Finite Group Databasebeta

← #16950597

Subgroups — #16950597

Order 4352.

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

normal subgroups only

This group has 614 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
1#14531376C211yesmaximal, characteristic
2#14528712C211yesmaximal, characteristic
3#14526910C214yesmaximal, complemented
4C17×#1785C211yesmaximal, characteristic
5C2×#10587612C411yescharacteristic
6C22×C4⋊Dic17C411yescharacteristic
7C2×#10587507C2212yes—
8C2×#10587504—21no—
9#11762773—21no—
10#11762716C2212yes—
11#11762712C412yes—
12#11762689—21no—
13#11762689—21no—
14#11762669—21no—
15#11762669—21no—
16C22×#41×C17C2211yescharacteristic
17C2×C17×#101C2212yes—
18C2×C17×#100—21no—
19C17×#407—21no—
20C2×#41×C17—24no—
21C2×#41×C17—42no—
22C2×#41×C17—42no—
23C22×Q8×C17C2311yescharacteristic
24C22×D4×C17C2×C411yescharacteristic
25C22×D4×C17D411yescharacteristic
26C22×D4×C17D411yescharacteristic
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C23×Dic17—21no—
31C2×M16×C17—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35#10587612—42no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—42no—
39C2×C4⋊Dic17C2×C412yes—
40C2×C4×Dic17C2×C412yes—
41C2×#63223—22no—
42C17×#101—24no—
43C17×#100—42no—
44C17×C22⋊C8—42no—
45#10587507—24no—
46#10587504—42no—
47#10587502—22no—
48#10587502—22no—
49C22⋊#3622—42no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C24×C17C22⋊C411yescharacteristic
53#41×C17—416no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17#4114yes—
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—22no—
80C22×Dic17—22no—
81C22×Dic17—42no—
82C22×Dic17—82no—
83M16×C17—42no—
84M16×C17—42no—
85C2×C8×C17—42no—
86C22⋊Dic17—42no—
87C22⋊Dic17—82no—
88C22⋊Dic17—82no—
89C4⋊Dic17—42no—
90C4⋊Dic17—42no—
91C4⋊Dic17—42no—
92C4×Dic17—42no—
93#63223—42no—
94#63223—42no—
95C2×#3622—42no—
96#12863—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17#9912yes—
112C23×C17C2×C22⋊C411yescharacteristic
113Q8×C17—44no—
114D4×C17—44no—
115D4×C17—44no—
116D4×C17—44no—
117C2×C4×C17—28no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—44no—
127C2×C4×C17—44no—
128C2×C4×C17#9912yes—
129C2×C4×C17#11712yes—
130C2×C4×C17#12312yes—
131C2×C4×C17#12312yes—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—84no—
139C8×C17—82no—
140#3622—82no—
141#2311—171no—
142#1785—171no—
143#1785D1711yescharacteristic
144#1394—174no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—42no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17#38111yescharacteristic
163C22×C17C2×#9911yescharacteristic
164C22×C17C2×#9911yescharacteristic
165C4×C17—22no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—22no—
169Dic17—82no—
170Dic17—82no—
171C22×#41C2×D1711yescharacteristic
172C2×#117—171no—
173C22×C4⋊C4—171no—
174C2×#101—172no—
175C2×#101C2×D1712yes—
176C2×#100—21no—
177C2×#100—341no—
178#407—21no—
179#407—341no—
180#391—172no—
181#390—172no—
182#386—341no—
183#386—341no—
184#380—341no—
185#380—341no—
186C2×C17—21no—
187C2×C17—21no—
188C2×C17—42no—
189C2×C17—81no—
190C2×C17—81no—
191C2×C17#181911yescharacteristic
192C2×C17#213411yescharacteristic
193C2×C17#213511yescharacteristic
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—42no—
197C22×Q8C22×D1711yescharacteristic
198C22×D4C4×D1711yescharacteristic
199C22×D4D6811yescharacteristic
200C22×D4C22⋊D1711yescharacteristic
201C23×C4—341no—
202C23×C4C4×D1711yescharacteristic
203C23×C4D6811yescharacteristic
204C23×C4C22⋊D1711yescharacteristic
205C2×M16—22no—
206C2×M16—342no—
207#117—682no—
208C2×C4⋊C4—172no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—682no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4—342no—
215C2×C42—172no—
216#101—24no—
217#101—344no—
218#100—42no—
219#100—682no—
220C22⋊C8—42no—
221C22⋊C8—682no—
222#95—342no—
223#95—342no—
224C17#1286311yescharacteristic, complemented, Sylow
225C24#6322711yescharacteristic
226C24#6322711yescharacteristic
227C24#6322711yescharacteristic
228#41—416no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8#6325414yes—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—342no—
251C22×C4—342no—
252C22×C4—682no—
253C22×C4—1362no—
254C22×C4#6322711yescharacteristic
255C22×C4C2×C4×D1711yescharacteristic
256C22×C4C2×D6811yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—42no—
259M16—42no—
260M16—682no—
261M16—682no—
262C2×C8—42no—
263C2×C8—682no—
264C4⋊C4—682no—
265C4⋊C4—682no—
266C4⋊C4—682no—
267C22⋊C4—682no—
268C22⋊C4—1362no—
269C22⋊C4—1362no—
270C42—682no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1058747712yes—
286C23C2×#6322711yescharacteristic
287Q8—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—682no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—1364no—
309C2×C4#1058747712yes—
310C2×C4#1058756312yes—
311C2×C4#1058757212yes—
312C2×C4#1058762112yes—
313C8—82no—
314C8—1362no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22#1176267011yescharacteristic
333C22C2×#1058747711yescharacteristic
334C22C2×#1058747711yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—1362no—
340C4—1362no—
341C2—21no—
342C2—21no—
343C2—42no—
344C2—81no—
345C2—81no—
346C2#1452885611yescharacteristic
347C2#1453047511yescharacteristic
348C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1785C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×Q8×C17C23
hypercenter#1785D17
hypocenterC17#12863
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #14526910C2semidirect
224. C17#12863semidirect

Maximal subgroups

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

subgroupindex
#145313762
#145287122
#145269102
C17×#17852
#1286317