Finite Group Databasebeta

← C17×#7486 #16865230

Subgroups — C17×#7486 #16865230

Order 4352.

286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C17×#2126C211yesmaximal, characteristic
2C17×#1884C211yesmaximal, characteristic
3C17×#1525C211yesmaximal, characteristic
4C17×#568C411yescharacteristic, complemented
5C17×#532C2211yescharacteristic
6C17×#421—21no—
7C17×#410—21no—
8C17×#410C411yescharacteristic, complemented
9C4×M16×C17—21no—
10C17×#354—21no—
11C17×#353C412yescomplemented
12C17×#339—21no—
13C17×#329—21no—
14C17×#329C412yescomplemented
15C2×#41×C17C2×C411yescharacteristic
16C2×#41×C17D411yescharacteristic
17C17×#131—42no—
18C2×M16×C17—21no—
19C2×M16×C17—41no—
20C2×M16×C17—41no—
21C17×C4⋊Q8Q811yescharacteristic
22C17×#127C2×C411yescharacteristic
23C17×#124C2×C411yescharacteristic
24C17×#124D411yescharacteristic
25C17×#121—41no—
26C4×D4×C17—41no—
27C17×#117—21no—
28C17×#117—41no—
29C2×C42×C17D411yescharacteristic
30C17×#104—42no—
31C17×#99—22no—
32C17×#99—41no—
33C17×#99—41no—
34C17×#97—22no—
35C4×C8×C17—41no—
36#41×C17—41no—
37#41×C17—41no—
38#41×C17—41no—
39#41×C17—41no—
40C2×Q8×C17C4⋊C411yescharacteristic
41C2×Q8×C17C4⋊C411yescharacteristic
42C2×D4×C17—21no—
43C2×D4×C17—21no—
44C2×D4×C17—41no—
45C2×D4×C17C4211yescharacteristic
46C2×D4×C17C22⋊C411yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17C22⋊C411yescharacteristic
51M16×C17—22no—
52M16×C17—44no—
53M16×C17—42no—
54M16×C17—41no—
55C2×C8×C17—41no—
56C2×C8×C17—42no—
57C2×C8×C17—42no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—81no—
61C22⋊C4×C17—41no—
62C22⋊C4×C17—41no—
63C22⋊C4×C17—42no—
64C22⋊C4×C17—41no—
65C42×C17—21no—
66C42×C17—41no—
67C42×C17—81no—
68C42×C17C22⋊C411yescharacteristic
69C42×C17C4⋊C411yescharacteristic
70#7486C1711yesmaximal, characteristic, direct product factor, Sylow
71C23×C17—21no—
72C23×C17—21no—
73C23×C17#9911yescharacteristic
74Q8×C17—41no—
75Q8×C17—41no—
76D4×C17—41no—
77D4×C17—41no—
78D4×C17—42no—
79D4×C17—82no—
80D4×C17—81no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—42no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—81no—
93C2×C4×C17—82no—
94C2×C4×C17#9511yescharacteristic
95C2×C4×C17#9911yescharacteristic
96C8×C17—44no—
97C8×C17—42no—
98#2126C2×C1711yescharacteristic
99#1884C2×C1711yescharacteristic
100#1525C2×C1711yescharacteristic
101C22×C17—21no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—81no—
106C22×C17—81no—
107C22×C17#34211yescharacteristic
108C4×C17—22no—
109C4×C17—21no—
110C4×C17—41no—
111C4×C17—41no—
112C4×C17—41no—
113C4×C17—161no—
114C4×C17—162no—
115C4×C17AΣL(1,𝔽16)11yescharacteristic
116C4×C17#35211yescharacteristic
117#568C4×C1711yescharacteristic, complemented
118#532C22×C1711yescharacteristic
119#421—21no—
120#410—21no—
121#410C4×C1711yescharacteristic, complemented
122C4×M16—21no—
123#354—21no—
124#353C4×C1712yescomplemented
125#339—21no—
126#329—21no—
127#329C4×C1712yescomplemented
128C2×C17—21no—
129C2×C17—41no—
130C2×C17—81no—
131C2×C17—81no—
132C2×C17#139011yescharacteristic
133C2×#41C2×C4×C1711yescharacteristic
134C2×#41D4×C1711yescharacteristic
135#131—42no—
136C2×M16—21no—
137C2×M16—41no—
138C2×M16—41no—
139C4⋊Q8Q8×C1711yescharacteristic
140#127C2×C4×C1711yescharacteristic
141#124C2×C4×C1711yescharacteristic
142#124D4×C1711yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—21no—
146#117—41no—
147C2×C42D4×C1711yescharacteristic
148#104—42no—
149#99—22no—
150#99—41no—
151#99—41no—
152#97—22no—
153C4×C8—41no—
154C17#748611yescharacteristic, direct product factor, Sylow
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8C4⋊C4×C1711yescharacteristic
160C2×Q8C4⋊C4×C1711yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C42×C1711yescharacteristic
165C2×D4C22⋊C4×C1711yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4C22⋊C4×C1711yescharacteristic
170M16—22no—
171M16—44no—
172M16—42no—
173M16—41no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—42no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42—81no—
187C42C22⋊C4×C1711yescharacteristic
188C42C4⋊C4×C1711yescharacteristic
189C23—21no—
190C23—21no—
191C23C17×#9911yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—82no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4C17×#9511yescharacteristic
213C2×C4C17×#9911yescharacteristic
214C8—44no—
215C8—42no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22C17×#34211yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—161no—
229C4—162no—
230C4C17×AΣL(1,𝔽16)11yescharacteristic
231C4C17×#35211yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2C17×#139011yescharacteristic
subgroupG/H
centerC4×C17AΣL(1,𝔽16)
commutatorC2×D4C42×C17
Frattini, cosocle#532C22×C17
socleC2×C17#1390
prime radicalC17×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C17×#532 ◁ G

Complements

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

NH ≅ G/Ntype
4. C17×#568C4semidirect
8. C17×#410C4semidirect
11. C17×#353C4semidirect
14. C17×#329C4semidirect
70. #7486C17direct (N × H)
117. #568C4×C17semidirect
121. #410C4×C17semidirect
124. #353C4×C17semidirect
127. #329C4×C17semidirect
154. C17#7486direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#21262
C17×#18842
C17×#15252
#748617