Finite Group Databasebeta

← #14527481

Subgroups — #14527481

Order 2176.

208 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 67 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#11763364C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic
3C4⋊C8×D17C211yesmaximal, characteristic
4#11762556C211yesmaximal, characteristic, complemented
5#11762555C211yesmaximal, characteristic, complemented
6#11762554C211yesmaximal, characteristic, complemented
7C17×#332C211yesmaximal, characteristic, complemented
8C17×C4⋊Q8C2211yescharacteristic
9#10587632—21no—
10Q8×Dic17—21no—
11C2×C8×D17—21no—
12#10587583—21no—
13#10587578C412yes—
14#10587576C412yes—
15#10587573—21no—
16C42×D17C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20#10587491—21no—
21#10587486—21no—
22#10587485—21no—
23C4⋊#3622C2211yescharacteristic
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26C2×C4×D17C2×C411yescharacteristic
27C2×C4×D17D411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×Dic34—21no—
30C2×C8×C17—21no—
31C2×C8×C17—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17C2×C412yes—
34C42×C17C2311yescharacteristic
35#63227—22no—
36C4⋊Dic17—21no—
37C4⋊Dic17—22no—
38C4⋊Dic17C2×C412yes—
39#63225—22no—
40C4×Dic17—22no—
41C4×Dic17C2×C411yescharacteristic
42C4×Dic17D411yescharacteristic
43C4×Dic17D411yescharacteristic
44C2×#3622—21no—
45C2×#3622—21no—
46C8×D17—42no—
47C22×D17C22⋊C411yescharacteristic
48Q8×C17—41no—
49Q8×C17—41no—
50C2×C4×C17—22no—
51C2×C4×C17C22×C411yescharacteristic
52C2×C4×C17C2×D411yescharacteristic
53C2×C4×C17C2×D411yescharacteristic
54C2×Dic17—22no—
55C2×Dic17C22⋊C411yescharacteristic
56C2×Dic17C22⋊C411yescharacteristic
57C2×Dic17C22⋊C411yescharacteristic
58C4×D17—22no—
59C4×D17—21no—
60C4×D17—21no—
61C4×D17—21no—
62C4×D17—21no—
63Dic34—41no—
64Dic34—41no—
65C8×C17—41no—
66C8×C17—41no—
67#3622—41no—
68#3622—41no—
69#1552—171nomaximal, Sylow
70C22×C17C2×C22⋊C411yescharacteristic
71C2×D17—21no—
72C2×D17—21no—
73C2×D17#10112yes—
74C4×C17—21no—
75C4×C17—42no—
76C4×C17#13511yescharacteristic
77C4×C17#13511yescharacteristic
78C4×C17AGL(1,ℤ/8)11yescharacteristic
79C4×C17#13711yescharacteristic
80Dic17—21no—
81Dic17—21no—
82Dic17—21no—
83Dic17—42no—
84#533—171no—
85#432—171no—
86C2×C4⋊C8—171no—
87#332—171no—
88#332—171no—
89#332—171no—
90#332D1711yescharacteristic, complemented
91C2×C17C2×#10111yescharacteristic
92C2×C17#41611yescharacteristic
93C2×C17#41711yescharacteristic
94D17—22no—
95C22×C8—341no—
96C4⋊Q8—171no—
97C4⋊Q8C2×D1711yescharacteristic
98#125—172no—
99C22⋊Q8—341no—
100C22⋊Q8—341no—
101C4×Q8—341no—
102C4×Q8—341no—
103#117—172no—
104C2×C42—171no—
105C4⋊C8—171no—
106C4⋊C8—171no—
107C4⋊C8—341no—
108C4⋊C8C2×D1711yescharacteristic
109C4⋊C8C2×D1711yescharacteristic
110C22⋊C8—341no—
111#97—341no—
112C17#155211yescharacteristic, complemented, Sylow
113C2×Q8—21no—
114C2×Q8—341no—
115C22×C4—171no—
116C22×C4—171no—
117C22×C4—171no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—341no—
121C2×C8—341no—
122C2×C8—682no—
123C4⋊C4—21no—
124C4⋊C4—172no—
125C4⋊C4—342no—
126C4⋊C4—342no—
127C4⋊C4—341no—
128C4⋊C4C4×D1712yes—
129C22⋊C4—342no—
130C42—171no—
131C42—171no—
132C42—171no—
133C42—342no—
134C42C22×D1711yescharacteristic
135C23—171no—
136Q8—41no—
137Q8—41no—
138Q8—681no—
139Q8—681no—
140C2×C4—22no—
141C2×C4—171no—
142C2×C4—171no—
143C2×C4—171no—
144C2×C4—342no—
145C2×C4—341no—
146C2×C4—341no—
147C2×C4—341no—
148C2×C4—341no—
149C2×C4—342no—
150C2×C4C2×C4×D1711yescharacteristic
151C2×C4D4×D1711yescharacteristic
152C2×C4D4×D1711yescharacteristic
153C8—41no—
154C8—41no—
155C8—681no—
156C8—681no—
157C22—172no—
158C22—341no—
159C22—341no—
160C22C22⋊C4×D1711yescharacteristic
161C4—21no—
162C4—42no—
163C4—341no—
164C4—341no—
165C4—341no—
166C4—682no—
167C4#1058759811yescharacteristic
168C4#1058760011yescharacteristic
169C4#1058760211yescharacteristic
170C4#1058760411yescharacteristic
171C2—342no—
172C2#101×D1711yescharacteristic
173C2#1176280211yescharacteristic
174C2#1176280911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×#332C2
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C22⋊C4
cosocleC42×C17C23
hypercenter#332D17
hypocenterC17#1552
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #11762556C2semidirect
5. #11762555C2semidirect
6. #11762554C2semidirect
7. C17×#332C2semidirect
90. #332D17semidirect
112. C17#1552semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117628652
C4⋊C8×D172
#117625562
#117625552
#117625542
C17×#3322
#155217