Finite Group Databasebeta

← C17×#12157 #16869901

Subgroups — C17×#12157 #16869901

Order 4352.

372 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2822C211yesmaximal, characteristic, complemented
2C17×#2040C211yesmaximal, characteristic
3C17×#1610C212yesmaximal
4C17×#1346C212yesmaximal, complemented
5C17×#1324C211yesmaximal, characteristic, complemented
6C17×#537C2211yescharacteristic
7C2×C17×#125—21no—
8C17×#480—22no—
9C2×C17×#104—21no—
10C17×#417—21no—
11C17×#400—22no—
12C17×#398—21no—
13C17×#398—22no—
14C17×#398C2212yes—
15C17×#382C2211yescharacteristic
16C17×#343C2211yescharacteristic
17C17×#327—22no—
18C17×#324C2212yes—
19C2×#41×C17—21no—
20C2×M16×C17—21no—
21C17×#125—42no—
22C17×#125—42no—
23C17×C22⋊Q8—41no—
24C17×#121—22no—
25C17×#121—41no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D412yes—
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C42×C17D411yescharacteristic
35C17×#107—42no—
36C17×#104—42no—
37C17×#103—41no—
38C17×#102—41no—
39C17×#102—42no—
40C17×C22⋊C8—22no—
41C17×#95—22no—
42C17×#95—21no—
43C17×#95—22no—
44C17×#95—22no—
45C17×#95D412yes—
46#41×C17—41no—
47#41×C17—41no—
48#41×C17—41no—
49#41×C17—41no—
50C2×Q8×C17—21no—
51C2×D4×C17—21no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—41no—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D412yes—
61M16×C17—41no—
62M16×C17—41no—
63C2×C8×C17—41no—
64C2×C8×C17—42no—
65C4⋊C4×C17—22no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—44no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—81no—
75C22⋊C4×C17—42no—
76C42×C17—21no—
77C42×C17D812yes—
78#12157C1711yesmaximal, characteristic, direct product factor, Sylow
79C23×C17—41no—
80C23×C17#12011yescharacteristic
81Q8×C17—41no—
82Q8×C17—41no—
83D4×C17—41no—
84D4×C17—41no—
85D4×C17—82no—
86D4×C17—81no—
87D4×C17—81no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—81no—
105C2×C4×C17—81no—
106C2×C4×C17C2×D811yescharacteristic
107C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
108C8×C17—81no—
109C8×C17—82no—
110#2822C2×C1711yescharacteristic, complemented
111#2040C2×C1711yescharacteristic
112#1610C2×C1712yes—
113#1346C2×C1712yescomplemented
114#1324C2×C1711yescharacteristic, complemented
115C22×C17—21no—
116C22×C17—21no—
117C22×C17—81no—
118C22×C17—81no—
119C22×C17—81no—
120C22×C17#44711yescharacteristic
121C22×C17#45411yescharacteristic
122C22×C17B(4,𝔽2)11yescharacteristic
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—42no—
126C4×C17—42no—
127C4×C17—82no—
128C4×C17—82no—
129C4×C17—81no—
130C4×C17—81no—
131#537C22×C1711yescharacteristic
132C2×#125—21no—
133#480—22no—
134C2×#104—21no—
135#417—21no—
136#400—22no—
137#398—21no—
138#398—22no—
139#398C22×C1712yes—
140#382C22×C1711yescharacteristic
141#343C22×C1711yescharacteristic
142#327—22no—
143#324C22×C1712yes—
144C2×C17—21no—
145C2×C17—21no—
146C2×C17—161no—
147C2×C17#159811yescharacteristic
148C2×C17#220111yescharacteristic
149C2×C17#220311yescharacteristic
150C2×#41—21no—
151C2×M16—21no—
152#125—42no—
153#125—42no—
154C22⋊Q8—41no—
155#121—22no—
156#121—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4D4×C1711yescharacteristic
163C2×C4⋊C4D4×C1712yes—
164C2×C4⋊C4C23×C1711yescharacteristic
165C2×C42D4×C1711yescharacteristic
166#107—42no—
167#104—42no—
168#103—41no—
169#102—41no—
170#102—42no—
171C22⋊C8—22no—
172#95—22no—
173#95—21no—
174#95—22no—
175#95—22no—
176#95D4×C1712yes—
177C17#1215711yescharacteristic, direct product factor, Sylow
178#41—41no—
179#41—41no—
180#41—41no—
181#41—41no—
182C2×Q8—21no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×D4×C1711yescharacteristic
192C22×C4C2×D4×C1712yes—
193M16—41no—
194M16—41no—
195C2×C8—42no—
196C2×C8—41no—
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—81no—
207C22⋊C4—42no—
208C42—21no—
209C42D8×C1712yes—
210C23—41no—
211C23C17×#12011yescharacteristic
212Q8—41no—
213Q8—41no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4C2×D8×C1711yescharacteristic
238C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22C17×#44711yescharacteristic
247C22C17×#45411yescharacteristic
248C22C17×B(4,𝔽2)11yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2C17×#159811yescharacteristic
261C2C17×#220111yescharacteristic
262C2C17×#220311yescharacteristic
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C17
prime radicalC17×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C4⋊C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#2822C2semidirect
4. C17×#1346C2semidirect
5. C17×#1324C2semidirect
78. #12157C17direct (N × H)
110. #2822C2×C17semidirect
113. #1346C2×C17semidirect
114. #1324C2×C17semidirect
177. C17#12157direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28222
C17×#20402
C17×#16102
C17×#13462
C17×#13242
#1215717