Finite Group Databasebeta

← #28335693

Subgroups — #28335693

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#20293280C211yesmaximal, characteristic
2#20292904C211yesmaximal, characteristic, complemented
3#20292829C211yesmaximal, characteristic, complemented
4#20292373C211yesmaximal, characteristic, complemented
5#20292356C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7#500×C107C211yesmaximal, characteristic, complemented
8C2×#13133819—21no—
9C2×#13133805—21no—
10#16812094—22no—
11#16812085—21no—
12#16812084—21no—
13C4×D428C2211yescharacteristic
14C4×D428C2211yescharacteristic
15C4⋊C4×D107—21no—
16#16812042—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812033—21no—
20#16812029—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23#107×C107—22no—
24C4×C8×C107C2211yescharacteristic
25#13133819—42no—
26#13133819—42no—
27C2×D428—22no—
28C2×D428D411yescharacteristic
29C2×D428D411yescharacteristic
30C2×C4×D107—22no—
31C2×C4×D107—21no—
32C2×C4×D107—21no—
33#13133805—42no—
34C2×C107⋊C8—21no—
35C2×C107⋊C8—21no—
36C4⋊Dic107D411yescharacteristic
37C4⋊Dic107D411yescharacteristic
38Dic107⋊C4—22no—
39(C2×D107)⋊C4—22no—
40(C2×D107)⋊C4—21no—
41(C2×D107)⋊C4—21no—
42C2×Q8×C107—21no—
43C2×Q8×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107—21no—
48C4⋊C4×C107D412yes—
49C42×C107C2311yescharacteristic
50C22×D107—21no—
51C22×D107—21no—
52Q8×C107—41no—
53Q8×C107—41no—
54Q8×C107—41no—
55Q8×C107—41no—
56C2×C4×C107—22no—
57C2×C4×C107—41no—
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C2×C4×C107C2×D411yescharacteristic
61C2×Dic107—21no—
62C2×Dic107—21no—
63D428—21no—
64D428—21no—
65D428—21no—
66D428—44no—
67D428D812yes—
68C4×D107—44no—
69C4×D107—41no—
70C4×D107—41no—
71C4×D107—41no—
72C4×D107—41no—
73C8×C107—22no—
74C107⋊C8—41no—
75C107⋊C8—41no—
76C22×C107#12011yescharacteristic
77C2×D107—22no—
78C2×D107—41no—
79C2×D107—41no—
80C2×D107—41no—
81C2×D107—41no—
82C2×D107—41no—
83C4×C107—21no—
84C4×C107—42no—
85C4×C107—81no—
86C4×C107C2×D811yescharacteristic
87C4×C107#13511yescharacteristic
88C4×C107AGL(1,ℤ/8)11yescharacteristic
89C4×C107#13711yescharacteristic
90Dic107—41no—
91Dic107—41no—
92C2×C107#44711yescharacteristic
93C2×C107#45211yescharacteristic
94C2×C107#45611yescharacteristic
95D107—42no—
96D107—81no—
97#1647—1071nomaximal, Sylow
98C107#164711yescharacteristic, complemented, Sylow
99#547—1071no—
100#500D10711yescharacteristic, complemented
101#474—1071no—
102#461—1071no—
103#332—1071no—
104D4⋊C8—1071no—
105D4⋊C8—1071no—
106C2×#41—2141no—
107C2×SD16—2141no—
108C4⋊Q8C2×D10711yescharacteristic
109C4⋊Q8C2×D10711yescharacteristic
110#123—2142no—
111C22⋊Q8—2141no—
112#121—2141no—
113C4×D4—1071no—
114C4×D4—1071no—
115C2×C4⋊C4—2141no—
116#107—22no—
117#107—2141no—
118C4⋊C8—1071no—
119C4⋊C8—1071no—
120#103—2141no—
121#102—2141no—
122C4×C8C2×D10711yescharacteristic
123#41—4282no—
124#41—4282no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—1071no—
128C2×D4—1071no—
129C2×D4—2142no—
130C22×C4—2142no—
131C22×C4—2141no—
132C22×C4—2141no—
133SD16—4282no—
134C2×C8—21no—
135C2×C8—2141no—
136C2×C8—2141no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—1071no—
141C4⋊C4—1071no—
142C4⋊C4—2142no—
143C4⋊C4C22⋊D10712yes—
144C22⋊C4—2142no—
145C22⋊C4—2141no—
146C22⋊C4—2141no—
147C42C22×D10711yescharacteristic
148C23—2141no—
149C23—2141no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—1072no—
155D4—4284no—
156D4—2141no—
157D4—2141no—
158D4—2141no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—4284no—
162C2×C4—4281no—
163C2×C4—4281no—
164C2×C4—4281no—
165C2×C4—4281no—
166C2×C4—2141no—
167C2×C4—2141no—
168C2×C4D4×D10711yescharacteristic
169C2×C4D4×D10711yescharacteristic
170C2×C4C2×C22⋊D10711yescharacteristic
171C8—22no—
172C8—4281no—
173C8—4281no—
174C22—4281no—
175C22—4281no—
176C22—4281no—
177C22—4281no—
178C22—4281no—
179C22—2142no—
180C22#1681207711yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—4281no—
185C4—4281no—
186C4D8×D10711yescharacteristic
187C4#1681213211yescharacteristic
188C4#1681213311yescharacteristic
189C4#1681214111yescharacteristic
190C2—8561no—
191C2—4282no—
192C2#2029275911yescharacteristic
193C2#2029278811yescharacteristic
194C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#500×C107C2
socleC22×C107#120
prime radical#20293280C2
hypercenter#500D107
hypocenterC107#1647
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #20292904C2semidirect
3. #20292829C2semidirect
4. #20292373C2semidirect
5. #20292356C2semidirect
6. #20292356C2semidirect
7. #500×C107C2semidirect
98. C107#1647semidirect
100. #500D107semidirect

Maximal subgroups

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

subgroupindex
#202932802
#202929042
#202928292
#202923732
#202923562
#500×C1072
#1647107