Finite Group Databasebeta

← #28335705

Subgroups — #28335705

Order 13696.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293355C211yesmaximal, characteristic
2#20293032C211yesmaximal, characteristic
3#20292913C211yesmaximal, characteristic
4#20292837C211yesmaximal, characteristic
5#20292372C211yesmaximal, characteristic
6#20292359C211yesmaximal, characteristic
7Q8⋊C8×C107C211yesmaximal, characteristic
8C2×#13133808—21no—
9#16812109C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812107—21no—
12#16812101—21no—
13#16812099—22no—
14C4×Dic214—21no—
15C4×Dic214C2211yescharacteristic
16#16812074—21no—
17#16812043—22no—
18#16812041—21no—
19C4⋊(C107⋊C8)C2211yescharacteristic
20#16812034—21no—
21#16812032—21no—
22C4×Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214D411yescharacteristic
28#13133808—42no—
29C2×C107⋊C8—21no—
30C4⋊Dic107—21no—
31C4⋊Dic107—21no—
32C4⋊Dic107—21no—
33C4⋊Dic107D411yescharacteristic
34C4⋊Dic107D412yes—
35Dic107⋊C4—22no—
36Dic107⋊C4—22no—
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39C4×Dic107—22no—
40C4×Dic107—21no—
41C2×Q8×C107D411yescharacteristic
42C2×C8×C107—21no—
43C2×C8×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D411yescharacteristic
46C42×C107—21no—
47C42×C107C2311yescharacteristic
48Q8×C107—21no—
49Q8×C107—21no—
50C2×C4×C107—21no—
51C2×C4×C107—21no—
52C2×C4×C107C2×D411yescharacteristic
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×Dic107—21no—
56C2×Dic107—21no—
57C2×Dic107—22no—
58C2×Dic107—41no—
59Dic214—21no—
60Dic214—41no—
61Dic214—41no—
62Dic214—41no—
63Dic214—41no—
64Dic214Q1612yescomplemented
65C8×C107—22no—
66C8×C107—41no—
67C107⋊C8—41no—
68C22×C107#12011yescharacteristic
69C4×C107—21no—
70C4×C107—41no—
71C4×C107—41no—
72C4×C107C2×Q1611yescharacteristic
73C4×C107#13511yescharacteristic
74C4×C107AGL(1,ℤ/8)11yescharacteristic
75C4×C107#13711yescharacteristic
76Dic107—22no—
77Dic107—42no—
78Dic107—41no—
79Dic107—81no—
80C2×C107#44911yescharacteristic
81C2×C107#45111yescharacteristic
82C2×C107#45611yescharacteristic
83#1648—1071nomaximal, Sylow
84C107#164811yescharacteristic, complemented, Sylow
85#557—1071no—
86#500—1071no—
87#475—1071no—
88#462—1071no—
89#332—1071no—
90Q8⋊C8—1071no—
91Q8⋊C8D10711yescharacteristic
92C2×Q16—2141no—
93C4⋊Q8—1071no—
94C4⋊Q8—1071no—
95C4⋊Q8—2141no—
96#125—2141no—
97#125—2142no—
98C4×Q8—1071no—
99C4×Q8—2141no—
100C4×Q8—2141no—
101C4×Q8C2×D10711yescharacteristic
102#107—2142no—
103#106—2141no—
104C4⋊C8—1071no—
105C4⋊C8C2×D10711yescharacteristic
106#103—2141no—
107#103—2141no—
108C4×C8C2×D10711yescharacteristic
109C2×Q8—1071no—
110C2×Q8—2141no—
111C2×Q8—2141no—
112C2×Q8D42811yescharacteristic
113Q16—4282no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—2141no—
117C4⋊C4—21no—
118C4⋊C4—1071no—
119C4⋊C4—1072no—
120C4⋊C4—2142no—
121C4⋊C4—2142no—
122C4⋊C4—2142no—
123C4⋊C4—2141no—
124C4⋊C4—2141no—
125C4⋊C4—2141no—
126C4⋊C4—2141no—
127C4⋊C4D42811yescharacteristic
128C42—21no—
129C42—2142no—
130C42—2141no—
131C42C22×D10711yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—1072no—
135Q8—4281no—
136Q8—4281no—
137Q8—4281no—
138Q8—4281no—
139Q8—2141no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—4281no—
143C2×C4—2141no—
144C2×C4—2142no—
145C2×C4—2141no—
146C2×C4D4×D10711yescharacteristic
147C2×C4D4×D10711yescharacteristic
148C2×C4C2×D42811yescharacteristic
149C8—22no—
150C8—41no—
151C8—4281no—
152C22#1681207911yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—8561no—
157C4—4282no—
158C4—4281no—
159C4—2142no—
160C4Q16×D10711yescharacteristic
161C4#1681213111yescharacteristic
162C4#1681213711yescharacteristic
163C4#1681213811yescharacteristic
164C2#2029277411yescharacteristic
165C2#2029278411yescharacteristic
166C2#2029281111yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingQ8⋊C8×C107C2
socle, prime radicalC22×C107#120
hypercenterQ8⋊C8D107
hypocenterC107#1648
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
64. Dic214Q16semidirect
84. C107#1648semidirect

Maximal subgroups

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

subgroupindex
#202933552
#202930322
#202929132
#202928372
#202923722
#202923592
Q8⋊C8×C1072
#1648107