Finite Group Databasebeta

← #28335516

Subgroups — #28335516

Order 13696.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293262C211yesmaximal, characteristic, complemented
2#20293003C211yesmaximal, characteristic, complemented
3#20292823C212yesmaximal, complemented
4#20292354C212yesmaximal, complemented
5#331×C107C211yesmaximal, characteristic, complemented
6C2×D4×D107—22no—
7C2×D856—22no—
8C2×#13133803—22no—
9#16812106C2211yescharacteristic, complemented
10#16812080—22no—
11#16812077—22no—
12C4×D428C2212yes—
13#16812031—22no—
14C4×C107⋊C8C2211yescharacteristic, complemented
15#127×C107C2211yescharacteristic
16C4⋊C8×C107C2212yescomplemented
17C23×D107—22no—
18C2×D428—21no—
19C2×D428—22no—
20C2×D428D412yes—
21C2×C22⋊D107—24no—
22D4×D107—44no—
23D4×D107—44no—
24C2×C4×D107—22no—
25D856—44no—
26#13133803—44no—
27C2×C107⋊C8—21no—
28C4⋊Dic107D412yes—
29C22⋊Dic107—22no—
30(C2×D107)⋊C4—22no—
31C2×D4×C107—22no—
32C2×D4×C107D412yes—
33C2×C8×C107—22no—
34C42×C107C2311yescharacteristic
35C23×C107—22no—
36C22×D107—22no—
37C22×D107—48no—
38C22×D107—41no—
39C22×D107—82no—
40D4×C107—42no—
41D4×C107—42no—
42D4×C107—42no—
43C2×C4×C107C2×D412yes—
44C2×C4×C107C2×D411yescharacteristic
45C22⋊D107—48no—
46C2×Dic107—22no—
47D428—22no—
48D428—42no—
49D428—42no—
50D428—42no—
51D428D814yescomplemented
52C4×D107—42no—
53C4×D107—42no—
54C8×C107—42no—
55C107⋊C8—22no—
56C22×C107—44no—
57C22×C107—42no—
58C22×C107#12011yescharacteristic
59C2×D107—24no—
60C2×D107—42no—
61C2×D107—42no—
62C2×D107—88no—
63C2×D107—82no—
64C2×D107—81no—
65C4×C107—21no—
66C4×C107C2×D812yes—
67C4×C107AGL(1,ℤ/8)12yes—
68Dic107—42no—
69C2×C107—82no—
70C2×C107#44712yes—
71C2×C107#45311yescharacteristic
72D107—44no—
73D107—161no—
74#1619—1071nomaximal, Sylow
75C107#161911yescharacteristic, complemented, Sylow
76D42—1071no—
77#493—1071no—
78#459—1072no—
79#331D10711yescharacteristic, complemented
80D4⋊C8—1072no—
81C22×D4—2142no—
82C2×D8—2142no—
83C2×D8—2142no—
84#127—1071no—
85#127C2×D10711yescharacteristic
86#121—2142no—
87#120—2142no—
88C4×D4—1072no—
89C4⋊C8C2×D10712yescomplemented
90#102—2142no—
91C4×C8—1071no—
92C24—2142no—
93C2×D4—22no—
94C2×D4—1072no—
95C2×D4—4284no—
96C2×D4—4284no—
97C2×D4—2144no—
98C2×D4—2142no—
99C2×D4—2141no—
100C2×D4D42812yes—
101C22×C4—2142no—
102D8—4284no—
103D8—4284no—
104C2×C8—22no—
105C2×C8—2141no—
106C4⋊C4—1072no—
107C22⋊C4—2142no—
108C22⋊C4—2142no—
109C42C22×D10711yescharacteristic
110C23—22no—
111C23—8562no—
112C23—4288no—
113C23—4281no—
114C23—2142no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—1074no—
119D4—4282no—
120D4—4288no—
121D4—4282no—
122D4—4282no—
123D4—2142no—
124C2×C4—4282no—
125C2×C4—4282no—
126C2×C4—2142no—
127C2×C4D4×D10712yes—
128C2×C4C2×D42811yescharacteristic
129C8—42no—
130C8—2142no—
131C22—44no—
132C22—42no—
133C22—8568no—
134C22—8562no—
135C22—8561no—
136C22—4282no—
137C22—4282no—
138C22—2144no—
139C22#1681207911yescharacteristic
140C4—21no—
141C4—4282no—
142C4D8×D10712yes—
143C4#1681213612yes—
144C2—82no—
145C2—4284no—
146C2—17121no—
147C2#2029276212yes—
148C2#2029279711yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107#120
hypercenter#331D107
hypocenterC107#1619
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #20293262C2semidirect
2. #20293003C2semidirect
3. #20292823C2semidirect
4. #20292354C2semidirect
5. #331×C107C2semidirect
9. #16812106C22semidirect
14. C4×C107⋊C8C22semidirect
16. C4⋊C8×C107C22semidirect
51. D428D8semidirect
75. C107#1619semidirect
79. #331D107semidirect
89. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932622
#202930032
#202928232
#202923542
#331×C1072
#1619107