Finite Group Databasebeta

← #16944833

Subgroups — #16944833

Order 4352.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533071C211yesmaximal, characteristic
2#14527114C211yesmaximal, characteristic
3#14526810C212yesmaximal, complemented
4#14526789C212yesmaximal, complemented
5C2×C17×#343C211yesmaximal, characteristic
6C2×#10587612C411yescharacteristic
7#11762773—21no—
8#11762748C411yescharacteristic
9#11762748C2211yescharacteristic
10#11762739C411yescharacteristic
11#11762739C2211yescharacteristic
12#11762690—21no—
13#11762690—21no—
14#11762682—21no—
15#11762672—21no—
16#11762672C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×M16×C17—21no—
20C22×C4⋊C4×C17C2211yescharacteristic
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C2×M16×C17—24no—
27C2×M16×C17—22no—
28C2×M16×C17—41no—
29C2×M16×C17—42no—
30C22×C8×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39#10587612—41no—
40#10587612—41no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C2×#63223—41no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17C22⋊C411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80M16×C17—24no—
81M16×C17—44no—
82M16×C17—42no—
83C2×C8×C17—44no—
84C2×C8×C17—42no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C22⋊Dic17—42no—
90C22⋊Dic17—41no—
91C22⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94#63225—42no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—42no—
98#63223—82no—
99C2×#3622—42no—
100#11764—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17#9911yescharacteristic
106C23×C17#9911yescharacteristic
107C23×C17C2×C22⋊C411yescharacteristic
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
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—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—81no—
128C2×C4×C17#10412yes—
129C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C17#13711yescharacteristic
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—42no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C8×C17—44no—
143#3622—82no—
144#2484—171no—
145C2×#343D1711yescharacteristic
146#1463—171no—
147#1353—172no—
148#1346—172no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17#35212yes—
159C22×C17C2×#9911yescharacteristic
160C22×C17#41711yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C4×C17—21no—
163C4×C17—22no—
164C4×C17—21no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169Dic17—81no—
170Dic17—81no—
171Dic17—81no—
172Dic17—81no—
173C22×M16—21no—
174C2×#117—171no—
175C22×C4⋊C4C2×D1711yescharacteristic
176#407—341no—
177#400—171no—
178#400—171no—
179#398—171no—
180#398—171no—
181#387—341no—
182#387—341no—
183#385—341no—
184#381—171no—
185#381—341no—
186#343—21no—
187#343C2×D1712yes—
188#324—172no—
189#324—341no—
190C2×C17—21no—
191C2×C17—21no—
192C2×C17—22no—
193C2×C17—41no—
194C2×C17#150411yescharacteristic
195C2×C17C2×#35211yescharacteristic
196C2×C17#213311yescharacteristic
197C23×C4C4×D1711yescharacteristic
198C23×C4D6811yescharacteristic
199C23×C4C22⋊D1711yescharacteristic
200C2×M16—24no—
201C2×M16—22no—
202C2×M16—41no—
203C2×M16—42no—
204C2×M16—681no—
205C22×C8—41no—
206#117—681no—
207#117—681no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4C4×D1711yescharacteristic
215C2×C4⋊C4D6811yescharacteristic
216C2×C4⋊C4C22⋊D1711yescharacteristic
217C2×C4⋊C4C22×D1711yescharacteristic
218C2×C22⋊C4—341no—
219C2×C22⋊C4—341no—
220C2×C42—341no—
221C22⋊C8—342no—
222C22⋊C8—681no—
223#95—171no—
224#95—171no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232C17#1176411yescharacteristic, complemented, Sylow
233C24#6322711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4#6322711yescharacteristic
250C22×C4#6322711yescharacteristic
251C22×C4#6322711yescharacteristic
252C22×C4C2×C4×D1711yescharacteristic
253C22×C4C2×D6811yescharacteristic
254C22×C4C2×C22⋊D1711yescharacteristic
255M16—24no—
256M16—44no—
257M16—42no—
258M16—1362no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—682no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C4⋊C4—682no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C22⋊C4—682no—
272C22⋊C4—681no—
273C22⋊C4—681no—
274C42—682no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1058747711yescharacteristic
280C23#1058747711yescharacteristic
281C23C2×#6322711yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—682no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4#1058747612yes—
314C2×C4#1058761811yescharacteristic
315C2×C4#1058762811yescharacteristic
316C8—44no—
317C8—1362no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1176260012yes—
328C22C2×#1058747711yescharacteristic
329C22#1176280811yescharacteristic
330C22C2×#1058747611yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—1361no—
339C4—1361no—
340C4—1361no—
341C4—1361no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#1452728611yescharacteristic
347C2C2×#1176260011yescharacteristic
348C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC2×C17×#343C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenterC2×#343D17
hypocenterC17#11764
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #14526810C2semidirect
4. #14526789C2semidirect
232. C17#11764semidirect

Maximal subgroups

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

subgroupindex
#145330712
#145271142
#145268102
#145267892
C2×C17×#3432
#1176417