Finite Group Databasebeta

← #14527992

Subgroups — #14527992

Order 2176.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#11763520C211yesmaximal, characteristic
2#11763159C211yesmaximal, characteristic, complemented
3#11763095C211yesmaximal, characteristic, complemented
4#11763015C211yesmaximal, characteristic
5#11762539C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic
7C17×#333C211yesmaximal, characteristic, complemented
8C17×C4⋊Q8C2211yescharacteristic
9#10587632—22no—
10C2×#63219—21no—
11#10587583—21no—
12#10587579—21no—
13C4⋊C4×D17—21no—
14#10587576—21no—
15C4×D68C2211yescharacteristic
16#10587555C2211yescharacteristic
17C4×Dic34C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20#10587506—22no—
21#10587488—21no—
22#10587487—21no—
23#10587487—21no—
24C4×#3622C2211yescharacteristic
25C2×Q8×C17D412yes—
26C2×D68D411yescharacteristic
27C2×C4×D17—21no—
28C2×C4×D17—22no—
29C2×Dic34—21no—
30C2×Dic34D411yescharacteristic
31C2×C8×C17—21no—
32C2×C8×C17—21no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17—21no—
35C42×C17C2311yescharacteristic
36#63227—22no—
37#63227—21no—
38C4⋊Dic17—22no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17D411yescharacteristic
42C4⋊Dic17D411yescharacteristic
43#63225—22no—
44#63225—21no—
45C4×Dic17—21no—
46C2×#3622—21no—
47#63219—42no—
48C22×D17—21no—
49Q8×C17—42no—
50C2×C4×C17—22no—
51C2×C4×C17C2×D411yescharacteristic
52C2×C4×C17C2×D411yescharacteristic
53C2×C4×C17C2×D411yescharacteristic
54C2×Dic17—21no—
55C2×Dic17—21no—
56C2×Dic17—21no—
57C2×Dic17—41no—
58D68—21no—
59D68SD1612yes—
60C4×D17—41no—
61C4×D17—41no—
62C4×D17—82no—
63Dic34—21no—
64Dic34—21no—
65Dic34—41no—
66Dic34—41no—
67C8×C17—41no—
68C8×C17—41no—
69#3622—41no—
70#1643—171nomaximal, Sylow
71C22×C17#12011yescharacteristic
72C2×D17—22no—
73C2×D17—41no—
74C2×D17—41no—
75C4×C17—21no—
76C4×C17—42no—
77C4×C17C2×SD1611yescharacteristic
78C4×C17#13511yescharacteristic
79C4×C17#13711yescharacteristic
80C4×C17#13711yescharacteristic
81Dic17—41no—
82Dic17—41no—
83Dic17—41no—
84Dic17—81no—
85#554—171no—
86#487—171no—
87#475—171no—
88#461—171no—
89#333D1711yescharacteristic, complemented
90Q8⋊C8—171no—
91D4⋊C8—171no—
92C2×C17#44811yescharacteristic
93C2×C17#45211yescharacteristic
94C2×C17#45511yescharacteristic
95D17—42no—
96C2×SD16—341no—
97C4⋊Q8—171no—
98C4⋊Q8C2×D1711yescharacteristic
99#125—341no—
100C22⋊Q8—341no—
101C22⋊Q8—342no—
102C4×Q8—171no—
103C4×D4—171no—
104C4×D4—341no—
105C2×C4⋊C4—341no—
106#106—341no—
107C4⋊C8C2×D1711yescharacteristic
108C4⋊C8C2×D1711yescharacteristic
109#103—341no—
110#103—341no—
111#103—342no—
112C4×C8—171no—
113C17#164311yescharacteristic, complemented, Sylow
114C2×Q8—171no—
115C2×Q8—341no—
116C2×Q8D6812yes—
117C2×D4—171no—
118C22×C4—342no—
119C22×C4—341no—
120SD16—682no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—341no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—171no—
127C4⋊C4—171no—
128C4⋊C4—342no—
129C4⋊C4—342no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—341no—
133C22⋊C4—342no—
134C22⋊C4—341no—
135C42—341no—
136C42C22×D1711yescharacteristic
137C23—341no—
138Q8—42no—
139Q8—341no—
140Q8—341no—
141Q8—681no—
142Q8—681no—
143D4—172no—
144D4—341no—
145C2×C4—22no—
146C2×C4—341no—
147C2×C4—341no—
148C2×C4—341no—
149C2×C4—681no—
150C2×C4—681no—
151C2×C4—681no—
152C2×C4—1362no—
153C2×C4C2×D6811yescharacteristic
154C2×C4D4×D1711yescharacteristic
155C2×C4D4×D1711yescharacteristic
156C8—41no—
157C8—41no—
158C8—681no—
159C22—342no—
160C22—681no—
161C22—681no—
162C22#1058756811yescharacteristic
163C4—21no—
164C4—42no—
165C4—681no—
166C4—681no—
167C4—681no—
168C4—1361no—
169C4SD16×D1711yescharacteristic
170C4#1058760211yescharacteristic
171C4#1058760411yescharacteristic
172C4#1058760411yescharacteristic
173C2—682no—
174C2#1176294911yescharacteristic
175C2#1176297411yescharacteristic
176C2#1176298911yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#333C2
socleC22×C17#120
prime radicalC2×D68D4
hypercenter#333D17
hypocenterC17#1643
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #11763159C2semidirect
3. #11763095C2semidirect
5. #11762539C2semidirect
7. C17×#333C2semidirect
89. #333D17semidirect
113. C17#1643semidirect

Maximal subgroups

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

subgroupindex
#117635202
#117631592
#117630952
#117630152
#117625392
#117625362
C17×#3332
#164317