Finite Group Databasebeta

← #16947397

Subgroups — #16947397

Order 4352.

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

normal subgroups only

This group has 332 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#14527825C211yesmaximal, characteristic
3#14527757C211yesmaximal, characteristic
4#14526814C211yesmaximal, characteristic
5#14526811C211yesmaximal, characteristic
6#14526730C211yesmaximal, characteristic
7C17×#2843C211yesmaximal, characteristic
8#11763440C2211yescharacteristic
9#11763144—21no—
10#11762965—21no—
11#11762849—21no—
12#11762578C2211yescharacteristic
13#11762569—21no—
14#11762545—21no—
15#11762545—21no—
16#11762535C2211yescharacteristic
17#11762535C2211yescharacteristic
18C17×#400—21no—
19C17×#400C2211yescharacteristic
20C17×#398—21no—
21C17×#398—21no—
22C17×#398—21no—
23C17×#395—21no—
24C17×#395C2211yescharacteristic
25C17×#384C2211yescharacteristic
26C17×#383—21no—
27C22×Q8×C17D411yescharacteristic
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×#63231—41no—
39#10587612—21no—
40#10587610—21no—
41#10587610—21no—
42C2×#63223—21no—
43#10587576—41no—
44#10587574—41no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95D411yescharacteristic
54C17×#95D411yescharacteristic
55#10587506—41no—
56C22⋊#3622—21no—
57C22⋊#3622—21no—
58#10587481—41no—
59#10587479—41no—
60#10587478—41no—
61#10587478—41no—
62C2×Q8×C17—41no—
63C2×Q8×C17—81no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C2×Dic34—21no—
74C2×Dic34—21no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—81no—
79C4⋊C4×C17—81no—
80C4⋊C4×C17—81no—
81C4⋊C4×C17—81no—
82C4⋊C4×C17—81no—
83C4⋊C4×C17—81no—
84C42×C17—21no—
85C42×C17Q1612yescomplemented
86C22⋊Dic17—41no—
87#63231—82no—
88C4⋊Dic17—21no—
89C4⋊Dic17—21no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92#63225—41no—
93#63225—41no—
94C4×Dic17—41no—
95#63223—22no—
96#63223—41no—
97C2×#3622—41no—
98C2×#3622—41no—
99C2×#3622—41no—
100#12259—171nomaximal, Sylow
101C23×C17#12011yescharacteristic
102Q8×C17—42no—
103Q8×C17—81no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17C2×Q1611yescharacteristic
128C2×C4×C17#13711yescharacteristic
129C2×Dic17—41no—
130C2×Dic17—41no—
131Dic34—81no—
132Dic34—81no—
133#3622—42no—
134#3622—81no—
135#3622—81no—
136#2843D1711yescharacteristic
137#2060—171no—
138#1617—171no—
139#1605—171no—
140#1354—171no—
141#1353—171no—
142#1325—171no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17#45111yescharacteristic
146C22×C17#45611yescharacteristic
147C22×C17B(4,𝔽2)11yescharacteristic
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—42no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—81no—
154C4×C17—81no—
155C4×C17—81no—
156C4×C17—81no—
157C4×C17—81no—
158Dic17—81no—
159Dic17—81no—
160#544—171no—
161#484—341no—
162#451—341no—
163#428—341no—
164#400—21no—
165#400C2×D1711yescharacteristic
166#398—21no—
167#398—21no—
168#398—21no—
169#395—21no—
170#395C2×D1711yescharacteristic
171#384C2×D1711yescharacteristic
172#383—21no—
173#343—171no—
174#339—341no—
175#328—341no—
176#328—341no—
177#324—171no—
178#324—171no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17#160511yescharacteristic
182C2×C17#220011yescharacteristic
183C2×C17#220311yescharacteristic
184C22×Q8C22⋊D1711yescharacteristic
185C2×Q16—681no—
186C2×M16—341no—
187C4⋊Q8—681no—
188#125—681no—
189C22⋊Q8—341no—
190C22⋊Q8—341no—
191#117—341no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D1711yescharacteristic
199C2×C4⋊C4C22⋊D1711yescharacteristic
200C2×C4⋊C4C22×D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202#107—681no—
203#107—681no—
204#106—681no—
205#103—681no—
206#103—681no—
207C22⋊C8—341no—
208C22⋊C8—341no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95C22⋊D1711yescharacteristic
218#95C22⋊D1711yescharacteristic
219C17#1225911yescharacteristic, complemented, Sylow
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—341no—
223C2×Q8—341no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4C2×C22⋊D1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233Q16—1362no—
234M16—342no—
235M16—681no—
236C2×C8—681no—
237C2×C8—681no—
238C2×C8—681no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—341no—
249C4⋊C4—341no—
250C4⋊C4—341no—
251C4⋊C4—341no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C22⋊C4—681no—
255C42—21no—
256C42—681no—
257C42#6323112yescomplemented
258C23#1058763911yescharacteristic
259Q8—42no—
260Q8—81no—
261Q8—1361no—
262Q8—1361no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4#1058762811yescharacteristic
289C2×C4C2×#6323111yescharacteristic
290C8—682no—
291C8—1361no—
292C8—1361no—
293C22—21no—
294C22—21no—
295C22#1176296511yescharacteristic
296C22#1176299211yescharacteristic
297C22#1176299611yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—1361no—
309C4—1361no—
310C2—21no—
311C2—21no—
312C2#1452775811yescharacteristic
313C2#1453079111yescharacteristic
314C2#1453080511yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2843C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2843D17
hypocenterC17#12259
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2843

Complements

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

NH ≅ G/Ntype
85. C42×C17Q16semidirect
219. C17#12259semidirect
257. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145301292
#145278252
#145277572
#145268142
#145268112
#145267302
C17×#28432
#1225917