Finite Group Databasebeta

← #28336434

Subgroups — #28336434

Order 13696.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16812156C211yesmaximal, characteristic
2#20292645C212yesmaximal, complemented
3#20292635C212yesmaximal, complemented
4#20292611C211yesmaximal, characteristic, complemented
5#410×C107C211yesmaximal, characteristic, complemented
6#16812156—24no—
7#16812156—24no—
8C2×#13133818C412yescomplemented
9C2×#13133818C2211yescharacteristic
10C2×#13133817—21no—
11C2×#13133817—22no—
12C22×Dic214C412yescomplemented
13C22×Dic214C2211yescharacteristic
14C2×Q8×D107—21no—
15C2×#13133802C2212yes—
16#16812037—24no—
17#16812032—24no—
18#16812028—21no—
19#16812026—21no—
20C2×#41×C107C2211yescharacteristic
21#117×C107C2212yes—
22#99×C107—22no—
23#13133818—22no—
24#13133818—24no—
25#13133818—24no—
26#13133818D414yes—
27#13133817—216no—
28#13133817—28no—
29C2×Dic214—22no—
30C2×Dic214—21no—
31C2×Dic214—22no—
32C2×Dic214—24no—
33C2×Dic214—24no—
34C2×Dic214C2×C411yescharacteristic
35C2×Dic214C2×C412yes—
36C2×Dic214D414yes—
37Q8×D107—28no—
38C2×D428C2×C411yescharacteristic
39C2×C22⋊D107—22no—
40C2×C22⋊D107—21no—
41C22×Dic107—22no—
42C22×Dic107—21no—
43C2×C4×D107—22no—
44C2×C4×D107—21no—
45#13133802—22no—
46#13133802—22no—
47C2×C107⋊C8—22no—
48#41×C107—22no—
49#41×C107—22no—
50C2×Q8×C107D411yescharacteristic
51C2×D4×C107D411yescharacteristic
52C2×D4×C107D412yes—
53C22×C4×C107C2×C412yes—
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—22no—
56C22⋊C4×C107—22no—
57C42×C107—22no—
58C23×C107C22⋊C412yes—
59C23×C107C2×D411yescharacteristic
60C22×D107—21no—
61Q8×C107—22no—
62D4×C107—24no—
63D4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—42no—
67C2×C4×C107C22⋊C412yes—
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D411yescharacteristic
72C22⋊D107—28no—
73C22⋊D107—24no—
74C2×Dic107—28no—
75C2×Dic107—22no—
76C2×Dic107—21no—
77C2×Dic107—24no—
78D428—21no—
79D428C22⋊C412yes—
80C4×D107—28no—
81C4×D107—24no—
82Dic214—28no—
83Dic214—22no—
84Dic214—21no—
85Dic214—24no—
86Dic214C22⋊C412yes—
87Dic214C22⋊C414yes—
88C107⋊C8—42no—
89C22×C107—22no—
90C22×C107—41no—
91C22×C107—42no—
92C22×C107C2×C22⋊C411yescharacteristic
93C22×C107#12012yes—
94C2×D107—22no—
95C2×D107—42no—
96C4×C107—22no—
97C4×C107—82no—
98C4×C107C2×C22⋊C411yescharacteristic
99C4×C107C2×C22⋊C411yescharacteristic
100C4×C107#12012yes—
101Dic107—22no—
102Dic107—24no—
103C2×C107—22no—
104C2×C107—21no—
105C2×C107—42no—
106C2×C107#37911yescharacteristic
107D107—42no—
108#1796—1071nomaximal, Sylow
109C107#179611yescharacteristic, complemented, Sylow
110C2×2−1+4—1071no—
111#421—1072no—
112#419—1072no—
113#413—1071no—
114#410D10711yescharacteristic, complemented
1152−1+4—2144no—
1162−1+4—2144no—
117C2×#41—1072no—
118C2×#41—1071no—
119C2×#41—2141no—
120C2×#41—2142no—
121C2×#41C2×D10711yescharacteristic
122C22×Q8—1072no—
123C22×Q8—1071no—
124C22×Q8—2141no—
125C2×M16—1072no—
126#117C2×D10712yes—
127#104—2144no—
128#103—2144no—
129#101—2141no—
130#100—2141no—
131#99—22no—
132#41—22no—
133#41—22no—
134#41—1074no—
135#41—2144no—
136#41—2144no—
137#41—21416no—
138#41—2148no—
139#41—2142no—
140C2×Q8—1072no—
141C2×Q8—1071no—
142C2×Q8—1074no—
143C2×Q8—2144no—
144C2×Q8—2144no—
145C2×Q8—2148no—
146C2×Q8—2142no—
147C2×Q8—2142no—
148C2×Q8—2141no—
149C2×Q8D42811yescharacteristic
150C2×D4—1071no—
151C2×D4—2142no—
152C2×D4—2141no—
153C2×D4D42811yescharacteristic
154C2×D4C22⋊D10712yes—
155C22×C4—2142no—
156C22×C4—2141no—
157C22×C4—2142no—
158C22×C4—2141no—
159C22×C4C4×D10712yes—
160C22×C4C22×D10711yescharacteristic
161M16—2142no—
162M16—2142no—
163C2×C8—2142no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—2141no—
168C23(C2×D107)⋊C412yes—
169C23C2×C22⋊D10711yescharacteristic
170Q8—22no—
171Q8—1074no—
172Q8—1072no—
173Q8—2148no—
174Q8—2142no—
175Q8—2141no—
176Q8—2144no—
177D4—24no—
178D4—22no—
179D4—1072no—
180D4—2148no—
181D4—2144no—
182D4—2141no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—2148no—
187C2×C4—2148no—
188C2×C4—2141no—
189C2×C4—2142no—
190C2×C4—2144no—
191C2×C4—2144no—
192C2×C4(C2×D107)⋊C412yes—
193C2×C4C2×C4×D10711yescharacteristic
194C2×C4D4×D10712yes—
195C2×C4D4×D10712yes—
196C2×C4C2×D42811yescharacteristic
197C8—4282no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—4282no—
202C22—2142no—
203C22C2×(C2×D107)⋊C411yescharacteristic
204C22#1681207712yes—
205C4—22no—
206C4—82no—
207C4—2144no—
208C4—2142no—
209C4C22⋊C4×D10711yescharacteristic
210C4C22⋊C4×D10711yescharacteristic
211C4#1681207912yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—4282no—
216C2#2029248211yescharacteristic
subgroupG/H
centerC2#20292482
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#379
prime radicalC2×#16812156C2
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1796
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20292645C2semidirect
3. #20292635C2semidirect
4. #20292611C2semidirect
5. #410×C107C2semidirect
8. C2×#13133818C4semidirect
12. C22×Dic214C4semidirect
109. C107#1796semidirect
114. #410D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
#202926452
#202926352
#202926112
#410×C1072
#1796107