Finite Group Databasebeta

← #28335722

Subgroups — #28335722

Order 13696.

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

normal subgroups only

This group has 234 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
2#20293023C211yesmaximal, characteristic, complemented
3#20292909C211yesmaximal, characteristic, complemented
4#20292372C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic
7#461×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9#16812109C2211yescharacteristic
10#16812108—21no—
11#16812088—21no—
12#16812086—22no—
13C4×Dic214C2211yescharacteristic
14D4×Dic107—21no—
15#16812040—21no—
16#16812040—22no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18#16812032—21no—
19C4×C107⋊C8C2211yescharacteristic
20C2×SD16×C107—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24#103×C107—21no—
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214—42no—
28C2×Dic214—42no—
29C2×Dic214D411yescharacteristic
30C22×Dic107—22no—
31C2×C107⋊C8—21no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107D411yescharacteristic
35C4⋊Dic107D412yes—
36Dic107⋊C4—22no—
37Dic107⋊C4—21no—
38C22⋊Dic107—22no—
39C4×Dic107—21no—
40C2×Q8×C107—21no—
41C2×D4×C107D411yescharacteristic
42C22×C4×C107—21no—
43SD16×C107—42no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107D411yescharacteristic
48C22⋊C4×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—21no—
51Q8×C107—41no—
52Q8×C107—41no—
53D4×C107—21no—
54D4×C107SD1612yes—
55C2×C4×C107—21no—
56C2×C4×C107—41no—
57C2×C4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107C2×D411yescharacteristic
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C2×Dic107—21no—
63C2×Dic107—21no—
64C2×Dic107—22no—
65C2×Dic107—44no—
66Dic214—21no—
67Dic214—44no—
68Dic214—41no—
69Dic214—41no—
70Dic214SD1612yescomplemented
71C8×C107—41no—
72C107⋊C8—22no—
73C107⋊C8—41no—
74C22×C107—22no—
75C22×C107—41no—
76C22×C107—41no—
77C22×C107#12011yescharacteristic
78C4×C107—21no—
79C4×C107—41no—
80C4×C107—81no—
81C4×C107C2×SD1611yescharacteristic
82C4×C107C2×SD1611yescharacteristic
83C4×C107AGL(1,ℤ/8)11yescharacteristic
84C4×C107#13711yescharacteristic
85Dic107—22no—
86Dic107—42no—
87Dic107—41no—
88C2×C107—42no—
89C2×C107#44811yescharacteristic
90C2×C107#45011yescharacteristic
91C2×C107#45611yescharacteristic
92#1651—1071nomaximal, Sylow
93C107#165111yescharacteristic, complemented, Sylow
94D4×Q8—1071no—
95#498—1071no—
96#475—1071no—
97#461D10711yescharacteristic
98#332—1071no—
99Q8⋊C8—1071no—
100D4⋊C8—1071no—
101C22×Q8—2141no—
102C2×SD16—21no—
103C4⋊Q8—1071no—
104C4⋊Q8—2141no—
105C4⋊Q8C2×D10711yescharacteristic
106C22⋊Q8—2142no—
107C22⋊Q8—2141no—
108C4×Q8—1071no—
109C4×D4—2141no—
110C4×D4C2×D10711yescharacteristic
111#106—2141no—
112#106—2142no—
113C4⋊C8—1071no—
114C4⋊C8C2×D10711yescharacteristic
115#103—21no—
116#103—2141no—
117C4×C8—1071no—
118C2×Q8—21no—
119C2×Q8—1071no—
120C2×Q8—4282no—
121C2×Q8—4282no—
122C2×Q8—2142no—
123C2×Q8—2141no—
124C2×D4C22⋊D10711yescharacteristic
125C22×C4—21no—
126C22×C4—2142no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—2141no—
130C2×C8—2141no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1071no—
134C4⋊C4—1072no—
135C4⋊C4—2142no—
136C4⋊C4—2141no—
137C4⋊C4—2141no—
138C4⋊C4C22⋊D10711yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2142no—
141C42—2141no—
142C42C22×D10711yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—1072no—
147Q8—4284no—
148Q8—4281no—
149Q8—4281no—
150Q8—2141no—
151D4—21no—
152D4#1313380612yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—4284no—
158C2×C4—2141no—
159C2×C4—2142no—
160C2×C4—2141no—
161C2×C4D4×D10711yescharacteristic
162C2×C4D4×D10711yescharacteristic
163C2×C4C2×C22⋊D10711yescharacteristic
164C8—41no—
165C8—4281no—
166C8—2142no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1681207711yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—4282no—
175C4—4281no—
176C4—2142no—
177C4SD16×D10711yescharacteristic
178C4C2×#1313380611yescharacteristic
179C4#1681213611yescharacteristic
180C4#1681213911yescharacteristic
181C2—42no—
182C2#2029276511yescharacteristic
183C2#2029277611yescharacteristic
184C2#2029280811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#461×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenter#461D107
hypocenterC107#1651
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20293023C2semidirect
3. #20292909C2semidirect
4. #20292372C2semidirect
5. #20292357C2semidirect
70. Dic214SD16semidirect
93. C107#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202930232
#202929092
#202923722
#202923572
D4⋊(C107⋊C8)2
#461×C1072
#1651107