Finite Group Databasebeta

← #16947172

Subgroups — #16947172

Order 4352.

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

normal subgroups only

This group has 336 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#14529945C211yesmaximal, characteristic
2#14527825C212yesmaximal
3#14526811C212yesmaximal
4#14526718C211yesmaximal, characteristic
5C17×#2809C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11763144—22no—
8#11762818—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C17×#400C2212yes—
14C17×#398—22no—
15C17×#391—21no—
16C17×#391C2211yescharacteristic
17C17×#389—21no—
18C17×#387—22no—
19C22×Q8×C17D411yescharacteristic
20C2×C4⋊C4×C17—22no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17D412yes—
25C2×C4⋊C4×C17C2311yescharacteristic
26C2×C42×C17—21no—
27C2×C42×C17D411yescharacteristic
28#10587612—21no—
29#10587610—22no—
30C2×#63223—21no—
31#10587576—41no—
32#10587574—41no—
33C17×#95—22no—
34C17×#95—22no—
35C17×#95—22no—
36C17×#95—22no—
37C17×#95D412yes—
38#10587506—41no—
39#10587506—41no—
40C22⋊#3622—22no—
41#10587481—42no—
42#10587478—42no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17Q1612yes—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17C2×D411yescharacteristic
52C22×C4×C17C2×D412yes—
53C2×Dic34—22no—
54C4⋊C4×C17—41no—
55C4⋊C4×C17—42no—
56C4⋊C4×C17—44no—
57C4⋊C4×C17—42no—
58C4⋊C4×C17—81no—
59C4⋊C4×C17—81no—
60C42×C17—41no—
61C42×C17—81no—
62C22⋊Dic17—41no—
63C4⋊Dic17—22no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66#63225—42no—
67C4×Dic17—41no—
68#63223—41no—
69#63223—41no—
70C2×#3622—42no—
71C2×#3622—41no—
72#12225—171nomaximal, Sylow
73C23×C17#12011yescharacteristic
74Q8×C17—42no—
75Q8×C17—81no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—41no—
82C2×C4×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17C2×Q1611yescharacteristic
93C2×C4×C17#13711yescharacteristic
94C2×Dic17—41no—
95C2×Dic17—41no—
96Dic34—82no—
97#3622—81no—
98#3622—82no—
99#2809D1711yescharacteristic
100#2028—171no—
101#1617—172no—
102#1353—172no—
103#1319—171no—
104C22×C17—21no—
105C22×C17—21no—
106C22×C17#45111yescharacteristic
107C22×C17#45511yescharacteristic
108C22×C17B(4,𝔽2)11yescharacteristic
109C4×C17—21no—
110C4×C17—21no—
111C4×C17—42no—
112C4×C17—42no—
113C4×C17—42no—
114C4×C17—82no—
115C4×C17—82no—
116Dic17—81no—
117Dic17—81no—
118#544—171no—
119#484—342no—
120#419—341no—
121#400C2×D1712yes—
122#398—22no—
123#391—21no—
124#391C2×D1711yescharacteristic
125#389—21no—
126#387—22no—
127#343—171no—
128#339—341no—
129#328—342no—
130#324—172no—
131C2×C17—21no—
132C2×C17—21no—
133C2×C17#160211yescharacteristic
134C2×C17#220111yescharacteristic
135C2×C17#220411yescharacteristic
136C22×Q8C22⋊D1711yescharacteristic
137C2×M16—341no—
138C4⋊Q8—681no—
139#125—681no—
140C22⋊Q8—342no—
141#117—341no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4C22⋊D1712yes—
147C2×C4⋊C4C22×D1711yescharacteristic
148C2×C42—21no—
149C2×C42C22⋊D1711yescharacteristic
150#107—682no—
151#103—682no—
152#103—681no—
153#103—681no—
154C22⋊C8—342no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95C22⋊D1712yes—
160C17#1222511yescharacteristic, complemented, Sylow
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—342no—
165C2×Q8#6323112yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C22⋊D1711yescharacteristic
171C22×C4C2×C22⋊D1712yes—
172M16—681no—
173M16—681no—
174C2×C8—682no—
175C2×C8—681no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—341no—
183C4⋊C4—341no—
184C4⋊C4—342no—
185C4⋊C4—682no—
186C22⋊C4—681no—
187C42—41no—
188C42—81no—
189C42—681no—
190C23#1058763911yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—1362no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—681no—
211C2×C4—681no—
212C2×C4#1058762811yescharacteristic
213C2×C4C2×#6323111yescharacteristic
214C8—1361no—
215C8—1362no—
216C22—21no—
217C22—21no—
218C22#1176296511yescharacteristic
219C22#1176298711yescharacteristic
220C22#1176299611yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—1361no—
229C4—1361no—
230C2—21no—
231C2—21no—
232C2#1452774111yescharacteristic
233C2#1453079711yescharacteristic
234C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2809C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2809D17
hypocenterC17#12225
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
160. C17#12225semidirect

Maximal subgroups

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

subgroupindex
#145299452
#145278252
#145268112
#145267182
C17×#28092
#1222517