Finite Group Databasebeta

← #28335662

Subgroups — #28335662

Order 13696.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293297C211yesmaximal, characteristic
2#20292973C211yesmaximal, characteristic, complemented
3#20292913C211yesmaximal, characteristic, complemented
4#20292833C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic, complemented
7#331×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9C2×#13133807—21no—
10#16812109C2211yescharacteristic
11#16812105—21no—
12#16812081—22no—
13#16812080—21no—
14C4×Dic214C2211yescharacteristic
15C4×D428C2211yescharacteristic
16D4×Dic107—21no—
17#16812041—21no—
18#16812034—21no—
19#16812034—21no—
20#16812030—22no—
21C4×C107⋊C8C2211yescharacteristic
22#127×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214—21no—
28C2×Dic214D411yescharacteristic
29C2×D428D411yescharacteristic
30C2×C22⋊D107—22no—
31C2×C22⋊D107—22no—
32C22×Dic107—22no—
33C2×C4×D107—21no—
34#13133807—42no—
35C2×C107⋊C8—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107D411yescharacteristic
39C4⋊Dic107D411yescharacteristic
40Dic107⋊C4—21no—
41C22⋊Dic107—22no—
42(C2×D107)⋊C4—21no—
43C4×Dic107—21no—
44C2×D4×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107D412yes—
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—22no—
51C22×D107—21no—
52D4×C107—42no—
53D4×C107—41no—
54D4×C107—41no—
55D4×C107—41no—
56D4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C22⋊D107—44no—
61C22⋊D107—82no—
62C2×Dic107—21no—
63C2×Dic107—21no—
64C2×Dic107—21no—
65C2×Dic107—44no—
66C2×Dic107—41no—
67D428—21no—
68D428—21no—
69C4×D107—41no—
70C4×D107—41no—
71Dic214—21no—
72Dic214—41no—
73Dic214—41no—
74Dic214SD1612yescomplemented
75C8×C107—41no—
76C8×C107—41no—
77C107⋊C8—41no—
78C22×C107—42no—
79C22×C107—42no—
80C22×C107—42no—
81C22×C107#12011yescharacteristic
82C2×D107—41no—
83C2×D107—41no—
84C2×D107—41no—
85C4×C107—21no—
86C4×C107C2×SD1611yescharacteristic
87C4×C107#13511yescharacteristic
88C4×C107AGL(1,ℤ/8)11yescharacteristic
89C4×C107AGL(1,ℤ/8)11yescharacteristic
90Dic107—22no—
91Dic107—41no—
92Dic107—41no—
93Dic107—81no—
94C2×C107—82no—
95C2×C107#44911yescharacteristic
96C2×C107#45011yescharacteristic
97C2×C107#45511yescharacteristic
98D107—81no—
99#1642—1071nomaximal, Sylow
100C107#164211yescharacteristic, complemented, Sylow
101#550—1071no—
102#486—1071no—
103#475—1071no—
104#461—1071no—
105#331D10711yescharacteristic, complemented
106Q8⋊C8—1071no—
107D4⋊C8—1071no—
108C2×#41—2141no—
109C2×SD16—2141no—
110C4⋊Q8—1071no—
111#127—2141no—
112#127C2×D10711yescharacteristic
113#121—2141no—
114#121—2142no—
115C4×Q8—1071no—
116C4×D4—1071no—
117C4×D4—2141no—
118#106—2141no—
119C4⋊C8C2×D10711yescharacteristic
120C4⋊C8C2×D10711yescharacteristic
121#103—2141no—
122#103—2141no—
123#102—2142no—
124C4×C8—1071no—
125#41—4282no—
126#41—4282no—
127C2×Q8—1071no—
128C2×Q8—2141no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—1071no—
132C2×D4—2142no—
133C2×D4—2142no—
134C2×D4D42812yes—
135C22×C4—2142no—
136C22×C4—2141no—
137SD16—4282no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2141no—
141C4⋊C4—1071no—
142C4⋊C4—1071no—
143C4⋊C4—2141no—
144C4⋊C4—2141no—
145C4⋊C4—2141no—
146C22⋊C4—2142no—
147C22⋊C4—2141no—
148C42—2141no—
149C42C22×D10711yescharacteristic
150C23—22no—
151C23—2141no—
152Q8—1072no—
153Q8—4281no—
154Q8—4281no—
155Q8—2141no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—8562no—
162D4—4284no—
163D4—2141no—
164D4—2141no—
165C2×C4—4281no—
166C2×C4—4284no—
167C2×C4—4281no—
168C2×C4—4281no—
169C2×C4—2141no—
170C2×C4—2141no—
171C2×C4—2141no—
172C2×C4D4×D10711yescharacteristic
173C2×C4D4×D10711yescharacteristic
174C2×C4C2×D42811yescharacteristic
175C8—41no—
176C8—41no—
177C8—4281no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—4281no—
182C22—4281no—
183C22—4281no—
184C22#1681207911yescharacteristic
185C4—21no—
186C4—8561no—
187C4—4281no—
188C4—4281no—
189C4—2142no—
190C4SD16×D10711yescharacteristic
191C4#1681213011yescharacteristic
192C4#1681213311yescharacteristic
193C4#1681213311yescharacteristic
194C2—82no—
195C2—8561no—
196C2#2029277311yescharacteristic
197C2#2029277911yescharacteristic
198C2#2029280711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107#120
prime radical#20293297C2
hypercenter#331D107
hypocenterC107#1642
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20292973C2semidirect
3. #20292913C2semidirect
4. #20292833C2semidirect
5. #20292357C2semidirect
6. #20292354C2semidirect
7. #331×C107C2semidirect
74. Dic214SD16semidirect
100. C107#1642semidirect
105. #331D107semidirect

Maximal subgroups

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

subgroupindex
#202932972
#202929732
#202929132
#202928332
#202923572
#202923542
#331×C1072
#1642107