Finite Group Databasebeta

← #14530438

Subgroups — #14530438

Order 2176.

234 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 73 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#11762793C211yesmaximal, characteristic
2#11762783C211yesmaximal, characteristic
3#11762599C214yesmaximal, complemented
4C17×#536C211yesmaximal, characteristic
5C2×#41×C17C2211yescharacteristic
6C22×Q8×C17C411yescharacteristic, complemented
7C17×C4⋊Q8—21no—
8C17×#124—21no—
9C17×#123C412yescomplemented
10C17×#123C2212yes—
11C17×C22⋊Q8—21no—
12C17×C22⋊Q8—21no—
13C17×#117C411yescharacteristic, complemented
14#10587612—21no—
15C2×#63223—21no—
16#10587507—21no—
17#10587505—21no—
18#10587505C2212yes—
19#10587504C2212yes—
20#41×C17—41no—
21#41×C17—41no—
22C2×Q8×C17—21no—
23C2×Q8×C17—21no—
24C2×Q8×C17—41no—
25C2×Q8×C17—41no—
26C2×Q8×C17D411yescharacteristic
27C2×D4×C17C2×C411yescharacteristic
28C2×D4×C17D411yescharacteristic
29C2×D4×C17C2311yescharacteristic
30C22×C4×C17C2×C411yescharacteristic
31C22×C4×C17C2×C412yes—
32C22×C4×C17D411yescharacteristic
33C22×C4×C17D411yescharacteristic
34C4⋊C4×C17—21no—
35C4⋊C4×C17—22no—
36C4⋊C4×C17—22no—
37C22⋊C4×C17—22no—
38C22⋊C4×C17—22no—
39C22⋊C4×C17C2×C412yes—
40C42×C17—21no—
41C22⋊Dic17—22no—
42C4⋊Dic17—41no—
43C4×Dic17—41no—
44#63223—22no—
45#63223—41no—
46C2×#3622—41no—
47C23×C17C22⋊C411yescharacteristic
48C23×C17C22×C411yescharacteristic
49C23×C17C2×D411yescharacteristic
50Q8×C17—41no—
51Q8×C17—42no—
52Q8×C17—41no—
53Q8×C17—41no—
54D4×C17—41no—
55D4×C17—41no—
56D4×C17—41no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—42no—
66C2×C4×C17C22⋊C411yescharacteristic
67C2×C4×C17C22⋊C411yescharacteristic
68C2×C4×C17C22⋊C411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C2×Dic17—42no—
71#3622—42no—
72#2123—171nomaximal, Sylow
73C22×C17—21no—
74C22×C17—21no—
75C22×C17—21no—
76C22×C17—41no—
77C22×C17—41no—
78C22×C17—41no—
79C22×C17C2×C22⋊C411yescharacteristic
80C4×C17—21no—
81C4×C17—21no—
82C4×C17—21no—
83C4×C17—42no—
84C4×C17—42no—
85C4×C17#9911yescharacteristic
86C4×C17#9911yescharacteristic
87Dic17—82no—
88#536D1711yescharacteristic
89#413—171no—
90#410—171no—
91#352—174no—
92C2×C17—21no—
93C2×C17—41no—
94C2×C17—41no—
95C2×C17—41no—
96C2×C17C2×#9911yescharacteristic
97C2×#41C2×D1711yescharacteristic
98C22×Q8Dic1711yescharacteristic, complemented
99C2×M16—341no—
100C4⋊Q8—21no—
101#124—21no—
102#123Dic1712yescomplemented
103#123C2×D1712yes—
104C22⋊Q8—21no—
105C22⋊Q8—21no—
106#117—341no—
107#117Dic1711yescharacteristic, complemented
108#101—341no—
109#100—172no—
110#99—172no—
111#99—341no—
112C17#212311yescharacteristic, complemented, Sylow
113#41—41no—
114#41—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8C22⋊D1711yescharacteristic
120C2×D4C2×Dic1711yescharacteristic
121C2×D4C22⋊D1711yescharacteristic
122C2×D4C22×D1711yescharacteristic
123C22×C4C2×Dic1711yescharacteristic
124C22×C4C2×Dic1712yes—
125C22×C4C22⋊D1711yescharacteristic
126C22×C4C22⋊D1711yescharacteristic
127M16—342no—
128M16—681no—
129C2×C8—681no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—681no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—342no—
137C22⋊C4C2×Dic1712yes—
138C42—21no—
139C42—681no—
140C23C22⋊Dic1711yescharacteristic
141C23C22×Dic1711yescharacteristic
142C23C2×C22⋊D1711yescharacteristic
143Q8—41no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—682no—
160C2×C4C22⋊Dic1711yescharacteristic
161C2×C4C22⋊Dic1711yescharacteristic
162C2×C4C22⋊Dic1711yescharacteristic
163C2×C4C2×C22⋊D1711yescharacteristic
164C8—682no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊Dic1711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—42no—
176C4—42no—
177C4—1362no—
178C4#1058750511yescharacteristic
179C4#1058750511yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1058750511yescharacteristic
subgroupG/H
centerC2C2×#10587505
commutatorC23×C17C22×C4
FrattiniC2×D4C22×D17
FittingC17×#536C2
socleC2×C17C2×#99
prime radicalC2×#41×C17C22
cosocleC2×D4×C17C23
hypercenter#536D17
hypocenterC17#2123
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
3. #11762599C2semidirect
6. C22×Q8×C17C4semidirect
9. C17×#123C4semidirect
13. C17×#117C4semidirect
98. C22×Q8Dic17semidirect
102. #123Dic17semidirect
107. #117Dic17semidirect
112. C17#2123semidirect

Maximal subgroups

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

subgroupindex
#117627932
#117627832
#117625992
C17×#5362
#212317