristic
166#121—2141no—
167#121—2141no—
168C4×D4—2141no—
169C4×D4—2141no—
170C2×C4⋊C4—1072no—
171C2×C4⋊C4—2141no—
172C2×C42—1071no—
173#103—1072no—
174#103—1072no—
175#103—1072no—
176#103C2×D10712yes—
177#102—1072no—
178#102—1072no—
179#102—1072no—
180#102C2×D10712yes—
181#97—1071no—
182#97C2×D10711yescharacteristic
183C4×C8—1072no—
184#41—2144no—
185#41—2144no—
186C2×Q8—21no—
187C2×Q8—1071no—
188C2×Q8—4284no—
189C2×Q8—2142no—
190C2×Q8—2141no—
191C2×Q8C22×D10711yescharacteristic
192C2×D4—1071no—
193C2×D4—2141no—
194C2×D4—2141no—
195C2×D4—2141no—
196C2×D4—2141no—
197C2×D4C22×D10711yescharacteristic
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—2141no—
202C22×C4—2142no—
203C22×C4—2141no—
204M16—2144no—
205C2×C8—1072no—
206C2×C8C22×D10712yes—
207C4⋊C4—21no—
208C4⋊C4—1072no—
209C4⋊C4—2142no—
210C4⋊C4—2142no—
211C4⋊C4—2141no—
212C4⋊C4—2141no—
213C4⋊C4—2141no—
214C4⋊C4C22×D10712yes—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C42—1071no—
220C42—1071no—
221C42—1071no—
222C42C22×D10711yescharacteristic
223C23—21no—
224C23—1071no—
225C23—2141no—
226Q8—22no—
227Q8—42no—
228Q8—4282no—
229Q8—4284no—
230Q8—2142no—
231D4—22no—
232D4—4282no—
233D4—4282no—
234D4—4282no—
235D4—4282no—
236D4—2142no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—1071no—
240C2×C4—1071no—
241C2×C4—1071no—
242C2×C4—1074no—
243C2×C4—4282no—
244C2×C4—4284no—
245C2×C4—4282no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4—2141no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—22no—
254C8—2142no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—1072no—
259C22—4281no—
260C22—4281no—
261C22—4281no—
262C22—2141no—
263C22—2141no—
264C22C2×D4×D10711yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—42no—
269C4—1072no—
270C4—1072no—
271C4—1072no—
272C4—4281no—
273C4—4282no—
274C4#41×D10712yes—
275C2—81no—
276C2—8561no—
277C2—2142no—
278C2#124×D10711yescharacteristic
279C2AGL(1,ℤ/8)×D10711yescharacteristic
280C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#489×C107C2
socleC22×C107C22×D4
prime radical#20293176C2
hypercenter#489D107
hypocenterC107#3141
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D107C2semidirect
3. #20292989C2semidirect
4. #20292987C2semidirect
5. #20292986C2semidirect
6. #20292976C2semidirect
7. #20292972C2semidirect
8. #20292627C2semidirect
9. #20292625C2semidirect
10. #20292579C2semidirect
11. #489×C107C2semidirect
141. C107#3141semidirect
147. #489D107semidirect

Maximal subgroups

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

subgroupindex
#202931762
C4⋊Q8×D1072
#202929892
#202929872
#202929862
#202929762
#202929722
#202926272
#202926252
#202925792
#489×C1072
#3141107