Finite Group Databasebeta

← #14544218

Subgroups — #14544218

Order 2176.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#11763827C211yesmaximal, characteristic
2#11763753C211yesmaximal, characteristic
3#11763663C214yesmaximal
4#11763364C211yesmaximal, characteristic
5C2×C4⋊#595—21no—
6C17×C4⋊Q8C411yescharacteristic, complemented
7#10587632C2212yes—
8Q8×Dic17C412yescomplemented
9#10587583—21no—
10#10587578—21no—
11#10587576—21no—
12#10587573—21no—
13C42×D17C2211yescharacteristic
14#10587555C411yescharacteristic, complemented
15(C2×#595)⋊C4—21no—
16(C2×#595)⋊C4C2212yes—
17#10587544—21no—
18#10587543C2212yes—
19C4×#3624—21no—
20C22×#595—22no—
21C2×Q8×C17C2×C412yes—
22C2×C4×D17D411yescharacteristic
23C2×C4×D17D411yescharacteristic
24C2×C4×D17C2311yescharacteristic
25C2×Dic34—21no—
26C2×#3624—22no—
27C4⋊#595—42no—
28C4⋊C4×C17—21no—
29C4⋊C4×C17—21no—
30C42×C17C2×C411yescharacteristic
31#63227—22no—
32C4⋊Dic17—21no—
33C4⋊Dic17—22no—
34C4⋊Dic17C2×C412yes—
35#63225—22no—
36C4×Dic17—22no—
37C4×Dic17C2×C411yescharacteristic
38C4×Dic17D411yescharacteristic
39C4×Dic17D411yescharacteristic
40C22×D17C2×D411yescharacteristic
41C2×#595—42no—
42C2×#595—42no—
43C2×#595—42no—
44Q8×C17—42no—
45C2×C4×C17—22no—
46C2×C4×C17C22⋊C411yescharacteristic
47C2×C4×C17C22⋊C411yescharacteristic
48C2×C4×C17C22×C411yescharacteristic
49C2×Dic17—22no—
50C2×Dic17C22⋊C411yescharacteristic
51C2×Dic17C22⋊C411yescharacteristic
52C2×Dic17C2×D411yescharacteristic
53C4×D17—41no—
54C4×D17—41no—
55C4×D17SD1612yes—
56C4×D17Q1612yes—
57Dic34—42no—
58#3624—42no—
59#1518—171nomaximal, Sylow
60C22×C17C2×C22⋊C411yescharacteristic
61C2×D17—21no—
62C2×D17—21no—
63C2×D17C2×SD1611yescharacteristic
64C2×D17C2×Q1611yescharacteristic
65#595—82no—
66C4×C17—21no—
67C4×C17—21no—
68C4×C17—42no—
69C4×C17#10312yes—
70Dic17—21no—
71Dic17—21no—
72Dic17—42no—
73Dic17#10312yes—
74#533—171no—
75#431—171no—
76#382—171no—
77#328—174no—
78C2×C17#41011yescharacteristic
79C2×C17C2×#10311yescharacteristic
80C2×C17#42111yescharacteristic
81D17—21no—
82D17—21no—
83C4⋊Q8—171no—
84C4⋊Q8#59511yescharacteristic, complemented
85#125—341no—
86C22⋊Q8—172no—
87C22⋊Q8—341no—
88C4×Q8—172no—
89C4×Q8—341no—
90#117—341no—
91C2×C4⋊C4—341no—
92C2×C42—171no—
93C4⋊C8—341no—
94C22⋊C8—172no—
95C4×C8—341no—
96#95—172no—
97#95—341no—
98C17#151811yescharacteristic, complemented, Sylow
99C2×Q8—341no—
100C2×Q8C2×#59512yes—
101C22×C4—171no—
102C22×C4—171no—
103C22×C4—171no—
104C22×C4—342no—
105C2×C8—342no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—172no—
109C4⋊C4—342no—
110C4⋊C4—342no—
111C4⋊C4—341no—
112C4⋊C4—682no—
113C22⋊C4—342no—
114C42—171no—
115C42—171no—
116C42—171no—
117C42—342no—
118C42C2×#59511yescharacteristic
119C23—171no—
120Q8—42no—
121Q8—682no—
122C2×C4—22no—
123C2×C4—171no—
124C2×C4—172no—
125C2×C4—172no—
126C2×C4—171no—
127C2×C4—171no—
128C2×C4—342no—
129C2×C4—681no—
130C2×C4—681no—
131C2×C4—682no—
132C2×C4—682no—
133C2×C4—682no—
134C2×C4C22⋊#59511yescharacteristic
135C2×C4C22⋊#59511yescharacteristic
136C2×C4C22×#59511yescharacteristic
137C8—682no—
138C22—171no—
139C22—171no—
140C22—341no—
141C22—341no—
142C22C2×C22⋊#59511yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—172no—
147C4—341no—
148C4—341no—
149C4—682no—
150C4—1362no—
151C4#1058754912yes—
152C2—341no—
153C2—341no—
154C2#1176378911yescharacteristic
155C2C2×#1058754911yescharacteristic
156C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C4⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4⋊Q8#595
hypocenterC17#1518
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
6. C17×C4⋊Q8C4semidirect
8. Q8×Dic17C4semidirect
14. #10587555C4semidirect
84. C4⋊Q8#595semidirect
98. C17#1518semidirect

Maximal subgroups

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

subgroupindex
#117638272
#117637532
#117636632
#117633642
#151817