Finite Group Databasebeta

← #14527936

Subgroups — #14527936

Order 2176.

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

normal subgroups only

This group has 234 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#11763161C211yesmaximal, characteristic, complemented
3#11763100C211yesmaximal, characteristic, complemented
4#11763100C211yesmaximal, characteristic, complemented
5#11762536C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic, complemented
7C17×#332C211yesmaximal, characteristic, complemented
8C2×#63256—21no—
9C17×C4⋊Q8C2211yescharacteristic
10#10587632—21no—
11#10587581—21no—
12#10587580—22no—
13C4⋊C4×D17—21no—
14C4×D68C2211yescharacteristic
15C4×D68C2211yescharacteristic
16#10587555C2211yescharacteristic
17C17×C4⋊C8C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19#10587492—21no—
20#10587492—21no—
21#10587488—21no—
22#10587488—21no—
23#10587481—22no—
24C4×#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26#63256—42no—
27#63256—42no—
28C2×D68—22no—
29C2×D68D411yescharacteristic
30C2×D68D411yescharacteristic
31C2×C4×D17—22no—
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×Dic34—21no—
35C2×C8×C17—21no—
36C2×C8×C17—21no—
37C4⋊C4×C17—21no—
38C4⋊C4×C17D412yes—
39C42×C17C2311yescharacteristic
40#63227—22no—
41#63227—21no—
42#63227—21no—
43C4⋊Dic17—21no—
44C4⋊Dic17—21no—
45C4⋊Dic17D411yescharacteristic
46C4⋊Dic17D411yescharacteristic
47#63225—22no—
48C2×#3622—21no—
49C22×D17—21no—
50C22×D17—21no—
51Q8×C17—41no—
52Q8×C17—41no—
53C2×C4×C17—22no—
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×C4×C17C2×D411yescharacteristic
57C2×Dic17—21no—
58C2×Dic17—21no—
59C2×Dic17—41no—
60D68—21no—
61D68—21no—
62D68—21no—
63D68—44no—
64D68SD1612yes—
65C4×D17—44no—
66C4×D17—41no—
67C4×D17—41no—
68C4×D17—41no—
69C4×D17—41no—
70Dic34—42no—
71C8×C17—41no—
72C8×C17—41no—
73#3622—41no—
74#1635—171nomaximal, Sylow
75C22×C17#12011yescharacteristic
76C2×D17—22no—
77C2×D17—41no—
78C2×D17—41no—
79C2×D17—41no—
80C2×D17—41no—
81C2×D17—41no—
82C4×C17—21no—
83C4×C17—42no—
84C4×C17C2×SD1611yescharacteristic
85C4×C17#13511yescharacteristic
86C4×C17AGL(1,ℤ/8)11yescharacteristic
87C4×C17#13711yescharacteristic
88Dic17—41no—
89Dic17—41no—
90Dic17—81no—
91#547—171no—
92#487—171no—
93#476—171no—
94#476—171no—
95#332D1711yescharacteristic, complemented
96D4⋊C8—171no—
97D4⋊C8—171no—
98C2×C17#45011yescharacteristic
99C2×C17#45211yescharacteristic
100C2×C17#45411yescharacteristic
101D17—42no—
102D17—81no—
103C2×#41—341no—
104C4⋊Q8—171no—
105C4⋊Q8C2×D1711yescharacteristic
106#123—342no—
107C22⋊Q8—341no—
108#121—341no—
109C4×D4—171no—
110C4×D4—171no—
111C2×C4⋊C4—341no—
112#106—341no—
113#106—341no—
114C4⋊C8C2×D1711yescharacteristic
115C4⋊C8C2×D1711yescharacteristic
116#103—342no—
117#102—341no—
118#102—341no—
119C4×C8—171no—
120C17#163511yescharacteristic, complemented, Sylow
121#41—682no—
122#41—682no—
123C2×Q8—21no—
124C2×Q8—341no—
125C2×D4—171no—
126C2×D4—171no—
127C2×D4—342no—
128C22×C4—342no—
129C22×C4—341no—
130C22×C4—341no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—341no—
134C4⋊C4—21no—
135C4⋊C4—171no—
136C4⋊C4—171no—
137C4⋊C4—342no—
138C4⋊C4—341no—
139C4⋊C4—341no—
140C4⋊C4D6812yes—
141C22⋊C4—342no—
142C22⋊C4—341no—
143C22⋊C4—341no—
144C42C22×D1711yescharacteristic
145C23—341no—
146C23—341no—
147Q8—41no—
148Q8—41no—
149Q8—682no—
150D4—172no—
151D4—341no—
152D4—341no—
153D4—341no—
154D4—684no—
155C2×C4—22no—
156C2×C4—341no—
157C2×C4—341no—
158C2×C4—684no—
159C2×C4—681no—
160C2×C4—681no—
161C2×C4—681no—
162C2×C4—681no—
163C2×C4—681no—
164C2×C4C2×D6811yescharacteristic
165C2×C4D4×D1711yescharacteristic
166C2×C4D4×D1711yescharacteristic
167C8—41no—
168C8—41no—
169C8—681no—
170C22—342no—
171C22—681no—
172C22—681no—
173C22—681no—
174C22—681no—
175C22—681no—
176C22#1058756811yescharacteristic
177C4—21no—
178C4—42no—
179C4—681no—
180C4—681no—
181C4—1361no—
182C4#1058759711yescharacteristic
183C4SD16×D1711yescharacteristic
184C4#1058760211yescharacteristic
185C4#1058760411yescharacteristic
186C2—682no—
187C2—1361no—
188C2#1176296111yescharacteristic
189C2#1176297411yescharacteristic
190C2#1176298411yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#332C2
socleC22×C17#120
prime radical#11763462C2
hypercenter#332D17
hypocenterC17#1635
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #11763161C2semidirect
3. #11763100C2semidirect
4. #11763100C2semidirect
5. #11762536C2semidirect
6. #11762536C2semidirect
7. C17×#332C2semidirect
95. #332D17semidirect
120. C17#1635semidirect

Maximal subgroups

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

subgroupindex
#117634622
#117631612
#117631002
#117625362
C17×#3322
#163517