Finite Group Databasebeta

← #28335614

Subgroups — #28335614

Order 13696.

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

normal subgroups only

This group has 234 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
2#20292978C211yesmaximal, characteristic, complemented
3#20292918C211yesmaximal, characteristic, complemented
4#20292917C211yesmaximal, characteristic, complemented
5#20292372C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9#16812109C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812093—22no—
12#16812089—21no—
13#16812083—21no—
14C4×D428C2211yescharacteristic
15C2×C4⋊Dic107—21no—
16#16812041—21no—
17#16812040—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812034—22no—
20#16812031—21no—
21#16812030—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×D428D411yescharacteristic
30C2×C22⋊D107—22no—
31C22×Dic107—22no—
32C2×C4×D107—21no—
33C2×C107⋊C8—21no—
34C4⋊Dic107—22no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107D412yes—
39C4⋊Dic107D411yescharacteristic
40(C2×D107)⋊C4—22no—
41(C2×D107)⋊C4—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—21no—
51D4×C107—21no—
52D4×C107SD1612yes—
53C2×C4×C107—21no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—44no—
60C2×Dic107—21no—
61C2×Dic107—22no—
62C2×Dic107—44no—
63C2×Dic107—41no—
64D428—21no—
65D428—21no—
66C4×D107—41no—
67C4×D107—41no—
68Dic214—41no—
69Dic214—41no—
70Dic214—42no—
71C8×C107—41no—
72C8×C107—41no—
73C107⋊C8—41no—
74C22×C107—22no—
75C22×C107—41no—
76C22×C107—41no—
77C22×C107#12011yescharacteristic
78C2×D107—41no—
79C2×D107—41no—
80C2×D107—41no—
81C4×C107—21no—
82C4×C107—41no—
83C4×C107C2×SD1611yescharacteristic
84C4×C107#13511yescharacteristic
85C4×C107AGL(1,ℤ/8)11yescharacteristic
86C4×C107#13711yescharacteristic
87Dic107—42no—
88Dic107—41no—
89Dic107—81no—
90C2×C107—42no—
91C2×C107#45011yescharacteristic
92C2×C107#45211yescharacteristic
93C2×C107#45411yescharacteristic
94D107—81no—
95#1635—1071nomaximal, Sylow
96C107#163511yescharacteristic, complemented, Sylow
97#547—1071no—
98#487—1071no—
99#476—1071no—
100#476—1071no—
101#332—1071no—
102D4⋊C8—1071no—
103D4⋊C8D10711yescharacteristic, complemented
104C2×#41—2141no—
105C4⋊Q8—1071no—
106C4⋊Q8—1071no—
107#123—2142no—
108C22⋊Q8—2141no—
109#121—2141no—
110C4×D4—1071no—
111C4×D4C2×D10711yescharacteristic
112C2×C4⋊C4—2141no—
113#106—2141no—
114#106—2141no—
115C4⋊C8—1071no—
116C4⋊C8C2×D10711yescharacteristic
117#103—2142no—
118#102—2141no—
119#102—2141no—
120C4×C8C2×D10711yescharacteristic
121#41—4282no—
122#41—4282no—
123C2×Q8—2141no—
124C2×Q8—2141no—
125C2×D4—1071no—
126C2×D4—2142no—
127C2×D4D42811yescharacteristic
128C22×C4—21no—
129C22×C4—2142no—
130C22×C4—2141no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2141no—
134C4⋊C4—1072no—
135C4⋊C4—1071no—
136C4⋊C4—2142no—
137C4⋊C4—2141no—
138C4⋊C4—2141no—
139C4⋊C4—2141no—
140C4⋊C4D42811yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—2142no—
143C22⋊C4—2141no—
144C42C22×D10711yescharacteristic
145C23—21no—
146C23—2141no—
147Q8—4282no—
148Q8—4281no—
149Q8—4281no—
150D4—21no—
151D4—4284no—
152D4—2141no—
153D4—2141no—
154D4#1313380712yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—4281no—
159C2×C4—4284no—
160C2×C4—4281no—
161C2×C4—4281no—
162C2×C4—2141no—
163C2×C4—2142no—
164C2×C4D4×D10711yescharacteristic
165C2×C4D4×D10711yescharacteristic
166C2×C4C2×D42811yescharacteristic
167C8—41no—
168C8—41no—
169C8—4281no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—4281no—
174C22—4281no—
175C22—4281no—
176C22#1681207911yescharacteristic
177C4—21no—
178C4—41no—
179C4—8561no—
180C4—4281no—
181C4—4282no—
182C4C2×#1313380711yescharacteristic
183C4#1681212811yescharacteristic
184C4#1681213711yescharacteristic
185C4#1681213811yescharacteristic
186C2—42no—
187C2—8561no—
188C2#2029278011yescharacteristic
189C2#2029279011yescharacteristic
190C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radical#20293281C2
hypercenterD4⋊C8D107
hypocenterC107#1635
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
2. #20292978C2semidirect
3. #20292918C2semidirect
4. #20292917C2semidirect
5. #20292372C2semidirect
6. #20292356C2semidirect
7. D4⋊C8×C107C2semidirect
96. C107#1635semidirect
103. D4⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202932812
#202929782
#202929182
#202929172
#202923722
#202923562
D4⋊C8×C1072
#1635107