Finite Group Databasebeta

← #28334987

Subgroups — #28334987

Order 13696.

242 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 73 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#20293161C211yesmaximal, characteristic
2C4×C4⋊Dic107C211yesmaximal, characteristic, complemented
3#20292365C212yesmaximal, complemented
4#20292362C212yesmaximal
5#432×C107C211yesmaximal, characteristic, complemented
6#16812103—21no—
7#16812101C411yescharacteristic, complemented
8#16812097C411yescharacteristic, complemented
9#16812092—21no—
10#16812088C2211yescharacteristic
11#16812083C2211yescharacteristic
12C4×Dic214C411yescharacteristic, complemented
13C4×D428C411yescharacteristic, complemented
14C4×C22⋊D107—21no—
15#16812066—21no—
16C2×C4⋊Dic107—21no—
17C2×C4×Dic107—21no—
18C107⋊#95C2212yes—
19C2×C42×C107C2211yescharacteristic
20C4⋊C8×C107—21no—
21C22⋊C8×C107C2212yes—
22#97×C107—21no—
23C2×Dic214C2×C411yescharacteristic
24C2×D428C2×C411yescharacteristic
25C2×C22⋊D107—21no—
26C22×Dic107—22no—
27C2×C4×D107—21no—
28C4⋊Dic107—24no—
29C4⋊Dic107C2×C411yescharacteristic
30C4⋊Dic107C2×C411yescharacteristic
31Dic107⋊C4—21no—
32Dic107⋊C4—21no—
33Dic107⋊C4—21no—
34Dic107⋊C4—21no—
35C22⋊Dic107—21no—
36(C2×D107)⋊C4—21no—
37(C2×D107)⋊C4—21no—
38(C2×D107)⋊C4—21no—
39(C2×D107)⋊C4—21no—
40C4×Dic107—21no—
41C4×Dic107—42no—
42C22×C4×C107D411yescharacteristic
43C22×C4×C107D411yescharacteristic
44C22×C4×C107C2311yescharacteristic
45C2×C8×C107—22no—
46C42×C107C2×C411yescharacteristic
47C42×C107C2×C411yescharacteristic
48C42×C107D411yescharacteristic
49C42×C107D411yescharacteristic
50C23×C107C2×D411yescharacteristic
51C22×D107—21no—
52C2×C4×C107—21no—
53C2×C4×C107—21no—
54C2×C4×C107—22no—
55C2×C4×C107—41no—
56C2×C4×C107C22⋊C411yescharacteristic
57C2×C4×C107C22⋊C411yescharacteristic
58C2×C4×C107C22⋊C411yescharacteristic
59C2×C4×C107C22⋊C411yescharacteristic
60C2×C4×C107C22×C411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—41no—
63C22⋊D107—41no—
64C2×Dic107—24no—
65C2×Dic107—21no—
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—42no—
69C2×Dic107—42no—
70D428—41no—
71C4×D107—42no—
72Dic214—41no—
73C8×C107—42no—
74C22×C107—21no—
75C22×C107—21no—
76C22×C107C2×C22⋊C411yescharacteristic
77C22×C107AGL(1,ℤ/8)11yescharacteristic
78C22×C107#13711yescharacteristic
79C2×D107—41no—
80C2×D107—41no—
81C2×D107—41no—
82C4×C107—21no—
83C4×C107—21no—
84C4×C107—21no—
85C4×C107—21no—
86C4×C107—21no—
87C4×C107#10412yes—
88Dic107—44no—
89Dic107—41no—
90Dic107—41no—
91Dic107—41no—
92C2×C107—21no—
93C2×C107—21no—
94C2×C107#41011yescharacteristic
95C2×C107#41711yescharacteristic
96C2×C107C2×#10411yescharacteristic
97D107—81no—
98#1509—1071nomaximal, Sylow
99C107#150911yescharacteristic, complemented, Sylow
100#529—1071no—
101#432D10711yescharacteristic, complemented
102C4×C4⋊C4—1071no—
103#328—1072no—
104#327—1072no—
105#126—2141no—
106#125—1071no—
107#124—1071no—
108#123—2141no—
109C22⋊Q8—1071no—
110#121—1071no—
111C4×Q8—1071no—
112C4×D4—1071no—
113C4×D4—2141no—
114#117—2141no—
115C2×C4⋊C4—2141no—
116C2×C42—2141no—
117C2×C42C2×D10711yescharacteristic
118C4⋊C8—21no—
119C22⋊C8C2×D10712yes—
120#97—21no—
121#95—1072no—
122C2×Q8—1071no—
123C2×D4—1071no—
124C2×D4—2141no—
125C22×C4—2142no—
126C22×C4—2141no—
127C22×C4D42811yescharacteristic
128C22×C4D42811yescharacteristic
129C22×C4C22×D10711yescharacteristic
130C2×C8—22no—
131C4⋊C4—1071no—
132C4⋊C4—1071no—
133C4⋊C4—2144no—
134C4⋊C4—2141no—
135C4⋊C4—2141no—
136C4⋊C4—2141no—
137C4⋊C4—2141no—
138C22⋊C4—2141no—
139C22⋊C4—2141no—
140C22⋊C4—2141no—
141C22⋊C4—2141no—
142C22⋊C4—2141no—
143C42—4282no—
144C42—2141no—
145C42C4×D10711yescharacteristic
146C42C4×D10711yescharacteristic
147C42C22⋊D10711yescharacteristic
148C42C22⋊D10711yescharacteristic
149C23—2141no—
150C23C2×D42811yescharacteristic
151Q8—4281no—
152D4—4281no—
153D4—4281no—
154D4—4281no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—4282no—
160C2×C4—4282no—
161C2×C4—4282no—
162C2×C4—2144no—
163C2×C4—2141no—
164C2×C4—2141no—
165C2×C4—2141no—
166C2×C4(C2×D107)⋊C411yescharacteristic
167C2×C4(C2×D107)⋊C411yescharacteristic
168C2×C4(C2×D107)⋊C411yescharacteristic
169C2×C4(C2×D107)⋊C411yescharacteristic
170C2×C4C2×C4×D10711yescharacteristic
171C2×C4C2×C22⋊D10711yescharacteristic
172C8—42no—
173C22—21no—
174C22—21no—
175C22—4281no—
176C22—4281no—
177C22—4281no—
178C22C2×(C2×D107)⋊C411yescharacteristic
179C22#1681213711yescharacteristic
180C22#1681214211yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—4284no—
187C4—4281no—
188C4—4281no—
189C4—4281no—
190C4#1681203512yes—
191C2—21no—
192C2—21no—
193C2—8561no—
194C2#2029260011yescharacteristic
195C2#2029262911yescharacteristic
196C2C2×#1681203511yescharacteristic
subgroupG/H
centerC22C2×(C2×D107)⋊C4
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#432×C107C2
socleC22×C107C2×C22⋊C4
prime radical#16812083C22
cosocleC22×C4×C107C23
hypercenter#432D107
hypocenterC107#1509
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Dic107C2semidirect
3. #20292365C2semidirect
5. #432×C107C2semidirect
7. #16812101C4semidirect
8. #16812097C4semidirect
12. C4×Dic214C4semidirect
13. C4×D428C4semidirect
99. C107#1509semidirect
101. #432D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
C4×C4⋊Dic1072
#202923652
#202923622
#432×C1072
#1509107