Finite Group Databasebeta

← #28347022

Subgroups — #28347022

Order 13696.

412 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 133 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#20293000C211yesmaximal, characteristic, complemented
4#20292979C211yesmaximal, characteristic, complemented
5#20292975C211yesmaximal, characteristic, complemented
6#20292857C214yesmaximal, complemented
7#20292714C214yesmaximal, complemented
8C4×C8×D107C211yesmaximal, characteristic, complemented
9#498×C107C211yesmaximal, characteristic, complemented
10C2×#13133819—21no—
11C2×#13133807C2214yes—
12C2×C8×D107C2212yes—
13#16812109C2211yescharacteristic
14#16812106C2211yescharacteristic
15#16812100—21no—
16#16812098—21no—
17#16812089C2212yes—
18#16812085—21no—
19#16812084—21no—
20#16812084C2212yes—
21Q8×Dic107—21no—
22#16812074C2212yes—
23#16812071—21no—
24#16812071C2212yes—
25#16812067—21no—
26C42×D107C2211yescharacteristic
27#16812032C2214yes—
28#16812029C2214yes—
29C4×C107⋊C8C2211yescharacteristic
30C8×Dic107C2212yes—
31C4⋊Q8×C107C2211yescharacteristic
32C4⋊Q8×C107C2211yescharacteristic
33#106×C107C2214yes—
34C4×C8×C107C2211yescharacteristic
35#13133819—44no—
36C2×Dic214C2312yes—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428—21no—
40C2×D428C2312yes—
41C2×C4×D107—22no—
42C2×C4×D107D411yescharacteristic
43C2×C4×D107D411yescharacteristic
44C2×C4×D107C2311yescharacteristic
45#13133807—28no—
46C2×C107⋊C8C2312yes—
47C8×D107—24no—
48C4⋊Dic107—21no—
49C4⋊Dic107—21no—
50C4⋊Dic107—22no—
51Dic107⋊C4—22no—
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—22no—
55C4×Dic107—22no—
56C4×Dic107D411yescharacteristic
57C4×Dic107D411yescharacteristic
58C4×Dic107C2311yescharacteristic
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C2×C8×C107C2312yes—
62C4⋊C4×C107—21no—
63C4⋊C4×C107—21no—
64C4⋊C4×C107C2312yes—
65C4⋊C4×C107C2312yes—
66C42×C107C2311yescharacteristic
67C22×D107—22no—
68C22×D107C2×D411yescharacteristic
69Q8×C107—42no—
70Q8×C107—42no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C2×Dic107—22no—
77C2×Dic107C2×D411yescharacteristic
78C2×Dic107C2×D411yescharacteristic
79C2×Dic107C2×D411yescharacteristic
80D428—24no—
81D428—44no—
82D428—42no—
83D428—42no—
84C4×D107—22no—
85C4×D107—22no—
86C4×D107—44no—
87C4×D107SD1614yescomplemented
88Dic214—24no—
89C8×C107#4114yescomplemented
90C107⋊C8—22no—
91C22×C107C22×D411yescharacteristic
92C2×D107—21no—
93C2×D107—21no—
94C2×D107—42no—
95C2×D107—42no—
96C2×D107—42no—
97C2×D107C2×SD1612yes—
98C4×C107—42no—
99C4×C107—42no—
100C4×C107C2×SD1612yes—
101C4×C107#13512yes—
102C4×C107C2×#4112yes—
103Dic107—21no—
104Dic107—21no—
105Dic107—42no—
106Dic107C2×SD1612yes—
107C2×C107#53211yescharacteristic
108C2×C107C22×SD1611yescharacteristic
109C2×C107C2×#13511yescharacteristic
110D107—22no—
111D107—82no—
112#3167—1071nomaximal, Sylow
113C107#316711yescharacteristic, complemented, Sylow
114#533—1071no—
115#532—1071no—
116#498D10711yescharacteristic, complemented
117#492—1071no—
118#487—1071no—
119#486—1071no—
120#465—1074no—
121C4×SD16—1074no—
122C2×C4×C8—1071no—
123C2×#41—2141no—
124C2×SD16—1074no—
125C22×C8—1072no—
126C4⋊Q8—1071no—
127C4⋊Q8C2×D10711yescharacteristic
128C4⋊Q8C2×D10711yescharacteristic
129#127—1071no—
130#125—2141no—
131#124—2141no—
132C22⋊Q8—1072no—
133C22⋊Q8—2141no—
134#121—1072no—
135#121—2141no—
136C4×Q8—1072no—
137C4×Q8—2141no—
138C4×D4—1072no—
139C4×D4—2141no—
140#117—2141no—
141C2×C42—1071no—
142#106C2×D10714yes—
143#103—1074no—
144#102—1074no—
145C4×C8—1072no—
146C4×C8—1071no—
147C4×C8C2×D10711yescharacteristic
148#41—4284no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—1072no—
152C2×D4—1072no—
153C2×D4—2142no—
154C2×D4—2141no—
155C2×D4—2141no—
156C22×C4—1071no—
157C22×C4—1071no—
158C22×C4—1071no—
159C22×C4—2142no—
160SD16—2148no—
161C2×C8—1072no—
162C2×C8—2144no—
163C2×C8C22×D10712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—2142no—
167C4⋊C4—2142no—
168C4⋊C4—2141no—
169C4⋊C4—2141no—
170C4⋊C4C22×D10712yes—
171C4⋊C4C22×D10712yes—
172C22⋊C4—2142no—
173C22⋊C4—2142no—
174C22⋊C4—2142no—
175C42—1071no—
176C42—1071no—
177C42—1071no—
178C42—2142no—
179C42C22×D10711yescharacteristic
180C23—1071no—
181C23—2142no—
182Q8—42no—
183Q8—42no—
184Q8—2144no—
185D4—4284no—
186D4—4282no—
187D4—4282no—
188D4—2144no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—1071no—
192C2×C4—1074no—
193C2×C4—1071no—
194C2×C4—1071no—
195C2×C4—4284no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4D4×D10711yescharacteristic
200C2×C4D4×D10711yescharacteristic
201C2×C4C23×D10711yescharacteristic
202C8—2142no—
203C8#1313381914yescomplemented
204C22—1072no—
205C22—4282no—
206C22—4282no—
207C22—4282no—
208C22—2141no—
209C22—2141no—
210C22C2×D4×D10711yescharacteristic
211C4—42no—
212C4—42no—
213C4—1072no—
214C4—4282no—
215C4—2141no—
216C4—2141no—
217C4SD16×D10712yes—
218C4#1681213012yes—
219C4C2×#1313381912yes—
220C2—8562no—
221C2—2142no—
222C2#2029317811yescharacteristic
223C2C2×SD16×D10711yescharacteristic
224C2C2×#1681213011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#498×C107C2
socleC22×C107C22×D4
prime radical#20293178C2
hypercenter#498D107
hypocenterC107#3167
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20293000C2semidirect
4. #20292979C2semidirect
5. #20292975C2semidirect
6. #20292857C2semidirect
7. #20292714C2semidirect
8. C4×C8×D107C2semidirect
9. #498×C107C2semidirect
87. C4×D107SD16semidirect
89. C8×C107#41semidirect
113. C107#3167semidirect
116. #498D107semidirect
203. C8#13133819semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931782
#202930002
#202929792
#202929752
#202928572
#202927142
C4×C8×D1072
#498×C1072
#3167107