Finite Group Databasebeta

← C2×C17×#2986 #16917990

Subgroups — C2×C17×#2986 #16917990

Order 4352.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C17×#533C211yesmaximal, characteristic
2C17×#2986C2116yesmaximal, direct product factor
3C2×C17×#446C214yesmaximal, complemented
4C2×C8×Q8×C17C214yesmaximal, complemented
5C2×C17×#433C214yesmaximal
6C2×C17×#431C212yesmaximal
7C17×#533C418yes—
8C17×#533C2218yes—
9C2×C17×C4⋊Q8C412yes—
10C2×C17×#125C412yes—
11C2×C17×C22⋊Q8C414yes—
12C2×C4×Q8×C17C2214yes—
13C2×C17×#117C2212yes—
14C22×C42×C17C2211yescharacteristic
15C17×#446C22132yescomplemented
16C8×Q8×C17C22132yescomplemented
17C17×#433C22132yes—
18C17×#431C22116yes—
19C2×C17×C4⋊C8C2212yes—
20C2×C17×C4⋊C8C2218yes—
21C2×C17×C4⋊C8C2212yes—
22C2×C17×C22⋊C8C2214yes—
23C2×C17×#97C2214yes—
24C2×C4×C8×C17C2214yes—
25C2×C4×C8×C17C2214yes—
26C22×Q8×C17C2×C414yes—
27C23×C4×C17C2×C412yes—
28C23×C4×C17C2311yescharacteristic
29C22×C8×C17C2314yes—
30C22×C8×C17C2314yes—
31C17×C4⋊Q8C2×C4116yes—
32C17×#125C2×C4116yes—
33C17×C22⋊Q8C2×C4132yes—
34C4×Q8×C17C2×C4116yes—
35C4×Q8×C17C23116yes—
36C17×#117C2×C418yes—
37C17×#117C2318yes—
38C2×C4⋊C4×C17C2×C414yes—
39C2×C4⋊C4×C17C2×C418yes—
40C2×C4⋊C4×C17C2×C414yes—
41C2×C22⋊C4×C17C2×C414yes—
42C2×C42×C17C2×C411yescharacteristic
43C2×C42×C17C2×C411yescharacteristic
44C2×C42×C17C2×C414yes—
45C2×C42×C17C2314yes—
46C2×C42×C17C2312yes—
47C2×C42×C17C2314yes—
48C17×C4⋊C8C2318yes—
49C17×C4⋊C8C2318yes—
50C17×C4⋊C8C23132yes—
51C17×C22⋊C8C23116yes—
52C17×#97C23116yes—
53C4×C8×C17C23116yes—
54C4×C8×C17C23116yes—
55C24×C17C22×C411yescharacteristic
56C2×Q8×C17—216no—
57C2×Q8×C17C22×C4116yes—
58C22×C4×C17—28no—
59C22×C4×C17—24no—
60C22×C4×C17C22×C418yes—
61C22×C4×C17C22×C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C22×C412yes—
64C22×C4×C17C22×C418yes—
65C22×C4×C17C22×C412yes—
66C22×C4×C17C2412yes—
67C22×C4×C17C2411yescharacteristic
68C2×C8×C17—28no—
69C2×C8×C17—28no—
70C2×C8×C17C2418yes—
71C2×C8×C17C2418yes—
72C4⋊C4×C17C22×C4132yes—
73C4⋊C4×C17C22×C4116yes—
74C4⋊C4×C17C22×C4116yes—
75C22⋊C4×C17C22×C4116yes—
76C42×C17C22×C414yes—
77C42×C17C22×C414yes—
78C42×C17C22×C414yes—
79C42×C17C22×C418yes—
80C42×C17C2414yes—
81C42×C17C2418yes—
82C2×#2986C1711yesmaximal, characteristic, direct product factor, Sylow
83C23×C17—22no—
84C23×C17—24no—
85C23×C17C23×C412yes—
86C23×C17C23×C411yescharacteristic
87Q8×C17—232no—
88C2×C4×C17—22no—
89C2×C4×C17—24no—
90C2×C4×C17—216no—
91C2×C4×C17—24no—
92C2×C4×C17—216no—
93C2×C4×C17—24no—
94C2×C4×C17#13118yes—
95C2×C4×C17C23×C412yes—
96C2×C4×C17C23×C414yes—
97C2×C4×C17C23×C4116yes—
98C2×C4×C17C23×C411yescharacteristic
99C2×C4×C17C23×C414yes—
100C2×C4×C172−1+414yes—
101C2×C4×C17C2511yescharacteristic
102C8×C17—28no—
103C8×C17—28no—
104C2×#533C2×C1711yescharacteristic
105#2986C2×C17116yesdirect product factor
106C2×#446C2×C1714yescomplemented
107C2×C8×Q8C2×C1714yescomplemented
108C2×#433C2×C1714yes—
109C2×#431C2×C1712yes—
110C22×C17—22no—
111C22×C17—24no—
112C22×C17—44no—
113C22×C17#51912yes—
114C22×C17C2×#13114yes—
115C22×C17C24×C411yescharacteristic
116C4×C17—22no—
117C4×C17—24no—
118C4×C17—216no—
119C4×C17#51912yes—
120C4×C17C2×#13118yes—
121C4×C17C2×2−1+412yes—
122#533C4×C1718yes—
123#533C22×C1718yes—
124C2×C4⋊Q8C4×C1712yes—
125C2×#125C4×C1712yes—
126C2×C22⋊Q8C4×C1714yes—
127C2×C4×Q8C22×C1714yes—
128C2×#117C22×C1712yes—
129C22×C42C22×C1711yescharacteristic
130#446C22×C17132yescomplemented
131C8×Q8C22×C17132yescomplemented
132#433C22×C17132yes—
133#431C22×C17116yes—
134C2×C4⋊C8C22×C1712yes—
135C2×C4⋊C8C22×C1712yes—
136C2×C4⋊C8C22×C1718yes—
137C2×C22⋊C8C22×C1714yes—
138C2×#97C22×C1714yes—
139C2×C4×C8C22×C1714yes—
140C2×C4×C8C22×C1714yes—
141C2×C17—42no—
142C2×C17#298614yesdirect product factor
143C2×C17C2×#51911yescharacteristic
144C2×C17C22×#13112yes—
145C22×Q8C2×C4×C1714yes—
146C23×C4C2×C4×C1712yes—
147C23×C4C23×C1711yescharacteristic
148C22×C8C23×C1714yes—
149C22×C8C23×C1714yes—
150C4⋊Q8C2×C4×C17116yes—
151#125C2×C4×C17116yes—
152C22⋊Q8C2×C4×C17132yes—
153C4×Q8C2×C4×C17116yes—
154C4×Q8C23×C17116yes—
155#117C2×C4×C1718yes—
156#117C23×C1718yes—
157C2×C4⋊C4C2×C4×C1714yes—
158C2×C4⋊C4C2×C4×C1718yes—
159C2×C4⋊C4C2×C4×C1714yes—
160C2×C22⋊C4C2×C4×C1714yes—
161C2×C42C2×C4×C1711yescharacteristic
162C2×C42C2×C4×C1711yescharacteristic
163C2×C42C2×C4×C1714yes—
164C2×C42C23×C1714yes—
165C2×C42C23×C1712yes—
166C2×C42C23×C1714yes—
167C4⋊C8C23×C1718yes—
168C4⋊C8C23×C1718yes—
169C4⋊C8C23×C17132yes—
170C22⋊C8C23×C17116yes—
171#97C23×C17116yes—
172C4×C8C23×C17116yes—
173C4×C8C23×C17116yes—
174C17C2×#298611yescharacteristic, direct product factor, Sylow
175C24C22×C4×C1711yescharacteristic
176C2×Q8—216no—
177C2×Q8C22×C4×C17116yes—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4C22×C4×C1711yescharacteristic
181C22×C4C22×C4×C1712yes—
182C22×C4C22×C4×C1712yes—
183C22×C4C22×C4×C1718yes—
184C22×C4C22×C4×C1718yes—
185C22×C4C22×C4×C1712yes—
186C22×C4C24×C1712yes—
187C22×C4C24×C1711yescharacteristic
188C2×C8—28no—
189C2×C8—28no—
190C2×C8C24×C1718yes—
191C2×C8C24×C1718yes—
192C4⋊C4C22×C4×C17116yes—
193C4⋊C4C22×C4×C17132yes—
194C4⋊C4C22×C4×C17116yes—
195C22⋊C4C22×C4×C17116yes—
196C42C22×C4×C1714yes—
197C42C22×C4×C1714yes—
198C42C22×C4×C1714yes—
199C42C22×C4×C1718yes—
200C42C24×C1714yes—
201C42C24×C1718yes—
202C23—22no—
203C23—24no—
204C23C23×C4×C1712yes—
205C23C23×C4×C1711yescharacteristic
206Q8—232no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—216no—
210C2×C4—24no—
211C2×C4—216no—
212C2×C4—24no—
213C2×C4C17×#13118yes—
214C2×C4C23×C4×C1712yes—
215C2×C4C23×C4×C1714yes—
216C2×C4C23×C4×C17116yes—
217C2×C4C23×C4×C1711yescharacteristic
218C2×C4C23×C4×C1714yes—
219C2×C4C17×2−1+414yes—
220C2×C4C25×C1711yescharacteristic
221C8—28no—
222C8—28no—
223C22—24no—
224C22—22no—
225C22—44no—
226C22C17×#51912yes—
227C22C2×C17×#13114yes—
228C22C24×C4×C1711yescharacteristic
229C4—24no—
230C4—216no—
231C4—22no—
232C4C17×#51912yes—
233C4C2×C17×#13118yes—
234C4C2×C17×2−1+412yes—
235C2—42no—
236C2C17×#298614yesdirect product factor
237C2C2×C17×#51911yescharacteristic
238C2C22×C17×#13112yes—
subgroupG/H
centerC22×C4×C17C24
commutatorC22C24×C4×C17
Frattini, cosocleC2×C4C25×C17
socleC23×C17C23×C4
prime radicalC24×C17C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2986C2direct (N × H)
3. C2×C17×#446C2semidirect
4. C2×C8×Q8×C17C2semidirect
15. C17×#446C22semidirect
16. C8×Q8×C17C22semidirect
82. C2×#2986C17direct (N × H)
105. #2986C2×C17direct (N × H)
106. C2×#446C2×C17semidirect
107. C2×C8×Q8C2×C17semidirect
130. #446C22×C17semidirect
131. C8×Q8C22×C17semidirect
142. C2×C17#2986direct (N × H)
174. C17C2×#2986direct (N × H)
236. C2C17×#2986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#5332
C17×#29862
C2×C17×#4462
C2×C8×Q8×C172
C2×C17×#4332
C2×C17×#4312
C2×#298617