Finite Group Databasebeta

← #28346803

Subgroups — #28346803

Order 13696.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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
2#20293176C211yesmaximal, characteristic
3#20292994C212yesmaximal, complemented
4#20292990C212yesmaximal, complemented
5#20292989C211yesmaximal, characteristic, complemented
6#20292987C211yesmaximal, characteristic, complemented
7#20292986C211yesmaximal, characteristic, complemented
8#20292622C212yesmaximal, complemented
9#20292620C212yesmaximal, complemented
10#97×D107C211yesmaximal, characteristic, complemented
11#489×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812107C2211yescharacteristic
17#16812105C2211yescharacteristic
18#16812100C2212yes—
19#16812097C2211yescharacteristic
20#16812089—21no—
21#16812085—21no—
22#16812082—21no—
23#16812082—21no—
24Q8×Dic107—21no—
25#16812074—21no—
26(C22⋊D107)⋊C4—21no—
27(C22⋊D107)⋊C4—21no—
28#16812067C2212yes—
29C42×D107C2211yescharacteristic
30#16812034C2212yes—
31#16812033C2212yes—
32#16812032C2212yes—
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812020C2211yescharacteristic
37#16812019C2212yes—
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×Dic214C2311yescharacteristic
47C2×D428—21no—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—21no—
52C22×Dic107—21no—
53C2×C4×D107—21no—
54C2×C4×D107D411yescharacteristic
55C2×C4×D107D411yescharacteristic
56C2×C4×D107C2311yescharacteristic
57C2×C107⋊C8C2312yes—
58C8×D107—24no—
59C4⋊Dic107—21no—
60C4⋊Dic107—22no—
61C4⋊Dic107C2312yes—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64Dic107⋊C4—22no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68C4×Dic107—22no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107—21no—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107C2312yes—
78C22⋊C4×C107—21no—
79C22⋊C4×C107—21no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C22×D107—21no—
83C22×D107C2×D411yescharacteristic
84Q8×C107—22no—
85Q8×C107—42no—
86D4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—42no—
93C22⋊D107—42no—
94C22⋊D107—42no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—42no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101D428—22no—
102D428—42no—
103C4×D107—41no—
104C4×D107—41no—
105C4×D107—42no—
106C4×D107#4114yes—
107Dic214—22no—
108Dic214—42no—
109C8×C107—22no—
110C107⋊C8—22no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107—41no—
114C22×C107C22×D411yescharacteristic
115C2×D107—21no—
116C2×D107—21no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—41no—
120C2×D107#12412yes—
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—41no—
124C4×C107—42no—
125C4×C107C2×#4112yes—
126Dic107—21no—
127Dic107—21no—
128Dic107—41no—
129Dic107—42no—
130Dic107#12412yes—
131C2×C107—81no—
132C2×C107C2×#12411yescharacteristic
133C2×C107#57511yescharacteristic
134C2×C107#57511yescharacteristic
135D107—22no—
136D107—81no—
137#3135—1071nomaximal, Sylow
138C107#313511yescharacteristic, complemented, Sylow
139#533—1071no—
140#532—1071no—
141#491—1072no—
142#490—1072no—
143#489—1071no—
144#489—1071no—
145#489—1071no—
146#489D10711yescharacteristic, complemented
147#416—1072no—
148#416—1072no—
149C2×#97—1071no—
150C2×#41—1071no—
151C2×#41—1071no—
152C22×C8—1072no—
153C4⋊Q8—1071no—
154C4⋊Q8—1071no—
155C4⋊Q8C2×D10711yescharacteristic
156#127—1071no—
157#125—1072no—
158#124—1071no—
159#124C2×D10711yescharacteristic
160C22⋊Q8—2141no—
161C22⋊Q8—2141no—
162#121—2141no—
163#121—2141no—
164C4×Q8—2141no—
165C4×Q8—2141no—
166C4×D4—2141no—
167C4×D4—2141no—
168#117—1072no—
169C2×C42—1071no—
170#103—1072no—
171#103—1072no—
172#103—1072no—
173#103C2×D10712yes—
174#102—1072no—
175#102—1072no—
176#102—1072no—
177#102C2×D10712yes—
178#97—1072no—
179#97—1071no—
180#97C2×D10711yescharacteristic
181#41—2144no—
182#41—2144no—
183C2×Q8—21no—
184C2×Q8—1071no—
185C2×Q8—2141no—
186C2×Q8C22×D10711yescharacteristic
187C2×D4—1071no—
188C2×D4—2141no—
189C2×D4—2141no—
190C2×D4—2141no—
191C2×D4—2141no—
192C2×D4C22×D10711yescharacteristic
193C22×C4—1071no—
194C22×C4—1071no—
195C22×C4—1071no—
196C22×C4—2141no—
197C22×C4—2141no—
198C2×C8—1072no—
199C2×C8—2144no—
200C2×C8C22×D10712yes—
201C4⋊C4—21no—
202C4⋊C4—1072no—
203C4⋊C4—2141no—
204C4⋊C4—2141no—
205C4⋊C4—2142no—
206C4⋊C4—2142no—
207C4⋊C4—2141no—
208C4⋊C4C22×D10712yes—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—2142no—
212C22⋊C4—2141no—
213C22⋊C4—2141no—
214C42—1071no—
215C42—1071no—
216C42—1071no—
217C42—2142no—
218C42C22×D10711yescharacteristic
219C23—21no—
220C23—1071no—
221C23—2141no—
222Q8—22no—
223Q8—42no—
224Q8—4282no—
225Q8—2142no—
226D4—22no—
227D4—4282no—
228D4—4282no—
229D4—4282no—
230D4—4282no—
231D4—2142no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—1071no—
235C2×C4—1074no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—4281no—
239C2×C4—4281no—
240C2×C4—4282no—
241C2×C4—4282no—
242C2×C4—2142no—
243C2×C4—2141no—
244C2×C4D4×D10711yescharacteristic
245C2×C4D4×D10711yescharacteristic
246C2×C4C23×D10711yescharacteristic
247C8—22no—
248C8—2142no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—1072no—
253C22—4281no—
254C22—4281no—
255C22—4281no—
256C22—2141no—
257C22—2141no—
258C22C2×D4×D10711yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—42no—
263C4—1072no—
264C4—4281no—
265C4—4282no—
266C4—2141no—
267C4—2141no—
268C4#41×D10712yes—
269C2—81no—
270C2—8561no—
271C2—2142no—
272C2#124×D10711yescharacteristic
273C2#2029342311yescharacteristic
274C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#489×C107C2
socleC22×C107C22×D4
prime radical#20293176C2
hypercenter#489D107
hypocenterC107#3135
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
1. #20293182C2semidirect
3. #20292994C2semidirect
4. #20292990C2semidirect
5. #20292989C2semidirect
6. #20292987C2semidirect
7. #20292986C2semidirect
8. #20292622C2semidirect
9. #20292620C2semidirect
10. #97×D107C2semidirect
11. #489×C107C2semidirect
138. C107#3135semidirect
146. #489D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931762
#202929942
#202929902
#202929892
#202929872
#202929862
#202926222
#202926202
#97×D1072
#489×C1072
#3135107