Finite Group Databasebeta

← #16954003

Subgroups — #16954003

Order 4352.

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

normal subgroups only

This group has 484 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×#2804C211yesmaximal, characteristic
6#11763284C2211yescharacteristic
7#11762849—21no—
8C2×#10587504C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762682—22no—
11#11762575—22no—
12#11762569—21no—
13C17×#400—21no—
14C17×#393—22no—
15C17×#391—21no—
16C17×#390C2211yescharacteristic
17C17×#386C2212yes—
18C17×#385—22no—
19C22×Q8×C17—21no—
20C22×D4×C17C2311yescharacteristic
21C23×C4×C17D412yes—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C22⋊C4×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17D412yes—
28C2×C42×C17D411yescharacteristic
29C2×C22⋊Dic17—22no—
30C2×C22⋊Dic17—22no—
31D4×Dic17—41no—
32D4×Dic17—41no—
33#10587612—21no—
34C2×#63225—42no—
35C2×#63223—21no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95—21no—
39C17×#95—22no—
40#10587505—22no—
41#10587505Q814yescomplemented
42#10587504—21no—
43#10587504Q812yescomplemented
44#10587502—42no—
45#10587479—41no—
46#10587478—41no—
47C24×C17C2×D412yes—
48C2×Q8×C17—42no—
49C2×Q8×C17—81no—
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—21no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—82no—
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—42no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—81no—
68C4⋊C4×C17—81no—
69C22⋊C4×C17—44no—
70C22⋊C4×C17—42no—
71C22⋊C4×C17—82no—
72C42×C17—41no—
73C22⋊Dic17—24no—
74C22⋊Dic17—42no—
75C22⋊Dic17—42no—
76C22⋊Dic17—41no—
77C4⋊Dic17—21no—
78C4⋊Dic17—21no—
79#63225—84no—
80C4×Dic17—41no—
81#63223—22no—
82#63223—41no—
83C2×#3622—41no—
84#13463—171nomaximal, Sylow
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17#12011yescharacteristic
90C23×C17C22⋊Q812yes—
91C23×C17C4⋊Q812yes—
92Q8×C17—84no—
93D4×C17—41no—
94D4×C17—42no—
95D4×C17—41no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—44no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—82no—
111C2×C4×C17C22⋊Q811yescharacteristic
112C2×C4×C17C4⋊Q811yescharacteristic
113C2×Dic17—44no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—82no—
117C2×Dic17—82no—
118C2×Dic17—82no—
119C2×Dic17—82no—
120#3622—42no—
121#2804D1711yescharacteristic
122#1867—172no—
123#1856—171no—
124#1393—171no—
125#1390—172no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—22no—
129C22×C17—22no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—82no—
135C22×C17#39511yescharacteristic
136C22×C17B(4,𝔽2)11yescharacteristic
137C22×C17#45811yescharacteristic
138C4×C17—21no—
139C4×C17—21no—
140C4×C17—42no—
141C4×C17—82no—
142C4×C17—81no—
143C4×C17—82no—
144Dic17—84no—
145Dic17—81no—
146Dic17—81no—
147#518—171no—
148#428—341no—
149C2×#100—171no—
150C2×#99—172no—
151#400—21no—
152#393—22no—
153#391—21no—
154#390C2×D1711yescharacteristic
155#386C2×D1712yes—
156#385—22no—
157#385—342no—
158#342—342no—
159#339—341no—
160C2×C17—21no—
161C2×C17—21no—
162C2×C17—42no—
163C2×C17—42no—
164C2×C17#206211yescharacteristic
165C2×C17#219811yescharacteristic
166C2×C17#219911yescharacteristic
167C22×Q8—21no—
168C22×D4C22×D1711yescharacteristic
169C23×C4C22⋊D1712yes—
170C2×M16—341no—
171C4×D4—681no—
172C4×D4—681no—
173#117—341no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—682no—
177C2×C4⋊C4C22⋊D1711yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—342no—
181C2×C22⋊C4—342no—
182C2×C22⋊C4C22⋊D1712yes—
183C2×C42C22⋊D1711yescharacteristic
184#107—681no—
185#106—681no—
186#100—172no—
187#100—341no—
188#99—174no—
189#99—342no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—22no—
194#95—682no—
195C17#1346311yescharacteristic, complemented, Sylow
196C24C2×C22⋊D1712yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D1712yes—
203C2×D4Q8×D1711yescharacteristic
204C2×D4#6325611yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—682no—
214C22×C4C2×C22⋊D1711yescharacteristic
215M16—342no—
216M16—681no—
217C2×C8—681no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—341no—
222C4⋊C4—341no—
223C4⋊C4—1364no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—344no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C22⋊C4—681no—
231C42—41no—
232C42—681no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1058763012yes—
238C23#1058763212yes—
239C23#1058763911yescharacteristic
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—684no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4—1362no—
263C2×C4—1362no—
264C2×C4—1362no—
265C2×C4—1362no—
266C2×C4#1058763011yescharacteristic
267C2×C4#1058763211yescharacteristic
268C8—682no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1176272811yescharacteristic
279C22#1176299611yescharacteristic
280C22#1176299911yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—82no—
287C4—1364no—
288C4—1361no—
289C4—1361no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#1453013611yescharacteristic
295C2#1453078011yescharacteristic
296C2#1453078511yescharacteristic
subgroupG/H
centerC22#11762996
commutator, prime radical, cosocleC22×D4×C17C23
FrattiniC22×D4C22×D17
FittingC17×#2804C2
socleC22×C17B(4,𝔽2)
hypercenter#2804D17
hypocenterC17#13463
LCSG ▷ C22×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2804

Complements

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

NH ≅ G/Ntype
41. #10587505Q8semidirect
43. #10587504Q8semidirect
195. C17#13463semidirect

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×#28042
#1346317