Finite Group Databasebeta

← #16948018

Subgroups — #16948018

Order 4352.

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

normal subgroups only

This group has 312 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#14541653C211yesmaximal, characteristic, complemented
2#14530816C211yesmaximal, characteristic
3#14528141C211yesmaximal, characteristic, complemented
4#14526842C211yesmaximal, characteristic, complemented
5#14526764C211yesmaximal, characteristic, complemented
6#14526746C211yesmaximal, characteristic, complemented
7C17×#2240C211yesmaximal, characteristic, complemented
8#11763520—21no—
9C2×#10587497—21no—
10#11763210—21no—
11#11763112—22no—
12#11763096—21no—
13#11763006—21no—
14C8×Dic34C2211yescharacteristic
15C4×D136C2211yescharacteristic
16#11762891—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—
29#10587632—42no—
30C2×#63229—41no—
31C2×D136D411yescharacteristic
32#10587583—41no—
33#10587579—22no—
34C4⋊C4×D17—41no—
35#10587576—22no—
36C4×D68—21no—
37C4×Dic34D411yescharacteristic
38C2×C16×C17—21no—
39C17×#107—21no—
40C17×#107D412yes—
41C4×C8×C17C2311yescharacteristic
42#10587503—41no—
43#10587497—42no—
44#10587492—21no—
45#10587489D411yescharacteristic
46#10587485—21no—
47C8×Dic17—21no—
48C2×#63214—21no—
49#10587480—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×D68—21no—
58C2×C4×D17—41no—
59C2×C4×D17—42no—
60C2×Dic34D811yescharacteristic
61D8×C17—42no—
62D8×C17—41no—
63D8×C17—41no—
64C2×C8×C17C2×D411yescharacteristic
65C2×C8×C17C2×D411yescharacteristic
66C4⋊C4×C17—22no—
67C4⋊C4×C17—41no—
68C42×C17C2×D411yescharacteristic
69#63229—82no—
70#63227—42no—
71#63227—41no—
72C4⋊Dic17—21no—
73C4⋊Dic17—42no—
74C4⋊Dic17D811yescharacteristic
75#63225—21no—
76#63225—42no—
77C4×Dic17—21no—
78C2×#3622—21no—
79C2×#3622—41no—
80D136—21no—
81D136SD1612yescomplemented
82C16×C17—41no—
83#63214—41no—
84#12355—171nomaximal, Sylow
85C23×C17—81no—
86C22×D17—41no—
87Q8×C17—161no—
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—
100D68—22no—
101D68—41no—
102C4×D17—81no—
103C4×D17—81no—
104C4×D17—162no—
105Dic34—21no—
106Dic34—21no—
107C8×C17—21no—
108C8×C17C2×SD1611yescharacteristic
109C8×C17#13711yescharacteristic
110#3622—41no—
111#3622—81no—
112#3380—171no—
113#2240D1711yescharacteristic, complemented
114#2206—171no—
115#1668—171no—
116#1365—171no—
117Q8⋊C16—171no—
118#1331—171no—
119C22×C17—161no—
120C22×C17—161no—
121C22×C17—161no—
122C22×C17#44711yescharacteristic
123C2×D17—42no—
124C2×D17—81no—
125C2×D17—81no—
126C4×C17—21no—
127C4×C17—82no—
128C4×C17#44811yescharacteristic
129C4×C17#45411yescharacteristic
130C4×C17#50811yescharacteristic
131C4×C17#50911yescharacteristic
132Dic17—41no—
133Dic17—41no—
134Dic17—81no—
135#554—341no—
136C2×D16—341no—
137#500C2×D1711yescharacteristic
138#499—341no—
139#493C2×D1711yescharacteristic
140#478—342no—
141#476—341no—
142#460—341no—
143C8×Q8—171no—
144C4×D8—171no—
145C4×D8—341no—
146C4⋊C16—171no—
147#357—22no—
148#357—341no—
149C4×C16C2×D1711yescharacteristic
150#335—171no—
151#331—341no—
152C2×C17—321no—
153C2×C17#162711yescharacteristic
154C2×C17#218611yescharacteristic
155C2×C17#219211yescharacteristic
156D17—82no—
157C2×SD16—681no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—171no—
161C4⋊Q8—21no—
162#127—21no—
163#125—342no—
164C22⋊Q8—681no—
165C22⋊Q8—682no—
166C4×Q8—171no—
167C4×D4—342no—
168C4×D4—341no—
169C2×C4⋊C4—681no—
170D16—682no—
171C2×C16—21no—
172C2×C16—341no—
173#107—21no—
174#107—171no—
175#107C22⋊D1712yes—
176#106—342no—
177C4⋊C8—171no—
178C4⋊C8—341no—
179C4⋊C8—341no—
180#102—342no—
181#102—341no—
182#102—681no—
183C4×C8—341no—
184C4×C8C22×D1711yescharacteristic
185C17#1235511yescharacteristic, complemented, Sylow
186C2×Q8—41no—
187C2×Q8—171no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—341no—
192C22×C4—682no—
193C22×C4—681no—
194SD16—1362no—
195D8—42no—
196D8—41no—
197D8—41no—
198D8—172no—
199D8—341no—
200C2×C8—341no—
201C2×C8—681no—
202C2×C8D4×D1711yescharacteristic
203C2×C8C2×C22⋊D1711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—171no—
207C4⋊C4—341no—
208C4⋊C4—341no—
209C4⋊C4—682no—
210C4⋊C4—682no—
211C22⋊C4—682no—
212C22⋊C4—681no—
213C42—341no—
214C42D4×D1711yescharacteristic
215C16—41no—
216C16—681no—
217C23—81no—
218C23—681no—
219Q8—161no—
220Q8—341no—
221Q8—341no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—161no—
227D4—342no—
228D4—681no—
229C2×C4—42no—
230C2×C4—2722no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—681no—
234C2×C4—1361no—
235C2×C4—1361no—
236C2×C4D8×D1711yescharacteristic
237C2×C4#1058759711yescharacteristic
238C2×C4#1058762311yescharacteristic
239C8—21no—
240C8—681no—
241C8—1361no—
242C8SD16×D1711yescharacteristic
243C8#1058760111yescharacteristic
244C22—161no—
245C22—161no—
246C22—161no—
247C22—682no—
248C22—1361no—
249C22—1361no—
250C22#1176294111yescharacteristic
251C4—21no—
252C4—82no—
253C4—681no—
254C4—681no—
255C4—1361no—
256C4#1176294611yescharacteristic
257C4#1176298111yescharacteristic
258C4#1176324011yescharacteristic
259C4#1176324611yescharacteristic
260C2—321no—
261C2—1362no—
262C2#1452788411yescharacteristic
263C2#1453070611yescharacteristic
264C2#1453074011yescharacteristic
subgroupG/H
centerC22#11762941
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2240C2
socleC22×C17#447
prime radical#14530816C2
hypercenter#2240D17
hypocenterC17#12355
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. #14541653C2semidirect
3. #14528141C2semidirect
4. #14526842C2semidirect
5. #14526764C2semidirect
6. #14526746C2semidirect
7. C17×#2240C2semidirect
81. D136SD16semidirect
113. #2240D17semidirect
185. C17#12355semidirect

Maximal subgroups

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

subgroupindex
#145416532
#145308162
#145281412
#145268422
#145267642
#145267462
C17×#22402
#1235517