Finite Group Databasebeta

← #16947348

Subgroups — #16947348

Order 4352.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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#14536852C211yesmaximal, characteristic
2#14530134C211yesmaximal, characteristic
3#14527794C212yesmaximal
4#14526791C212yesmaximal
5C17×#1324C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7C2×#10587556—21no—
8#11763140—22no—
9#11762849—21no—
10#11762749—22no—
11#11762741—22no—
12#11762739—21no—
13#11762739C2212yes—
14#11762578C2211yescharacteristic
15#11762569—21no—
16#11762547—22no—
17C17×#382C2211yescharacteristic
18C17×#324C2212yes—
19C2×C4⋊C4×C17D411yescharacteristic
20C2×C4⋊C4×C17C2311yescharacteristic
21C2×C42×C17D411yescharacteristic
22#10587612—21no—
23C2×C4⋊Dic17—21no—
24C2×C4⋊Dic17—21no—
25C2×C4⋊Dic17—21no—
26C2×C4⋊Dic17D412yes—
27#10587610—22no—
28C2×#63225—22no—
29C2×#63225—22no—
30C2×#63223—21no—
31#10587576—41no—
32#10587574—41no—
33#10587556—42no—
34#10587556—42no—
35C17×C22⋊C8—22no—
36C17×#95—22no—
37#10587502—22no—
38#10587502—21no—
39#10587502—22no—
40#10587502D412yes—
41#10587488—42no—
42#10587487—42no—
43#10587479—41no—
44#10587478—41no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C2×Dic34—22no—
52C2×C8×C17—42no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—41no—
55C42×C17—21no—
56C42×C17SD1612yes—
57C22⋊Dic17—41no—
58C4⋊Dic17—21no—
59C4⋊Dic17—21no—
60C4⋊Dic17—22no—
61C4⋊Dic17—42no—
62C4⋊Dic17—42no—
63C4⋊Dic17—42no—
64C4⋊Dic17—81no—
65#63225—44no—
66#63225—42no—
67#63225—42no—
68#63225—42no—
69C4×Dic17—41no—
70#63223—22no—
71#63223—41no—
72C2×#3622—41no—
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—41no—
84C2×C4×C17C2×SD1611yescharacteristic
85C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
86C2×Dic17—42no—
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—42no—
91C2×Dic17—41no—
92C2×Dic17—41no—
93C2×Dic17—42no—
94Dic34—82no—
95C8×C17—82no—
96#3622—42no—
97#2822—171no—
98#2061—171no—
99#1611—172no—
100#1346—172no—
101#1324D1711yescharacteristic
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—81no—
112Dic17—82no—
113Dic17—82no—
114Dic17—81no—
115Dic17—81no—
116#544—171no—
117C2×#125—341no—
118#483—342no—
119#428—341no—
120#400—342no—
121#398—172no—
122#398—342no—
123#398—341no—
124#382C2×D1711yescharacteristic
125#343—171no—
126#339—341no—
127#328—342no—
128#324C2×D1712yes—
129C2×C17—21no—
130C2×C17—21no—
131C2×C17#160411yescharacteristic
132C2×C17#220111yescharacteristic
133C2×C17#220311yescharacteristic
134C2×M16—341no—
135C4⋊Q8—681no—
136#125—682no—
137#125—681no—
138#125—682no—
139C22⋊Q8—342no—
140#117—341no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4—342no—
144C2×C4⋊C4—341no—
145C2×C4⋊C4—342no—
146C2×C4⋊C4—341no—
147C2×C4⋊C4D6811yescharacteristic
148C2×C4⋊C4C22×D1711yescharacteristic
149C2×C42D6811yescharacteristic
150#107—681no—
151#106—681no—
152#106—682no—
153#103—682no—
154C22⋊C8—22no—
155#95—22no—
156#95—172no—
157#95—342no—
158#95—341no—
159#95—342no—
160C17#1225111yescharacteristic, complemented, Sylow
161C2×Q8—342no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—342no—
165C22×C4—342no—
166C22×C4C2×D6811yescharacteristic
167C22×C4D4×D1712yes—
168M16—342no—
169M16—681no—
170C2×C8—42no—
171C2×C8—681no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—341no—
175C4⋊C4—342no—
176C4⋊C4—341no—
177C4⋊C4—682no—
178C4⋊C4—684no—
179C4⋊C4—682no—
180C4⋊C4—682no—
181C4⋊C4—682no—
182C4⋊C4—682no—
183C4⋊C4—682no—
184C4⋊C4—1361no—
185C22⋊C4—681no—
186C42—21no—
187C42—681no—
188C42#6321912yes—
189C23#1058756811yescharacteristic
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—41no—
200C2×C4—682no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4—681no—
206C2×C4—681no—
207C2×C4—682no—
208C2×C4C2×#6321911yescharacteristic
209C2×C4#1058759411yescharacteristic
210C8—82no—
211C8—682no—
212C22—21no—
213C22—21no—
214C22#1176296211yescharacteristic
215C22#1176299411yescharacteristic
216C22#1176299711yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—1362no—
223C4—1362no—
224C4—1361no—
225C4—1361no—
226C2—21no—
227C2—21no—
228C2#1452775511yescharacteristic
229C2#1453079911yescharacteristic
230C2#1453080711yescharacteristic
subgroupG/H
centerC22#11762997
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1324C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#1324D17
hypocenterC17#12251
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

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
#145368522
#145301342
#145277942
#145267912
C17×#13242
#1225117