Finite Group Databasebeta

← #16953991

Subgroups — #16953991

Order 4352.

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

normal subgroups only

This group has 540 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#14529064C212yesmaximal
2#14528993C211yesmaximal, characteristic
3#14526909C211yesmaximal, characteristic
4#14526902C212yesmaximal
5C17×#2683C211yesmaximal, characteristic
6#11763284C2211yescharacteristic
7#11762849—21no—
8C2×#10587504C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762682—22no—
11#11762575—22no—
12#11762569—21no—
13C2×C17×C22⋊Q8—22no—
14C17×#396—22no—
15C17×#395—21no—
16C17×#390C2211yescharacteristic
17C17×#389—21no—
18C17×#386C2212yes—
19C22×Q8×C17—21no—
20C22×D4×C17C2311yescharacteristic
21C23×C4×C17D412yes—
22C17×C22⋊Q8—48no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17D412yes—
31C2×C42×C17D411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34D4×Dic17—41no—
35D4×Dic17—41no—
36#10587612—21no—
37C2×#63225—42no—
38C2×#63223—21no—
39C17×#95—21no—
40C17×#95—22no—
41#10587505—22no—
42#10587505Q814yescomplemented
43#10587504—21no—
44#10587504Q812yescomplemented
45#10587502—42no—
46#10587479—41no—
47#10587478—41no—
48C24×C17C2×D412yes—
49C2×Q8×C17—44no—
50C2×Q8×C17—81no—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C2×D4×C17C2×Q811yescharacteristic
54C2×D4×C17C2×Q812yes—
55C2×D4×C17#4111yescharacteristic
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—44no—
61C22×C4×C17—42no—
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—42no—
66C4⋊C4×C17—24no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—44no—
69C4⋊C4×C17—44no—
70C4⋊C4×C17—81no—
71C22⋊C4×C17—44no—
72C22⋊C4×C17—44no—
73C22⋊C4×C17—42no—
74C42×C17—41no—
75C22⋊Dic17—24no—
76C22⋊Dic17—42no—
77C22⋊Dic17—42no—
78C22⋊Dic17—41no—
79C4⋊Dic17—21no—
80C4⋊Dic17—21no—
81#63225—84no—
82C4×Dic17—41no—
83#63223—22no—
84#63223—41no—
85C2×#3622—41no—
86#13461—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—42no—
90C23×C17—42no—
91C23×C17#12011yescharacteristic
92C23×C17C22⋊Q812yes—
93C23×C17C4⋊Q812yes—
94Q8×C17—84no—
95D4×C17—41no—
96D4×C17—42no—
97D4×C17—41no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—24no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—44no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—82no—
113C2×C4×C17C22⋊Q811yescharacteristic
114C2×C4×C17C4⋊Q811yescharacteristic
115C2×Dic17—44no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—82no—
119C2×Dic17—82no—
120C2×Dic17—82no—
121C2×Dic17—82no—
122#3622—42no—
123#2683D1711yescharacteristic
124#1867—172no—
125#1856—171no—
126#1393—171no—
127#1390—172no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—22no—
131C22×C17—22no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—82no—
137C22×C17#39511yescharacteristic
138C22×C17B(4,𝔽2)11yescharacteristic
139C22×C17#45811yescharacteristic
140C4×C17—21no—
141C4×C17—21no—
142C4×C17—42no—
143C4×C17—44no—
144C4×C17—82no—
145C4×C17—81no—
146Dic17—84no—
147Dic17—81no—
148Dic17—81no—
149C2×C22⋊Q8—22no—
150#518—171no—
151#428—341no—
152C2×#100—171no—
153C2×#99—172no—
154#396—22no—
155#395—21no—
156#390C2×D1711yescharacteristic
157#389—21no—
158#386C2×D1712yes—
159#385—342no—
160#342—342no—
161#339—341no—
162C2×C17—21no—
163C2×C17—21no—
164C2×C17—42no—
165C2×C17—42no—
166C2×C17#206211yescharacteristic
167C2×C17#219611yescharacteristic
168C2×C17#220111yescharacteristic
169C22×Q8—21no—
170C22×D4C22×D1711yescharacteristic
171C23×C4C22⋊D1712yes—
172C2×M16—341no—
173C22⋊Q8—48no—
174C4×D4—681no—
175C4×D4—681no—
176#117—341no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—682no—
182C2×C4⋊C4C22⋊D1711yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—342no—
186C2×C22⋊C4—342no—
187C2×C22⋊C4C22⋊D1712yes—
188C2×C42C22⋊D1711yescharacteristic
189#107—681no—
190#106—681no—
191#100—172no—
192#100—341no—
193#99—174no—
194#99—342no—
195#95—21no—
196#95—22no—
197#95—682no—
198C17#1346111yescharacteristic, complemented, Sylow
199C24C2×C22⋊D1712yes—
200C2×Q8—44no—
201C2×Q8—81no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4Q8×D1712yes—
205C2×D4Q8×D1711yescharacteristic
206C2×D4#6325611yescharacteristic
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—342no—
214C22×C4—342no—
215C22×C4—682no—
216C22×C4C2×C22⋊D1711yescharacteristic
217M16—342no—
218M16—681no—
219C2×C8—681no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—81no—
225C4⋊C4—341no—
226C4⋊C4—341no—
227C4⋊C4—1364no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—344no—
232C22⋊C4—682no—
233C22⋊C4—682no—
234C22⋊C4—681no—
235C42—41no—
236C42—681no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1058763012yes—
242C23#1058763212yes—
243C23#1058763911yescharacteristic
244Q8—84no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—684no—
264C2×C4—681no—
265C2×C4—681no—
266C2×C4—1362no—
267C2×C4—1362no—
268C2×C4—1362no—
269C2×C4—1362no—
270C2×C4#1058763011yescharacteristic
271C2×C4#1058763211yescharacteristic
272C8—682no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1176272811yescharacteristic
283C22#1176299611yescharacteristic
284C22#1176299911yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—44no—
289C4—82no—
290C4—81no—
291C4—1364no—
292C4—1361no—
293C4—1361no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#1453013611yescharacteristic
299C2#1453076811yescharacteristic
300C2#1453079711yescharacteristic
subgroupG/H
centerC22#11762996
commutator, prime radical, cosocleC22×D4×C17C23
FrattiniC22×D4C22×D17
FittingC17×#2683C2
socleC22×C17B(4,𝔽2)
hypercenter#2683D17
hypocenterC17#13461
LCSG ▷ C22×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
42. #10587505Q8semidirect
44. #10587504Q8semidirect
198. C17#13461semidirect

Maximal subgroups

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

subgroupindex
#145290642
#145289932
#145269092
#145269022
C17×#26832
#1346117