Finite Group Databasebeta

← #28335695

Subgroups — #28335695

Order 13696.

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

normal subgroups only

This group has 240 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#20293282C211yesmaximal, characteristic
2#20293030C211yesmaximal, characteristic, complemented
3#20292829C211yesmaximal, characteristic, complemented
4#20292372C211yesmaximal, characteristic, complemented
5D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic, complemented
7#474×C107C211yesmaximal, characteristic, complemented
8C2×#13133818—21no—
9C2×#13133805—21no—
10#16812109C2211yescharacteristic
11#16812091—22no—
12#16812088—21no—
13#16812080—21no—
14C4×D428C2211yescharacteristic
15C4⋊C4×D107—21no—
16#16812042—22no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18#16812033—21no—
19C4×C107⋊C8C2211yescharacteristic
20C4⋊Q8×C107C2211yescharacteristic
21C4×D4×C107C2211yescharacteristic
22#107×C107—21no—
23C4⋊C8×C107C2211yescharacteristic
24#102×C107—21no—
25#13133818—42no—
26#13133818—42no—
27C2×Dic214—21no—
28C2×D428D411yescharacteristic
29C2×C22⋊D107—22no—
30C2×C4×D107—22no—
31C2×C4×D107—21no—
32#13133805—42no—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107D411yescharacteristic
37C4⋊Dic107D412yes—
38Dic107⋊C4—22no—
39C22⋊Dic107—22no—
40(C2×D107)⋊C4—21no—
41C2×Q8×C107—21no—
42C2×D4×C107D411yescharacteristic
43C22×C4×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107D411yescharacteristic
48C22⋊C4×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—21no—
51C22×D107—21no—
52Q8×C107—41no—
53Q8×C107—41no—
54D4×C107—21no—
55D4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107—41no—
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C2×C4×C107C2×D411yescharacteristic
63C22⋊D107—44no—
64C2×Dic107—21no—
65C2×Dic107—22no—
66D428—21no—
67D428D812yes—
68C4×D107—44no—
69C4×D107—41no—
70C4×D107—41no—
71Dic214—41no—
72Dic214—41no—
73C8×C107—41no—
74C107⋊C8—22no—
75C107⋊C8—41no—
76C22×C107—41no—
77C22×C107—41no—
78C22×C107—41no—
79C22×C107#12011yescharacteristic
80C2×D107—22no—
81C2×D107—41no—
82C2×D107—41no—
83C4×C107—21no—
84C4×C107—41no—
85C4×C107—81no—
86C4×C107C2×D811yescharacteristic
87C4×C107#13511yescharacteristic
88C4×C107AGL(1,ℤ/8)11yescharacteristic
89C4×C107#13711yescharacteristic
90Dic107—42no—
91Dic107—41no—
92C2×C107—81no—
93C2×C107#44711yescharacteristic
94C2×C107#45211yescharacteristic
95C2×C107#45611yescharacteristic
96D107—42no—
97#1647—1071nomaximal, Sylow
98C107#164711yescharacteristic, complemented, Sylow
99#547—1071no—
100#500—1071no—
101#474D10711yescharacteristic, complemented
102#461—1071no—
103#332—1071no—
104D4⋊C8—1071no—
105D4⋊C8—1071no—
106C2×#41—2141no—
107C2×SD16—2141no—
108C4⋊Q8—1071no—
109C4⋊Q8C2×D10711yescharacteristic
110#123—2142no—
111C22⋊Q8—2141no—
112#121—2141no—
113C4×D4—1071no—
114C4×D4C2×D10711yescharacteristic
115C2×C4⋊C4—2141no—
116#107—21no—
117#107—2142no—
118C4⋊C8—1071no—
119C4⋊C8C2×D10711yescharacteristic
120#103—2141no—
121#102—21no—
122C4×C8—1071no—
123#41—4282no—
124#41—4282no—
125C2×Q8—21no—
126C2×Q8—2141no—
127C2×D4—1071no—
128C2×D4—2142no—
129C2×D4C22⋊D10711yescharacteristic
130C22×C4—21no—
131C22×C4—2142no—
132C22×C4—2141no—
133SD16—4282no—
134C2×C8—21no—
135C2×C8—2141no—
136C2×C8—2141no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—1071no—
140C4⋊C4—1072no—
141C4⋊C4—2142no—
142C4⋊C4—2141no—
143C4⋊C4C22⋊D10711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2142no—
146C22⋊C4—2141no—
147C42C22×D10711yescharacteristic
148C23—21no—
149C23—2141no—
150Q8—41no—
151Q8—41no—
152Q8—4281no—
153Q8—4281no—
154D4—21no—
155D4—21no—
156D4—1072no—
157D4—4284no—
158D4—2141no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—4284no—
164C2×C4—4281no—
165C2×C4—4281no—
166C2×C4—2141no—
167C2×C4—2142no—
168C2×C4D4×D10711yescharacteristic
169C2×C4D4×D10711yescharacteristic
170C2×C4C2×C22⋊D10711yescharacteristic
171C8—41no—
172C8—4281no—
173C8—2142no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—4281no—
178C22—4281no—
179C22—2142no—
180C22#1681207711yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—4282no—
185C4—4281no—
186C4D8×D10711yescharacteristic
187C4#1681212911yescharacteristic
188C4#1681213311yescharacteristic
189C4#1681214011yescharacteristic
190C2—81no—
191C2—4282no—
192C2#2029275911yescharacteristic
193C2#2029278711yescharacteristic
194C2#2029280811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#474×C107C2
socleC22×C107#120
prime radical#20293282C2
hypercenter#474D107
hypocenterC107#1647
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20293030C2semidirect
3. #20292829C2semidirect
4. #20292372C2semidirect
5. D4⋊(C107⋊C8)C2semidirect
6. #20292354C2semidirect
7. #474×C107C2semidirect
98. C107#1647semidirect
101. #474D107semidirect

Maximal subgroups

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

subgroupindex
#202932822
#202930302
#202928292
#202923722
D4⋊(C107⋊C8)2
#202923542
#474×C1072
#1647107