Finite Group Databasebeta

← #14528016

Subgroups — #14528016

Order 2176.

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

normal subgroups only

This group has 240 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#11763462C211yesmaximal, characteristic
2#11763086C211yesmaximal, characteristic, complemented
3#11763011C211yesmaximal, characteristic, complemented
4#11762555C211yesmaximal, characteristic, complemented
5#11762538C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic, complemented
7C17×#500C211yesmaximal, characteristic, complemented
8C2×#63256—21no—
9C17×C4⋊Q8C2211yescharacteristic
10C17×C4⋊Q8C2211yescharacteristic
11#10587632—21no—
12C2×#63230—21no—
13#10587581—21no—
14#10587580—22no—
15C4⋊C4×D17—21no—
16C4×D68C2211yescharacteristic
17C4×D68C2211yescharacteristic
18C17×#107—22no—
19C4×C8×C17C2211yescharacteristic
20#10587506—21no—
21#10587480—21no—
22#10587478—21no—
23C4⋊#3622C2211yescharacteristic
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26C2×Q8×C17—21no—
27#63256—42no—
28#63256—42no—
29C2×D68—22no—
30C2×D68D411yescharacteristic
31C2×D68D411yescharacteristic
32C2×C4×D17—22no—
33C2×C4×D17—21no—
34C2×C4×D17—21no—
35C2×C8×C17—21no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17—21no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17D412yes—
40C42×C17C2311yescharacteristic
41#63230—42no—
42#63227—22no—
43#63227—21no—
44#63227—21no—
45C4⋊Dic17D411yescharacteristic
46C4⋊Dic17D411yescharacteristic
47#63225—22no—
48C2×#3622—21no—
49C2×#3622—21no—
50C22×D17—21no—
51C22×D17—21no—
52Q8×C17—41no—
53Q8×C17—41no—
54Q8×C17—41no—
55Q8×C17—41no—
56C2×C4×C17—22no—
57C2×C4×C17—41no—
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C2×C4×C17C2×D411yescharacteristic
61C2×Dic17—21no—
62C2×Dic17—21no—
63D68—21no—
64D68—21no—
65D68—21no—
66D68—44no—
67D68D812yes—
68C4×D17—44no—
69C4×D17—41no—
70C4×D17—41no—
71C4×D17—41no—
72C4×D17—41no—
73C8×C17—22no—
74#3622—41no—
75#3622—41no—
76#1647—171nomaximal, Sylow
77C22×C17#12011yescharacteristic
78C2×D17—22no—
79C2×D17—41no—
80C2×D17—41no—
81C2×D17—41no—
82C2×D17—41no—
83C2×D17—41no—
84C4×C17—21no—
85C4×C17—42no—
86C4×C17—81no—
87C4×C17C2×D811yescharacteristic
88C4×C17#13511yescharacteristic
89C4×C17AGL(1,ℤ/8)11yescharacteristic
90C4×C17#13711yescharacteristic
91Dic17—41no—
92Dic17—41no—
93#547—171no—
94#500D1711yescharacteristic, complemented
95#474—171no—
96#461—171no—
97#332—171no—
98D4⋊C8—171no—
99D4⋊C8—171no—
100C2×C17#44711yescharacteristic
101C2×C17#45211yescharacteristic
102C2×C17#45611yescharacteristic
103D17—42no—
104D17—81no—
105C2×#41—341no—
106C2×SD16—341no—
107C4⋊Q8C2×D1711yescharacteristic
108C4⋊Q8C2×D1711yescharacteristic
109#123—342no—
110C22⋊Q8—341no—
111#121—341no—
112C4×D4—171no—
113C4×D4—171no—
114C2×C4⋊C4—341no—
115#107—22no—
116#107—341no—
117C4⋊C8—171no—
118C4⋊C8—171no—
119#103—341no—
120#102—341no—
121C4×C8C2×D1711yescharacteristic
122C17#164711yescharacteristic, complemented, Sylow
123#41—682no—
124#41—682no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—171no—
128C2×D4—171no—
129C2×D4—342no—
130C22×C4—342no—
131C22×C4—341no—
132C22×C4—341no—
133SD16—682no—
134C2×C8—21no—
135C2×C8—341no—
136C2×C8—341no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—171no—
141C4⋊C4—171no—
142C4⋊C4—342no—
143C4⋊C4C22⋊D1712yes—
144C22⋊C4—342no—
145C22⋊C4—341no—
146C22⋊C4—341no—
147C42C22×D1711yescharacteristic
148C23—341no—
149C23—341no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—172no—
155D4—341no—
156D4—341no—
157D4—341no—
158D4—684no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—341no—
162C2×C4—341no—
163C2×C4—684no—
164C2×C4—681no—
165C2×C4—681no—
166C2×C4—681no—
167C2×C4—681no—
168C2×C4D4×D1711yescharacteristic
169C2×C4D4×D1711yescharacteristic
170C2×C4C2×C22⋊D1711yescharacteristic
171C8—22no—
172C8—681no—
173C8—681no—
174C22—342no—
175C22—681no—
176C22—681no—
177C22—681no—
178C22—681no—
179C22—681no—
180C22#1058762311yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—681no—
185C4—681no—
186C4D8×D1711yescharacteristic
187C4#1058759711yescharacteristic
188C4#1058760411yescharacteristic
189C4#1058760511yescharacteristic
190C2—682no—
191C2—1361no—
192C2#1176294111yescharacteristic
193C2#1176297011yescharacteristic
194C2#1176299111yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#500C2
socleC22×C17#120
prime radical#11763462C2
hypercenter#500D17
hypocenterC17#1647
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #11763086C2semidirect
3. #11763011C2semidirect
4. #11762555C2semidirect
5. #11762538C2semidirect
6. #11762538C2semidirect
7. C17×#500C2semidirect
94. #500D17semidirect
122. C17#1647semidirect

Maximal subgroups

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

subgroupindex
#117634622
#117630862
#117630112
#117625552
#117625382
C17×#5002
#164717