Finite Group Databasebeta

← C17×#1468 #14524474

Subgroups — C17×#1468 #14524474

Order 2176.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C17×#123C211yesmaximal, characteristic
2C17×#407C211yesmaximal, characteristic, complemented
3C2×C17×C22⋊C8C211yesmaximal, characteristic, complemented
4C17×#324C212yesmaximal, complemented
5C17×#323C212yesmaximal, complemented
6C22×D4×C17C411yescharacteristic
7C23×C4×C17C2211yescharacteristic
8C2×M16×C17—21no—
9C22×C8×C17—21no—
10C17×#123—21no—
11C17×#123—21no—
12C17×#123—21no—
13C17×#123—21no—
14C2×C4⋊C4×C17C411yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C411yescharacteristic
17C2×C22⋊C4×C17C411yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C17×C22⋊C8—21no—
20C17×C22⋊C8—21no—
21C17×C22⋊C8C2212yes—
22C17×C22⋊C8C2212yes—
23C24×C17C2×C411yescharacteristic
24C24×C17C2×C411yescharacteristic
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17—42no—
28C22×C4×C17—42no—
29C22×C4×C17C2×C411yescharacteristic
30C22×C4×C17C2×C411yescharacteristic
31C22×C4×C17C2×C411yescharacteristic
32C22×C4×C17C2×C411yescharacteristic
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D411yescharacteristic
37C22×C4×C17C2311yescharacteristic
38M16×C17—42no—
39C2×C8×C17—22no—
40C2×C8×C17—22no—
41C2×C8×C17—42no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17—21no—
44C4⋊C4×C17—21no—
45C4⋊C4×C17—21no—
46C22⋊C4×C17—21no—
47C22⋊C4×C17—21no—
48C22⋊C4×C17—21no—
49C22⋊C4×C17—21no—
50C22⋊C4×C17—21no—
51C22⋊C4×C17—21no—
52C23×C17—22no—
53C23×C17—22no—
54C23×C17—21no—
55C23×C17—22no—
56C23×C17—41no—
57C23×C17—41no—
58C23×C17—41no—
59C23×C17C22⋊C411yescharacteristic
60C23×C17C22⋊C411yescharacteristic
61C23×C17C22×C411yescharacteristic
62D4×C17—42no—
63D4×C17—42no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—41no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17—41no—
76C2×C4×C17C22⋊C411yescharacteristic
77C2×C4×C17C22⋊C411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C8×C17—42no—
81C8×C17—42no—
82#1468C1711yesmaximal, characteristic, direct product factor, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—44no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17—82no—
95C22×C17#9912yes—
96C22×C17C2×C22⋊C411yescharacteristic
97C22×C17#13112yes—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—41no—
103C4×C17—41no—
104C4×C17—41no—
105C2×#123C2×C1711yescharacteristic
106#407C2×C1711yescharacteristic, complemented
107C2×C22⋊C8C2×C1711yescharacteristic, complemented
108#324C2×C1712yescomplemented
109#323C2×C1712yescomplemented
110C2×C17—22no—
111C2×C17—22no—
112C2×C17—41no—
113C2×C17—81no—
114C2×C17#40811yescharacteristic
115C2×C17C2×#9911yescharacteristic
116C2×C17#41311yescharacteristic
117C22×D4C4×C1711yescharacteristic
118C23×C4C22×C1711yescharacteristic
119C2×M16—21no—
120C22×C8—21no—
121#123—21no—
122#123—21no—
123#123—21no—
124#123—21no—
125C2×C4⋊C4C4×C1711yescharacteristic
126C2×C4⋊C4C22×C1711yescharacteristic
127C2×C22⋊C4C4×C1711yescharacteristic
128C2×C22⋊C4C4×C1711yescharacteristic
129C2×C22⋊C4C22×C1711yescharacteristic
130C22⋊C8—21no—
131C22⋊C8—21no—
132C22⋊C8C22×C1712yes—
133C22⋊C8C22×C1712yes—
134C17#146811yescharacteristic, direct product factor, Sylow
135C24C2×C4×C1711yescharacteristic
136C24C2×C4×C1711yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—42no—
141C22×C4C2×C4×C1711yescharacteristic
142C22×C4C2×C4×C1711yescharacteristic
143C22×C4C2×C4×C1711yescharacteristic
144C22×C4C2×C4×C1711yescharacteristic
145C22×C4D4×C1711yescharacteristic
146C22×C4D4×C1711yescharacteristic
147C22×C4D4×C1711yescharacteristic
148C22×C4D4×C1711yescharacteristic
149C22×C4C23×C1711yescharacteristic
150M16—42no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊C4×C1711yescharacteristic
172C23C22⋊C4×C1711yescharacteristic
173C23C22×C4×C1711yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4C22⋊C4×C1711yescharacteristic
189C2×C4C22⋊C4×C1711yescharacteristic
190C2×C4C2×D4×C1711yescharacteristic
191C2×C4C2×D4×C1711yescharacteristic
192C8—42no—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—82no—
206C22C17×#9912yes—
207C22C2×C22⋊C4×C1711yescharacteristic
208C22C17×#13112yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C17×#40811yescharacteristic
221C2C2×C17×#9911yescharacteristic
222C2C17×#41311yescharacteristic
subgroupG/H
center, socleC22×C17C2×C22⋊C4
commutatorC23C22×C4×C17
Frattini, cosocleC22×C4C23×C17
prime radicalC22×D4×C17C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#407C2semidirect
3. C2×C17×C22⋊C8C2semidirect
4. C17×#324C2semidirect
5. C17×#323C2semidirect
82. #1468C17direct (N × H)
106. #407C2×C17semidirect
107. C2×C22⋊C8C2×C17semidirect
108. #324C2×C17semidirect
109. #323C2×C17semidirect
134. C17#1468direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#1232
C17×#4072
C2×C17×C22⋊C82
C17×#3242
C17×#3232
#146817