Finite Group Databasebeta

← #28335552

Subgroups — #28335552

Order 13696.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293281C211yesmaximal, characteristic, complemented
2#20292974C211yesmaximal, characteristic, complemented
3#20292823C211yesmaximal, characteristic, complemented
4#20292822C211yesmaximal, characteristic, complemented
5#20292372C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9C2×D856—21no—
10C2×#13133803—21no—
11#16812109C2211yescharacteristic
12#16812106C2211yescharacteristic
13#16812093—22no—
14#16812089—21no—
15#16812083—21no—
16C4×D428C2211yescharacteristic
17C2×C4⋊Dic107—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812031—22no—
20#16812031—21no—
21#16812029—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428D411yescharacteristic
32C2×C22⋊D107—22no—
33C22×Dic107—22no—
34C2×C4×D107—21no—
35D856—42no—
36#13133803—42no—
37C2×C107⋊C8—21no—
38C4⋊Dic107—22no—
39C4⋊Dic107—21no—
40C4⋊Dic107D411yescharacteristic
41C4⋊Dic107D412yes—
42(C2×D107)⋊C4—22no—
43(C2×D107)⋊C4—21no—
44C2×D4×C107D411yescharacteristic
45C22×C4×C107—21no—
46C2×C8×C107—21no—
47C2×C8×C107—21no—
48C4⋊C4×C107D411yescharacteristic
49C22⋊C4×C107—21no—
50C42×C107C2311yescharacteristic
51C23×C107—21no—
52C22×D107—21no—
53C22×D107—41no—
54D4×C107—21no—
55D4×C107D812yescomplemented
56C2×C4×C107—21no—
57C2×C4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107C2×D411yescharacteristic
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—44no—
63C2×Dic107—21no—
64C2×Dic107—22no—
65C2×Dic107—44no—
66D428—21no—
67D428—21no—
68D428—42no—
69D428—41no—
70D428—41no—
71D428—41no—
72D428—41no—
73C4×D107—41no—
74C4×D107—41no—
75Dic214—41no—
76Dic214—41no—
77C8×C107—41no—
78C8×C107—41no—
79C107⋊C8—41no—
80C22×C107—22no—
81C22×C107—41no—
82C22×C107—41no—
83C22×C107#12011yescharacteristic
84C2×D107—41no—
85C2×D107—41no—
86C2×D107—41no—
87C2×D107—81no—
88C2×D107—81no—
89C2×D107—81no—
90C4×C107—21no—
91C4×C107—41no—
92C4×C107C2×D811yescharacteristic
93C4×C107#13511yescharacteristic
94C4×C107AGL(1,ℤ/8)11yescharacteristic
95C4×C107AGL(1,ℤ/8)11yescharacteristic
96Dic107—42no—
97Dic107—41no—
98C2×C107—42no—
99C2×C107#44711yescharacteristic
100C2×C107#44911yescharacteristic
101C2×C107#45411yescharacteristic
102D107—81no—
103D107—161no—
104#1625—1071nomaximal, Sylow
105C107#162511yescharacteristic, complemented, Sylow
106#547—1071no—
107#486—1071no—
108#459—1071no—
109#459—1071no—
110#332—1071no—
111D4⋊C8—1071no—
112D4⋊C8D10711yescharacteristic, complemented
113C2×#41—2141no—
114C2×D8—2141no—
115C2×D8—2141no—
116C4⋊Q8—1071no—
117#127—1071no—
118#123—2142no—
119C22⋊Q8—2141no—
120#121—2141no—
121C4×D4—1071no—
122C4×D4C2×D10711yescharacteristic
123C2×C4⋊C4—2141no—
124C4⋊C8—1071no—
125C4⋊C8C2×D10711yescharacteristic
126#102—2141no—
127#102—2142no—
128#102—2141no—
129C4×C8C2×D10711yescharacteristic
130#41—4282no—
131#41—4282no—
132C2×Q8—2141no—
133C2×D4—1071no—
134C2×D4—2142no—
135C2×D4—2141no—
136C2×D4—2141no—
137C2×D4—2141no—
138C2×D4D42811yescharacteristic
139C22×C4—21no—
140C22×C4—2142no—
141C22×C4—2141no—
142D8—4282no—
143D8—4282no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—2141no—
147C4⋊C4—1071no—
148C4⋊C4—1072no—
149C4⋊C4—2142no—
150C4⋊C4—2141no—
151C4⋊C4D42811yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2142no—
154C22⋊C4—2141no—
155C42C22×D10711yescharacteristic
156C23—21no—
157C23—4281no—
158C23—2141no—
159Q8—4281no—
160Q8—4281no—
161D4—21no—
162D4—4282no—
163D4—4284no—
164D4—4281no—
165D4—4281no—
166D4—4281no—
167D4—4281no—
168D4—2141no—
169D4—2141no—
170D4D85612yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—4284no—
175C2×C4—4281no—
176C2×C4—4281no—
177C2×C4—2141no—
178C2×C4—2142no—
179C2×C4D4×D10711yescharacteristic
180C2×C4D4×D10711yescharacteristic
181C2×C4C2×D42811yescharacteristic
182C8—41no—
183C8—41no—
184C8—4281no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—8561no—
189C22—8561no—
190C22—8561no—
191C22—4281no—
192C22—4281no—
193C22—4281no—
194C22#1681207911yescharacteristic
195C4—21no—
196C4—41no—
197C4—4281no—
198C4—4282no—
199C4C2×D85611yescharacteristic
200C4#1681213011yescharacteristic
201C4#1681213311yescharacteristic
202C4#1681213711yescharacteristic
203C2—42no—
204C2—8561no—
205C2—17121no—
206C2#2029276311yescharacteristic
207C2#2029277311yescharacteristic
208C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
hypercenterD4⋊C8D107
hypocenterC107#1625
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20293281C2semidirect
2. #20292974C2semidirect
3. #20292823C2semidirect
4. #20292822C2semidirect
5. #20292372C2semidirect
6. #20292356C2semidirect
7. D4⋊C8×C107C2semidirect
55. D4×C107D8semidirect
105. C107#1625semidirect
112. D4⋊C8D107semidirect
170. D4D856semidirect

Maximal subgroups

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

subgroupindex
#202932812
#202929742
#202928232
#202928222
#202923722
#202923562
D4⋊C8×C1072
#1625107