Finite Group Databasebeta

← C17×#7491 #16865235

Subgroups — C17×#7491 #16865235

Order 4352.

276 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 70 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×#356C211yesmaximal, characteristic
2C17×#1886C211yesmaximal, characteristic
3C2×C17×#333C211yesmaximal, characteristic
4C2×C17×C4⋊Q8C2211yescharacteristic
5C17×#519C411yescharacteristic
6C2×C17×C4⋊C8—21no—
7C17×#419—21no—
8C2×C17×#101—21no—
9C2×C17×#101C411yescharacteristic
10C17×#356—21no—
11C17×#356C412yescomplemented
12C17×#339—21no—
13C17×#333—21no—
14C17×#333C412yescomplemented
15C22×Q8×C17C2×C411yescharacteristic
16C22×Q8×C17D411yescharacteristic
17C2×M16×C17—21no—
18C2×M16×C17—41no—
19C22×C8×C17—41no—
20C17×C4⋊Q8—41no—
21C17×C4⋊Q8C2×C411yescharacteristic
22C17×C4⋊Q8C2×C411yescharacteristic
23C17×C4⋊Q8D411yescharacteristic
24C17×C4⋊Q8Q811yescharacteristic
25C4×Q8×C17—42no—
26C17×#117—21no—
27C17×#117—41no—
28C2×C4⋊C4×C17—41no—
29C2×C42×C17D411yescharacteristic
30C17×C4⋊C8—22no—
31C17×C4⋊C8—41no—
32C17×#103—42no—
33C17×#101—22no—
34C17×#101—41no—
35C17×#101—41no—
36C2×Q8×C17—21no—
37C2×Q8×C17—21no—
38C2×Q8×C17—41no—
39C2×Q8×C17—41no—
40C2×Q8×C17—41no—
41C2×Q8×C17—41no—
42C2×Q8×C17C4211yescharacteristic
43C2×Q8×C17C22⋊C411yescharacteristic
44C2×Q8×C17C4⋊C411yescharacteristic
45C2×Q8×C17C4⋊C411yescharacteristic
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C22⋊C411yescharacteristic
50M16×C17—41no—
51M16×C17—42no—
52M16×C17—41no—
53M16×C17—81no—
54C2×C8×C17—42no—
55C2×C8×C17—41no—
56C2×C8×C17—82no—
57C2×C8×C17—81no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—41no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—41no—
65C42×C17—21no—
66C42×C17—41no—
67C42×C17—42no—
68C42×C17C22⋊C411yescharacteristic
69C42×C17C4⋊C411yescharacteristic
70#7491C1711yesmaximal, characteristic, direct product factor, Sylow
71C23×C17#9911yescharacteristic
72Q8×C17—41no—
73Q8×C17—41no—
74Q8×C17—41no—
75Q8×C17—41no—
76Q8×C17—42no—
77Q8×C17—81no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17#9511yescharacteristic
94C2×C4×C17#9911yescharacteristic
95C8×C17—82no—
96C8×C17—81no—
97C8×C17—82no—
98C2×#356C2×C1711yescharacteristic
99#1886C2×C1711yescharacteristic
100C2×#333C2×C1711yescharacteristic
101C22×C17—22no—
102C22×C17#34211yescharacteristic
103C22×C17AΣL(1,𝔽16)11yescharacteristic
104C22×C17#35211yescharacteristic
105C4×C17—21no—
106C4×C17—22no—
107C4×C17—21no—
108C4×C17—21no—
109C4×C17—21no—
110C4×C17—41no—
111C4×C17—41no—
112C4×C17—41no—
113C4×C17—41no—
114C4×C17—82no—
115C2×C4⋊Q8C22×C1711yescharacteristic
116#519C4×C1711yescharacteristic
117C2×C4⋊C8—21no—
118#419—21no—
119C2×#101—21no—
120C2×#101C4×C1711yescharacteristic
121#356—21no—
122#356C4×C1712yescomplemented
123#339—21no—
124#333—21no—
125#333C4×C1712yescomplemented
126C2×C17—21no—
127C2×C17—21no—
128C2×C17#139011yescharacteristic
129C2×C17#140512yes—
130C22×Q8C2×C4×C1711yescharacteristic
131C22×Q8D4×C1711yescharacteristic
132C2×M16—21no—
133C2×M16—41no—
134C22×C8—41no—
135C4⋊Q8—41no—
136C4⋊Q8C2×C4×C1711yescharacteristic
137C4⋊Q8C2×C4×C1711yescharacteristic
138C4⋊Q8D4×C1711yescharacteristic
139C4⋊Q8Q8×C1711yescharacteristic
140C4×Q8—42no—
141#117—21no—
142#117—41no—
143C2×C4⋊C4—41no—
144C2×C42D4×C1711yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—41no—
147#103—42no—
148#101—22no—
149#101—41no—
150#101—41no—
151C17#749111yescharacteristic, direct product factor, Sylow
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C42×C1711yescharacteristic
159C2×Q8C22⋊C4×C1711yescharacteristic
160C2×Q8C4⋊C4×C1711yescharacteristic
161C2×Q8C4⋊C4×C1711yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22⋊C4×C1711yescharacteristic
166M16—42no—
167M16—41no—
168M16—41no—
169M16—81no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—82no—
173C2×C8—81no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42—42no—
184C42C22⋊C4×C1711yescharacteristic
185C42C4⋊C4×C1711yescharacteristic
186C23C17×#9911yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4C17×#9511yescharacteristic
209C2×C4C17×#9911yescharacteristic
210C8—82no—
211C8—81no—
212C8—82no—
213C22—22no—
214C22C17×#34211yescharacteristic
215C22C17×AΣL(1,𝔽16)11yescharacteristic
216C22C17×#35211yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—82no—
227C2—21no—
228C2—21no—
229C2C17×#139011yescharacteristic
230C2C17×#140512yes—
subgroupG/H
center, socleC22×C17AΣL(1,𝔽16)
commutatorC2×Q8C42×C17
Frattini, cosocleC2×C4⋊Q8C22×C17
prime radicalC23×C17#99
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C22×C4×C17 ◁ C2×C17×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
11. C17×#356C4semidirect
14. C17×#333C4semidirect
70. #7491C17direct (N × H)
122. #356C4×C17semidirect
125. #333C4×C17semidirect
151. C17#7491direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#3562
C17×#18862
C2×C17×#3332
#749117