Finite Group Databasebeta

← #14527913

Subgroups — #14527913

Order 2176.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#11763481C211yesmaximal, characteristic
2#11763182C211yesmaximal, characteristic, complemented
3#11763104C211yesmaximal, characteristic, complemented
4#11763100C211yesmaximal, characteristic, complemented
5#11762552C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic, complemented
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#63256—21no—
9C4×Q8×C17C2211yescharacteristic
10C2×#63219—22no—
11#10587581—22no—
12#10587581—21no—
13#10587560—21no—
14#10587560C2211yescharacteristic
15C4×D68—21no—
16C4×D68C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C4×C8×C17C2211yescharacteristic
20#10587506—21no—
21#10587492—21no—
22#10587488—21no—
23#10587478—21no—
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17D411yescharacteristic
26#63256—42no—
27#63256—42no—
28C2×D68—22no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68—22no—
32C2×D68D411yescharacteristic
33C2×D68D412yes—
34C2×C4×D17—22no—
35C2×C4×D17—21no—
36C2×Dic34—21no—
37C2×C8×C17—21no—
38C2×C8×C17—21no—
39C4⋊C4×C17—21no—
40C4⋊C4×C17D411yescharacteristic
41C42×C17—21no—
42C42×C17C2311yescharacteristic
43#63227—22no—
44#63227—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17—21no—
47C4⋊Dic17D411yescharacteristic
48C2×#3622—21no—
49#63219—44no—
50C22×D17—21no—
51C22×D17—22no—
52Q8×C17—21no—
53Q8×C17SD1612yescomplemented
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17C2×D411yescharacteristic
57C2×C4×C17C2×D411yescharacteristic
58C2×C4×C17C2×D411yescharacteristic
59C2×Dic17—21no—
60C2×Dic17—41no—
61D68—21no—
62D68—21no—
63D68—42no—
64D68—44no—
65D68—41no—
66D68—41no—
67D68—41no—
68D68—41no—
69D68—82no—
70C4×D17—44no—
71C4×D17—41no—
72C4×D17—41no—
73Dic34—42no—
74C8×C17—22no—
75C8×C17—41no—
76#3622—41no—
77#1631—171nomaximal, Sylow
78C22×C17#12011yescharacteristic
79C2×D17—42no—
80C2×D17—41no—
81C2×D17—41no—
82C2×D17—42no—
83C2×D17—42no—
84C2×D17—41no—
85C4×C17—22no—
86C4×C17—21no—
87C4×C17—41no—
88C4×C17C2×SD1611yescharacteristic
89C4×C17#13511yescharacteristic
90C4×C17AGL(1,ℤ/8)11yescharacteristic
91C4×C17#13711yescharacteristic
92Dic17—41no—
93Dic17—81no—
94#550—171no—
95#492—171no—
96#477—171no—
97#476—171no—
98#331—171no—
99Q8⋊C8D1711yescharacteristic, complemented
100D4⋊C8—171no—
101C2×C17#45011yescharacteristic
102C2×C17#45211yescharacteristic
103C2×C17#45311yescharacteristic
104D17—81no—
105D17—82no—
106C2×#41—341no—
107C2×SD16—342no—
108C4⋊Q8—171no—
109#127—171no—
110#127—341no—
111#121—342no—
112#121—341no—
113C4×Q8C2×D1711yescharacteristic
114C4×D4—171no—
115C4×D4—341no—
116#107—341no—
117#106—341no—
118C4⋊C8—171no—
119C4⋊C8C2×D1711yescharacteristic
120#103—341no—
121#102—341no—
122C4×C8C2×D1711yescharacteristic
123C17#163111yescharacteristic, complemented, Sylow
124#41—682no—
125#41—682no—
126C2×Q8—341no—
127C2×Q8D6811yescharacteristic
128C2×D4—171no—
129C2×D4—172no—
130C2×D4—342no—
131C2×D4—342no—
132C2×D4—341no—
133C2×D4—341no—
134C22×C4—342no—
135C22×C4—341no—
136SD16—684no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—341no—
140C4⋊C4—21no—
141C4⋊C4—171no—
142C4⋊C4—341no—
143C4⋊C4—341no—
144C4⋊C4D6811yescharacteristic
145C22⋊C4—342no—
146C22⋊C4—341no—
147C42—21no—
148C42C22×D1711yescharacteristic
149C23—341no—
150C23—342no—
151Q8—21no—
152Q8—682no—
153Q8#6321912yescomplemented
154D4—341no—
155D4—341no—
156D4—682no—
157D4—684no—
158D4—681no—
159D4—681no—
160D4—681no—
161D4—681no—
162D4—1362no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—341no—
166C2×C4—684no—
167C2×C4—681no—
168C2×C4—681no—
169C2×C4—681no—
170C2×C4C2×D6811yescharacteristic
171C2×C4D4×D1711yescharacteristic
172C2×C4D4×D1711yescharacteristic
173C8—22no—
174C8—41no—
175C8—681no—
176C22—682no—
177C22—681no—
178C22—681no—
179C22—682no—
180C22—682no—
181C22—681no—
182C22#1058756811yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—681no—
187C4—1361no—
188C4C2×#6321911yescharacteristic
189C4#1058759411yescharacteristic
190C4#1058760211yescharacteristic
191C4#1058760411yescharacteristic
192C2—1361no—
193C2—1362no—
194C2#1176296211yescharacteristic
195C2#1176297411yescharacteristic
196C2#1176297811yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17#120
prime radical#11763481C2
hypercenterQ8⋊C8D17
hypocenterC17#1631
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11763182C2semidirect
3. #11763104C2semidirect
4. #11763100C2semidirect
5. #11762552C2semidirect
6. #11762538C2semidirect
7. C17×Q8⋊C8C2semidirect
53. Q8×C17SD16semidirect
99. Q8⋊C8D17semidirect
123. C17#1631semidirect
153. Q8#63219semidirect

Maximal subgroups

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

subgroupindex
#117634812
#117631822
#117631042
#117631002
#117625522
#117625382
C17×Q8⋊C82
#163117