Finite Group Databasebeta

← #16922635

Subgroups — #16922635

Order 4352.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587610C211yesmaximal, characteristic
2#14528764C211yesmaximal, characteristic
3#14526609C212yesmaximal, complemented
4#14526608C212yesmaximal, complemented
5C17×#1817C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7C2×C4×Dic34C411yescharacteristic
8C2×C4⋊#3622—21no—
9C2×C22⋊#3622—21no—
10#11762715C2211yescharacteristic
11#11762707C2211yescharacteristic
12#11762693—21no—
13#11762682—21no—
14#11762669C411yescharacteristic
15C4×#63225—21no—
16C4×C4⋊Dic17C411yescharacteristic
17C4×C22⋊Dic17—21no—
18#11762564C2212yes—
19C22×C42×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#336—21no—
22C17×#336C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×C8×C17—22no—
28C2×C42×C17—42no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38#10587610—22no—
39#10587610—22no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C22×#3622—22no—
46C4×Dic34—42no—
47C17×C22⋊C8—42no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—42no—
54C4⋊#3622—42no—
55C24×C17C22⋊C411yescharacteristic
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17SD1612yes—
69C22×C4×C17Q1612yes—
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C2×Dic34—21no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80C2×Dic34M1612yes—
81C2×C8×C17—42no—
82C2×C8×C17—42no—
83C2×C8×C17—42no—
84C42×C17—22no—
85C42×C17—22no—
86C42×C17—41no—
87C42×C17—41no—
88C22⋊Dic17—24no—
89C22⋊Dic17—42no—
90C4⋊Dic17—21no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17M1612yes—
94#63225—24no—
95#63225—42no—
96#63225—42no—
97C4×Dic17—42no—
98C4×Dic17—42no—
99C2×#3622—42no—
100C2×#3622—42no—
101C2×#3622—42no—
102#7648—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17#10312yes—
110C23×C17C2×C22⋊C411yescharacteristic
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
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—42no—
125C2×C4×C17—42no—
126C2×C4×C17—44no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17#9912yes—
130C2×C4×C17#10312yes—
131C2×C4×C17C2×M1611yescharacteristic
132C2×C4×C17C2×M1611yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
134C2×C4×C17C2×Q1611yescharacteristic
135C2×C4×C17#13512yes—
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148Dic34—42no—
149Dic34—42no—
150C8×C17—82no—
151#3622—82no—
152C4×C22⋊Q8—171no—
153#1817D1711yescharacteristic
154#1797—171no—
155#1272—172no—
156#1271—172no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17#40711yescharacteristic
165C22×C17C2×#9911yescharacteristic
166C22×C17#41311yescharacteristic
167C22×C17C2×#10311yescharacteristic
168C22×C17#41611yescharacteristic
169C22×C17#42111yescharacteristic
170C22×C17#42111yescharacteristic
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—22no—
174C4×C17—22no—
175C4×C17—21no—
176C4×C17—22no—
177C4×C17—21no—
178C4×C17—41no—
179Dic17—42no—
180Dic17—42no—
181Dic17—81no—
182Dic17—81no—
183C2×C22⋊Q8—171no—
184C2×C4×Q8—171no—
185C22×C42C2×D1711yescharacteristic
186C2×C4⋊C8—341no—
187C2×C22⋊C8—21no—
188C2×C22⋊C8—341no—
189#391—171no—
190#389—171no—
191#387—341no—
192#385—341no—
193#380—171no—
194C4×C4⋊C4—171no—
195C4×C4⋊C4—341no—
196C4×C22⋊C4—341no—
197#336—21no—
198#336—172no—
199#336C2×D1712yes—
200C2×C17—42no—
201C2×C17#146511yescharacteristic
202C2×C17#148711yescharacteristic
203C2×C17#148811yescharacteristic
204C2×C17#151611yescharacteristic
205C2×C17#152011yescharacteristic
206C2×C17#153511yescharacteristic
207C2×C17#154211yescharacteristic
208C22×Q8—171no—
209C23×C4C4×D1711yescharacteristic
210C23×C4D6811yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C22×C8—22no—
213C22×C8—342no—
214C22⋊Q8—342no—
215C22⋊Q8—342no—
216C4×Q8—682no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C42—42no—
226C2×C42—341no—
227C2×C42—341no—
228C2×C42C4×D1711yescharacteristic
229C2×C42D6811yescharacteristic
230C2×C42C22⋊D1711yescharacteristic
231C2×C42C22×D1711yescharacteristic
232C4⋊C8—682no—
233C22⋊C8—42no—
234C22⋊C8—682no—
235#95—341no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240C17#764811yescharacteristic, complemented, Sylow
241C24#6322711yescharacteristic
242C2×Q8—172no—
243C2×Q8—341no—
244C2×Q8—681no—
245C2×Q8—681no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4#6321912yes—
260C22×C4Dic6812yes—
261C22×C4#6322711yescharacteristic
262C22×C4#6322711yescharacteristic
263C22×C4#6322711yescharacteristic
264C22×C4C2×C4×D1711yescharacteristic
265C22×C4C2×D6811yescharacteristic
266C22×C4C2×C22⋊D1711yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—682no—
271C2×C8—682no—
272C2×C8—682no—
273C4⋊C4—172no—
274C4⋊C4—344no—
275C4⋊C4—341no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—682no—
279C4⋊C4—682no—
280C22⋊C4—344no—
281C22⋊C4—682no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—682no—
287C42—682no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1058748712yes—
295C23C2×#6322711yescharacteristic
296Q8—682no—
297Q8—682no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—342no—
317C2×C4—342no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4#1058747712yes—
329C2×C4#1058748712yes—
330C2×C4C2×#6321911yescharacteristic
331C2×C4C2×Dic6811yescharacteristic
332C2×C4M16×D1711yescharacteristic
333C2×C4M16×D1711yescharacteristic
334C2×C4#1058763612yes—
335C8—82no—
336C8—1362no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1176277211yescharacteristic
345C22C2×#1058747711yescharacteristic
346C22#1176279111yescharacteristic
347C22C2×#1058748711yescharacteristic
348C22#1176280311yescharacteristic
349C22#1176282511yescharacteristic
350C22#1176282711yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—682no—
360C4—682no—
361C4—1361no—
362C4—1361no—
363C2—42no—
364C2#1452712011yescharacteristic
365C2#1452721411yescharacteristic
366C2#1452721611yescharacteristic
367C2#1452733811yescharacteristic
368C2#1452735711yescharacteristic
369C2#1452741611yescharacteristic
370C2#1452743911yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1817C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1817D17
hypocenterC17#7648
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #14526609C2semidirect
4. #14526608C2semidirect
240. C17#7648semidirect

Maximal subgroups

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

subgroupindex
C4×#105876102
#145287642
#145266092
#145266082
C17×#18172
#764817