Finite Group Databasebeta

← C17×#13368 #16871112

Subgroups — C17×#13368 #16871112

Order 4352.

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3162C211yesmaximal, characteristic
2C17×#2141C212yesmaximal
3C17×#1971C211yesmaximal, characteristic
4C17×#1386C212yesmaximal
5C17×#1377C211yesmaximal, characteristic
6C22×Q16×C17C2211yescharacteristic
7C17×#533—21no—
8C2×C17×M32C2212yes—
9C17×#501—24no—
10C17×#427—21no—
11C17×#427C2212yes—
12C2×C17×#106C2211yescharacteristic
13C2×C17×#103—21no—
14C2×C17×#97C2211yescharacteristic
15C17×#391—21no—
16C17×#358—24no—
17C22×Q8×C17—21no—
18C2×Q16×C17—22no—
19C2×Q16×C17—22no—
20C2×Q16×C17—21no—
21C2×Q16×C17Q812yescomplemented
22C22×C8×C17D411yescharacteristic
23C22×C8×C17C2311yescharacteristic
24C17×C4⋊Q8—22no—
25C17×#125—22no—
26C17×C22⋊Q8—42no—
27C4×Q8×C17—42no—
28C17×#117—22no—
29C2×C4⋊C4×C17—21no—
30C2×C42×C17D411yescharacteristic
31C17×M32—22no—
32C17×M32—22no—
33C2×C16×C17—22no—
34C17×#107—22no—
35C17×#107D414yes—
36C17×#106—22no—
37C17×#106—22no—
38C17×#106—22no—
39C17×#103—42no—
40C17×#97—22no—
41C17×#95—42no—
42C2×Q8×C17—22no—
43C2×Q8×C17—42no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C22×C4×C17—41no—
50C22×C4×C17—41no—
51C22×C4×C17D812yes—
52C22×C4×C17C2×D411yescharacteristic
53Q16×C17—24no—
54Q16×C17—42no—
55C2×C8×C17—21no—
56C2×C8×C17—21no—
57C2×C8×C17SD1612yes—
58C2×C8×C17C2×D412yes—
59C2×C8×C17C2×Q811yescharacteristic
60C2×C8×C17#4112yes—
61C2×C8×C17#4111yescharacteristic
62C4⋊C4×C17—24no—
63C4⋊C4×C17—44no—
64C4⋊C4×C17—42no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—41no—
67C22⋊C4×C17—42no—
68C42×C17—21no—
69C42×C17—21no—
70C42×C17—42no—
71C16×C17—42no—
72#13368C1711yesmaximal, characteristic, direct product factor, Sylow
73C23×C17C2×D811yescharacteristic
74Q8×C17—42no—
75Q8×C17—42no—
76Q8×C17—84no—
77Q8×C17—81no—
78Q8×C17—81no—
79C2×C4×C17—22no—
80C2×C4×C17—42no—
81C2×C4×C17—44no—
82C2×C4×C17—42no—
83C2×C4×C17—81no—
84C2×C4×C17—81no—
85C2×C4×C17—82no—
86C2×C4×C17—81no—
87C2×C4×C17#12011yescharacteristic
88C2×C4×C17C22⋊Q811yescharacteristic
89C2×C4×C17#12411yescharacteristic
90C2×C4×C17C2×SD1611yescharacteristic
91C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C17#13711yescharacteristic
93C8×C17—22no—
94C8×C17C22⋊Q812yes—
95C8×C17#12412yes—
96#3162C2×C1711yescharacteristic
97#2141C2×C1712yes—
98#1971C2×C1711yescharacteristic
99#1386C2×C1712yes—
100#1377C2×C1711yescharacteristic
101C22×C17—22no—
102C22×C17#44711yescharacteristic
103C22×C17#47411yescharacteristic
104C22×C17#48611yescharacteristic
105C4×C17—42no—
106C4×C17—84no—
107C4×C17—82no—
108C4×C17—161no—
109C4×C17#39311yescharacteristic
110C4×C17#44811yescharacteristic
111C4×C17#47611yescharacteristic
112C4×C17#48911yescharacteristic
113C22×Q16C22×C1711yescharacteristic
114#533—21no—
115C2×M32C22×C1712yes—
116#501—24no—
117#427—21no—
118#427C22×C1712yes—
119C2×#106C22×C1711yescharacteristic
120C2×#103—21no—
121C2×#97C22×C1711yescharacteristic
122#391—21no—
123#358—24no—
124C2×C17—21no—
125C2×C17—21no—
126C2×C17#202311yescharacteristic
127C2×C17#219112yes—
128C22×Q8—21no—
129C2×Q16—22no—
130C2×Q16—22no—
131C2×Q16—21no—
132C2×Q16Q8×C1712yescomplemented
133C22×C8D4×C1711yescharacteristic
134C22×C8C23×C1711yescharacteristic
135C4⋊Q8—22no—
136#125—22no—
137C22⋊Q8—42no—
138C4×Q8—42no—
139#117—22no—
140C2×C4⋊C4—21no—
141C2×C42D4×C1711yescharacteristic
142M32—22no—
143M32—22no—
144C2×C16—22no—
145#107—22no—
146#107D4×C1714yes—
147#106—22no—
148#106—22no—
149#106—22no—
150#103—42no—
151#97—22no—
152#95—42no—
153C17#1336811yescharacteristic, direct product factor, Sylow
154C2×Q8—22no—
155C2×Q8—42no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4D8×C1712yes—
164C22×C4C2×D4×C1711yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8SD16×C1712yes—
170C2×C8C2×D4×C1712yes—
171C2×C8C2×Q8×C1711yescharacteristic
172C2×C8#41×C1712yes—
173C2×C8#41×C1711yescharacteristic
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—42no—
184C23C2×D8×C1711yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—84no—
188Q8—81no—
189Q8—81no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4C17×#12011yescharacteristic
199C2×C4C17×C22⋊Q811yescharacteristic
200C2×C4C17×#12411yescharacteristic
201C2×C4C2×SD16×C1711yescharacteristic
202C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
203C2×C4C17×#13711yescharacteristic
204C8—22no—
205C8C17×C22⋊Q812yes—
206C8C17×#12412yes—
207C22—22no—
208C22C17×#44711yescharacteristic
209C22C17×#47411yescharacteristic
210C22C17×#48611yescharacteristic
211C4—42no—
212C4—84no—
213C4—82no—
214C4—161no—
215C4C17×#39311yescharacteristic
216C4C17×#44811yescharacteristic
217C4C17×#47611yescharacteristic
218C4C17×#48911yescharacteristic
219C2—21no—
220C2—21no—
221C2C17×#202311yescharacteristic
222C2C17×#219112yes—
subgroupG/H
center, socleC22×C17#447
commutator, Frattini, cosocleC22×C8C23×C17
prime radicalC23×C17C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C2×C17×#97 ◁ G

Complements

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

NH ≅ G/Ntype
21. C2×Q16×C17Q8semidirect
72. #13368C17direct (N × H)
132. C2×Q16Q8×C17semidirect
153. C17#13368direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#31622
C17×#21412
C17×#19712
C17×#13862
C17×#13772
#1336817