Finite Group Databasebeta

← #28336427

Subgroups — #28336427

Order 13696.

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

normal subgroups only

This group has 548 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
1#20293463C211yesmaximal, characteristic
2#20292645C212yesmaximal, complemented
3#20292621C212yesmaximal, complemented
4#20292594C211yesmaximal, characteristic, complemented
5#410×C107C211yesmaximal, characteristic, complemented
6#16812157C412yescomplemented
7#16812156—22no—
8#16812155—22no—
9#16812154—22no—
10#16812154C412yescomplemented
11C2×#13133818—22no—
12C2×#13133818C2212yes—
13#41×D107—24no—
14C2×#13133802C2211yescharacteristic, complemented
15C22×C107⋊C8C2211yescharacteristic, complemented
16#16812037—24no—
17#16812032—22no—
18#16812029—22no—
19C22⋊(C107⋊C8)—22no—
20C2×#41×C107C2211yescharacteristic
21#117×C107C2212yescomplemented
22#99×C107—22no—
23#13133819—24no—
24#13133818—24no—
25#13133818—24no—
26#13133818—24no—
27#13133818—22no—
28#13133818—22no—
29#13133818C2×C414yes—
30#13133818D414yes—
31#13133817—28no—
32#13133817—24no—
33C2×Dic214—22no—
34C2×Dic214D412yes—
35Q8×D107—24no—
36C2×D428—22no—
37C2×D428D412yes—
38C2×C22⋊D107—24no—
39C2×C22⋊D107—22no—
40D4×D107—28no—
41D4×D107—24no—
42C2×C4×D107—24no—
43C2×C4×D107—22no—
44#13133802—22no—
45C2×C107⋊C8—22no—
46C2×C107⋊C8—21no—
47C2×C107⋊C8—21no—
48#41×C107—22no—
49#41×C107—22no—
50C2×Q8×C107C2×C411yescharacteristic
51C2×D4×C107C2×C411yescharacteristic
52C2×D4×C107D412yes—
53C22×C4×C107D412yes—
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—22no—
56C22⋊C4×C107—22no—
57C42×C107—22no—
58C23×C107C22⋊C412yes—
59C23×C107C2×D411yescharacteristic
60C22×D107—24no—
61C22×D107—22no—
62Q8×C107—22no—
63D4×C107—24no—
64D4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107—42no—
68C2×C4×C107C22⋊C412yes—
69C2×C4×C107C22×C411yescharacteristic
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D412yes—
72C2×C4×C107C2×D411yescharacteristic
73C22⋊D107—28no—
74C22⋊D107—24no—
75C22⋊D107—24no—
76C2×Dic107—24no—
77C2×Dic107—22no—
78D428—24no—
79D428—22no—
80D428C22⋊C414yes—
81C4×D107—28no—
82C4×D107—24no—
83C4×D107—24no—
84Dic214—24no—
85Dic214—22no—
86Dic214C22⋊C414yes—
87C107⋊C8—41no—
88C107⋊C8—41no—
89C22×C107—22no—
90C22×C107—41no—
91C22×C107—42no—
92C22×C107C2×C22⋊C411yescharacteristic
93C22×C107#12012yes—
94C2×D107—24no—
95C2×D107—82no—
96C2×D107—84no—
97C4×C107—22no—
98C4×C107—82no—
99C4×C107C2×C22⋊C412yes—
100C4×C107#12011yescharacteristic
101C4×C107#12011yescharacteristic
102Dic107—24no—
103C2×C107—22no—
104C2×C107—21no—
105C2×C107—42no—
106C2×C107#37911yescharacteristic
107D107—44no—
108#1794—1071nomaximal, Sylow
109C107#179411yescharacteristic, complemented, Sylow
110#585—1071no—
111#421—1072no—
112#416—1072no—
113#410D10711yescharacteristic, complemented
114#408—1071no—
1152−1+4—1072no—
1162−1+4—2142no—
117H(5,𝔽2)—1072no—
118H(5,𝔽2)—2142no—
119H(5,𝔽2)—2142no—
120C2×#41—1072no—
121C2×#41—2144no—
122C2×#41—2142no—
123C2×#41C2×D10711yescharacteristic
124C2×M16—1071no—
125C22×C8—1071no—
126#117C2×D10712yescomplemented
127#104—2144no—
128#103—2142no—
129#102—2142no—
130#99—22no—
131C22⋊C8—2142no—
132#41—22no—
133#41—22no—
134#41—1074no—
135#41—1074no—
136#41—2144no—
137#41—2144no—
138#41—2144no—
139#41—2148no—
140#41—2144no—
141#41—2144no—
142#41—2142no—
143#41—2142no—
144C2×Q8—1072no—
145C2×Q8—2142no—
146C2×Q8—2144no—
147C2×Q8C4×D10711yescharacteristic
148C2×D4—1072no—
149C2×D4—2142no—
150C2×D4—2144no—
151C2×D4—2148no—
152C2×D4—2144no—
153C2×D4—2142no—
154C2×D4C4×D10711yescharacteristic
155C2×D4C22⋊D10712yes—
156C22×C4—2144no—
157C22×C4—2142no—
158C22×C4D42812yes—
159C22×C4C22×D10711yescharacteristic
160M16—2142no—
161C2×C8—2142no—
162C2×C8—2141no—
163C2×C8—2141no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—2144no—
168C23—2142no—
169C23(C2×D107)⋊C412yes—
170C23C2×C22⋊D10711yescharacteristic
171Q8—22no—
172Q8—1074no—
173Q8—2144no—
174Q8—2142no—
175D4—24no—
176D4—22no—
177D4—1074no—
178D4—2144no—
179D4—2148no—
180D4—2144no—
181D4—2144no—
182D4—2142no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—2148no—
187C2×C4—2144no—
188C2×C4—2142no—
189C2×C4—2144no—
190C2×C4—2144no—
191C2×C4(C2×D107)⋊C412yes—
192C2×C4C2×C4×D10711yescharacteristic
193C2×C4D4×D10712yes—
194C2×C4D4×D10712yes—
195C2×C4C2×D42811yescharacteristic
196C8—4281no—
197C8—4281no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—8562no—
202C22—8564no—
203C22—2144no—
204C22C2×(C2×D107)⋊C411yescharacteristic
205C22#1681207712yes—
206C4—22no—
207C4—82no—
208C4—2144no—
209C4C22⋊C4×D10712yes—
210C4#1681207911yescharacteristic
211C4#1681207911yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—4284no—
216C2#2029248211yescharacteristic
subgroupG/H
centerC4#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#379
prime radical#20293463C2
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1794
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20292645C2semidirect
3. #20292621C2semidirect
4. #20292594C2semidirect
5. #410×C107C2semidirect
6. #16812157C4semidirect
10. #16812154C4semidirect
14. C2×#13133802C22semidirect
15. C22×C107⋊C8C22semidirect
21. #117×C107C22semidirect
109. C107#1794semidirect
113. #410D107semidirect
126. #117C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
#202926452
#202926212
#202925942
#410×C1072
#1794107