Finite Group Databasebeta

← #28346858

Subgroups — #28346858

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#20293176C211yesmaximal, characteristic
2#125×D107C211yesmaximal, characteristic, complemented
3#20292994C212yesmaximal, complemented
4#20292990C212yesmaximal, complemented
5#20292984C211yesmaximal, characteristic, complemented
6#20292982C211yesmaximal, characteristic, complemented
7#20292981C211yesmaximal, characteristic, complemented
8#20292627C212yesmaximal, complemented
9#20292625C212yesmaximal, complemented
10C4×#13133801C211yesmaximal, characteristic, complemented
11#488×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812107C2211yescharacteristic
16#16812105C2211yescharacteristic
17#16812101C2211yescharacteristic
18#16812100C2212yes—
19#16812099—22no—
20#16812097C2211yescharacteristic
21#16812082—22no—
22(C22⋊D107)⋊C4—22no—
23C4⋊C4×D107—22no—
24C4⋊C4×D107C2212yes—
25C42×D107C2211yescharacteristic
26#16812034C2212yes—
27#16812033C2212yes—
28#16812032C2212yes—
29#16812031C2212yes—
30#16812030C2212yes—
31#16812029C2212yes—
32#16812019C2212yes—
33C4×C107⋊C8C2211yescharacteristic
34#125×C107C2211yescharacteristic
35#124×C107C2211yescharacteristic
36#103×C107C2212yes—
37#102×C107C2212yes—
38C4×C8×C107C2211yescharacteristic
39#13133819—24no—
40#13133817—24no—
41C2×Dic214C2311yescharacteristic
42C2×D428—21no—
43C2×D428C2311yescharacteristic
44C2×C22⋊D107—21no—
45C2×C22⋊D107—22no—
46C22×Dic107—21no—
47C2×C4×D107—22no—
48C2×C4×D107—21no—
49C2×C4×D107D412yes—
50C2×C4×D107C2311yescharacteristic
51#13133801—24no—
52C2×C107⋊C8C2312yes—
53C4⋊Dic107—22no—
54C4⋊Dic107C2312yes—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59(C2×D107)⋊C4—22no—
60C4×Dic107D412yes—
61C4×Dic107C2311yescharacteristic
62C2×Q8×C107C2311yescharacteristic
63C2×D4×C107C2311yescharacteristic
64C2×C8×C107C2312yes—
65C4⋊C4×C107—22no—
66C4⋊C4×C107C2312yes—
67C22⋊C4×C107—22no—
68C42×C107C2311yescharacteristic
69C23×C107—21no—
70C22×D107—21no—
71C22×D107C2×D411yescharacteristic
72Q8×C107—22no—
73D4×C107—22no—
74C2×C4×C107—21no—
75C2×C4×C107—22no—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—44no—
79C22⋊D107—42no—
80C2×Dic107—21no—
81C2×Dic107—22no—
82C2×Dic107—42no—
83C2×Dic107C2×D412yes—
84C2×Dic107C2×D411yescharacteristic
85D428—22no—
86D428—42no—
87C4×D107—42no—
88C4×D107—44no—
89C4×D107—42no—
90C4×D107#4114yes—
91Dic214—22no—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—41no—
95C22×C107—42no—
96C22×C107C22×D411yescharacteristic
97C2×D107—22no—
98C2×D107—41no—
99C2×D107—42no—
100C2×D107#12412yes—
101C4×C107—22no—
102C4×C107—41no—
103C4×C107—42no—
104C4×C107C2×#4112yes—
105Dic107—22no—
106Dic107—41no—
107Dic107—42no—
108Dic107#12412yes—
109C2×C107—81no—
110C2×C107C2×#12411yescharacteristic
111C2×C107#57512yes—
112D107—22no—
113D107—81no—
114#3142—1071nomaximal, Sylow
115C107#314211yescharacteristic, complemented, Sylow
116#532—1071no—
117C2×#125—1071no—
118#491—1072no—
119#490—1072no—
120#488—1071no—
121#488—1071no—
122#488—1071no—
123#488D10711yescharacteristic, complemented
124#417—1072no—
125#417—1072no—
126C4×M16—1071no—
127C2×#41—1071no—
128C2×#41—1071no—
129C2×M16—1072no—
130C4⋊Q8—1071no—
131#127—1071no—
132#125—1072no—
133#125—1071no—
134#125—2142no—
135#125C2×D10711yescharacteristic
136#124—1071no—
137#124C2×D10711yescharacteristic
138#121—2142no—
139C4×D4—2142no—
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—2142no—
142C2×C42—1071no—
143#103—1072no—
144#103—1072no—
145#103—1072no—
146#103C2×D10712yes—
147#102—1072no—
148#102—1072no—
149#102—1072no—
150#102C2×D10712yes—
151#97—1072no—
152C4×C8—1071no—
153C4×C8C2×D10711yescharacteristic
154#41—2144no—
155#41—2144no—
156C2×Q8—1071no—
157C2×Q8C22×D10711yescharacteristic
158C2×D4—1071no—
159C2×D4—2141no—
160C2×D4—2142no—
161C2×D4—2141no—
162C2×D4C22×D10711yescharacteristic
163C22×C4—1071no—
164C22×C4—1072no—
165C22×C4—2141no—
166C22×C4—2142no—
167C22×C4—2141no—
168M16—2144no—
169C2×C8—1072no—
170C2×C8C22×D10712yes—
171C4⋊C4—22no—
172C4⋊C4—1072no—
173C4⋊C4—2142no—
174C4⋊C4—2142no—
175C4⋊C4—2142no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C4⋊C4C22×D10712yes—
179C22⋊C4—22no—
180C22⋊C4—2142no—
181C42—1072no—
182C42—1071no—
183C42C22×D10711yescharacteristic
184C23—21no—
185C23—1071no—
186C23—2141no—
187Q8—22no—
188Q8—2142no—
189D4—22no—
190D4—4282no—
191D4—4282no—
192D4—4284no—
193D4—2142no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—1071no—
197C2×C4—1074no—
198C2×C4—1072no—
199C2×C4—4282no—
200C2×C4—4282no—
201C2×C4—4284no—
202C2×C4—4282no—
203C2×C4—2141no—
204C2×C4—2142no—
205C2×C4D4×D10712yes—
206C2×C4C23×D10711yescharacteristic
207C8—22no—
208C8—2142no—
209C22—41no—
210C22—42no—
211C22—1072no—
212C22—4281no—
213C22—4282no—
214C22—2142no—
215C22C2×D4×D10711yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1072no—
220C4—4281no—
221C4—4282no—
222C4—2142no—
223C4#41×D10712yes—
224C2—81no—
225C2—8561no—
226C2—2142no—
227C2#124×D10711yescharacteristic
228C2#2029342412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#488×C107C2
socleC22×C107C22×D4
prime radical#20293176C2
hypercenter#488D107
hypocenterC107#3142
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D107C2semidirect
3. #20292994C2semidirect
4. #20292990C2semidirect
5. #20292984C2semidirect
6. #20292982C2semidirect
7. #20292981C2semidirect
8. #20292627C2semidirect
9. #20292625C2semidirect
10. C4×#13133801C2semidirect
11. #488×C107C2semidirect
115. C107#3142semidirect
123. #488D107semidirect

Maximal subgroups

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

subgroupindex
#202931762
#125×D1072
#202929942
#202929902
#202929842
#202929822
#202929812
#202926272
#202926252
C4×#131338012
#488×C1072
#3142107