Finite Group Databasebeta

← #28349084

Subgroups — #28349084

Order 13696.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293343C211yesmaximal, characteristic
2#20293337C211yesmaximal, characteristic, complemented
3#20293030C211yesmaximal, characteristic, complemented
4#20292966C212yesmaximal, complemented
5#20292942C212yesmaximal, complemented
6#20292918C211yesmaximal, characteristic, complemented
7#20292876C212yesmaximal, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9#20292704C211yesmaximal, characteristic, complemented
10#20292668C212yesmaximal, complemented
11#477×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2211yescharacteristic
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812101—21no—
16#16812098—22no—
17#16812087—22no—
18#16812085C2212yes—
19#16812084—21no—
20#16812084C2212yes—
21C4×D428—21no—
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812067—21no—
25#16812067C2212yes—
26C4⋊C4×D107C2212yes—
27#16812042C2212yes—
28#16812042C2212yes—
29#16812041C2212yes—
30#16812040C2211yescharacteristic
31#16812033C2211yescharacteristic
32#16812031C2212yes—
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#107×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×D428—22no—
43C2×D428D411yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C4×D107—21no—
46C2×C4×D107—22no—
47C2×C4×D107C2312yes—
48#13133805—22no—
49#13133801—24no—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C4⋊Dic107—21no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55C4⋊Dic107C2312yes—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—21no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4D412yes—
63C4×Dic107—22no—
64C2×Q8×C107—21no—
65C2×Q8×C107C2311yescharacteristic
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2312yes—
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C22×D107—21no—
74C22×D107C2×D412yes—
75Q8×C107—21no—
76Q8×C107—21no—
77Q8×C107—81no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2411yescharacteristic
84C2×Dic107—21no—
85C2×Dic107—22no—
86C2×Dic107C2×D412yes—
87D428—22no—
88D428—21no—
89D428—21no—
90D428—82no—
91C4×D107—22no—
92C4×D107—22no—
93C4×D107—42no—
94C4×D107—82no—
95Dic214—81no—
96C8×C107—22no—
97C107⋊C8—21no—
98C107⋊C8#4112yes—
99C22×C107C22×D411yescharacteristic
100C2×D107—22no—
101C2×D107—22no—
102C2×D107—22no—
103C2×D107—41no—
104C2×D107—41no—
105C2×D107—41no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—41no—
109C4×C107—42no—
110C4×C107#13712yes—
111C4×C107C2×#4111yescharacteristic
112C4×C1072−1+411yescharacteristic
113Dic107—22no—
114Dic107—41no—
115Dic107—42no—
116C2×C107#54711yescharacteristic
117C2×C107C2×#13711yescharacteristic
118C2×C107#57611yescharacteristic
119D107—42no—
120D107—81no—
121#3357—1071nomaximal, Sylow
122C107#335711yescharacteristic, complemented, Sylow
123#555—1071no—
124#554—1071no—
125#500—1071no—
126#485—1072no—
127#480—1072no—
128#477D10711yescharacteristic, complemented
129#476—1071no—
130#468—1072no—
131C4×SD16—1071no—
132#436—1071no—
133#428—1072no—
134C2×SD16—1071no—
135C2×M16—1072no—
136C4⋊Q8—1071no—
137C4⋊Q8C2×D10711yescharacteristic
138#125—2141no—
139#124—2142no—
140C22⋊Q8—1072no—
141C22⋊Q8—2142no—
142#121—1072no—
143#121—2141no—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1071no—
146C4×D4—1071no—
147C4×D4—2141no—
148#117—1072no—
149#117—2141no—
150C2×C4⋊C4—1072no—
151#107—1072no—
152#107—1072no—
153#107C2×D10712yes—
154#106—1072no—
155#106—1071no—
156C4⋊C8C2×D10711yescharacteristic
157#103—1071no—
158#103C2×D10712yes—
159#102—1072no—
160#102—1071no—
161C22⋊C8—1072no—
162C4×C8—1071no—
163C2×Q8—21no—
164C2×Q8—2141no—
165C2×Q8C22×D10711yescharacteristic
166C2×D4—1071no—
167C2×D4—1071no—
168C2×D4—2142no—
169C22×C4—1072no—
170C22×C4—2142no—
171C22×C4—2141no—
172SD16—2142no—
173M16—2144no—
174C2×C8—1071no—
175C2×C8—1071no—
176C2×C8C22×D10712yes—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—1071no—
180C4⋊C4—1072no—
181C4⋊C4—1071no—
182C4⋊C4—2142no—
183C4⋊C4—2142no—
184C4⋊C4—2141no—
185C4⋊C4C22×D10711yescharacteristic
186C4⋊C4C22×D10712yes—
187C22⋊C4—1072no—
188C22⋊C4—2142no—
189C22⋊C4—2142no—
190C22⋊C4—2142no—
191C22⋊C4—2141no—
192C42—21no—
193C42—2142no—
194C42C22×D10711yescharacteristic
195C23—1072no—
196C23—2141no—
197Q8—21no—
198Q8—21no—
199Q8—81no—
200Q8—8561no—
201D4—8562no—
202D4—2142no—
203D4—2141no—
204D4—2141no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—8562no—
209C2×C4—1072no—
210C2×C4—4282no—
211C2×C4—2141no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4D4×D10711yescharacteristic
216C2×C4D4×D10711yescharacteristic
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—1072no—
220C8—2141no—
221C22—4281no—
222C22—4281no—
223C22—4281no—
224C22—2142no—
225C22—2142no—
226C22—2142no—
227C22C2×D4×D10711yescharacteristic
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—4281no—
233C4—4282no—
234C4—2142no—
235C4#1681214112yes—
236C4#41×D10711yescharacteristic
237C4#1681215711yescharacteristic
238C2—8561no—
239C2—4282no—
240C2#2029327911yescharacteristic
241C2C2×#1681214111yescharacteristic
242C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#477×C107C2
socleC22×C107C22×D4
prime radical#20293343C2
hypercenter#477D107
hypocenterC107#3357
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20293337C2semidirect
3. #20293030C2semidirect
4. #20292966C2semidirect
5. #20292942C2semidirect
6. #20292918C2semidirect
7. #20292876C2semidirect
8. C4×#13133805C2semidirect
9. #20292704C2semidirect
10. #20292668C2semidirect
11. #477×C107C2semidirect
122. C107#3357semidirect
128. #477D107semidirect

Maximal subgroups

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

subgroupindex
#202933432
#202933372
#202930302
#202929662
#202929422
#202929182
#202928762
C4×#131338052
#202927042
#202926682
#477×C1072
#3357107