Finite Group Databasebeta

← #16951163

Subgroups — #16951163

Order 4352.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#11762608C211yesmaximal, characteristic
2#14529178C211yesmaximal, characteristic
3#14528998C212yesmaximal
4#14526911C212yesmaximal
5C17×#2369C211yesmaximal, characteristic
6#11763287C2211yescharacteristic
7#11762849—22no—
8#11762818—21no—
9C2×#10587507C2211yescharacteristic
10C2×#10587507C2212yes—
11#11762608C414yescomplemented
12#11762569—21no—
13#11762569—22no—
14C2×C17×C4⋊Q8C2211yescharacteristic
15C17×#391—21no—
16C17×#391C2212yes—
17C17×#389—22no—
18C4×C4⋊C4×C17—22no—
19C4×C4⋊C4×C17—21no—
20C22×Q8×C17D411yescharacteristic
21C22×Q8×C17C2311yescharacteristic
22C17×C4⋊Q8—22no—
23C17×C4⋊Q8C2×C412yes—
24C17×C4⋊Q8Q812yes—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17D412yes—
30C2×C42×C17—22no—
31C2×C42×C17—21no—
32C2×C42×C17D411yescharacteristic
33C2×C42×C17D412yes—
34Q8×Dic17—41no—
35Q8×Dic17—41no—
36#10587612—21no—
37#10587612—22no—
38C2×#63223—22no—
39C2×#63223—21no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95—22no—
43#10587507—22no—
44#10587507—22no—
45#10587507C2×C414yes—
46#10587506—41no—
47#10587506—41no—
48#10587479—42no—
49#10587478—42no—
50C2×Q8×C17—21no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17C4⋊C412yes—
56C2×Q8×C17C22×C411yescharacteristic
57C2×Q8×C17C2×Q811yescharacteristic
58C2×Q8×C17#4112yes—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D412yes—
65C4⋊C4×C17—24no—
66C4⋊C4×C17—24no—
67C4⋊C4×C17—24no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—81no—
71C42×C17—21no—
72C42×C17—42no—
73C42×C17—44no—
74C42×C17—81no—
75C42×C17C4⋊C412yes—
76C22⋊Dic17—41no—
77C4⋊Dic17—22no—
78C4⋊Dic17—22no—
79C4⋊Dic17—21no—
80C4⋊Dic17—21no—
81C4×Dic17—42no—
82C4×Dic17—41no—
83#63223—24no—
84#63223—42no—
85#63223—41no—
86#63223—41no—
87C2×#3622—42no—
88C2×#3622—41no—
89#12973—171nomaximal, Sylow
90C23×C17#12011yescharacteristic
91Q8×C17—41no—
92Q8×C17—41no—
93Q8×C17—42no—
94Q8×C17—42no—
95Q8×C17—81no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—24no—
104C2×C4×C17—22no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17C2×C4⋊C411yescharacteristic
113C2×C4×C17C4×D412yes—
114C2×C4×C17C22⋊Q812yes—
115C2×C4×C17#12311yescharacteristic
116C2×Dic17—41no—
117C2×Dic17—41no—
118#3622—44no—
119#3622—81no—
120#2369D1711yescharacteristic
121C2×#356—171no—
122#1886—171no—
123#1857—172no—
124#1394—172no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17#38511yescharacteristic
128C22×C17B(4,𝔽2)11yescharacteristic
129C22×C17#45811yescharacteristic
130C4×C17—24no—
131C4×C17—22no—
132C4×C17—21no—
133C4×C17—21no—
134C4×C17—22no—
135C4×C17—44no—
136C4×C17—42no—
137C4×C17—82no—
138Dic17—81no—
139Dic17—81no—
140C2×C4⋊Q8C2×D1711yescharacteristic
141#519—171no—
142#428—342no—
143#419—341no—
144C2×#101—172no—
145C2×#101—171no—
146#391—21no—
147#391C2×D1712yes—
148#389—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—21no—
151#356—174no—
152#339—342no—
153#339—341no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17#186711yescharacteristic
157C2×C17#220411yescharacteristic
158C2×C17#220511yescharacteristic
159C22×Q8C22⋊D1711yescharacteristic
160C22×Q8C22×D1711yescharacteristic
161C2×M16—342no—
162C2×M16—341no—
163C4⋊Q8—22no—
164C4⋊Q8Dic3412yes—
165C4⋊Q8C4×D1712yes—
166C4×Q8—681no—
167C4×Q8—681no—
168#117—341no—
169#117—342no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C22⋊D1712yes—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42C22⋊D1711yescharacteristic
178C2×C42C22⋊D1712yes—
179#107—682no—
180#106—682no—
181#103—681no—
182#103—681no—
183#101—174no—
184#101—342no—
185#101—342no—
186#95—22no—
187#95—22no—
188#95—22no—
189C17#1297311yescharacteristic, complemented, Sylow
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8#6322512yes—
196C2×Q8C2×Dic3411yescharacteristic
197C2×Q8C2×C4×D1711yescharacteristic
198C2×Q8#6325212yes—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4C2×C22⋊D1711yescharacteristic
204C22×C4C2×C22⋊D1712yes—
205M16—344no—
206M16—682no—
207M16—681no—
208M16—681no—
209C2×C8—682no—
210C2×C8—681no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—341no—
218C4⋊C4—341no—
219C4⋊C4—342no—
220C4⋊C4—342no—
221C22⋊C4—681no—
222C42—21no—
223C42—42no—
224C42—44no—
225C42—81no—
226C42—682no—
227C42—681no—
228C42#6322512yes—
229C23#1058763911yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—681no—
252C2×C4—681no—
253C2×C4C2×#6322511yescharacteristic
254C2×C4#1058761012yes—
255C2×C4C4×C22⋊D1712yes—
256C2×C4#1058761511yescharacteristic
257C8—684no—
258C8—1361no—
259C22—21no—
260C22—21no—
261C22#1176268211yescharacteristic
262C22#1176299611yescharacteristic
263C22#1176299911yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—1361no—
273C4—1361no—
274C2—21no—
275C2—21no—
276C2#1452906511yescharacteristic
277C2#1453081011yescharacteristic
278C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutatorC2×Q8×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2369C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
cosocleC22×Q8×C17C23
hypercenter#2369D17
hypocenterC17#12973
LCSG ▷ C2×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2369

Complements

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

NH ≅ G/Ntype
11. #11762608C4semidirect
189. C17#12973semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145291782
#145289982
#145269112
C17×#23692
#1297317