Finite Group Databasebeta

← #28335573

Subgroups — #28335573

Order 13696.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293347C211yesmaximal, characteristic, complemented
2#20292974C211yesmaximal, characteristic, complemented
3#20292828C211yesmaximal, characteristic, complemented
4#20292822C211yesmaximal, characteristic
5#20292372C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133807—21no—
9C2×#13133803—21no—
10#16812109C2211yescharacteristic
11#16812106C2211yescharacteristic
12#16812102—22no—
13#16812096—21no—
14#16812088—21no—
15C4×Dic214C2211yescharacteristic
16#16812074—21no—
17#16812068—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812031—22no—
20#16812031—21no—
21#16812029—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30#13133807—42no—
31#13133803—42no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107D411yescharacteristic
35C4⋊Dic107D412yes—
36Dic107⋊C4—22no—
37Dic107⋊C4—21no—
38C22⋊Dic107—22no—
39C22⋊Dic107—22no—
40C4×Dic107—22no—
41C4×Dic107—21no—
42C2×D4×C107D411yescharacteristic
43C22×C4×C107—21no—
44C2×C8×C107—21no—
45C2×C8×C107—21no—
46C4⋊C4×C107D411yescharacteristic
47C22⋊C4×C107—21no—
48C42×C107C2311yescharacteristic
49C23×C107—21no—
50C22×D107—41no—
51D4×C107—21no—
52D4×C107—21no—
53C2×C4×C107—21no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—21no—
60C2×Dic107—21no—
61C2×Dic107—22no—
62D428—42no—
63D428—41no—
64D428—41no—
65D428—41no—
66D428—41no—
67Dic214—21no—
68Dic214—21no—
69Dic214—41no—
70Dic214—41no—
71C8×C107—41no—
72C8×C107—41no—
73C107⋊C8—41no—
74C22×C107—41no—
75C22×C107—41no—
76C22×C107—41no—
77C22×C107#12011yescharacteristic
78C2×D107—81no—
79C2×D107—81no—
80C2×D107—81no—
81C4×C107—21no—
82C4×C107—41no—
83C4×C107#13511yescharacteristic
84C4×C107#13511yescharacteristic
85C4×C107AGL(1,ℤ/8)11yescharacteristic
86C4×C107AGL(1,ℤ/8)11yescharacteristic
87Dic107—41no—
88Dic107—42no—
89Dic107—41no—
90C2×C107—81no—
91C2×C107#44911yescharacteristic
92C2×C107#44911yescharacteristic
93C2×C107#45411yescharacteristic
94D107—161no—
95#1628—1071nomaximal, Sylow
96C107#162811yescharacteristic, complemented, Sylow
97#556—1071no—
98#486—1071no—
99#460—1071no—
100#459—1071no—
101#332—1071no—
102Q8⋊C8—1071no—
103D4⋊C8D10711yescharacteristic, complemented
104C2×SD16—2141no—
105C2×D8—2141no—
106C4⋊Q8—1071no—
107#127—1071no—
108#126—2142no—
109#124—2141no—
110C22⋊Q8—2141no—
111C4×Q8—1071no—
112C4×Q8—2141no—
113C4×D4C2×D10711yescharacteristic
114#117—2141no—
115C4⋊C8—1071no—
116C4⋊C8C2×D10711yescharacteristic
117#102—2141no—
118#102—2142no—
119#102—2141no—
120C4×C8C2×D10711yescharacteristic
121C2×Q8—1071no—
122C2×Q8—2141no—
123C2×D4—2141no—
124C2×D4—2141no—
125C2×D4—2141no—
126C2×D4D42811yescharacteristic
127C22×C4—21no—
128SD16—4282no—
129D8—4282no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—2141no—
133C4⋊C4—1071no—
134C4⋊C4—1072no—
135C4⋊C4—2142no—
136C4⋊C4—2141no—
137C4⋊C4—2141no—
138C4⋊C4D42811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2142no—
141C22⋊C4—2142no—
142C42—2142no—
143C42—2141no—
144C42C22×D10711yescharacteristic
145C23—21no—
146C23—4281no—
147Q8—4281no—
148Q8—4281no—
149Q8—2141no—
150Q8—2141no—
151D4—21no—
152D4—21no—
153D4—4282no—
154D4—4281no—
155D4—4281no—
156D4—4281no—
157D4—4281no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—2141no—
162C2×C4—2142no—
163C2×C4—2141no—
164C2×C4D4×D10711yescharacteristic
165C2×C4D4×D10711yescharacteristic
166C2×C4C2×D42811yescharacteristic
167C8—41no—
168C8—41no—
169C8—4281no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—8561no—
174C22—8561no—
175C22—8561no—
176C22#1681207911yescharacteristic
177C4—21no—
178C4—41no—
179C4—4281no—
180C4—4282no—
181C4—4281no—
182C4#1681213011yescharacteristic
183C4#1681213111yescharacteristic
184C4#1681213311yescharacteristic
185C4#1681213711yescharacteristic
186C2—81no—
187C2—17121no—
188C2#2029277311yescharacteristic
189C2#2029277411yescharacteristic
190C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radical#20292822C2
hypercenterD4⋊C8D107
hypocenterC107#1628
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. #20293347C2semidirect
2. #20292974C2semidirect
3. #20292828C2semidirect
5. #20292372C2semidirect
6. #20292359C2semidirect
7. D4⋊C8×C107C2semidirect
96. C107#1628semidirect
103. D4⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202933472
#202929742
#202928282
#202928222
#202923722
#202923592
D4⋊C8×C1072
#1628107