Finite Group Databasebeta

← #16947379

Subgroups — #16947379

Order 4352.

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

normal subgroups only

This group has 442 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#14530129C211yesmaximal, characteristic
2#14527757C212yesmaximal
3#14526814C212yesmaximal
4#14526730C211yesmaximal, characteristic
5C17×#2551C211yesmaximal, characteristic
6#11763440C2211yescharacteristic
7#11762965—22no—
8#11762849—21no—
9#11762578C2211yescharacteristic
10#11762569—21no—
11#11762545—22no—
12#11762535C2212yes—
13C2×C17×C4⋊Q8—22no—
14C17×#395—21no—
15C17×#395C2212yes—
16C17×#389—22no—
17C17×#384C2211yescharacteristic
18C4×C4⋊C4×C17—21no—
19C22×Q8×C17—21no—
20C22×Q8×C17D412yes—
21C17×C4⋊Q8—48no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C42×C17—22no—
30C2×C42×C17D411yescharacteristic
31C2×#63231—42no—
32#10587612—21no—
33#10587610—22no—
34C2×#63223—21no—
35#10587576—41no—
36#10587574—41no—
37C17×#95—21no—
38C17×#95—22no—
39C17×#95D412yes—
40#10587506—42no—
41C22⋊#3622—22no—
42#10587479—41no—
43#10587478—41no—
44C2×Q8×C17—44no—
45C2×Q8×C17—44no—
46C2×Q8×C17—42no—
47C2×Q8×C17—81no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—21no—
52C22×C4×C17C2×D411yescharacteristic
53C22×C4×C17C2×D412yes—
54C2×Dic34—22no—
55C4⋊C4×C17—24no—
56C4⋊C4×C17—24no—
57C4⋊C4×C17—41no—
58C4⋊C4×C17—41no—
59C4⋊C4×C17—44no—
60C4⋊C4×C17—44no—
61C4⋊C4×C17—44no—
62C42×C17—21no—
63C42×C17—44no—
64C42×C17Q1612yescomplemented
65C22⋊Dic17—41no—
66#63231—84no—
67C4⋊Dic17—21no—
68C4⋊Dic17—21no—
69C4⋊Dic17—22no—
70#63225—42no—
71C4×Dic17—41no—
72#63223—22no—
73#63223—41no—
74C2×#3622—41no—
75C2×#3622—42no—
76#12256—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—44no—
79Q8×C17—82no—
80Q8×C17—84no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—24no—
87C2×C4×C17—24no—
88C2×C4×C17—22no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—41no—
96C2×C4×C17C2×Q1611yescharacteristic
97C2×C4×C17#13711yescharacteristic
98C2×Dic17—41no—
99C2×Dic17—41no—
100Dic34—82no—
101#3622—42no—
102#3622—82no—
103#2551D1711yescharacteristic
104#2060—171no—
105#1605—172no—
106#1354—172no—
107#1325—171no—
108C22×C17—21no—
109C22×C17—21no—
110C22×C17#45111yescharacteristic
111C22×C17#45611yescharacteristic
112C22×C17B(4,𝔽2)11yescharacteristic
113C4×C17—24no—
114C4×C17—21no—
115C4×C17—21no—
116C4×C17—44no—
117C4×C17—42no—
118C4×C17—44no—
119C4×C17—81no—
120Dic17—81no—
121Dic17—81no—
122#544—171no—
123C2×C4⋊Q8—22no—
124#451—342no—
125#428—341no—
126#395—21no—
127#395C2×D1712yes—
128#389—22no—
129#384C2×D1711yescharacteristic
130C4×C4⋊C4—21no—
131#343—171no—
132#339—341no—
133#328—342no—
134#324—172no—
135C2×C17—21no—
136C2×C17—21no—
137C2×C17#160511yescharacteristic
138C2×C17#219611yescharacteristic
139C2×C17#220511yescharacteristic
140C22×Q8—21no—
141C22×Q8C22⋊D1712yes—
142C2×Q16—682no—
143C2×M16—341no—
144C4⋊Q8—48no—
145C4⋊Q8—681no—
146#125—681no—
147C22⋊Q8—342no—
148#117—341no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C22⋊D1711yescharacteristic
155C2×C4⋊C4C22×D1711yescharacteristic
156C2×C42—22no—
157C2×C42C22⋊D1711yescharacteristic
158#107—681no—
159#106—681no—
160#103—682no—
161C22⋊C8—342no—
162#95—21no—
163#95—22no—
164#95C22⋊D1712yes—
165C17#1225611yescharacteristic, complemented, Sylow
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—342no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D1711yescharacteristic
176C22×C4C2×C22⋊D1712yes—
177Q16—1364no—
178M16—342no—
179M16—681no—
180C2×C8—681no—
181C2×C8—682no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—342no—
190C4⋊C4—341no—
191C4⋊C4—341no—
192C4⋊C4—682no—
193C22⋊C4—681no—
194C42—21no—
195C42—44no—
196C42—681no—
197C42#6323112yescomplemented
198C23#1058763911yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—1362no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—681no—
219C2×C4—681no—
220C2×C4#1058762811yescharacteristic
221C2×C4C2×#6323111yescharacteristic
222C8—682no—
223C8—1362no—
224C22—21no—
225C22—21no—
226C22#1176296511yescharacteristic
227C22#1176299211yescharacteristic
228C22#1176299611yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—1361no—
237C4—1361no—
238C2—21no—
239C2—21no—
240C2#1452775811yescharacteristic
241C2#1453076811yescharacteristic
242C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2551C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2551D17
hypocenterC17#12256
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
64. C42×C17Q16semidirect
165. C17#12256semidirect
197. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145301292
#145277572
#145268142
#145267302
C17×#25512
#1225617