Finite Group Databasebeta

← #14544064

Subgroups — #14544064

Order 2176.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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#11763780C211yesmaximal, characteristic
2#11763777C211yesmaximal, characteristic
3#11763655C214yesmaximal, complemented
4C2×#10587632C211yesmaximal, characteristic
5C22×Q8×C17C411yescharacteristic
6C22×C4×D17C2211yescharacteristic
7C2×#63243—21no—
8C2×#63242—21no—
9#10587632—21no—
10#10587632—21no—
11#10587632—21no—
12#10587632—21no—
13C2×#63227C412yes—
14C2×C4⋊Dic17C411yescharacteristic
15C2×#63225C2212yes—
16C22⋊#3624C2212yes—
17C22⋊#3624C2212yes—
18#10587543—21no—
19#10587543—21no—
20C23×D17C2×C411yescharacteristic
21C2×Q8×C17—21no—
22C2×Q8×C17—21no—
23C2×Q8×C17—42no—
24C22×C4×C17C2×C412yes—
25C22×C4×C17C2×C411yescharacteristic
26C22×Dic17C2×C412yes—
27C22×Dic17C2311yescharacteristic
28C2×C4×D17—42no—
29C2×C4×D17D411yescharacteristic
30C2×C4×D17D411yescharacteristic
31C2×C4×D17D411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×#3624—22no—
34C2×#3624—22no—
35#63243—42no—
36#63242—42no—
37#63227—22no—
38#63227—22no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41#63225—22no—
42#63225—22no—
43C23×C17C22×C411yescharacteristic
44C22×D17—22no—
45C22×D17—22no—
46C22×D17—41no—
47C22×D17C22⋊C411yescharacteristic
48C22×D17C22⋊C411yescharacteristic
49Q8×C17—42no—
50Q8×C17—42no—
51C2×C4×C17—22no—
52C2×C4×C17—22no—
53C2×C4×C17—42no—
54C2×C4×C17C22⋊C411yescharacteristic
55C2×C4×C17C22⋊C411yescharacteristic
56C2×Dic17—22no—
57C2×Dic17—24no—
58C2×Dic17—41no—
59C2×Dic17C2×D411yescharacteristic
60C2×Dic17C2×D411yescharacteristic
61C4×D17—22no—
62C4×D17—22no—
63C4×D17—41no—
64C4×D17—41no—
65#3624—42no—
66#3624—42no—
67#1472—171nomaximal, Sylow
68C22×C17—21no—
69C22×C17—21no—
70C22×C17C2×C22⋊C411yescharacteristic
71C22×C17#13112yes—
72C2×D17—21no—
73C2×D17—21no—
74C2×D17—21no—
75C2×D17—21no—
76C2×D17—21no—
77C2×D17—44no—
78C2×D17#10112yes—
79C4×C17—21no—
80C4×C17—42no—
81C4×C17#9912yes—
82Dic17—21no—
83Dic17—21no—
84Dic17—42no—
85C2×C22⋊Q8—171no—
86#407—171no—
87C2×C22⋊C8—171no—
88#324—174no—
89C2×C17—22no—
90C2×C17#40811yescharacteristic
91C2×C17C2×#9911yescharacteristic
92C2×C17C2×#10111yescharacteristic
93D17—22no—
94D17—41no—
95C22×Q8#59511yescharacteristic
96C23×C4—171no—
97C2×M16—341no—
98C22×C8—341no—
99C22⋊Q8—341no—
100C22⋊Q8—341no—
101C22⋊Q8—341no—
102C22⋊Q8—341no—
103C2×C4⋊C4—172no—
104C2×C4⋊C4—171no—
105C2×C22⋊C4—172no—
106C22⋊C8—172no—
107C22⋊C8—172no—
108C22⋊C8—341no—
109C22⋊C8—341no—
110C17#147211yescharacteristic, complemented, Sylow
111C24—171no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C22×C4—171no—
116C22×C4—171no—
117C22×C4—171no—
118C22×C4—171no—
119C22×C4—171no—
120C22×C4—172no—
121C22×C4—682no—
122C22×C4C2×#59512yes—
123C22×C4C2×#59511yescharacteristic
124M16—682no—
125C2×C8—342no—
126C2×C8—342no—
127C2×C8—682no—
128C4⋊C4—342no—
129C4⋊C4—342no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C22⋊C4—342no—
133C22⋊C4—342no—
134C23—171no—
135C23—171no—
136C23—342no—
137C23—342no—
138C23—681no—
139C23C22×#59511yescharacteristic
140Q8—42no—
141Q8—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—171no—
146C2×C4—171no—
147C2×C4—342no—
148C2×C4—342no—
149C2×C4—344no—
150C2×C4—342no—
151C2×C4—681no—
152C2×C4—681no—
153C2×C4—681no—
154C2×C4C22⋊#59511yescharacteristic
155C2×C4C22⋊#59511yescharacteristic
156C8—682no—
157C8—682no—
158C22—21no—
159C22—21no—
160C22—172no—
161C22—341no—
162C22—341no—
163C22—341no—
164C22—341no—
165C22—341no—
166C22—684no—
167C22#1058768012yes—
168C22C2×C22⋊#59511yescharacteristic
169C4—21no—
170C4—42no—
171C4—341no—
172C4—341no—
173C4—682no—
174C4#1058755112yes—
175C2—22no—
176C2—342no—
177C2—681no—
178C2#1176378411yescharacteristic
179C2C2×#1058755111yescharacteristic
180C2D17⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC22×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17C2×C4
cosocleC22×Dic17C23
hypercenterC22×Q8#595
hypocenterC17#1472
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #11763655C2semidirect
110. C17#1472semidirect

Maximal subgroups

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

subgroupindex
#117637802
#117637772
#117636552
C2×#105876322
#147217