Finite Group Databasebeta

← #28347018

Subgroups — #28347018

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#20293175C211yesmaximal, characteristic
3#20293025C211yesmaximal, characteristic, complemented
4#20292977C211yesmaximal, characteristic, complemented
5#20292973C211yesmaximal, characteristic, complemented
6#20292853C214yesmaximal, complemented
7SD16×Dic107C214yesmaximal, complemented
8C4×C8×D107C211yesmaximal, characteristic, complemented
9#492×C107C211yesmaximal, characteristic, complemented
10C2×#13133817—21no—
11C2×C8×D107C2212yes—
12#16812109C2211yescharacteristic
13#16812109C2211yescharacteristic
14#16812100—21no—
15#16812096—21no—
16#16812089—21no—
17#16812085C2212yes—
18#16812080—21no—
19#16812080C2212yes—
20Q8×Dic107C2212yes—
21#16812074—21no—
22D4×Dic107—21no—
23D4×Dic107C2212yes—
24#16812067—21no—
25C42×D107C2211yescharacteristic
26#16812041C2214yes—
27#16812033C2214yes—
28#16812030C2214yes—
29C4×C107⋊C8C2211yescharacteristic
30C8×Dic107C2212yes—
31C2×SD16×C107C2214yes—
32C4⋊Q8×C107C2211yescharacteristic
33#127×C107C2211yescharacteristic
34C4×C8×C107C2211yescharacteristic
35#13133817—44no—
36C2×Dic214—21no—
37C2×Dic214—21no—
38C2×C22⋊D107—22no—
39C22×Dic107—22no—
40C2×C4×D107D411yescharacteristic
41C2×C4×D107D411yescharacteristic
42C2×C4×D107C2311yescharacteristic
43C2×C107⋊C8C2312yes—
44C8×D107—24no—
45C4⋊Dic107—21no—
46C4⋊Dic107—21no—
47C4⋊Dic107—22no—
48C4⋊Dic107C2312yes—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—22no—
51C22⋊Dic107—22no—
52C22⋊Dic107—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—22no—
55C4×Dic107D411yescharacteristic
56C4×Dic107D411yescharacteristic
57C4×Dic107C2311yescharacteristic
58C2×Q8×C107C2312yes—
59C2×D4×C107—21no—
60C2×D4×C107—21no—
61C2×D4×C107C2312yes—
62SD16×C107—28no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107—21no—
65C4⋊C4×C107—21no—
66C42×C107C2311yescharacteristic
67C23×C107—22no—
68C22×D107C2×D411yescharacteristic
69Q8×C107—24no—
70D4×C107—24no—
71D4×C107—42no—
72D4×C107—42no—
73C2×C4×C107—22no—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2411yescharacteristic
77C22⋊D107—44no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—44no—
81C2×Dic107C2×D411yescharacteristic
82C2×Dic107C2×D411yescharacteristic
83C2×Dic107C2×D411yescharacteristic
84C4×D107—22no—
85C4×D107—22no—
86C4×D107SD1614yescomplemented
87Dic214—42no—
88Dic214—42no—
89C8×C107#4114yescomplemented
90C107⋊C8—22no—
91C22×C107—42no—
92C22×C107—42no—
93C22×C107—42no—
94C22×C107C22×D411yescharacteristic
95C2×D107—21no—
96C2×D107—21no—
97C2×D107C2×SD1612yes—
98C4×C107—42no—
99C4×C107C2×SD1612yes—
100C4×C107#13512yes—
101C4×C107C2×#4112yes—
102Dic107—21no—
103Dic107—21no—
104Dic107—42no—
105Dic107—42no—
106Dic107C2×SD1612yes—
107C2×C107—82no—
108C2×C107#53211yescharacteristic
109C2×C107C22×SD1611yescharacteristic
110C2×C107C2×#13511yescharacteristic
111D107—22no—
112#3167—1071nomaximal, Sylow
113C107#316711yescharacteristic, complemented, Sylow
114#533—1071no—
115#532—1071no—
116#498—1071no—
117#492D10711yescharacteristic, complemented
118#487—1071no—
119#486—1071no—
120#465—1074no—
121C4×SD16—1074no—
122C2×C4×C8—1071no—
123C2×#41—2141no—
124C2×SD16C2×D10714yes—
125C22×C8—1072no—
126C4⋊Q8—1071no—
127C4⋊Q8—1071no—
128C4⋊Q8C2×D10711yescharacteristic
129#127C2×D10711yescharacteristic
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#106—1074no—
143#103—1074no—
144#102—1074no—
145C4×C8—1072no—
146C4×C8—1071no—
147C4×C8C2×D10711yescharacteristic
148#41—4284no—
149C2×Q8—2141no—
150C2×Q8—2141no—
151C2×Q8C22×D10712yes—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—2142no—
155C2×D4C22×D10712yes—
156C22×C4—1071no—
157C22×C4—1071no—
158C22×C4—1071no—
159C22×C4—2142no—
160SD16—28no—
161C2×C8—1072no—
162C2×C8—2144no—
163C2×C8C22×D10712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—1072no—
167C4⋊C4—1072no—
168C4⋊C4—2142no—
169C4⋊C4—2142no—
170C4⋊C4—2141no—
171C4⋊C4—2141no—
172C22⋊C4—2142no—
173C22⋊C4—2142no—
174C22⋊C4—2142no—
175C42—1071no—
176C42—1071no—
177C42—1071no—
178C42—2142no—
179C42C22×D10711yescharacteristic
180C23—22no—
181C23—1071no—
182Q8—24no—
183Q8—4282no—
184Q8—4282no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—4284no—
189C2×C4—22no—
190C2×C4—1071no—
191C2×C4—1074no—
192C2×C4—1071no—
193C2×C4—1071no—
194C2×C4—4284no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4D4×D10711yescharacteristic
200C2×C4D4×D10711yescharacteristic
201C2×C4C23×D10711yescharacteristic
202C8—2142no—
203C8#1313381714yescomplemented
204C22—42no—
205C22—42no—
206C22—42no—
207C22—1072no—
208C22—2141no—
209C22—2141no—
210C22C2×D4×D10711yescharacteristic
211C4—42no—
212C4—1072no—
213C4—4282no—
214C4—4282no—
215C4—2141no—
216C4—2141no—
217C4SD16×D10712yes—
218C4#1681213012yes—
219C4C2×#1313381712yes—
220C2—82no—
221C2—2142no—
222C2#2029317511yescharacteristic
223C2C2×SD16×D10711yescharacteristic
224C2C2×#1681213011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
prime radical#20293175C2
hypercenter#492D107
hypocenterC107#3167
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20293025C2semidirect
4. #20292977C2semidirect
5. #20292973C2semidirect
6. #20292853C2semidirect
7. SD16×Dic107C2semidirect
8. C4×C8×D107C2semidirect
9. #492×C107C2semidirect
86. C4×D107SD16semidirect
89. C8×C107#41semidirect
113. C107#3167semidirect
117. #492D107semidirect
203. C8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931752
#202930252
#202929772
#202929732
#202928532
SD16×Dic1072
C4×C8×D1072
#492×C1072
#3167107