Finite Group Databasebeta

← #14543885

Subgroups — #14543885

Order 2176.

162 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 40 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#11763795C211yesmaximal, characteristic
2#11763788C211yesmaximal, characteristic
3#11762777C211yesmaximal, characteristic
4C2×#63252C2211yescharacteristic
5#10587677—21no—
6D17⋊M16—21no—
7C2×M16×C17C411yescharacteristic, complemented
8C22×#3622C411yescharacteristic, complemented
9#10587553—21no—
10#10587551—21no—
11#10587545C412yescomplemented
12#10587539C412yes—
13#10587491—21no—
14#10587491—21no—
15C22×C4×C17C2×C411yescharacteristic
16C2×C22⋊D17D411yescharacteristic
17C2×C22⋊D17Q811yescharacteristic
18#63252—41no—
19#63252—41no—
20C2×D68C2×C411yescharacteristic
21C2×C4×D17D411yescharacteristic
22C2×C4×D17D411yescharacteristic
23C2×Dic34C2×C411yescharacteristic
24C22⋊#595—22no—
25#63247—22no—
26C4⋊#595—41no—
27C4×#595—41no—
28#63243—41no—
29#63242—41no—
30M16×C17—41no—
31C2×C8×C17—21no—
32C2×#3622—21no—
33C2×#3622—41no—
34C23×C17C4⋊C411yescharacteristic
35C22×D17C22⋊C411yescharacteristic
36C22×D17C4⋊C411yescharacteristic
37C2×#595—42no—
38C2×C4×C17—21no—
39C2×C4×C17—21no—
40C2×C4×C17C4211yescharacteristic
41C2×C4×C17C22⋊C411yescharacteristic
42C22⋊D17—41no—
43C22⋊D17—41no—
44C2×Dic17C22⋊C411yescharacteristic
45C2×Dic17C4⋊C411yescharacteristic
46D68—41no—
47C4×D17—21no—
48C4×D17—21no—
49C4×D17—21no—
50C4×D17—21no—
51Dic34—41no—
52#3624—42no—
53C8×C17—41no—
54#3622—41no—
55#1395—171nomaximal, Sylow
56C22×C17—21no—
57C22×C17—41no—
58C22×C17#9511yescharacteristic
59C2×D17—21no—
60C2×D17—21no—
61C2×D17—41no—
62C2×D17—41no—
63#595—82no—
64C4×C17—21no—
65C4×C17#9911yescharacteristic
66C4×C17#9911yescharacteristic
67Dic17—21no—
68Dic17—21no—
69#413—171no—
70#410—171no—
71#408—171no—
72C2×C17—21no—
73C2×C17—41no—
74C2×C17#34211yescharacteristic
75D17—41no—
76D17—41no—
77C2×#41—171no—
78C2×M16—341no—
79C2×M16#59511yescharacteristic, complemented
80C22×C8—171no—
81#117—341no—
82#101—172no—
83#100—341no—
84#99—172no—
85#99—341no—
86C22⋊C8—341no—
87C22⋊C8—341no—
88C17#139511yescharacteristic, complemented, Sylow
89#41—681no—
90#41—681no—
91C2×Q8—171no—
92C2×D4—171no—
93C2×D4—171no—
94C2×D4—171no—
95C22×C4—171no—
96C22×C4—171no—
97C22×C4C2×#59511yescharacteristic
98M16—41no—
99M16—342no—
100M16—681no—
101C2×C8—21no—
102C2×C8—341no—
103C2×C8—681no—
104C2×C8—681no—
105C4⋊C4—681no—
106C22⋊C4—342no—
107C42—681no—
108C23—171no—
109C23—171no—
110C23C4⋊#59511yescharacteristic
111Q8—681no—
112D4—681no—
113D4—681no—
114D4—681no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—171no—
118C2×C4—171no—
119C2×C4—341no—
120C2×C4—341no—
121C2×C4—341no—
122C2×C4—341no—
123C2×C4—682no—
124C2×C4C4×#59511yescharacteristic
125C2×C4C22⋊#59511yescharacteristic
126C8—41no—
127C8—682no—
128C8—681no—
129C22—21no—
130C22—41no—
131C22—341no—
132C22—341no—
133C22—681no—
134C22—681no—
135C22(C2×#595)⋊C411yescharacteristic
136C4—21no—
137C4—341no—
138C4—341no—
139C4—1362no—
140C4#1058753911yescharacteristic
141C4#1058755111yescharacteristic
142C2—21no—
143C2—41no—
144C2—681no—
145C2—681no—
146C2#1176368711yescharacteristic
subgroupG/H
centerC4#10587551
commutatorC2×C4×C17C42
FrattiniC22×C4C2×#595
FittingC2×M16×C17C4
socleC2×C17#342
prime radical, cosocleC2×#63252C22
hypercenterC2×M16#595
hypocenterC17#1395
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
7. C2×M16×C17C4semidirect
8. C22×#3622C4semidirect
11. #10587545C4semidirect
79. C2×M16#595semidirect
88. C17#1395semidirect

Maximal subgroups

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

subgroupindex
#117637952
#117637882
#117627772
#139517