Finite Group Databasebeta

← #16947402

Subgroups — #16947402

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#14537056C211yesmaximal, characteristic
2#14527758C212yesmaximal
3#14526813C212yesmaximal
4#14526728C211yesmaximal, characteristic
5C17×#2061C211yesmaximal, characteristic
6#11762964—22no—
7#11762751—22no—
8#11762748—21no—
9#11762730—22no—
10#11762729C2212yes—
11#11762675—21no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#428—21no—
16C17×#382C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×Dic34D412yes—
20C2×M16×C17—21no—
21C17×C4⋊Q8—41no—
22C17×#125—41no—
23C17×C22⋊Q8—22no—
24C17×#117—21no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×#63231—42no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×#63225—22no—
33C17×#107—41no—
34C17×#106—41no—
35C17×#95—22no—
36#10587502—22no—
37#10587502—22no—
38#10587502—21no—
39#10587502—22no—
40#10587502D412yes—
41C22⋊#3622—22no—
42#10587481—42no—
43C2×Q8×C17—22no—
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17C2×D412yes—
47C22×C4×C17C2×D411yescharacteristic
48C22×Dic17—22no—
49C22×Dic17—22no—
50C2×Dic34—42no—
51C2×Dic34—82no—
52M16×C17—22no—
53M16×C17—41no—
54C2×C8×C17—41no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—21no—
57C4⋊C4×C17—21no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—42no—
61C22⋊C4×C17—41no—
62C42×C17—21no—
63C42×C17—41no—
64C42×C17Q1612yescomplemented
65#63231—84no—
66C4⋊Dic17—81no—
67C4⋊Dic17—81no—
68C4⋊Dic17—81no—
69#63225—82no—
70C2×#3622—42no—
71#12260—171nomaximal, Sylow
72C23×C17#12011yescharacteristic
73Q8×C17—82no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—41no—
79C2×C4×C17—41no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17C2×Q1611yescharacteristic
86C2×C4×C17#13711yescharacteristic
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—42no—
91C2×Dic17—42no—
92C2×Dic17—42no—
93Dic34—44no—
94Dic34—82no—
95C8×C17—42no—
96#3622—82no—
97#2844—171no—
98#2061D1711yescharacteristic
99#1605—172no—
100#1354—172no—
101#1324—171no—
102C22×C17—21no—
103C22×C17—21no—
104C22×C17#45111yescharacteristic
105C22×C17#45611yescharacteristic
106C22×C17B(4,𝔽2)11yescharacteristic
107C4×C17—21no—
108C4×C17—21no—
109C4×C17—42no—
110C4×C17—42no—
111C4×C17—81no—
112C4×C17—81no—
113C4×C17—81no—
114Dic17—82no—
115Dic17—82no—
116#544C2×D1711yescharacteristic
117#451—342no—
118#428—21no—
119#400—342no—
120#400—341no—
121#395—172no—
122#395—342no—
123#382—341no—
124#382C2×D1711yescharacteristic
125#343C2×D1711yescharacteristic
126#339—21no—
127#328—342no—
128#324—172no—
129C2×C17—21no—
130C2×C17—21no—
131C2×C17#160511yescharacteristic
132C2×C17#219911yescharacteristic
133C2×C17#220311yescharacteristic
134C22×Q8—172no—
135C2×Q16—682no—
136C2×M16—21no—
137C4⋊Q8—41no—
138#125—41no—
139C22⋊Q8—22no—
140#117—21no—
141C2×C4⋊C4—341no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4—341no—
144C2×C4⋊C4—342no—
145C2×C4⋊C4C22⋊D1711yescharacteristic
146C2×C4⋊C4C22×D1711yescharacteristic
147C2×C42C22⋊D1711yescharacteristic
148#107—41no—
149#106—41no—
150#103—682no—
151C22⋊C8—342no—
152#95—22no—
153#95—172no—
154#95—342no—
155#95—342no—
156#95—341no—
157#95—342no—
158C17#1226011yescharacteristic, complemented, Sylow
159C2×Q8—22no—
160C2×Q8—682no—
161C2×Q8—1362no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—342no—
165C22×C4—342no—
166C22×C4D4×D1712yes—
167C22×C4C2×C22⋊D1711yescharacteristic
168Q16—1364no—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—682no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—1362no—
180C4⋊C4—1361no—
181C4⋊C4—1361no—
182C4⋊C4—1361no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42#6323112yescomplemented
187C23#1058762311yescharacteristic
188Q8—82no—
189Q8—684no—
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—41no—
201C2×C4—41no—
202C2×C4—682no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4—682no—
206C2×C4—682no—
207C2×C4—682no—
208C2×C4C2×#6323111yescharacteristic
209C2×C4#1058763711yescharacteristic
210C8—42no—
211C8—1362no—
212C22—21no—
213C22—21no—
214C22#1176296411yescharacteristic
215C22#1176299111yescharacteristic
216C22#1176299511yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—1362no—
225C4—1362no—
226C2—21no—
227C2—21no—
228C2#1452775711yescharacteristic
229C2#1453078411yescharacteristic
230C2#1453080411yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2061C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2061D17
hypocenterC17#12260
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
64. C42×C17Q16semidirect
158. C17#12260semidirect
186. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145277582
#145268132
#145267282
C17×#20612
#1226017