Finite Group Databasebeta

← #28346785

Subgroups — #28346785

Order 13696.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293178C211yesmaximal, characteristic
3#20292983C214yesmaximal, complemented
4#20292977C211yesmaximal, characteristic, complemented
5#20292975C211yesmaximal, characteristic, complemented
6#20292974C211yesmaximal, characteristic, complemented
7#20292622C214yesmaximal, complemented
8C4×C8×D107C211yesmaximal, characteristic, complemented
9#487×C107C211yesmaximal, characteristic, complemented
10C2×#13133819C2212yes—
11C2×C8×D107C2212yes—
12#16812109C2211yescharacteristic
13#16812106C2211yescharacteristic
14#16812100C2212yes—
15#16812098C2212yes—
16#16812089—21no—
17#16812085—21no—
18#16812084—21no—
19#16812084—21no—
20Q8×Dic107—21no—
21#16812074—21no—
22#16812071—21no—
23#16812071—21no—
24#16812067C2212yes—
25C42×D107C2211yescharacteristic
26#16812033C2214yes—
27#16812031C2214yes—
28#16812029C2214yes—
29C4×C107⋊C8C2211yescharacteristic
30C8×Dic107C2212yes—
31C4⋊Q8×C107C2211yescharacteristic
32C4⋊Q8×C107C2211yescharacteristic
33#103×C107C2214yes—
34C4×C8×C107C2211yescharacteristic
35#13133819—28no—
36C2×Dic214—21no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428—21no—
40C2×D428C2312yes—
41C2×C4×D107—22no—
42C2×C4×D107D411yescharacteristic
43C2×C4×D107D411yescharacteristic
44C2×C4×D107C2311yescharacteristic
45C2×C107⋊C8C2312yes—
46C8×D107—24no—
47C4⋊Dic107—21no—
48C4⋊Dic107—22no—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—22no—
55C4×Dic107D411yescharacteristic
56C4×Dic107D411yescharacteristic
57C4×Dic107C2311yescharacteristic
58C2×Q8×C107—21no—
59C2×Q8×C107C2312yes—
60C2×C8×C107C2312yes—
61C4⋊C4×C107—21no—
62C4⋊C4×C107—21no—
63C4⋊C4×C107—21no—
64C4⋊C4×C107C2312yes—
65C42×C107C2311yescharacteristic
66C22×D107—22no—
67C22×D107C2×D411yescharacteristic
68Q8×C107—24no—
69Q8×C107—42no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2411yescharacteristic
75C2×Dic107—22no—
76C2×Dic107C2×D411yescharacteristic
77C2×Dic107C2×D411yescharacteristic
78C2×Dic107C2×D411yescharacteristic
79D428—24no—
80D428—44no—
81D428—42no—
82D428—42no—
83C4×D107—22no—
84C4×D107—22no—
85C4×D107—44no—
86C4×D107#4114yes—
87Dic214—42no—
88C8×C107—22no—
89C107⋊C8—22no—
90C22×C107C22×D411yescharacteristic
91C2×D107—21no—
92C2×D107—21no—
93C2×D107—42no—
94C2×D107—42no—
95C2×D107—42no—
96C2×D107#12412yes—
97C4×C107—42no—
98C4×C107—42no—
99C4×C107#13512yes—
100C4×C107#13512yes—
101C4×C107C2×#4112yes—
102Dic107—21no—
103Dic107—21no—
104Dic107—42no—
105Dic107#12412yes—
106C2×C107C2×#12411yescharacteristic
107C2×C107C2×#13511yescharacteristic
108C2×C107C2×#13511yescharacteristic
109D107—22no—
110D107—82no—
111#3131—1071nomaximal, Sylow
112C107#313111yescharacteristic, complemented, Sylow
113#533—1071no—
114#532—1071no—
115#488—1074no—
116#487—1071no—
117#487D10711yescharacteristic, complemented
118#486—1071no—
119#486—1071no—
120#416—1074no—
121C2×C4×C8—1071no—
122C2×#41—1072no—
123C22×C8—1072no—
124C4⋊Q8—1071no—
125C4⋊Q8C2×D10711yescharacteristic
126C4⋊Q8C2×D10711yescharacteristic
127#127—1071no—
128#125—1072no—
129#124—1072no—
130C22⋊Q8—2141no—
131C22⋊Q8—2141no—
132#121—2141no—
133#121—2141no—
134C4×Q8—2141no—
135C4×Q8—2141no—
136C4×D4—2141no—
137C4×D4—2141no—
138#117—1072no—
139C2×C42—1071no—
140#103—1074no—
141#103C2×D10714yes—
142#102—1074no—
143#102—1074no—
144C4×C8—1072no—
145C4×C8—1071no—
146C4×C8C2×D10711yescharacteristic
147#41—2148no—
148C2×Q8—21no—
149C2×Q8—2141no—
150C2×Q8C22×D10712yes—
151C2×D4—1072no—
152C2×D4—2142no—
153C2×D4—2141no—
154C2×D4—2141no—
155C22×C4—1071no—
156C22×C4—1071no—
157C22×C4—1071no—
158C22×C4—2142no—
159C2×C8—1072no—
160C2×C8—2144no—
161C2×C8C22×D10712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—1072no—
166C4⋊C4—2142no—
167C4⋊C4—2142no—
168C4⋊C4—2141no—
169C4⋊C4C22×D10712yes—
170C22⋊C4—2142no—
171C22⋊C4—2142no—
172C22⋊C4—2142no—
173C42—1071no—
174C42—1071no—
175C42—1071no—
176C42—2142no—
177C42C22×D10711yescharacteristic
178C23—1071no—
179C23—2142no—
180Q8—24no—
181Q8—42no—
182Q8—4282no—
183D4—4284no—
184D4—4282no—
185D4—4282no—
186D4—2144no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—1071no—
190C2×C4—1074no—
191C2×C4—1071no—
192C2×C4—1071no—
193C2×C4—4284no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4D4×D10711yescharacteristic
198C2×C4D4×D10711yescharacteristic
199C2×C4C23×D10711yescharacteristic
200C8—22no—
201C8—2142no—
202C22—1072no—
203C22—4282no—
204C22—4282no—
205C22—4282no—
206C22—2141no—
207C22—2141no—
208C22C2×D4×D10711yescharacteristic
209C4—42no—
210C4—42no—
211C4—1072no—
212C4—4282no—
213C4—2141no—
214C4—2141no—
215C4#1681213012yes—
216C4#1681213212yes—
217C4#41×D10712yes—
218C2—8562no—
219C2—2142no—
220C2#124×D10711yescharacteristic
221C2C2×#1681213011yescharacteristic
222C2C2×#1681213211yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#487×C107C2
socleC22×C107C22×D4
prime radical#20293178C2
hypercenter#487D107
hypocenterC107#3131
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. #20293182C2semidirect
3. #20292983C2semidirect
4. #20292977C2semidirect
5. #20292975C2semidirect
6. #20292974C2semidirect
7. #20292622C2semidirect
8. C4×C8×D107C2semidirect
9. #487×C107C2semidirect
112. C107#3131semidirect
117. #487D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931782
#202929832
#202929772
#202929752
#202929742
#202926222
C4×C8×D1072
#487×C1072
#3131107