Finite Group Databasebeta

← #28338409

Subgroups — #28338409

Order 13696.

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

normal subgroups only

This group has 226 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#20293427C211yesmaximal, characteristic, complemented
2#20293065C212yesmaximal, complemented
3#20292880C211yesmaximal, characteristic, complemented
4#20292438C212yesmaximal, complemented
5#349×C107C211yesmaximal, characteristic, complemented
6C2×#13133818—21no—
7#16812142—22no—
8#16812137—22no—
9#16812131—21no—
10#16812131C2212yes—
11C2×D856C2211yescharacteristic, complemented
12#16812083—21no—
13#16812054—22no—
14D1712—22no—
15#16812041C2211yescharacteristic, complemented
16#16812031—21no—
17C2×M16×C107C2211yescharacteristic
18M32×C107C2212yescomplemented
19#13133818—22no—
20#13133818—22no—
21#13133818—41no—
22#13133818—41no—
23C2×Dic214—21no—
24C2×D428—21no—
25C2×D428—21no—
26C2×C22⋊D107—41no—
27C2×C22⋊D107—41no—
28C2×C4×D107—41no—
29Dic428—21no—
30Dic428D412yescomplemented
31#13133807—22no—
32#13133807—22no—
33D856—21no—
34D856—22no—
35D856D412yescomplemented
36C4⋊Dic107—21no—
37(C2×D107)⋊C4—41no—
38C22×C4×C107D411yescharacteristic
39M16×C107—22no—
40C2×C8×C107D411yescharacteristic
41C2×C8×C107C2311yescharacteristic
42C16×C107—22no—
43C23×C107D811yescharacteristic
44C22×D107—41no—
45C22×D107—41no—
46C2×C4×C107—21no—
47C2×C4×C107—21no—
48C2×C4×C107D811yescharacteristic
49C2×C4×C107C2×D411yescharacteristic
50C22⋊D107—42no—
51C22⋊D107—42no—
52C22⋊D107—82no—
53C2×Dic107—41no—
54C2×Dic107—41no—
55D428—22no—
56D428—41no—
57D428—41no—
58D428—41no—
59C4×D107—42no—
60C4×D107—42no—
61Dic214—22no—
62Dic214—41no—
63C8×C107—21no—
64C8×C107C2×D412yes—
65C22×C107—21no—
66C22×C107—41no—
67C22×C107C2×D811yescharacteristic
68C2×D107—42no—
69C2×D107—82no—
70C2×D107—81no—
71C2×D107—81no—
72C2×D107—81no—
73C4×C107—21no—
74C4×C107#12011yescharacteristic
75C4×C107AGL(1,ℤ/8)11yescharacteristic
76Dic107—42no—
77Dic107—81no—
78C2×C107—21no—
79C2×C107—41no—
80C2×C107#44711yescharacteristic
81D107—82no—
82D107—161no—
83#2190—1071nomaximal, Sylow
84C107#219011yescharacteristic, complemented, Sylow
85#575—1071no—
86#509—1072no—
87#469—1071no—
88#360—1072no—
89#349D10711yescharacteristic, complemented
90C2×#41—2141no—
91#137—2142no—
92AGL(1,ℤ/8)—2142no—
93#135—1072no—
94#135—2141no—
95C2×D8—1071no—
96C2×M16C2×D10711yescharacteristic
97#121—2141no—
98SD32—2142no—
99D16—2142no—
100M32C2×D10712yescomplemented
101#106—1071no—
102#102—2141no—
103#41—4281no—
104#41—4281no—
105#41—2142no—
106#41—2142no—
107C2×Q8—2141no—
108C2×D4—4281no—
109C2×D4—4281no—
110C2×D4—2141no—
111C2×D4—2141no—
112C22×C4—4281no—
113C22×C4D42811yescharacteristic
114Q16—1072no—
115Q16—2141no—
116SD16—2142no—
117SD16—2142no—
118D8—1072no—
119D8—2141no—
120D8—2142no—
121M16—22no—
122C2×C8D42811yescharacteristic
123C2×C8C22×D10711yescharacteristic
124C4⋊C4—2141no—
125C22⋊C4—4281no—
126C16—22no—
127C23—4281no—
128C23—4281no—
129C23D85611yescharacteristic
130Q8—4281no—
131Q8—2142no—
132D4—8562no—
133D4—4281no—
134D4—4282no—
135D4—4282no—
136D4—4281no—
137D4—4281no—
138D4—2142no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—4282no—
142C2×C4—4281no—
143C2×C4—4281no—
144C2×C4—4282no—
145C2×C4D85611yescharacteristic
146C2×C4C2×D42811yescharacteristic
147C8—21no—
148C8D4×D10712yes—
149C22—21no—
150C22—41no—
151C22—8561no—
152C22—8561no—
153C22—8562no—
154C22—8561no—
155C22—4282no—
156C22C2×D85611yescharacteristic
157C4—21no—
158C4—8561no—
159C4—4282no—
160C4#1681207911yescharacteristic
161C4#1681213711yescharacteristic
162C2—21no—
163C2—41no—
164C2—8562no—
165C2—17121no—
166C2#2029276311yescharacteristic
subgroupG/H
centerC2#20292763
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#349×C107C2
socleC2×C107#447
hypercenter#349D107
hypocenterC107#2190
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #20293427C2semidirect
2. #20293065C2semidirect
3. #20292880C2semidirect
4. #20292438C2semidirect
5. #349×C107C2semidirect
11. C2×D856C22semidirect
15. #16812041C22semidirect
18. M32×C107C22semidirect
30. Dic428D4semidirect
35. D856D4semidirect
84. C107#2190semidirect
89. #349D107semidirect
100. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934272
#202930652
#202928802
#202924382
#349×C1072
#2190107