Finite Group Databasebeta

← #28335563

Subgroups — #28335563

Order 13696.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic214C211yesmaximal, characteristic, complemented
2#20292973C211yesmaximal, characteristic, complemented
3#20292824C211yesmaximal, characteristic, complemented
4#20292372C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
7#459×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9C2×#13133806—21no—
10#16812109C2211yescharacteristic
11#16812108—21no—
12#16812088—21no—
13#16812086—22no—
14C4×Dic214C2211yescharacteristic
15D4×Dic107—21no—
16C4⋊(C107⋊C8)C2211yescharacteristic
17#16812030—22no—
18#16812030—21no—
19C4×C107⋊C8C2211yescharacteristic
20C2×D8×C107—21no—
21#127×C107C2211yescharacteristic
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24#102×C107—21no—
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214—42no—
28C2×Dic214—42no—
29C2×Dic214D411yescharacteristic
30C22×Dic107—22no—
31#13133806—42no—
32C2×C107⋊C8—21no—
33C2×C107⋊C8—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107D411yescharacteristic
36C4⋊Dic107D412yes—
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39C22⋊Dic107—22no—
40C4×Dic107—21no—
41C2×D4×C107—21no—
42C2×D4×C107—21no—
43C2×D4×C107—21no—
44C2×D4×C107D411yescharacteristic
45C22×C4×C107—21no—
46D8×C107—42no—
47C2×C8×C107—21no—
48C4⋊C4×C107D411yescharacteristic
49C22⋊C4×C107—21no—
50C42×C107C2311yescharacteristic
51C23×C107—21no—
52C23×C107—41no—
53D4×C107—21no—
54D4×C107—42no—
55D4×C107—41no—
56D4×C107—41no—
57D4×C107—41no—
58D4×C107—41no—
59D4×C107SD1612yescomplemented
60C2×C4×C107—21no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—22no—
69C2×Dic107—44no—
70Dic214—21no—
71Dic214—44no—
72Dic214—41no—
73Dic214—41no—
74Dic214D812yescomplemented
75C8×C107—41no—
76C107⋊C8—41no—
77C107⋊C8—41no—
78C22×C107—22no—
79C22×C107—41no—
80C22×C107—41no—
81C22×C107—81no—
82C22×C107—81no—
83C22×C107—81no—
84C22×C107#12011yescharacteristic
85C4×C107—21no—
86C4×C107—41no—
87C4×C107C2×D811yescharacteristic
88C4×C107C2×SD1611yescharacteristic
89C4×C107AGL(1,ℤ/8)11yescharacteristic
90C4×C107#13711yescharacteristic
91Dic107—22no—
92Dic107—42no—
93Dic107—41no—
94C2×C107—42no—
95C2×C107—161no—
96C2×C107#44711yescharacteristic
97C2×C107#44811yescharacteristic
98C2×C107#45411yescharacteristic
99#1627—1071nomaximal, Sylow
100C107#162711yescharacteristic, complemented, Sylow
101D4×Q8—1071no—
102#486—1071no—
103#460—1071no—
104#459D10711yescharacteristic
105#332—1071no—
106Q8⋊C8—1071no—
107D4⋊C8—1071no—
108C22×Q8—2141no—
109C2×SD16—2141no—
110C2×D8—21no—
111C4⋊Q8—1071no—
112C4⋊Q8—2141no—
113#127C2×D10711yescharacteristic
114C22⋊Q8—2142no—
115C22⋊Q8—2141no—
116C4×Q8—1071no—
117C4×D4—2141no—
118C4×D4C2×D10711yescharacteristic
119C4⋊C8—1071no—
120C4⋊C8C2×D10711yescharacteristic
121#102—21no—
122#102—2141no—
123#102—2142no—
124C4×C8—1071no—
125C2×Q8—1071no—
126C2×Q8—4282no—
127C2×Q8—4282no—
128C2×Q8—2142no—
129C2×Q8—2141no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D10711yescharacteristic
134C22×C4—21no—
135C22×C4—2142no—
136SD16—4282no—
137D8—42no—
138C2×C8—21no—
139C2×C8—2141no—
140C2×C8—2141no—
141C4⋊C4—1071no—
142C4⋊C4—1072no—
143C4⋊C4—2142no—
144C4⋊C4—2141no—
145C4⋊C4—2141no—
146C4⋊C4C22⋊D10711yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—2142no—
149C42—2141no—
150C42C22×D10711yescharacteristic
151C23—21no—
152C23—41no—
153Q8—1072no—
154Q8—4284no—
155Q8—4281no—
156Q8—4281no—
157Q8—2141no—
158D4—21no—
159D4—42no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4#1313380612yescomplemented
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—4284no—
169C2×C4—2141no—
170C2×C4—2142no—
171C2×C4—2141no—
172C2×C4D4×D10711yescharacteristic
173C2×C4D4×D10711yescharacteristic
174C2×C4C2×C22⋊D10711yescharacteristic
175C8—41no—
176C8—4281no—
177C8—4281no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22#1681207711yescharacteristic
185C4—21no—
186C4—41no—
187C4—4282no—
188C4—4281no—
189C4—2142no—
190C4D8×D10711yescharacteristic
191C4C2×#1313380611yescharacteristic
192C4#1681213311yescharacteristic
193C4#1681213911yescharacteristic
194C2—42no—
195C2—161no—
196C2#2029275911yescharacteristic
197C2#2029276511yescharacteristic
198C2#2029279811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting, prime radical#459×C107C2
socleC22×C107#120
hypercenter#459D107
hypocenterC107#1627
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. D4×Dic214C2semidirect
2. #20292973C2semidirect
3. #20292824C2semidirect
4. #20292372C2semidirect
5. #20292357C2semidirect
6. D4⋊(C107⋊C8)C2semidirect
59. D4×C107SD16semidirect
74. Dic214D8semidirect
100. C107#1627semidirect
164. D4#13133806semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202929732
#202928242
#202923722
#202923572
D4⋊(C107⋊C8)2
#459×C1072
#1627107