Finite Group Databasebeta

← #28338415

Subgroups — #28338415

Order 13696.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#20293429C211yesmaximal, characteristic, complemented
2C2×#16812050C211yesmaximal, characteristic, complemented
3#20292894C211yesmaximal, characteristic, complemented
4#20292441C211yesmaximal, characteristic, complemented
5#20292434C211yesmaximal, characteristic, complemented
6#20292412C211yesmaximal, characteristic, complemented
7#509×C107C211yesmaximal, characteristic, complemented
8#16812155—21no—
9#16812136—22no—
10#16812135—21no—
11#16812134—21no—
12#16812132—21no—
13C2×D856C2211yescharacteristic
14D8×D107—21no—
15#16812115C2211yescharacteristic
16#16812050—22no—
17C2×C107⋊C16C2211yescharacteristic, complemented
18#16812045C2211yescharacteristic, complemented
19#16812026—21no—
20#135×C107C2211yescharacteristic
21C2×D8×C107C2211yescharacteristic, complemented
22SD32×C107—21no—
23D16×C107—21no—
24M32×C107C2211yescharacteristic, complemented
25#13133819—22no—
26#13133818—41no—
27#13133818D411yescharacteristic
28C2×D428—21no—
29C2×D428—41no—
30D4×D107—22no—
31D4×D107—42no—
32#13133806—41no—
33#13133805—22no—
34D856—21no—
35D856D412yes—
36#13133803—22no—
37#13133803—41no—
38#13133802—21no—
39#13133802D411yescharacteristic
40#13133801—21no—
41C8×D107—21no—
42C107⋊C16—21no—
43#41×C107—21no—
44C2×D4×C107—21no—
45Q16×C107D411yescharacteristic
46SD16×C107—21no—
47D8×C107—21no—
48D8×C107—21no—
49D8×C107D411yescharacteristic
50C2×C8×C107C2311yescharacteristic
51C16×C107—21no—
52C23×C107—41no—
53C22×D107—42no—
54C22×D107—41no—
55Q8×C107—21no—
56D4×C107—21no—
57D4×C107—41no—
58D4×C107—41no—
59D4×C107—41no—
60C2×C4×C107—41no—
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—21no—
63C22⋊D107—42no—
64D428—22no—
65D428—42no—
66D428—41no—
67D428D811yescharacteristic
68C4×D107—21no—
69C4×D107—42no—
70Dic214D811yescharacteristic
71C8×C107C2×D411yescharacteristic
72C8×C107C2×D411yescharacteristic
73C107⋊C8—21no—
74C107⋊C8—41no—
75C22×C107—41no—
76C22×C107—81no—
77C22×C107—161no—
78C22×C107AGL(1,ℤ/8)11yescharacteristic
79C2×D107—21no—
80C2×D107—42no—
81C2×D107—162no—
82C2×D107—161no—
83C4×C107—41no—
84C4×C107#12011yescharacteristic
85C4×C107C2×D811yescharacteristic
86Dic107—21no—
87C2×C107—21no—
88C2×C107—81no—
89C2×C107—161no—
90C2×C107#44711yescharacteristic
91D107—41no—
92D107—82no—
93#2192—1071nomaximal, Sylow
94C107#219211yescharacteristic, complemented, Sylow
95#576—1071no—
96#509D10711yescharacteristic, complemented
97C2×D16—1071no—
98#472—1071no—
99#361—1071no—
100#359—1071no—
101#350—1071no—
102H(5,𝔽2)—2141no—
103AGL(1,ℤ/8)—2141no—
104AGL(1,ℤ/8)—2142no—
105AGL(1,ℤ/8)—2141no—
106#135—2141no—
107#135C2×D10711yescharacteristic
108C2×D8—1071no—
109C2×D8—2141no—
110C2×D8C2×D10711yescharacteristic, complemented
111#131—1071no—
112SD32—21no—
113D16—21no—
114D16—2142no—
115M32C2×D10711yescharacteristic, complemented
116C2×C16—1071no—
117#108—1071no—
118#100—2141no—
119#41—21no—
120#41—1071no—
121#41—4281no—
122#41—2142no—
123C2×D4—21no—
124C2×D4—4281no—
125C2×D4—4282no—
126C2×D4—2142no—
127C2×D4—2141no—
128Q16C22⋊D10711yescharacteristic
129SD16—21no—
130SD16—4281no—
131SD16—2142no—
132D8—21no—
133D8—21no—
134D8—1072no—
135D8—4281no—
136D8—2142no—
137D8—2141no—
138D8C22⋊D10711yescharacteristic
139M16—1071no—
140M16—2141no—
141M16—2141no—
142C2×C8—2141no—
143C2×C8C22×D10711yescharacteristic
144C16—21no—
145C16—2141no—
146C23—41no—
147C23—4282no—
148C23—4281no—
149Q8—21no—
150Q8—1071no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—1071no—
156D4—4282no—
157D4—4282no—
158D4—4281no—
159D4—2141no—
160D4—2142no—
161C2×C4—41no—
162C2×C4—4282no—
163C2×C4—2141no—
164C2×C4D4×D10711yescharacteristic
165C8—4281no—
166C8—2141no—
167C8D4×D10711yescharacteristic
168C8C2×C22⋊D10711yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—4282no—
173C22—17121no—
174C22—17122no—
175C22—2141no—
176C22#1681213311yescharacteristic
177C4—41no—
178C4—2141no—
179C4#1681207711yescharacteristic
180C4D8×D10711yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—8562no—
185C2—4281no—
186C2#2029275911yescharacteristic
subgroupG/H
centerC2#20292759
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#509×C107C2
socleC2×C107#447
hypercenter#509D107
hypocenterC107#2192
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20293429C2semidirect
2. C2×#16812050C2semidirect
3. #20292894C2semidirect
4. #20292441C2semidirect
5. #20292434C2semidirect
6. #20292412C2semidirect
7. #509×C107C2semidirect
17. C2×C107⋊C16C22semidirect
18. #16812045C22semidirect
21. C2×D8×C107C22semidirect
24. M32×C107C22semidirect
94. C107#2192semidirect
96. #509D107semidirect
110. C2×D8C2×D107semidirect
115. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934292
C2×#168120502
#202928942
#202924412
#202924342
#202924122
#509×C1072
#2192107