Finite Group Databasebeta

← C17×#1601 #14524607

Subgroups — C17×#1601 #14524607

Order 2176.

308 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 68 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×#534C211yesmaximal, characteristic
2C17×#448C212yesmaximal, complemented
3C17×#393C211yesmaximal, characteristic, complemented
4C17×#327C212yesmaximal, complemented
5C17×#323C211yesmaximal, characteristic, complemented
6C22×Q8×C17C2211yescharacteristic, complemented
7C22×D4×C17—21no—
8C2×SD16×C17—22no—
9C17×#123—22no—
10C17×#121—21no—
11C17×#121C2212yes—
12C17×#120—22no—
13C2×C22⋊C4×C17—21no—
14C2×C22⋊C4×C17C2211yescharacteristic
15C17×#103—22no—
16C17×C22⋊C8C2212yescomplemented
17C17×#95—21no—
18C17×#95C2211yescharacteristic, complemented
19C24×C17—21no—
20C24×C17D411yescharacteristic
21C2×Q8×C17—22no—
22C2×Q8×C17—42no—
23C2×D4×C17—22no—
24C2×D4×C17—22no—
25C2×D4×C17—21no—
26C2×D4×C17—42no—
27C2×D4×C17—42no—
28C2×D4×C17D412yes—
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C22×C4×C17—21no—
32C22×C4×C17D411yescharacteristic
33C22×C4×C17C2311yescharacteristic
34SD16×C17—44no—
35C2×C8×C17—22no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17D412yes—
38C22⋊C4×C17—22no—
39C22⋊C4×C17—22no—
40C22⋊C4×C17—22no—
41C22⋊C4×C17—42no—
42C23×C17—21no—
43C23×C17—21no—
44C23×C17—22no—
45C23×C17—41no—
46C23×C17—41no—
47C23×C17—42no—
48C23×C17—81no—
49C23×C17—81no—
50C23×C17SD1612yescomplemented
51C23×C17C2×D411yescharacteristic
52Q8×C17—24no—
53Q8×C17—42no—
54D4×C17—42no—
55D4×C17—44no—
56D4×C17—42no—
57D4×C17—42no—
58D4×C17—81no—
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—41no—
63C2×C4×C17—42no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17—41no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17C2×D412yes—
71C8×C17—42no—
72#1601C1711yesmaximal, characteristic, direct product factor, Sylow
73C22×C17—21no—
74C22×C17—21no—
75C22×C17—22no—
76C22×C17—42no—
77C22×C17—41no—
78C22×C17—41no—
79C22×C17—42no—
80C22×C17—42no—
81C22×C17—42no—
82C22×C17—81no—
83C22×C17—84no—
84C22×C17#12011yescharacteristic
85C22×C17C2×SD1611yescharacteristic
86C22×C17AGL(1,ℤ/8)11yescharacteristic
87C4×C17—22no—
88C4×C17—42no—
89C4×C17—41no—
90C4×C17—42no—
91C4×C17—81no—
92#534C2×C1711yescharacteristic
93#448C2×C1712yescomplemented
94#393C2×C1711yescharacteristic, complemented
95#327C2×C1712yescomplemented
96#323C2×C1711yescharacteristic, complemented
97C2×C17—21no—
98C2×C17—21no—
99C2×C17—42no—
100C2×C17—82no—
101C2×C17#45011yescharacteristic
102C2×C17#45511yescharacteristic
103C2×C17B(4,𝔽2)11yescharacteristic
104C22×Q8C22×C1711yescharacteristic, complemented
105C22×D4—21no—
106C2×SD16—22no—
107#123—22no—
108#121—21no—
109#121C22×C1712yes—
110#120—22no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4C22×C1711yescharacteristic
113#103—22no—
114C22⋊C8C22×C1712yescomplemented
115#95—21no—
116#95C22×C1711yescharacteristic, complemented
117C17#160111yescharacteristic, direct product factor, Sylow
118C24—21no—
119C24D4×C1711yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4D4×C1712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4D4×C1711yescharacteristic
132C22×C4C23×C1711yescharacteristic
133SD16—44no—
134C2×C8—22no—
135C4⋊C4—21no—
136C4⋊C4D4×C1712yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—42no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23SD16×C1712yescomplemented
150C23C2×D4×C1711yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4C2×D4×C1712yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22C17×#12011yescharacteristic
183C22C2×SD16×C1711yescharacteristic
184C22C17×AGL(1,ℤ/8)11yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—42no—
189C4—81no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2C17×#45011yescharacteristic
195C2C17×#45511yescharacteristic
196C2C17×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C17#120
commutator, Frattini, cosocleC22×C4C23×C17
prime radicalC17×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C22×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#448C2semidirect
3. C17×#393C2semidirect
4. C17×#327C2semidirect
5. C17×#323C2semidirect
6. C22×Q8×C17C22semidirect
16. C17×C22⋊C8C22semidirect
18. C17×#95C22semidirect
50. C23×C17SD16semidirect
72. #1601C17direct (N × H)
93. #448C2×C17semidirect
94. #393C2×C17semidirect
95. #327C2×C17semidirect
96. #323C2×C17semidirect
104. C22×Q8C22×C17semidirect
114. C22⋊C8C22×C17semidirect
116. #95C22×C17semidirect
117. C17#1601direct (N × H)
149. C23SD16×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#5342
C17×#4482
C17×#3932
C17×#3272
C17×#3232
#160117