Finite Group Databasebeta

← #16947346

Subgroups — #16947346

Order 4352.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#14530134C211yesmaximal, characteristic
2#14527791C212yesmaximal
3#14526790C212yesmaximal
4#14526728C211yesmaximal, characteristic
5C17×#2822C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11763139—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C2×C17×#125—21no—
14C17×#400—22no—
15C17×#398—21no—
16C17×#398—22no—
17C17×#398C2212yes—
18C17×#382C2211yescharacteristic
19C17×#125—42no—
20C17×#125—42no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17D412yes—
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C42×C17D411yescharacteristic
30#10587612—21no—
31#10587610—22no—
32C2×#63223—21no—
33#10587576—41no—
34#10587574—41no—
35C17×#95—22no—
36C17×#95—21no—
37C17×#95—22no—
38C17×#95—22no—
39C17×#95D412yes—
40C22⋊#3622—22no—
41#10587481—42no—
42#10587479—41no—
43#10587479—42no—
44#10587478—41no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17C2×D411yescharacteristic
50C22×C4×C17C2×D412yes—
51C2×Dic34—22no—
52C4⋊C4×C17—41no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—42no—
55C4⋊C4×C17—42no—
56C4⋊C4×C17—44no—
57C4⋊C4×C17—42no—
58C4⋊C4×C17—42no—
59C4⋊C4×C17—42no—
60C4⋊C4×C17—81no—
61C42×C17—21no—
62C42×C17SD1612yes—
63C22⋊Dic17—41no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—22no—
67#63225—42no—
68C4×Dic17—41no—
69#63223—22no—
70#63223—41no—
71C2×#3622—41no—
72C2×#3622—42no—
73#12251—171nomaximal, Sylow
74C23×C17#12011yescharacteristic
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—41no—
90C2×C4×C17C2×SD1611yescharacteristic
91C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
92C2×Dic17—41no—
93C2×Dic17—41no—
94Dic34—82no—
95#3622—42no—
96#3622—82no—
97#2822D1711yescharacteristic
98#2061—171no—
99#1611—172no—
100#1346—172no—
101#1324—171no—
102C22×C17—21no—
103C22×C17—21no—
104C22×C17#45011yescharacteristic
105C22×C17#45611yescharacteristic
106C22×C17B(4,𝔽2)11yescharacteristic
107C4×C17—21no—
108C4×C17—21no—
109C4×C17—42no—
110C4×C17—42no—
111C4×C17—82no—
112C4×C17—82no—
113C4×C17—81no—
114Dic17—81no—
115Dic17—81no—
116#544—171no—
117C2×#125—21no—
118#483—342no—
119#428—341no—
120#400—22no—
121#398—21no—
122#398—22no—
123#398C2×D1712yes—
124#382C2×D1711yescharacteristic
125#343—171no—
126#339—341no—
127#328—342no—
128#324—172no—
129C2×C17—21no—
130C2×C17—21no—
131C2×C17#160411yescharacteristic
132C2×C17#220111yescharacteristic
133C2×C17#220311yescharacteristic
134C2×M16—341no—
135C4⋊Q8—681no—
136#125—42no—
137#125—42no—
138#125—681no—
139C22⋊Q8—342no—
140#117—341no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4C22⋊D1711yescharacteristic
147C2×C4⋊C4C22⋊D1712yes—
148C2×C4⋊C4C22×D1711yescharacteristic
149C2×C42C22⋊D1711yescharacteristic
150#107—681no—
151#106—681no—
152#106—682no—
153#103—682no—
154C22⋊C8—342no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95—22no—
159#95C22⋊D1712yes—
160C17#1225111yescharacteristic, complemented, Sylow
161C2×Q8—342no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D1711yescharacteristic
167C22×C4C2×C22⋊D1712yes—
168M16—342no—
169M16—681no—
170C2×C8—681no—
171C2×C8—682no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—341no—
182C4⋊C4—342no—
183C4⋊C4—341no—
184C4⋊C4—682no—
185C22⋊C4—681no—
186C42—21no—
187C42—681no—
188C42#6322912yes—
189C23#1058763911yescharacteristic
190Q8—1362no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—681no—
207C2×C4—681no—
208C2×C4#1058761811yescharacteristic
209C2×C4C2×#6322911yescharacteristic
210C8—682no—
211C8—1362no—
212C22—21no—
213C22—21no—
214C22#1176296011yescharacteristic
215C22#1176299211yescharacteristic
216C22#1176299611yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—1361no—
225C4—1361no—
226C2—21no—
227C2—21no—
228C2#1452775311yescharacteristic
229C2#1453079711yescharacteristic
230C2#1453080511yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2822C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2822D17
hypocenterC17#12251
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
160. C17#12251semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145277912
#145267902
#145267282
C17×#28222
#1225117