Finite Group Databasebeta

← #28338407

Subgroups — #28338407

Order 13696.

226 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293062C212yesmaximal, complemented
2#20292880C211yesmaximal, characteristic, complemented
3#20292437C212yesmaximal, complemented
4#20292409C211yesmaximal, characteristic, complemented
5#575×C107C211yesmaximal, characteristic, complemented
6C2×D856C2211yescharacteristic, complemented
7#16812083—21no—
8#16812052—22no—
9#16812050—22no—
10#16812049C2212yescomplemented
11#16812041C2211yescharacteristic, complemented
12#16812031—21no—
13C2×#41×C107—21no—
14#137×C107—22no—
15AGL(1,ℤ/8)×C107—22no—
16#135×C107—21no—
17#135×C107C2212yes—
18C2×M16×C107C2211yescharacteristic
19C2×D428—21no—
20C2×C22⋊D107—41no—
21D856—22no—
22C4⋊Dic107—21no—
23(C2×D107)⋊C4—41no—
24C107⋊C16—22no—
25#41×C107—22no—
26#41×C107—22no—
27#41×C107—41no—
28#41×C107—41no—
29C2×Q8×C107—21no—
30C2×D4×C107—21no—
31C2×D4×C107—41no—
32C22×C4×C107—41no—
33C22×C4×C107D411yescharacteristic
34Q16×C107—21no—
35Q16×C107D412yescomplemented
36SD16×C107—22no—
37SD16×C107—22no—
38D8×C107—21no—
39D8×C107D412yescomplemented
40M16×C107—22no—
41C2×C8×C107D411yescharacteristic
42C2×C8×C107C2311yescharacteristic
43C23×C107—41no—
44C23×C107D811yescharacteristic
45C22×D107—41no—
46Q8×C107—22no—
47Q8×C107—41no—
48D4×C107—22no—
49D4×C107—41no—
50D4×C107—42no—
51D4×C107—42no—
52C2×C4×C107—21no—
53C2×C4×C107—21no—
54C2×C4×C107—42no—
55C2×C4×C107—41no—
56C2×C4×C107—42no—
57C2×C4×C107D811yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—82no—
60C2×Dic107—41no—
61D428—41no—
62D428—41no—
63C8×C107—21no—
64C8×C107C2×D412yes—
65C22×C107—21no—
66C22×C107—41no—
67C22×C107—42no—
68C22×C107—82no—
69C22×C107C2×D811yescharacteristic
70C2×D107—81no—
71C2×D107—81no—
72C2×D107—81no—
73C4×C107—21no—
74C4×C107—42no—
75C4×C107#12011yescharacteristic
76C4×C107AGL(1,ℤ/8)11yescharacteristic
77Dic107—81no—
78C2×C107—21no—
79C2×C107—41no—
80C2×C107—82no—
81C2×C107#44711yescharacteristic
82D107—161no—
83#2190—1071nomaximal, Sylow
84C107#219011yescharacteristic, complemented, Sylow
85#575D10711yescharacteristic, complemented
86#509—1072no—
87#469—1071no—
88#360—1072no—
89#349—1071no—
90C2×#41—21no—
91#137—22no—
92AGL(1,ℤ/8)—22no—
93#135—21no—
94#135C2×D10712yes—
95C2×D8—1071no—
96C2×M16C2×D10711yescharacteristic
97#121—2141no—
98SD32—2142no—
99D16—2142no—
100M32—1072no—
101#106—1071no—
102#102—2141no—
103#41—22no—
104#41—22no—
105#41—41no—
106#41—41no—
107C2×Q8—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—4281no—
111C2×D4—2141no—
112C22×C4—41no—
113C22×C4C22⋊D10711yescharacteristic
114Q16—21no—
115Q16C22⋊D10712yescomplemented
116SD16—22no—
117SD16—22no—
118D8—21no—
119D8—2142no—
120D8C22⋊D10712yescomplemented
121M16—22no—
122C2×C8C22⋊D10711yescharacteristic
123C2×C8C22×D10711yescharacteristic
124C4⋊C4—2141no—
125C22⋊C4—4281no—
126C16—2142no—
127C23—41no—
128C23—4281no—
129C23#1313380311yescharacteristic
130Q8—22no—
131Q8—41no—
132D4—22no—
133D4—41no—
134D4—42no—
135D4—42no—
136D4—8562no—
137D4—4281no—
138D4—4281no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—4281no—
145C2×C4#1313380311yescharacteristic
146C2×C4C2×C22⋊D10711yescharacteristic
147C8—21no—
148C8C2×C22⋊D10712yes—
149C22—21no—
150C22—41no—
151C22—42no—
152C22—82no—
153C22—8561no—
154C22—8561no—
155C22—8561no—
156C22C2×#1313380311yescharacteristic
157C4—21no—
158C4—42no—
159C4—8561no—
160C4#1681207811yescharacteristic
161C4#1681213511yescharacteristic
162C2—21no—
163C2—41no—
164C2—82no—
165C2—17121no—
166C2#2029276111yescharacteristic
subgroupG/H
centerC2#20292761
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#575×C107C2
socleC2×C107#447
hypercenter#575D107
hypocenterC107#2190
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20293062C2semidirect
2. #20292880C2semidirect
3. #20292437C2semidirect
4. #20292409C2semidirect
5. #575×C107C2semidirect
6. C2×D856C22semidirect
10. #16812049C22semidirect
11. #16812041C22semidirect
35. Q16×C107D4semidirect
39. D8×C107D4semidirect
84. C107#2190semidirect
85. #575D107semidirect
115. Q16C22⋊D107semidirect
120. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202930622
#202928802
#202924372
#202924092
#575×C1072
#2190107