Finite Group Databasebeta

← C17×#1507 #14524513

Subgroups — C17×#1507 #14524513

Order 2176.

346 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 96 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×#121C211yesmaximal, characteristic
2C17×#407C211yesmaximal, characteristic, complemented
3C4×C22⋊C4×C17C211yesmaximal, characteristic, complemented
4C17×#327C214yesmaximal, complemented
5C22×D4×C17—21no—
6C22×D4×C17C411yescharacteristic, complemented
7C23×C4×C17C2211yescharacteristic
8C2×M16×C17—21no—
9C17×#121—21no—
10C17×#121—21no—
11C17×#121C412yescomplemented
12C17×#121C2212yes—
13C2×C4⋊C4×C17C411yescharacteristic, complemented
14C2×C22⋊C4×C17—21no—
15C2×C22⋊C4×C17—21no—
16C2×C42×C17—21no—
17C17×C22⋊C8—21no—
18C17×C22⋊C8C2212yescomplemented
19C17×#95C2212yescomplemented
20C24×C17—21no—
21C24×C17D411yescharacteristic
22C2×D4×C17—22no—
23C2×D4×C17—22no—
24C2×D4×C17—21no—
25C2×D4×C17—42no—
26C2×D4×C17—44no—
27C2×D4×C17C2×C412yes—
28C22×C4×C17—21no—
29C22×C4×C17—22no—
30C22×C4×C17—42no—
31C22×C4×C17C2×C411yescharacteristic
32C22×C4×C17C2×C411yescharacteristic
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×C4×C17D411yescharacteristic
36C22×C4×C17C2311yescharacteristic
37M16×C17—42no—
38C2×C8×C17—22no—
39C4⋊C4×C17—21no—
40C4⋊C4×C17C2×C412yes—
41C22⋊C4×C17—24no—
42C22⋊C4×C17—22no—
43C22⋊C4×C17—22no—
44C42×C17—42no—
45C23×C17—22no—
46C23×C17—22no—
47C23×C17—22no—
48C23×C17—41no—
49C23×C17—42no—
50C23×C17—42no—
51C23×C17—42no—
52C23×C17C22⋊C411yescharacteristic
53C23×C17C22⋊C411yescharacteristic
54C23×C17C2×D411yescharacteristic
55D4×C17—42no—
56D4×C17—44no—
57D4×C17—44no—
58D4×C17—42no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—24no—
63C2×C4×C17—22no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17—42no—
68C2×C4×C17—42no—
69C2×C4×C17—42no—
70C2×C4×C17C22⋊C411yescharacteristic
71C2×C4×C17C22⋊C411yescharacteristic
72C2×C4×C17C22×C411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C8×C17—42no—
75#1507C1711yesmaximal, characteristic, direct product factor, Sylow
76C22×C17—21no—
77C22×C17—21no—
78C22×C17—21no—
79C22×C17—21no—
80C22×C17—21no—
81C22×C17—21no—
82C22×C17—21no—
83C22×C17—44no—
84C22×C17—42no—
85C22×C17—42no—
86C22×C17—42no—
87C22×C17—84no—
88C22×C17#10412yes—
89C22×C17C2×C22⋊C411yescharacteristic
90C22×C17AGL(1,ℤ/8)12yes—
91C4×C17—21no—
92C4×C17—21no—
93C4×C17—21no—
94C4×C17—44no—
95C4×C17—42no—
96C4×C17#9912yes—
97C2×#121C2×C1711yescharacteristic
98#407C2×C1711yescharacteristic, complemented
99C4×C22⋊C4C2×C1711yescharacteristic, complemented
100#327C2×C1714yescomplemented
101C2×C17—22no—
102C2×C17—22no—
103C2×C17—41no—
104C2×C17—82no—
105C2×C17C2×#9911yescharacteristic
106C2×C17#41811yescharacteristic
107C2×C17C2×#10411yescharacteristic
108C22×D4—21no—
109C22×D4C4×C1711yescharacteristic, complemented
110C23×C4C22×C1711yescharacteristic
111C2×M16—21no—
112#121—21no—
113#121—21no—
114#121C4×C1712yescomplemented
115#121C22×C1712yes—
116C2×C4⋊C4C4×C1711yescharacteristic, complemented
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C42—21no—
120C22⋊C8—21no—
121C22⋊C8C22×C1712yescomplemented
122#95C22×C1712yescomplemented
123C17#150711yescharacteristic, direct product factor, Sylow
124C24—21no—
125C24D4×C1711yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×C4×C1712yes—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4C2×C4×C1711yescharacteristic
136C22×C4C2×C4×C1711yescharacteristic
137C22×C4D4×C1711yescharacteristic
138C22×C4D4×C1711yescharacteristic
139C22×C4D4×C1711yescharacteristic
140C22×C4C23×C1711yescharacteristic
141M16—42no—
142C2×C8—22no—
143C4⋊C4—21no—
144C4⋊C4C2×C4×C1712yes—
145C22⋊C4—24no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊C4×C1711yescharacteristic
157C23C22⋊C4×C1711yescharacteristic
158C23C2×D4×C1711yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4C22⋊C4×C1711yescharacteristic
175C2×C4C22⋊C4×C1711yescharacteristic
176C2×C4C22×C4×C1711yescharacteristic
177C2×C4C2×D4×C1711yescharacteristic
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22C17×#10412yes—
192C22C2×C22⋊C4×C1711yescharacteristic
193C22C17×AGL(1,ℤ/8)12yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—42no—
199C4C17×#9912yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×C17×#9911yescharacteristic
205C2C17×#41811yescharacteristic
206C2C2×C17×#10411yescharacteristic
subgroupG/H
center, socleC22×C17C2×C22⋊C4
commutatorC2×C4C22×C4×C17
Frattini, cosocleC22×C4C23×C17
prime radicalC2×C17×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ 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. C4×C22⋊C4×C17C2semidirect
4. C17×#327C2semidirect
6. C22×D4×C17C4semidirect
11. C17×#121C4semidirect
13. C2×C4⋊C4×C17C4semidirect
18. C17×C22⋊C8C22semidirect
19. C17×#95C22semidirect
75. #1507C17direct (N × H)
98. #407C2×C17semidirect
99. C4×C22⋊C4C2×C17semidirect
100. #327C2×C17semidirect
109. C22×D4C4×C17semidirect
114. #121C4×C17semidirect
116. C2×C4⋊C4C4×C17semidirect
121. C22⋊C8C22×C17semidirect
122. #95C22×C17semidirect
123. C17#1507direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#1212
C17×#4072
C4×C22⋊C4×C172
C17×#3272
#150717