Finite Group Databasebeta

← C17×#12974 #16870718

Subgroups — C17×#12974 #16870718

Order 4352.

408 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2980C211yesmaximal, characteristic
2C17×#2132C211yesmaximal, characteristic
3C17×#1884C211yesmaximal, characteristic
4C17×#1858C212yesmaximal, complemented
5C17×#1395C212yesmaximal, complemented
6C17×#568C2211yescharacteristic
7C17×#532C2211yescharacteristic
8C17×#435—22no—
9C17×#435—22no—
10C17×#432—21no—
11C17×#430—22no—
12C17×#421—21no—
13C17×#413C2212yes—
14C17×#410C2211yescharacteristic
15C17×#408—21no—
16C17×#408C2212yes—
17C17×#356C412yescomplemented
18C17×#355C412yescomplemented
19C17×#341—22no—
20C17×#339—21no—
21C2×#41×C17D411yescharacteristic
22C2×#41×C17C2311yescharacteristic
23C17×#131—42no—
24C2×M16×C17—21no—
25C2×M16×C17—21no—
26C2×M16×C17—22no—
27C2×M16×C17—22no—
28C2×M16×C17D412yes—
29C22×C8×C17—21no—
30C22×C8×C17D412yes—
31C17×C4⋊Q8C2×C411yescharacteristic
32C17×#127Q811yescharacteristic
33C17×#124C2×C411yescharacteristic
34C17×#124Q811yescharacteristic
35C17×#121—22no—
36C4×D4×C17—22no—
37C17×#117—21no—
38C2×C42×C17D411yescharacteristic
39C17×#108—44no—
40C17×C4⋊C8—21no—
41C17×C4⋊C8—21no—
42C17×#104—42no—
43C17×#101C2×C414yes—
44C17×#100—22no—
45C17×#99—22no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48C17×C22⋊C8—22no—
49C17×#97—22no—
50#41×C17—41no—
51#41×C17—41no—
52#41×C17—41no—
53#41×C17—41no—
54C2×Q8×C17C4⋊C411yescharacteristic
55C2×Q8×C17C22×C411yescharacteristic
56C2×D4×C17—22no—
57C2×D4×C17—21no—
58C2×D4×C17C4⋊C411yescharacteristic
59C2×D4×C17C2×Q811yescharacteristic
60C2×D4×C17#4112yes—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D412yes—
65M16×C17—22no—
66M16×C17—24no—
67M16×C17—24no—
68M16×C17—42no—
69M16×C17—41no—
70M16×C17—42no—
71M16×C17—81no—
72C2×C8×C17—22no—
73C2×C8×C17—22no—
74C2×C8×C17—41no—
75C2×C8×C17—42no—
76C2×C8×C17—42no—
77C2×C8×C17—42no—
78C2×C8×C17—81no—
79C4⋊C4×C17—22no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—41no—
86C42×C17C4⋊C411yescharacteristic
87C42×C17C4⋊C411yescharacteristic
88#12974C1711yesmaximal, characteristic, direct product factor, Sylow
89C23×C17—21no—
90C23×C17C22⋊Q812yes—
91C23×C17#12311yescharacteristic
92Q8×C17—41no—
93Q8×C17—41no—
94D4×C17—41no—
95D4×C17—42no—
96D4×C17—41no—
97D4×C17—81no—
98D4×C17—82no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—81no—
109C2×C4×C17—81no—
110C2×C4×C17C2×C4⋊C411yescharacteristic
111C2×C4×C17C4×D412yes—
112C2×C4×C17#12011yescharacteristic
113C8×C17—44no—
114C8×C17—42no—
115C8×C17—42no—
116C8×C17—42no—
117#2980C2×C1711yescharacteristic
118#2132C2×C1711yescharacteristic
119#1884C2×C1711yescharacteristic
120#1858C2×C1712yescomplemented
121#1395C2×C1712yescomplemented
122C22×C17—21no—
123C22×C17—22no—
124C22×C17—41no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—41no—
128C22×C17#38511yescharacteristic
129C4×C17—22no—
130C4×C17—21no—
131C4×C17—41no—
132C4×C17—41no—
133C4×C17—41no—
134C4×C17—161no—
135C4×C17B(4,𝔽2)11yescharacteristic
136C4×C17#45811yescharacteristic
137#568C22×C1711yescharacteristic
138#532C22×C1711yescharacteristic
139#435—22no—
140#435—22no—
141#432—21no—
142#430—22no—
143#421—21no—
144#413C22×C1712yes—
145#410C22×C1711yescharacteristic
146#408—21no—
147#408C22×C1712yes—
148#356C4×C1712yescomplemented
149#355C4×C1712yescomplemented
150#341—22no—
151#339—21no—
152C2×C17—21no—
153C2×C17—41no—
154C2×C17—42no—
155C2×C17—81no—
156C2×C17#186711yescharacteristic
157C2×#41D4×C1711yescharacteristic
158C2×#41C23×C1711yescharacteristic
159#131—42no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16D4×C1712yes—
165C22×C8—21no—
166C22×C8D4×C1712yes—
167C4⋊Q8C2×C4×C1711yescharacteristic
168#127Q8×C1711yescharacteristic
169#124C2×C4×C1711yescharacteristic
170#124Q8×C1711yescharacteristic
171#121—22no—
172C4×D4—22no—
173#117—21no—
174C2×C42D4×C1711yescharacteristic
175#108—44no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178#104—42no—
179#101C2×C4×C1714yes—
180#100—22no—
181#99—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185#97—22no—
186C17#1297411yescharacteristic, direct product factor, Sylow
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊C4×C1711yescharacteristic
192C2×Q8C22×C4×C1711yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊C4×C1711yescharacteristic
196C2×D4C2×Q8×C1711yescharacteristic
197C2×D4#41×C1712yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×D4×C1711yescharacteristic
201C22×C4C2×D4×C1712yes—
202M16—22no—
203M16—24no—
204M16—24no—
205M16—42no—
206M16—41no—
207M16—42no—
208M16—81no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—81no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—41no—
221C42—21no—
222C42—41no—
223C42C4⋊C4×C1711yescharacteristic
224C42C4⋊C4×C1711yescharacteristic
225C23—21no—
226C23C17×C22⋊Q812yes—
227C23C17×#12311yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—81no—
234D4—82no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4C2×C4⋊C4×C1711yescharacteristic
247C2×C4C4×D4×C1712yes—
248C2×C4C17×#12011yescharacteristic
249C8—44no—
250C8—42no—
251C8—42no—
252C8—42no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22C17×#38511yescharacteristic
260C4—22no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—161no—
266C4C17×B(4,𝔽2)11yescharacteristic
267C4C17×#45811yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2C17×#186711yescharacteristic
subgroupG/H
centerC4×C17B(4,𝔽2)
commutatorC2×Q8C22×C4×C17
Frattini, cosocleC2×#41C23×C17
socleC2×C17#1867
prime radicalC17×#532C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G

Complements

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

NH ≅ G/Ntype
4. C17×#1858C2semidirect
5. C17×#1395C2semidirect
17. C17×#356C4semidirect
18. C17×#355C4semidirect
88. #12974C17direct (N × H)
120. #1858C2×C17semidirect
121. #1395C2×C17semidirect
148. #356C4×C17semidirect
149. #355C4×C17semidirect
186. C17#12974direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#29802
C17×#21322
C17×#18842
C17×#18582
C17×#13952
#1297417