Finite Group Databasebeta

← #16944871

Subgroups — #16944871

Order 4352.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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#14530600C211yesmaximal, characteristic
2#14528539C211yesmaximal, characteristic
3#14526778C212yesmaximal
4#14526772C212yesmaximal
5C17×#3177C211yesmaximal, characteristic
6#11762764—21no—
7#11762630—21no—
8#11762591C2212yes—
9#11762587—21no—
10#11762567C2211yescharacteristic
11#11762566C2211yescharacteristic
12#11762565—21no—
13C17×#541—21no—
14C17×#499C411yescharacteristic, complemented
15C17×#494C411yescharacteristic, complemented
16C17×#491—21no—
17C17×#470C2212yes—
18C17×#467—21no—
19C17×#441C412yescomplemented
20C4×Q16×C17—21no—
21C17×#405C2211yescharacteristic
22C2×Q16×C17—21no—
23C2×Q16×C17C2×C412yes—
24C2×M16×C17C2311yescharacteristic
25C22×C8×C17D411yescharacteristic
26C17×C4⋊Q8—21no—
27C17×#126—42no—
28C17×#125—21no—
29C17×#125—42no—
30C17×C22⋊Q8—22no—
31C4×Q8×C17—22no—
32C17×#117D411yescharacteristic
33#10587612—21no—
34C2×#63223—21no—
35C17×#107—21no—
36C17×#106C2×C412yes—
37C17×#103—22no—
38C17×#103—22no—
39C17×#97C2×C411yescharacteristic
40C17×#97D411yescharacteristic
41C4×C8×C17C2×C411yescharacteristic
42C4×C8×C17D411yescharacteristic
43#10587486—22no—
44C8×Dic17—22no—
45#10587483—22no—
46C2×Q8×C17—22no—
47C22×C4×C17C2×D411yescharacteristic
48Q16×C17—42no—
49Q16×C17—41no—
50Q16×C17—41no—
51M16×C17—41no—
52C2×C8×C17—21no—
53C2×C8×C17—21no—
54C2×C8×C17C22⋊C411yescharacteristic
55C2×C8×C17C22⋊C411yescharacteristic
56C2×C8×C17C22×C411yescharacteristic
57C2×C8×C17C2×D411yescharacteristic
58C4⋊C4×C17—21no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—42no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—21no—
65C22⋊C4×C17—42no—
66C42×C17—42no—
67C42×C17C22⋊C411yescharacteristic
68C42×C17C22⋊C411yescharacteristic
69C22⋊Dic17—41no—
70C4⋊Dic17—41no—
71C4×Dic17—22no—
72#63223—41no—
73#63223—41no—
74C2×#3622—22no—
75#63214—42no—
76#11771—171nomaximal, Sylow
77C23×C17AGL(1,ℤ/8)11yescharacteristic
78Q8×C17—42no—
79Q8×C17—42no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17C2×C22⋊C411yescharacteristic
87C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
88C2×Dic17—42no—
89C8×C17—21no—
90C8×C17—21no—
91C8×C17—21no—
92C8×C17#10411yescharacteristic
93C8×C17#10411yescharacteristic
94#3622—42no—
95#3177D1711yescharacteristic
96#2166—171no—
97#1747—171no—
98#1342—172no—
99#1340—172no—
100C22×C17—21no—
101C22×C17—41no—
102C22×C17#41811yescharacteristic
103C4×C17—21no—
104C4×C17—41no—
105C4×C17—41no—
106C4×C17—82no—
107C4×C17—82no—
108C4×C17#41011yescharacteristic
109C4×C17C2×#10411yescharacteristic
110Dic17—82no—
111#541—21no—
112#499Dic1711yescharacteristic, complemented
113#494Dic1711yescharacteristic, complemented
114#491—21no—
115#470C2×D1712yes—
116#467—21no—
117#441Dic1712yescomplemented
118C4×Q16—21no—
119#405—341no—
120#405C2×D1711yescharacteristic
121#364—341no—
122#349—172no—
123#347—341no—
124#338—171no—
125H(3,ℤ/4)—171no—
126H(3,ℤ/4)—341no—
127C2×C17—21no—
128C2×C17—41no—
129C2×C17#150511yescharacteristic
130C2×Q16—21no—
131C2×Q16C2×Dic1712yes—
132C2×M16—341no—
133C2×M16C22×D1711yescharacteristic
134C22×C8C22⋊D1711yescharacteristic
135C4⋊Q8—21no—
136#126—42no—
137#125—21no—
138#125—42no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117—341no—
142#117C22⋊D1711yescharacteristic
143M32—342no—
144#107—21no—
145#106C2×Dic1712yes—
146#103—22no—
147#103—22no—
148#97—342no—
149#97C2×Dic1711yescharacteristic
150#97C22⋊D1711yescharacteristic
151C4×C8—342no—
152C4×C8C2×Dic1711yescharacteristic
153C4×C8C22⋊D1711yescharacteristic
154C17#1177111yescharacteristic, complemented, Sylow
155C2×Q8—22no—
156C22×C4C2×C22⋊D1711yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—681no—
162M16—681no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—342no—
166C2×C8C22⋊Dic1711yescharacteristic
167C2×C8C22⋊Dic1711yescharacteristic
168C2×C8C22×Dic1711yescharacteristic
169C2×C8C2×C22⋊D1711yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—681no—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—681no—
180C42—42no—
181C42—342no—
182C42C22⋊Dic1711yescharacteristic
183C42C22⋊Dic1711yescharacteristic
184C16—682no—
185C23#1058761811yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—682no—
195C2×C4#1058761811yescharacteristic
196C2×C4C2×C22⋊Dic1711yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—682no—
201C8#1058750811yescharacteristic
202C8#1058750811yescharacteristic
203C22—21no—
204C22—41no—
205C22#1176281411yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—1362no—
212C4#1176278311yescharacteristic
213C4C2×#1058750811yescharacteristic
214C2—21no—
215C2—41no—
216C2#1452729211yescharacteristic
subgroupG/H
centerC2#14527292
commutatorC2×C8×C17C22×C4
FrattiniC2×M16C22×D17
FittingC17×#3177C2
socleC2×C17#1505
prime radicalC23×C17AGL(1,ℤ/8)
cosocleC2×M16×C17C23
hypercenter#3177D17
hypocenterC17#11771
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
14. C17×#499C4semidirect
15. C17×#494C4semidirect
19. C17×#441C4semidirect
112. #499Dic17semidirect
113. #494Dic17semidirect
117. #441Dic17semidirect
154. C17#11771semidirect

Maximal subgroups

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

subgroupindex
#145306002
#145285392
#145267782
#145267722
C17×#31772
#1177117