Finite Group Databasebeta

← #28338420

Subgroups — #28338420

Order 13696.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293432C211yesmaximal, characteristic, complemented
2#20293065C211yesmaximal, characteristic, complemented
3C2×D1712C211yesmaximal, characteristic, complemented
4#20292897C211yesmaximal, characteristic, complemented
5#20292441C211yesmaximal, characteristic, complemented
6#20292435C211yesmaximal, characteristic, complemented
7#350×C107C211yesmaximal, characteristic, complemented
8#16812155—21no—
9#16812137—22no—
10#16812137—21no—
11#16812134—21no—
12#16812131—21no—
13#16812131C2211yescharacteristic
14C2×D856—21no—
15C2×D856C2211yescharacteristic
16C2×D856C2211yescharacteristic, complemented
17#16812054—21no—
18D1712—22no—
19D1712—21no—
20#16812045C2211yescharacteristic, complemented
21#16812025—21no—
22#131×C107C2211yescharacteristic
23M32×C107C2211yescharacteristic, complemented
24C2×C16×C107C2211yescharacteristic, complemented
25#13133819—41no—
26#13133818—22no—
27#13133818—21no—
28C2×D428—22no—
29C2×D428—21no—
30C2×D428—21no—
31D4×D107—42no—
32D4×D107—41no—
33Dic428D411yescharacteristic
34#13133807—22no—
35#13133807—21no—
36#13133805—41no—
37D856—22no—
38D856—21no—
39D856—21no—
40D856—21no—
41D856D411yescharacteristic
42D856D412yes—
43#13133803—41no—
44#13133802—21no—
45#41×C107D411yescharacteristic
46M16×C107—21no—
47M16×C107D411yescharacteristic
48C2×C8×C107—21no—
49C2×C8×C107C2311yescharacteristic
50C16×C107—21no—
51C16×C107—21no—
52C22×D107—42no—
53C22×D107—41no—
54C22×D107—41no—
55Q8×C107D811yescharacteristic
56D4×C107—21no—
57D4×C107D811yescharacteristic
58C2×C4×C107—21no—
59C2×C4×C107C2×D411yescharacteristic
60C22⋊D107—42no—
61C22⋊D107—41no—
62D428—21no—
63D428—22no—
64D428—42no—
65D428—41no—
66D428—41no—
67D428—41no—
68C4×D107—42no—
69C4×D107—41no—
70Dic214—21no—
71C8×C107—21no—
72C8×C107C2×D411yescharacteristic
73C8×C107C2×D411yescharacteristic
74C107⋊C8—41no—
75C22×C107—21no—
76C22×C107AGL(1,ℤ/8)11yescharacteristic
77C2×D107—42no—
78C2×D107—41no—
79C2×D107—81no—
80C2×D107—161no—
81C2×D107—162no—
82C2×D107—161no—
83C4×C107—21no—
84C4×C107#12011yescharacteristic
85C4×C107C2×D811yescharacteristic
86Dic107—41no—
87C2×C107—21no—
88C2×C107—41no—
89C2×C107#44711yescharacteristic
90D107—82no—
91D107—81no—
92D107—161no—
93#2192—1071nomaximal, Sylow
94C107#219211yescharacteristic, complemented, Sylow
95#576—1071no—
96#509—1071no—
97C2×D16—1071no—
98#472—1071no—
99#361—1071no—
100#359—1071no—
101#350D10711yescharacteristic, complemented
102H(5,𝔽2)—2141no—
103AGL(1,ℤ/8)—2142no—
104AGL(1,ℤ/8)—2141no—
105AGL(1,ℤ/8)—2141no—
106#135—1071no—
107#135—2141no—
108C2×D8—1071no—
109C2×D8—1071no—
110C2×D8—2141no—
111#131C2×D10711yescharacteristic
112SD32—2141no—
113D16—2142no—
114D16—2141no—
115M32C2×D10711yescharacteristic, complemented
116C2×C16C2×D10711yescharacteristic, complemented
117#108—1071no—
118#100—2141no—
119#41—4281no—
120#41—2142no—
121#41—2141no—
122#41D42811yescharacteristic
123C2×D4—4282no—
124C2×D4—4281no—
125C2×D4—2142no—
126C2×D4—2141no—
127C2×D4—2141no—
128Q16—1071no—
129SD16—4281no—
130SD16—2142no—
131SD16—2141no—
132D8—1072no—
133D8—1071no—
134D8—4281no—
135D8—2142no—
136D8—2141no—
137D8—2141no—
138D8—2141no—
139M16—21no—
140M16—2141no—
141M16D42811yescharacteristic
142C2×C8—21no—
143C2×C8C22×D10711yescharacteristic
144C16—21no—
145C16—21no—
146C23—4282no—
147C23—4281no—
148C23—4281no—
149Q8—2141no—
150Q8D85611yescharacteristic
151D4—21no—
152D4—4282no—
153D4—4282no—
154D4—4281no—
155D4—4281no—
156D4—4281no—
157D4—4281no—
158D4—2142no—
159D4—2141no—
160D4D85611yescharacteristic
161C2×C4—21no—
162C2×C4—4282no—
163C2×C4—4281no—
164C2×C4C2×D42811yescharacteristic
165C8—21no—
166C8—4281no—
167C8D4×D10711yescharacteristic
168C8D4×D10711yescharacteristic
169C22—21no—
170C22—8561no—
171C22—4282no—
172C22—4281no—
173C22—17121no—
174C22—17121no—
175C22—17122no—
176C22#1681213711yescharacteristic
177C4—21no—
178C4—4281no—
179C4#1681207911yescharacteristic
180C4C2×D85611yescharacteristic
181C2—21no—
182C2—41no—
183C2—8562no—
184C2—8561no—
185C2—17121no—
186C2#2029276311yescharacteristic
subgroupG/H
centerC2#20292763
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#350×C107C2
socleC2×C107#447
hypercenter#350D107
hypocenterC107#2192
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
1. #20293432C2semidirect
2. #20293065C2semidirect
3. C2×D1712C2semidirect
4. #20292897C2semidirect
5. #20292441C2semidirect
6. #20292435C2semidirect
7. #350×C107C2semidirect
16. C2×D856C22semidirect
20. #16812045C22semidirect
23. M32×C107C22semidirect
24. C2×C16×C107C22semidirect
94. C107#2192semidirect
101. #350D107semidirect
115. M32C2×D107semidirect
116. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934322
#202930652
C2×D17122
#202928972
#202924412
#202924352
#350×C1072
#2192107