Finite Group Databasebeta

← #14527488

Subgroups — #14527488

Order 2176.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#11763360C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic, complemented
3#11762861C211yesmaximal, characteristic, complemented
4#11762555C211yesmaximal, characteristic, complemented
5#11762552C211yesmaximal, characteristic, complemented
6#11762552C211yesmaximal, characteristic, complemented
7C17×#332C211yesmaximal, characteristic, complemented
8C2×#63256—21no—
9C17×C4⋊Q8C2211yescharacteristic
10#10587633—21no—
11#10587581—21no—
12#10587581C412yes—
13#10587579—21no—
14#10587579C412yes—
15#10587560C2211yescharacteristic
16C42×D17C2211yescharacteristic
17C17×C4⋊C8C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19#10587491—21no—
20#10587491—21no—
21#10587486—21no—
22C8×Dic17—21no—
23C4⋊#3622C2211yescharacteristic
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26#63256—42no—
27#63256—42no—
28C2×D68—22no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68C2×C412yes—
32C2×C4×D17—22no—
33C2×C4×D17C2×C411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C8×C17—21no—
37C2×C8×C17—21no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17C2×C412yes—
40C42×C17C2311yescharacteristic
41#63227—22no—
42#63227—22no—
43C4×Dic17C2×C411yescharacteristic
44C4×Dic17D411yescharacteristic
45C4×Dic17D411yescharacteristic
46C2×#3622—21no—
47C2×#3622—21no—
48C22×D17—22no—
49C22×D17C22⋊C411yescharacteristic
50Q8×C17—41no—
51Q8×C17—41no—
52C2×C4×C17—22no—
53C2×C4×C17C22×C411yescharacteristic
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×Dic17C22⋊C411yescharacteristic
57C2×Dic17C22⋊C411yescharacteristic
58C2×Dic17C22⋊C411yescharacteristic
59D68—42no—
60D68—44no—
61D68—41no—
62D68—41no—
63D68—41no—
64D68—41no—
65C4×D17—21no—
66C4×D17—21no—
67C4×D17—21no—
68C4×D17—21no—
69C4×D17—41no—
70C4×D17—44no—
71C8×C17—41no—
72C8×C17—41no—
73#3622—41no—
74#3622—41no—
75#1553—171nomaximal, Sylow
76C22×C17C2×C22⋊C411yescharacteristic
77C2×D17—21no—
78C2×D17—21no—
79C2×D17—21no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—42no—
83C4×C17—21no—
84C4×C17—42no—
85C4×C17#13511yescharacteristic
86C4×C17#13511yescharacteristic
87C4×C17AGL(1,ℤ/8)11yescharacteristic
88C4×C17AGL(1,ℤ/8)11yescharacteristic
89Dic17—21no—
90Dic17—21no—
91Dic17—21no—
92#532—171no—
93#432—171no—
94#431—171no—
95#332—171no—
96#332D1711yescharacteristic, complemented
97#331—171no—
98#331—171no—
99C2×C17#41311yescharacteristic
100C2×C17#41611yescharacteristic
101C2×C17#41811yescharacteristic
102D17—41no—
103D17—82no—
104C2×#41—341no—
105C4⋊Q8C2×D1711yescharacteristic
106#127—171no—
107#124—341no—
108#121—172no—
109#121—341no—
110C4×D4—172no—
111C4×D4—341no—
112C2×C42—171no—
113C4⋊C8—171no—
114C4⋊C8—171no—
115C4⋊C8C2×D1711yescharacteristic
116C4⋊C8C2×D1711yescharacteristic
117C22⋊C8—341no—
118C22⋊C8—341no—
119#97—341no—
120C4×C8—341no—
121C17#155311yescharacteristic, complemented, Sylow
122#41—682no—
123#41—682no—
124C2×Q8—21no—
125C2×D4—172no—
126C2×D4—342no—
127C2×D4—341no—
128C2×D4—341no—
129C22×C4—171no—
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—342no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—341no—
136C2×C8—341no—
137C4⋊C4—21no—
138C4⋊C4C4×D1712yes—
139C22⋊C4—342no—
140C22⋊C4—342no—
141C42—171no—
142C42—171no—
143C42—171no—
144C42C22×D1711yescharacteristic
145C23—171no—
146C23—342no—
147Q8—41no—
148Q8—41no—
149D4—682no—
150D4—684no—
151D4—681no—
152D4—681no—
153D4—681no—
154D4—681no—
155C2×C4—22no—
156C2×C4—171no—
157C2×C4—171no—
158C2×C4—171no—
159C2×C4—341no—
160C2×C4—341no—
161C2×C4—341no—
162C2×C4—341no—
163C2×C4—681no—
164C2×C4—684no—
165C2×C4C2×C4×D1711yescharacteristic
166C2×C4D4×D1711yescharacteristic
167C2×C4D4×D1711yescharacteristic
168C8—41no—
169C8—41no—
170C8—681no—
171C8—681no—
172C22—341no—
173C22—341no—
174C22—341no—
175C22—682no—
176C22—682no—
177C22—682no—
178C22C22⋊C4×D1711yescharacteristic
179C4—21no—
180C4—42no—
181C4—341no—
182C4—341no—
183C4—341no—
184C4#1058759711yescharacteristic
185C4#1058760011yescharacteristic
186C4#1058760211yescharacteristic
187C4#1058760511yescharacteristic
188C2—681no—
189C2—1362no—
190C2#1176279211yescharacteristic
191C2#1176280411yescharacteristic
192C2#1176281211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×#332C2
socleC22×C17C2×C22⋊C4
prime radical#11763360C2
cosocleC42×C17C23
hypercenter#332D17
hypocenterC17#1553
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #11762865C2semidirect
3. #11762861C2semidirect
4. #11762555C2semidirect
5. #11762552C2semidirect
6. #11762552C2semidirect
7. C17×#332C2semidirect
96. #332D17semidirect
121. C17#1553semidirect

Maximal subgroups

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

subgroupindex
#117633602
#117628652
#117628612
#117625552
#117625522
C17×#3322
#155317