Finite Group Databasebeta

← #16945136

Subgroups — #16945136

Order 4352.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#14532043C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic
3#14526812C212yesmaximal, complemented
4#14526791C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C2×#10587612C411yescharacteristic
7#11762833—21no—
8C2×#10587504—21no—
9#11762748C2211yescharacteristic
10#11762745C411yescharacteristic
11#11762739C2211yescharacteristic
12#11762737—21no—
13#11762734C411yescharacteristic
14#11762723—21no—
15C4×#63225—21no—
16C4×C22⋊Dic17—21no—
17C4×C22⋊Dic17C411yescharacteristic
18#11762578C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C2×M16×C17—41no—
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37#10587612—41no—
38#10587612—41no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×#63223—22no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—41no—
48#10587504—42no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55C4⋊#3622—41no—
56C4⋊#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78M16×C17—82no—
79C2×C8×C17—42no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C42×C17—21no—
84C42×C17—21no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91#63225—42no—
92#63225—41no—
93#63225—41no—
94C4×Dic17—42no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97#63223—42no—
98#63223—42no—
99C2×#3622—42no—
100#11830—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112D4×C17—42no—
113D4×C17—42no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—22no—
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—81no—
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C17#13711yescharacteristic
131C2×Dic17—22no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C8×C17—82no—
144#3622—82no—
145#2395—171no—
146#1800—171no—
147#1465D1711yescharacteristic
148#1353—172no—
149#1346—172no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—42no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17C2×#9911yescharacteristic
163C22×C17#41711yescharacteristic
164C22×C17#42111yescharacteristic
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—41no—
171C4×C17#35212yes—
172Dic17—42no—
173Dic17—81no—
174Dic17—81no—
175Dic17—81no—
176C2×C4×D4C2×D1711yescharacteristic
177C2×#117—171no—
178#423—341no—
179C2×#100—341no—
180#407—21no—
181#400—171no—
182#399—171no—
183#398—171no—
184#397—171no—
185#397—341no—
186#394—341no—
187C4×C4⋊C4—341no—
188C4×C22⋊C4—171no—
189C4×C22⋊C4—341no—
190#343—172no—
191#324C2×D1712yes—
192#323—21no—
193C2×C17—21no—
194C2×C17—21no—
195C2×C17—81no—
196C2×C17—81no—
197C2×C17#151311yescharacteristic
198C2×C17C2×#35211yescharacteristic
199C2×C17#213211yescharacteristic
200C22×D4D6811yescharacteristic
201C23×C4C4×D1711yescharacteristic
202C23×C4C22⋊D1711yescharacteristic
203C2×M16—41no—
204C2×M16—342no—
205C4×D4—41no—
206C4×D4—41no—
207#117—681no—
208#117—681no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4C22×D1711yescharacteristic
214C2×C22⋊C4—341no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4C4×D1711yescharacteristic
219C2×C22⋊C4C22⋊D1711yescharacteristic
220C2×C42—341no—
221C2×C42—341no—
222C2×C42D6811yescharacteristic
223C4⋊C8—681no—
224C4⋊C8—681no—
225#100—682no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228#95—171no—
229#95—171no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234C17#1183011yescharacteristic, complemented, Sylow
235C24#6322711yescharacteristic
236C24#6322711yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4#6322711yescharacteristic
252C22×C4#6322711yescharacteristic
253C22×C4C2×C4×D1711yescharacteristic
254C22×C4C2×D6811yescharacteristic
255C22×C4C2×C22⋊D1711yescharacteristic
256M16—82no—
257M16—682no—
258M16—682no—
259C2×C8—42no—
260C2×C8—682no—
261C4⋊C4—41no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—682no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—682no—
272C22⋊C4—682no—
273C42—21no—
274C42—21no—
275C42—682no—
276C42—681no—
277C42—681no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×#6322711yescharacteristic
289D4—42no—
290D4—42no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—342no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4#1058747711yescharacteristic
317C2×C4#1058747711yescharacteristic
318C2×C4#1058759411yescharacteristic
319C2×C4#1058759511yescharacteristic
320C8—82no—
321C8—1362no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1058747711yescharacteristic
335C22#1176281111yescharacteristic
336C22#1176282511yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—682no—
344C4—1361no—
345C4—1361no—
346C4—1361no—
347C4#1176259812yes—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—81no—
352C2#1452732811yescharacteristic
353C2C2×#1176259811yescharacteristic
354C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11830
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526812C2semidirect
4. #14526791C2semidirect
234. C17#11830semidirect

Maximal subgroups

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

subgroupindex
#145320432
#145287762
#145268122
#145267912
C17×#14652
#1183017