Finite Group Databasebeta

← #16948004

Subgroups — #16948004

Order 4352.

298 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 67 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#14541677C211yesmaximal, characteristic, complemented
2#14530821C211yesmaximal, characteristic
3#14528141C211yesmaximal, characteristic
4#14526839C211yesmaximal, characteristic, complemented
5#14526764C211yesmaximal, characteristic, complemented
6#14526749C211yesmaximal, characteristic, complemented
7C17×#2240C211yesmaximal, characteristic
8#11763534—21no—
9C2×#10587498—21no—
10#11763210—21no—
11#11763115—22no—
12#11763091—21no—
13#11763006—21no—
14C8×Dic34C2211yescharacteristic
15C4×Dic68C2211yescharacteristic
16#11762903—21no—
17C4⋊#63214C2211yescharacteristic
18#11762610—21no—
19#11762562C2211yescharacteristic
20#11762553—21no—
21C17×#500C2211yescharacteristic
22C17×#493C2211yescharacteristic
23C17×#357—22no—
24C4×C16×C17C2211yescharacteristic
25C2×D8×C17—21no—
26C2×D8×C17—21no—
27C17×C4⋊Q8—21no—
28C17×#127—21no—
29Q8×Dic17—41no—
30#10587630—41no—
31C2×#63229—41no—
32C2×Dic68D411yescharacteristic
33#10587576—22no—
34#10587575—42no—
35#10587573—22no—
36C4×Dic34—21no—
37C4×Dic34D411yescharacteristic
38C2×C16×C17—21no—
39C17×#107—21no—
40C17×#107D412yes—
41C4×C8×C17C2311yescharacteristic
42#10587503—41no—
43#10587498—42no—
44#10587489D411yescharacteristic
45#10587487—21no—
46#10587485—21no—
47C8×Dic17—21no—
48C2×#63214—21no—
49#10587481—22no—
50#10587479—22no—
51C4⋊#3622—21no—
52C4⋊#3622D411yescharacteristic
53C2×Q8×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×Dic34—21no—
58C2×Dic34D811yescharacteristic
59D8×C17—42no—
60D8×C17—41no—
61D8×C17—41no—
62C2×C8×C17C2×D411yescharacteristic
63C2×C8×C17C2×D411yescharacteristic
64C4⋊C4×C17—22no—
65C4⋊C4×C17—41no—
66C42×C17C2×D411yescharacteristic
67#63229—82no—
68C4⋊Dic17—21no—
69C4⋊Dic17—42no—
70C4⋊Dic17D811yescharacteristic
71#63225—21no—
72#63225—42no—
73#63225—42no—
74#63225—41no—
75C4×Dic17—21no—
76C4×Dic17—42no—
77C4×Dic17—41no—
78C2×#3622—21no—
79C2×#3622—41no—
80Dic68—21no—
81Dic68SD1612yescomplemented
82C16×C17—41no—
83#63214—41no—
84#12353—171nomaximal, Sylow
85C23×C17—81no—
86Q8×C17—81no—
87Q8×C17—81no—
88D4×C17—81no—
89D4×C17—81no—
90D4×C17—81no—
91D4×C17—81no—
92D4×C17—161no—
93C2×C4×C17—42no—
94C2×C4×C17#12011yescharacteristic
95C2×C4×C17C2×D811yescharacteristic
96C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—41no—
100C2×Dic17—41no—
101Dic34—22no—
102Dic34—21no—
103Dic34—21no—
104Dic34—41no—
105C8×C17—21no—
106C8×C17C2×SD1611yescharacteristic
107C8×C17#13711yescharacteristic
108#3622—41no—
109#3622—81no—
110#3383—171no—
111#2240D1711yescharacteristic
112#2207—171no—
113#1668—171no—
114#1364—171no—
115Q8⋊C16—171no—
116#1332—171no—
117C22×C17—161no—
118C22×C17—161no—
119C22×C17—161no—
120C22×C17#44711yescharacteristic
121C4×C17—21no—
122C4×C17—82no—
123C4×C17#44811yescharacteristic
124C4×C17#45411yescharacteristic
125C4×C17#50811yescharacteristic
126C4×C17#50911yescharacteristic
127Dic17—42no—
128Dic17—41no—
129Dic17—41no—
130Dic17—81no—
131#557—341no—
132C2×SD32—341no—
133#500C2×D1711yescharacteristic
134#499—341no—
135#493C2×D1711yescharacteristic
136#479—342no—
137#475—341no—
138#460—341no—
139C8×Q8—171no—
140C4×Q16—171no—
141C4×Q16—341no—
142C4⋊C16—171no—
143#357—22no—
144#357—341no—
145C4×C16C2×D1711yescharacteristic
146#335—171no—
147#331—341no—
148C2×C17—321no—
149C2×C17#162711yescharacteristic
150C2×C17#218611yescharacteristic
151C2×C17#219311yescharacteristic
152C2×Q16—171no—
153C2×SD16—681no—
154C2×D8—21no—
155C2×D8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—681no—
158#127—21no—
159#125—342no—
160#125—682no—
161C4×Q8—171no—
162C4×Q8—342no—
163C4×Q8—341no—
164C4×Q8—681no—
165SD32—682no—
166C2×C16—21no—
167C2×C16—341no—
168#107—21no—
169#107—171no—
170#107C22⋊D1712yes—
171#106—342no—
172C4⋊C8—171no—
173C4⋊C8—341no—
174C4⋊C8—341no—
175#103—342no—
176#103—341no—
177#102—681no—
178C4×C8—341no—
179C4×C8C22×D1711yescharacteristic
180C17#1235311yescharacteristic, complemented, Sylow
181C2×Q8—41no—
182C2×Q8—171no—
183C2×Q8—341no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187Q16—172no—
188Q16—341no—
189SD16—1362no—
190D8—42no—
191D8—41no—
192D8—41no—
193C2×C8—341no—
194C2×C8—681no—
195C2×C8D4×D1711yescharacteristic
196C2×C8C2×C22⋊D1711yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—171no—
200C4⋊C4—341no—
201C4⋊C4—341no—
202C4⋊C4—682no—
203C4⋊C4—682no—
204C4⋊C4—682no—
205C4⋊C4—681no—
206C42—341no—
207C42—682no—
208C42—681no—
209C42D4×D1711yescharacteristic
210C16—41no—
211C16—681no—
212C23—81no—
213Q8—81no—
214Q8—81no—
215Q8—342no—
216Q8—341no—
217Q8—341no—
218Q8—681no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—81no—
223D4—161no—
224C2×C4—42no—
225C2×C4—341no—
226C2×C4—341no—
227C2×C4—681no—
228C2×C4—681no—
229C2×C4D8×D1711yescharacteristic
230C2×C4#1058759711yescharacteristic
231C2×C4#1058762311yescharacteristic
232C8—21no—
233C8—681no—
234C8—1361no—
235C8SD16×D1711yescharacteristic
236C8#1058760111yescharacteristic
237C22—161no—
238C22—161no—
239C22—161no—
240C22#1176294111yescharacteristic
241C4—21no—
242C4—82no—
243C4—682no—
244C4—681no—
245C4—681no—
246C4—1361no—
247C4#1176294611yescharacteristic
248C4#1176298111yescharacteristic
249C4#1176323811yescharacteristic
250C4#1176324311yescharacteristic
251C2—321no—
252C2#1452788411yescharacteristic
253C2#1453070411yescharacteristic
254C2#1453074711yescharacteristic
subgroupG/H
centerC22#11762941
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2240C2
socleC22×C17#447
prime radicalC17×#493C22
hypercenter#2240D17
hypocenterC17#12353
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. #14541677C2semidirect
4. #14526839C2semidirect
5. #14526764C2semidirect
6. #14526749C2semidirect
81. Dic68SD16semidirect
180. C17#12353semidirect

Maximal subgroups

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

subgroupindex
#145416772
#145308212
#145281412
#145268392
#145267642
#145267492
C17×#22402
#1235317