Finite Group Databasebeta

← C17×#7494 #16865238

Subgroups — C17×#7494 #16865238

Order 4352.

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

normal subgroups only

This group has 270 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×#1536C211yesmaximal, characteristic
4C17×#568C411yescharacteristic, complemented
5C17×#532C2211yescharacteristic
6C17×#432—21no—
7C17×#421—21no—
8C17×#410—21no—
9C17×#410C411yescharacteristic, complemented
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—
20C17×C4⋊Q8Q811yescharacteristic
21C17×#127C2×C411yescharacteristic
22C17×#124C2×C411yescharacteristic
23C17×#124D411yescharacteristic
24C17×#121—41no—
25C4×D4×C17—41no—
26C17×#117—21no—
27C17×#117—41no—
28C2×C42×C17D411yescharacteristic
29C17×C4⋊C8—41no—
30C17×#104—42no—
31C17×#99—22no—
32C17×#99—41no—
33C17×#99—41no—
34C17×C22⋊C8—41no—
35C17×#97—22no—
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—41no—
53M16×C17—42no—
54C2×C8×C17—41no—
55C2×C8×C17—42no—
56C2×C8×C17—42no—
57C4⋊C4×C17—41no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—81no—
60C22⋊C4×C17—41no—
61C22⋊C4×C17—41no—
62C22⋊C4×C17—42no—
63C22⋊C4×C17—41no—
64C42×C17—21no—
65C42×C17—41no—
66C42×C17—81no—
67C42×C17C22⋊C411yescharacteristic
68C42×C17C4⋊C411yescharacteristic
69#7494C1711yesmaximal, characteristic, direct product factor, Sylow
70C23×C17—21no—
71C23×C17—21no—
72C23×C17#9911yescharacteristic
73Q8×C17—41no—
74Q8×C17—41no—
75D4×C17—41no—
76D4×C17—41no—
77D4×C17—42no—
78D4×C17—81no—
79D4×C17—82no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—41no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—81no—
91C2×C4×C17—81no—
92C2×C4×C17—82no—
93C2×C4×C17#9511yescharacteristic
94C2×C4×C17#9911yescharacteristic
95C8×C17—42no—
96C8×C17—82no—
97#2126C2×C1711yescharacteristic
98#1884C2×C1711yescharacteristic
99#1536C2×C1711yescharacteristic
100C22×C17—21no—
101C22×C17—41no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—81no—
105C22×C17—81no—
106C22×C17#34211yescharacteristic
107C4×C17—21no—
108C4×C17—41no—
109C4×C17—41no—
110C4×C17—41no—
111C4×C17—41no—
112C4×C17—161no—
113C4×C17—162no—
114C4×C17AΣL(1,𝔽16)11yescharacteristic
115C4×C17#35211yescharacteristic
116#568C4×C1711yescharacteristic, complemented
117#532C22×C1711yescharacteristic
118#432—21no—
119#421—21no—
120#410—21no—
121#410C4×C1711yescharacteristic, complemented
122#354—21no—
123#353C4×C1712yescomplemented
124#339—21no—
125#329—21no—
126#329C4×C1712yescomplemented
127C2×C17—21no—
128C2×C17—41no—
129C2×C17—81no—
130C2×C17—81no—
131C2×C17#139011yescharacteristic
132C2×#41C2×C4×C1711yescharacteristic
133C2×#41D4×C1711yescharacteristic
134#131—42no—
135C2×M16—21no—
136C2×M16—41no—
137C4⋊Q8Q8×C1711yescharacteristic
138#127C2×C4×C1711yescharacteristic
139#124C2×C4×C1711yescharacteristic
140#124D4×C1711yescharacteristic
141#121—41no—
142C4×D4—41no—
143#117—21no—
144#117—41no—
145C2×C42D4×C1711yescharacteristic
146C4⋊C8—41no—
147#104—42no—
148#99—22no—
149#99—41no—
150#99—41no—
151C22⋊C8—41no—
152#97—22no—
153C17#749411yescharacteristic, direct product factor, Sylow
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8C4⋊C4×C1711yescharacteristic
159C2×Q8C4⋊C4×C1711yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C42×C1711yescharacteristic
164C2×D4C22⋊C4×C1711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C22⋊C4×C1711yescharacteristic
169M16—22no—
170M16—41no—
171M16—42no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42—81no—
185C42C22⋊C4×C1711yescharacteristic
186C42C4⋊C4×C1711yescharacteristic
187C23—21no—
188C23—21no—
189C23C17×#9911yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4C17×#9511yescharacteristic
211C2×C4C17×#9911yescharacteristic
212C8—42no—
213C8—82no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22C17×#34211yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—161no—
227C4—162no—
228C4C17×AΣL(1,𝔽16)11yescharacteristic
229C4C17×#35211yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2C17×#139011yescharacteristic
subgroupG/H
center, socleC2×C17#1390
commutatorC2×D4C42×C17
Frattini, cosocle#532C22×C17
prime radicalC17×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×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
9. C17×#410C4semidirect
11. C17×#353C4semidirect
14. C17×#329C4semidirect
69. #7494C17direct (N × H)
116. #568C4×C17semidirect
121. #410C4×C17semidirect
123. #353C4×C17semidirect
126. #329C4×C17semidirect
153. C17#7494direct (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×#15362
#749417