Finite Group Databasebeta

← #28335766

Subgroups — #28335766

Order 13696.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293369C211yesmaximal, characteristic
2#20293019C211yesmaximal, characteristic, complemented
3#20292989C211yesmaximal, characteristic, complemented
4#20292367C211yesmaximal, characteristic, complemented
5#20292366C211yesmaximal, characteristic, complemented
6#20292366C211yesmaximal, characteristic, complemented
7#489×C107C211yesmaximal, characteristic, complemented
8C2×#13133808—21no—
9C2×#13133805—21no—
10#16812098—21no—
11#16812097C2211yescharacteristic
12#16812097C2211yescharacteristic
13#16812095—21no—
14#16812095—21no—
15#16812079—21no—
16#16812079—21no—
17#16812077—21no—
18#16812032—21no—
19#16812029—21no—
20#16812020C2211yescharacteristic, complemented
21#16812020C2211yescharacteristic, complemented
22C4⋊Q8×C107C2211yescharacteristic, complemented
23#124×C107C2211yescharacteristic
24#103×C107—21no—
25#102×C107—21no—
26#97×C107C2211yescharacteristic, complemented
27C23×D107—21no—
28C2×Dic214D411yescharacteristic
29C2×Dic214D411yescharacteristic
30C2×D428—21no—
31C2×D428D411yescharacteristic
32C2×D428D411yescharacteristic
33C2×C22⋊D107—21no—
34C2×C22⋊D107—21no—
35#13133808—42no—
36#13133805—42no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C22⋊Dic107—21no—
40(C2×D107)⋊C4—21no—
41(C2×D107)⋊C4—21no—
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44(C2×D107)⋊C4—21no—
45(C2×D107)⋊C4—21no—
46C4×Dic107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107D411yescharacteristic
49C2×D4×C107D411yescharacteristic
50C2×C8×C107—21no—
51C4⋊C4×C107—21no—
52C4⋊C4×C107—21no—
53C22⋊C4×C107—21no—
54C22⋊C4×C107—21no—
55C42×C107C2311yescharacteristic
56C23×C107—21no—
57C22×D107—21no—
58C22×D107—21no—
59C22×D107—81no—
60C22×D107—81no—
61C22×D107—81no—
62Q8×C107—41no—
63Q8×C107—41no—
64Q8×C107—41no—
65D4×C107—41no—
66C2×C4×C107—21no—
67C2×C4×C107—41no—
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C22⋊D107—81no—
72C22⋊D107—81no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75D428—41no—
76D428—41no—
77D428—81no—
78Dic214—41no—
79Dic214—41no—
80C8×C107—41no—
81C107⋊C8—22no—
82C107⋊C8—41no—
83C22×C107—41no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107#12011yescharacteristic
87C2×D107—41no—
88C2×D107—41no—
89C2×D107—41no—
90C2×D107—41no—
91C2×D107—41no—
92C2×D107—41no—
93C2×D107—84no—
94C4×C107—21no—
95C4×C107—21no—
96C4×C107—21no—
97C4×C107—41no—
98C4×C107—81no—
99Dic107—41no—
100Dic107—41no—
101C2×C107—81no—
102C2×C107#45311yescharacteristic
103C2×C107#45511yescharacteristic
104C2×C107#45511yescharacteristic
105D107—81no—
106D107—81no—
107#1659—1071nomaximal, Sylow
108C107#165911yescharacteristic, complemented, Sylow
109H(3,𝔽4)—1071no—
110#497—1071no—
111#489—1071no—
112#489D10711yescharacteristic, complemented
113#329—1071no—
114#329—1071no—
115#329—1071no—
116C2×Q16—2141no—
117C2×SD16—2141no—
118C4⋊Q8C2×D10711yescharacteristic, complemented
119#124—1071no—
120#124—1071no—
121#124—2141no—
122#124—2141no—
123#124—2141no—
124#124C2×D10711yescharacteristic
125#120—2141no—
126#120—2141no—
127#120—2141no—
128#103—21no—
129#103—2141no—
130#102—21no—
131#102—2141no—
132#97—1071no—
133#97—1071no—
134#97C2×D10711yescharacteristic, complemented
135C24—2141no—
136C2×Q8—21no—
137C2×Q8—1071no—
138C2×Q8—1071no—
139C2×Q8C22⋊D10711yescharacteristic
140C2×D4—1071no—
141C2×D4—1071no—
142C2×D4—2141no—
143C2×D4—2141no—
144C2×D4—2141no—
145C2×D4C22⋊D10711yescharacteristic
146Q16—4282no—
147SD16—4282no—
148C2×C8—21no—
149C2×C8—2141no—
150C2×C8—2141no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—2141no—
156C22⋊C4—2141no—
157C22⋊C4—2141no—
158C22⋊C4—2141no—
159C22⋊C4—2141no—
160C22⋊C4—2141no—
161C22⋊C4—2141no—
162C42—2141no—
163C42C22×D10711yescharacteristic
164C23—21no—
165C23—8561no—
166C23—8561no—
167C23—8561no—
168C23—2141no—
169C23—2141no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—4281no—
174Q8—4281no—
175D4—41no—
176D4—8561no—
177D4—8561no—
178D4—8561no—
179D4—4281no—
180D4—4281no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—2141no—
184C2×C4—2141no—
185C2×C4D4×D10711yescharacteristic
186C2×C4D4×D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C8—41no—
189C8—4281no—
190C8—2142no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—8564no—
195C22—4281no—
196C22—4281no—
197C22—4281no—
198C22—4281no—
199C22—4281no—
200C22—4281no—
201C22#1681207711yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—4281no—
208C4—4281no—
209C2—81no—
210C2—8561no—
211C2—8561no—
212C2#2029279311yescharacteristic
213C2#2029280311yescharacteristic
214C2#2029280411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#489×C107C2
socleC22×C107#120
prime radical#20293369C2
hypercenter#489D107
hypocenterC107#1659
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #20293019C2semidirect
3. #20292989C2semidirect
4. #20292367C2semidirect
5. #20292366C2semidirect
6. #20292366C2semidirect
7. #489×C107C2semidirect
20. #16812020C22semidirect
21. #16812020C22semidirect
22. C4⋊Q8×C107C22semidirect
26. #97×C107C22semidirect
108. C107#1659semidirect
112. #489D107semidirect
118. C4⋊Q8C2×D107semidirect
134. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933692
#202930192
#202929892
#202923672
#202923662
#489×C1072
#1659107