Finite Group Databasebeta

← #16954010

Subgroups — #16954010

Order 4352.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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#14536777C211yesmaximal, characteristic
2#14529001C212yesmaximal
3#14528998C211yesmaximal, characteristic
4#14526910C212yesmaximal
5C17×#1394C211yesmaximal, characteristic
6#11763287C2211yescharacteristic
7#11762851—22no—
8#11762849—21no—
9C2×#10587507C2211yescharacteristic
10#11762730—22no—
11#11762729—21no—
12#11762716C2212yes—
13#11762715—21no—
14#11762713—22no—
15#11762570—22no—
16#11762569—21no—
17C2×C17×#101C2212yes—
18C17×#391C2211yescharacteristic
19C22×Q8×C17C2311yescharacteristic
20C22×Dic34—21no—
21C22×Dic34—21no—
22C2×M16×C17—22no—
23C2×C4⋊C4×C17D411yescharacteristic
24C2×C42×C17D411yescharacteristic
25Q8×Dic17—41no—
26Q8×Dic17—41no—
27#10587612—21no—
28#10587612—22no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C2×C4×Dic17D412yes—
33C2×#63223—21no—
34C17×#101—22no—
35C17×#101Q814yescomplemented
36C17×#95—22no—
37#10587507—21no—
38#10587507Q812yescomplemented
39#10587502—21no—
40#10587502—22no—
41#10587502—22no—
42#10587502—22no—
43#10587489—42no—
44#10587488—42no—
45#10587479—41no—
46#10587478—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17C2×Q812yes—
50C2×Q8×C17C2×Q811yescharacteristic
51C2×Q8×C17#4111yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C2×D411yescharacteristic
55C22×C4×C17C2×D412yes—
56C22×Dic17—22no—
57C22×Dic17—22no—
58C2×Dic34—24no—
59C2×Dic34—81no—
60C2×Dic34—81no—
61C2×Dic34—82no—
62C2×Dic34—81no—
63M16×C17—24no—
64M16×C17—42no—
65C2×C8×C17—42no—
66C4⋊C4×C17—41no—
67C42×C17—41no—
68C22⋊Dic17—41no—
69C4⋊Dic17—21no—
70C4⋊Dic17—21no—
71C4⋊Dic17—22no—
72C4⋊Dic17—22no—
73C4⋊Dic17—42no—
74C4⋊Dic17—81no—
75#63225—82no—
76C4×Dic17—41no—
77C4×Dic17—42no—
78C4×Dic17—42no—
79#63223—22no—
80#63223—41no—
81C2×#3622—41no—
82#13464—171nomaximal, Sylow
83C23×C17#12011yescharacteristic
84Q8×C17—42no—
85Q8×C17—41no—
86Q8×C17—41no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—41no—
96C2×C4×C17C22⋊Q812yes—
97C2×C4×C17C22⋊Q811yescharacteristic
98C2×C4×C17C4⋊Q811yescharacteristic
99C2×C4×C17C4⋊Q812yes—
100C2×Dic17—24no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17—42no—
108Dic34—84no—
109Dic34—84no—
110C8×C17—44no—
111#3622—42no—
112#2811—171no—
113#1857—172no—
114#1857—171no—
115#1394—172no—
116#1394D1711yescharacteristic
117C22×C17—21no—
118C22×C17—21no—
119C22×C17#39511yescharacteristic
120C22×C17B(4,𝔽2)11yescharacteristic
121C22×C17#45811yescharacteristic
122C4×C17—21no—
123C4×C17—22no—
124C4×C17—21no—
125C4×C17—22no—
126C4×C17—42no—
127C4×C17—81no—
128Dic17—44no—
129Dic17—82no—
130Dic17—81no—
131Dic17—81no—
132#519—171no—
133#428—342no—
134#428—341no—
135C2×#101—171no—
136C2×#101C2×D1712yes—
137#395—342no—
138#395—341no—
139#391—172no—
140#391—342no—
141#391—341no—
142#391C2×D1711yescharacteristic
143#339—342no—
144#339—341no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17#206211yescharacteristic
148C2×C17#220211yescharacteristic
149C2×C17#220511yescharacteristic
150C22×Q8—341no—
151C22×Q8—341no—
152C22×Q8C22×D1711yescharacteristic
153C2×M16—22no—
154C2×M16—341no—
155C4×Q8—681no—
156C4×Q8—681no—
157#117—341no—
158#117—342no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—342no—
161C2×C4⋊C4—341no—
162C2×C4⋊C4D6811yescharacteristic
163C2×C42—172no—
164C2×C42D6811yescharacteristic
165#107—681no—
166#107—682no—
167#106—681no—
168#106—682no—
169#101—22no—
170#101—172no—
171#101—341no—
172#101Dic3414yescomplemented
173#95—22no—
174#95—342no—
175#95—341no—
176#95—342no—
177#95—342no—
178C17#1346411yescharacteristic, complemented, Sylow
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—344no—
182C2×Q8—1361no—
183C2×Q8—1361no—
184C2×Q8—1362no—
185C2×Q8—1361no—
186C2×Q8C2×Dic3412yes—
187C2×Q8#6325411yescharacteristic
188C2×Q8Q8×D1711yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—342no—
192C22×C4—342no—
193C22×C4C2×D6811yescharacteristic
194C22×C4D4×D1712yes—
195M16—24no—
196M16—42no—
197M16—342no—
198M16—681no—
199C2×C8—42no—
200C2×C8—681no—
201C4⋊C4—41no—
202C4⋊C4—341no—
203C4⋊C4—342no—
204C4⋊C4—341no—
205C4⋊C4—342no—
206C4⋊C4—682no—
207C4⋊C4—1362no—
208C4⋊C4—1361no—
209C22⋊C4—681no—
210C42—41no—
211C42—682no—
212C42—681no—
213C42—682no—
214C23#1058756811yescharacteristic
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8—1364no—
219Q8—1364no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—344no—
230C2×C4—682no—
231C2×C4—682no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—681no—
235C2×C4—681no—
236C2×C4—682no—
237C2×C4#1058755511yescharacteristic
238C2×C4#1058756412yes—
239C2×C4#1058757412yes—
240C2×C4#1058758311yescharacteristic
241C8—44no—
242C8—682no—
243C22—21no—
244C22—21no—
245C22#1176273011yescharacteristic
246C22#1176299711yescharacteristic
247C22#1176300011yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—81no—
254C4—684no—
255C4—1362no—
256C4—1361no—
257C4—1361no—
258C2—21no—
259C2—21no—
260C2#1453013811yescharacteristic
261C2#1453080211yescharacteristic
262C2#1453081511yescharacteristic
subgroupG/H
centerC22#11762997
commutator, cosocleC22×Q8×C17C23
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#1394D17
hypocenterC17#13464
LCSG ▷ C22×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
35. C17×#101Q8semidirect
38. #10587507Q8semidirect
172. #101Dic34semidirect
178. C17#13464semidirect

Maximal subgroups

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

subgroupindex
#145367772
#145290012
#145289982
#145269102
C17×#13942
#1346417