Finite Group Databasebeta

← #17951281

Subgroups — #17951281

Order 4352.

874 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 156 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
1D17⋊#407C212yesmaximal
2#14543666C214yesmaximal, complemented
3C2×#11762772C211yesmaximal, characteristic
4C2×D17⋊M16—22no—
5#11763781C414yes—
6#11763780—22no—
7#11763780C414yes—
8#11763779—22no—
9#11763777—22no—
10#11763777C2214yes—
11C23×C4×D17C2211yescharacteristic
12C22×#63223C411yescharacteristic
13#11762772—22no—
14#11762772C412yes—
15#11762772C2212yes—
16C2×#10587491—22no—
17C22×M16×C17C411yescharacteristic
18C24×D17Q811yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17D411yescharacteristic
21C22×C4×D17—42no—
22C22×C4×D17—42no—
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17C2×C412yes—
25C22×C4×D17C2×C412yes—
26C22×C4×D17D412yes—
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D412yes—
29C22×C4×D17Q811yescharacteristic
30C22×C4×D17C2311yescharacteristic
31C2×#63247—24no—
32D17⋊M16—44no—
33D17⋊M16—44no—
34C2×#63243—24no—
35C2×#63242—24no—
36C2×M16×C17—21no—
37C2×M16×C17—42no—
38C2×M16×C17C2×C412yes—
39C22×C8×C17—21no—
40C2×#63223—21no—
41C2×#63223—42no—
42C2×#63223C2×C412yes—
43C22×#3622—21no—
44C22⋊#3624—24no—
45C22⋊#3624—24no—
46#10587543—24no—
47#10587543C2×C418yes—
48#10587491—24no—
49#10587491—24no—
50C24×C17C4⋊C411yescharacteristic
51C23×D17—44no—
52C23×D17—42no—
53C23×D17C4⋊C412yes—
54C23×D17C4⋊C412yes—
55C23×D17C4⋊C411yescharacteristic
56C23×D17C2×Q811yescharacteristic
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17C4211yescharacteristic
60C22×C4×C17C4211yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C4⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×Dic17—42no—
66C22×Dic17C22⋊C412yes—
67C22×Dic17C22⋊C412yes—
68C22×Dic17C4⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C2×C4×D17—24no—
71C2×C4×D17—24no—
72C2×C4×D17—44no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C2×C4×D17—42no—
76C2×C4×D17—42no—
77C2×C4×D17—42no—
78C2×C4×D17—42no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—44no—
82C2×C4×D17C4212yes—
83C2×C4×D17C22⋊C412yes—
84C2×C4×D17C22⋊C412yes—
85C2×C4×D17C22⋊C412yes—
86C2×C4×D17C4⋊C412yes—
87C2×C4×D17C4⋊C412yes—
88C2×C4×D17C22×C412yes—
89C2×C4×D17C2×D412yes—
90#63247—48no—
91C2×#3624—28no—
92#63243—48no—
93#63242—44no—
94#63242—44no—
95M16×C17—42no—
96M16×C17—42no—
97C2×C8×C17—22no—
98C2×C8×C17—42no—
99#63223—42no—
100#63223—42no—
101C2×#3622—22no—
102C2×#3622—42no—
103#8055—171nomaximal, Sylow
104C23×C17—22no—
105C23×C17—42no—
106C23×C17#9511yescharacteristic
107C23×C17#9511yescharacteristic
108C23×C17C2×C4⋊C411yescharacteristic
109C22×D17—24no—
110C22×D17—24no—
111C22×D17—24no—
112C22×D17—24no—
113C22×D17—42no—
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—416no—
120C22×D17#9512yes—
121C22×D17C2×C4⋊C412yes—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17#9511yescharacteristic
130C2×C4×C17#9511yescharacteristic
131C2×C4×C17#9912yes—
132C2×C4×C17#9912yes—
133C2×C4×C17C2×C4211yescharacteristic
134C2×C4×C17C2×C22⋊C411yescharacteristic
135C2×Dic17—24no—
136C2×Dic17—42no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17#9512yes—
140C2×Dic17C2×C22⋊C412yes—
141C4×D17—24no—
142C4×D17—24no—
143C4×D17—22no—
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—22no—
147C4×D17—42no—
148C4×D17—42no—
149C4×D17—42no—
150C4×D17—42no—
151#3624—48no—
152C8×C17—42no—
153#3622—42no—
154C2×#407—171no—
155C2×#407—172no—
156#1298—174no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—44no—
163C22×C17#34212yes—
164C22×C17C2×#9511yescharacteristic
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—22no—
168C2×D17—22no—
169C2×D17—22no—
170C2×D17—44no—
171C2×D17—48no—
172C2×D17—48no—
173C2×D17—44no—
174C2×D17#43014yes—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17#34212yes—
178C4×C17C2×#9911yescharacteristic
179C4×C17C2×#9911yescharacteristic
180Dic17—22no—
181Dic17—22no—
182C24×C4—171no—
183C22×M16—171no—
184C22×M16—342no—
185C22×M16#59511yescharacteristic
186#407—174no—
187#407—174no—
188#407—172no—
189#407—172no—
190#407—342no—
191#407—342no—
192#407—342no—
193C2×C22⋊C8—174no—
194C2×C22⋊C8—342no—
195C2×C22⋊C8—342no—
196C2×C17—22no—
197C2×C17—41no—
198C2×C17—41no—
199C2×C17#173812yes—
200C2×C17C2×#34211yescharacteristic
201D17—24no—
202D17—42no—
203C25—171no—
204C23×C4—171no—
205C23×C4—172no—
206C23×C4—172no—
207C23×C4—172no—
208C23×C4—171no—
209C23×C4—171no—
210C23×C4—172no—
211C23×C4—171no—
212C23×C4—171no—
213C23×C4—682no—
214C23×C4—682no—
215C23×C4C2×#59511yescharacteristic
216C2×M16—21no—
217C2×M16—42no—
218C2×M16—172no—
219C2×M16—344no—
220C2×M16—344no—
221C2×M16—341no—
222C2×M16—684no—
223C2×M16—684no—
224C2×M16—682no—
225C2×M16C2×#59512yes—
226C22×C8—21no—
227C22×C8—344no—
228C22×C8—341no—
229C22⋊C8—178no—
230C22⋊C8—344no—
231C22⋊C8—344no—
232C22⋊C8—344no—
233C22⋊C8—344no—
234C22⋊C8—344no—
235C17#805511yescharacteristic, complemented, Sylow
236C24—171no—
237C24—172no—
238C24—172no—
239C24—171no—
240C24—682no—
241C24—684no—
242C24C4⋊#59511yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—172no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—172no—
249C22×C4—172no—
250C22×C4—172no—
251C22×C4—172no—
252C22×C4—172no—
253C22×C4—172no—
254C22×C4—172no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—344no—
258C22×C4—344no—
259C22×C4—682no—
260C22×C4—681no—
261C22×C4—681no—
262C22×C4—682no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—684no—
266C22×C4—682no—
267C22×C4—682no—
268C22×C4—684no—
269C22×C4—682no—
270C22×C4C4×#59511yescharacteristic
271C22×C4C4×#59511yescharacteristic
272C22×C4C4⋊#59511yescharacteristic
273C22×C4C22⋊#59511yescharacteristic
274C22×C4C22⋊#59511yescharacteristic
275C22×C4C22×#59511yescharacteristic
276M16—42no—
277M16—42no—
278M16—682no—
279M16—688no—
280M16—688no—
281M16—682no—
282C2×C8—22no—
283C2×C8—42no—
284C2×C8—348no—
285C2×C8—342no—
286C2×C8—684no—
287C2×C8—684no—
288C2×C8—682no—
289C23—22no—
290C23—42no—
291C23—172no—
292C23—172no—
293C23—344no—
294C23—344no—
295C23—344no—
296C23—344no—
297C23—681no—
298C23—681no—
299C23—682no—
300C23—682no—
301C23—682no—
302C23—682no—
303C23—6816no—
304C23(C2×#595)⋊C411yescharacteristic
305C23(C2×#595)⋊C411yescharacteristic
306C23C2×C4⋊#59511yescharacteristic
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—172no—
315C2×C4—172no—
316C2×C4—344no—
317C2×C4—342no—
318C2×C4—342no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—344no—
322C2×C4—344no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—681no—
328C2×C4—682no—
329C2×C4—681no—
330C2×C4#1058753912yes—
331C2×C4(C2×#595)⋊C411yescharacteristic
332C2×C4(C2×#595)⋊C411yescharacteristic
333C2×C4#1058755112yes—
334C2×C4C2×C4×#59511yescharacteristic
335C2×C4C2×C22⋊#59511yescharacteristic
336C8—42no—
337C8—688no—
338C8—682no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22—174no—
346C22—342no—
347C22—342no—
348C22—342no—
349C22—342no—
350C22—342no—
351C22—688no—
352C22—684no—
353C22—688no—
354C22—684no—
355C22#1176368712yes—
356C22C2×(C2×#595)⋊C411yescharacteristic
357C4—21no—
358C4—21no—
359C4—342no—
360C4—342no—
361C4#1176368712yes—
362C4C2×#1058755111yescharacteristic
363C4C2×#1058753911yescharacteristic
364C2—22no—
365C2—41no—
366C2—41no—
367C2—344no—
368C2—682no—
369C2#1454447712yes—
370C2C2×#1176368711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC22×M16×C17C4
socleC22×C17C2×#95
prime radicalC24×D17Q8
cosocleC22×C4×D17C23
hypercenterC22×M16#595
hypocenterC17#8055
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. #14543666C2semidirect
235. C17#8055semidirect

Maximal subgroups

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

subgroupindex
D17⋊#4072
#145436662
C2×#117627722
#805517