Finite Group Databasebeta

← #28349115

Subgroups — #28349115

Order 13696.

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

normal subgroups only

This group has 350 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#20293351C211yesmaximal, characteristic, complemented
2#20293338C211yesmaximal, characteristic, complemented
3#20292983C212yesmaximal, complemented
4#20292936C212yesmaximal, complemented
5#20292914C211yesmaximal, characteristic, complemented
6#20292845C212yesmaximal
7#20292829C211yesmaximal, characteristic, complemented
8C8×Dic214C211yesmaximal, characteristic, complemented
9#20292712C212yesmaximal, complemented
10C4×D856C211yesmaximal, characteristic
11#487×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2212yes—
13C2×D856C2211yescharacteristic
14#16812104—22no—
15#16812100C2212yes—
16#16812098C2212yes—
17#16812089—21no—
18#16812089—21no—
19#16812085—22no—
20Q8×Dic107C2212yes—
21C4×Dic214C2211yescharacteristic
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812071C2212yes—
25#16812067—21no—
26C4⋊C4×D107—21no—
27#16812043C2211yescharacteristic
28#16812040C2212yes—
29C4⋊(C107⋊C8)C2211yescharacteristic
30Dic107⋊C8C2212yes—
31#16812033C2212yes—
32#16812031C2211yescharacteristic
33#16812031C2211yescharacteristic
34#16812029C2212yes—
35C8×Dic107C2212yes—
36C4⋊Q8×C107C2211yescharacteristic
37C4⋊Q8×C107C2211yescharacteristic
38#103×C107C2212yes—
39#103×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41C2×Dic214D411yescharacteristic
42C2×D428C2311yescharacteristic
43C2×D428C2311yescharacteristic
44C2×C4×D107—21no—
45C2×C4×D107—21no—
46C2×C4×D107—22no—
47#13133805—24no—
48D856—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—22no—
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—22no—
56Dic107⋊C4D412yes—
57(C2×D107)⋊C4—21no—
58(C2×D107)⋊C4—21no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62C4×Dic107—22no—
63C4×Dic107C2312yes—
64C2×Q8×C107—21no—
65C2×Q8×C107C2312yes—
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C4⋊C4×C107—21no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2312yes—
72C42×C107C2311yescharacteristic
73C22×D107—21no—
74C22×D107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77Q8×C107—81no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107C2×D412yes—
86C2×Dic107C2×D412yes—
87D428—21no—
88D428—21no—
89D428—21no—
90D428#4112yes—
91C4×D107—42no—
92C4×D107—42no—
93C4×D107—82no—
94Dic214—22no—
95C8×C107—21no—
96C8×C107—21no—
97C107⋊C8—22no—
98C22×C107C22×D411yescharacteristic
99C2×D107—22no—
100C2×D107—41no—
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C2×D107—41no—
105C4×C107—21no—
106C4×C107—42no—
107C4×C107—42no—
108C4×C107#13512yes—
109C4×C107C2×#4111yescharacteristic
110C4×C1072−1+411yescharacteristic
111Dic107—22no—
112Dic107—22no—
113Dic107—41no—
114Dic107—41no—
115C2×C107#54811yescharacteristic
116C2×C107C2×#13511yescharacteristic
117C2×C107#57711yescharacteristic
118D107—42no—
119D107—81no—
120#3360—1071nomaximal, Sylow
121C107#336011yescharacteristic, complemented, Sylow
122#556—1071no—
123#554—1071no—
124#488—1072no—
125#487D10711yescharacteristic, complemented
126#479—1072no—
127#476—1071no—
128#463—1072no—
129#461—1071no—
130C8×Q8—1071no—
131C4×SD16—1072no—
132C4×D8—1071no—
133C2×SD16—1072no—
134C2×D8—1071no—
135C4⋊Q8C2×D10711yescharacteristic
136C4⋊Q8C2×D10711yescharacteristic
137#126—2142no—
138#125—1072no—
139#124—1072no—
140C22⋊Q8—2141no—
141C22⋊Q8—2141no—
142C22⋊Q8—2142no—
143C4×Q8—1071no—
144C4×Q8—1072no—
145C4×D4—1072no—
146C4×D4—1071no—
147C4×D4—1071no—
148#117—2141no—
149C2×C4⋊C4—2141no—
150#107—1071no—
151#106—1072no—
152C4⋊C8—1072no—
153C4⋊C8—1071no—
154#103—1072no—
155#103C2×D10712yes—
156#103C2×D10712yes—
157#102—1071no—
158#102—1071no—
159#102—1072no—
160C4×C8—1072no—
161C4×C8C2×D10711yescharacteristic
162C2×Q8—21no—
163C2×Q8—1071no—
164C2×Q8C22×D10712yes—
165C2×D4—1071no—
166C2×D4—1071no—
167C22×C4—2142no—
168C22×C4—2141no—
169C22×C4—2141no—
170SD16—2144no—
171D8—2142no—
172C2×C8—1072no—
173C2×C8C22×D10711yescharacteristic
174C2×C8C22×D10711yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—1071no—
179C4⋊C4—1072no—
180C4⋊C4—1071no—
181C4⋊C4—1071no—
182C4⋊C4—2142no—
183C4⋊C4—2142no—
184C4⋊C4—2142no—
185C4⋊C4C22×D10712yes—
186C22⋊C4—2142no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C22⋊C4—2141no—
190C22⋊C4—2141no—
191C42—1072no—
192C42—2142no—
193C42C22×D10711yescharacteristic
194C23—2141no—
195C23—2141no—
196Q8—22no—
197Q8—22no—
198Q8—81no—
199Q8—2142no—
200D4—1072no—
201D4—2141no—
202D4—2141no—
203D4—2141no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—8562no—
207C2×C4—1072no—
208C2×C4—1072no—
209C2×C4—4282no—
210C2×C4—4282no—
211C2×C4—2141no—
212C2×C4—2141no—
213C2×C4D4×D10711yescharacteristic
214C2×C4D4×D10711yescharacteristic
215C2×C4C23×D10711yescharacteristic
216C8—21no—
217C8—21no—
218C8—2142no—
219C22—4281no—
220C22—4281no—
221C22—4281no—
222C22—4281no—
223C22—4281no—
224C22—2142no—
225C22C2×D4×D10711yescharacteristic
226C4—21no—
227C4—42no—
228C4—42no—
229C4—4281no—
230C4—4281no—
231C4—2142no—
232C4—2142no—
233C4#1681213212yes—
234C4#41×D10711yescharacteristic
235C4#1681215611yescharacteristic
236C2—8561no—
237C2—4282no—
238C2#2029328511yescharacteristic
239C2C2×#1681213211yescharacteristic
240C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#487×C107C2
socleC22×C107C22×D4
prime radicalC2×D856C22
hypercenter#487D107
hypocenterC107#3360
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #20293351C2semidirect
2. #20293338C2semidirect
3. #20292983C2semidirect
4. #20292936C2semidirect
5. #20292914C2semidirect
7. #20292829C2semidirect
8. C8×Dic214C2semidirect
9. #20292712C2semidirect
11. #487×C107C2semidirect
121. C107#3360semidirect
125. #487D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202933382
#202929832
#202929362
#202929142
#202928452
#202928292
C8×Dic2142
#202927122
C4×D8562
#487×C1072
#3360107