Finite Group Databasebeta

← #16932201

Subgroups — #16932201

Order 4352.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#14537203C211yesmaximal, characteristic
2#14534361C211yesmaximal, characteristic
3#14527230C212yesmaximal, complemented
4#14526650C212yesmaximal, complemented
5C17×#1464C211yesmaximal, characteristic
6C22×#63223—21no—
7C2×C4×Dic34C2211yescharacteristic
8#11762765—24no—
9C4×#63223—21no—
10C4×#63223C2212yes—
11#11762739C411yescharacteristic
12#11762734C411yescharacteristic
13#11762729C411yescharacteristic
14#11762721C411yescharacteristic
15#11762697—21no—
16#11762693—21no—
17#11762690—21no—
18#11762682—21no—
19#11762677C2211yescharacteristic
20#11762539—24no—
21C2×C17×#117C2211yescharacteristic
22C17×#432C2212yes—
23C17×#324—21no—
24C17×#323—21no—
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C17×#117—22no—
28C17×#117—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C2×#63223—22no—
43C2×#63223—24no—
44C2×#63223—24no—
45C2×#63223—22no—
46C22×#3622—22no—
47C4×Dic34—22no—
48C4×Dic34—22no—
49C17×C4⋊C8—22no—
50C17×C22⋊C8—22no—
51C17×#97—22no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502C2×C411yescharacteristic
59#10587502C2×C411yescharacteristic
60#10587474—22no—
61#10587474—22no—
62C4×#3622—22no—
63C4×#3622D414yes—
64C24×C17C22⋊C411yescharacteristic
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C2×Dic34—22no—
80C2×Dic34—42no—
81C2×C8×C17—42no—
82C4⋊C4×C17—22no—
83C4⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—22no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17C2×D412yes—
89C42×C17#4112yes—
90C22⋊Dic17—42no—
91C22⋊Dic17—42no—
92C4⋊Dic17—22no—
93#63225—42no—
94#63225—42no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—42no—
98#63223—28no—
99#63223—24no—
100#63223—24no—
101C2×#3622—28no—
102C2×#3622—24no—
103#9491—171nomaximal, Sylow
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17#10412yes—
109C23×C17C2×C22⋊C411yescharacteristic
110C2×C4×C17—24no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
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—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17#10412yes—
124C2×C4×C17C4×D412yes—
125C2×C4×C17#12011yescharacteristic
126C2×C4×C17#12112yes—
127C2×C4×C17#12311yescharacteristic
128C2×Dic17—22no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140Dic34—24no—
141Dic34—42no—
142C8×C17—82no—
143#3622—28no—
144#2873—171no—
145#2599—171no—
146#1491—172no—
147#1464D1711yescharacteristic
148#1288—172no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—81no—
156C22×C17#38611yescharacteristic
157C22×C17C2×#10411yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C4×C17—22no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—21no—
163C4×C17—22no—
164C4×C17—41no—
165C4×C17—41no—
166Dic17—42no—
167Dic17—81no—
168Dic17—81no—
169Dic17—81no—
170C22×M16—341no—
171C2×C4×Q8—171no—
172C2×#117C2×D1711yescharacteristic
173#432C2×D1712yes—
174#405—344no—
175C4×M16—172no—
176C4×M16—341no—
177#398—171no—
178#397—171no—
179#395—171no—
180#393—171no—
181#388—341no—
182#387—341no—
183#387—341no—
184#385—341no—
185#383—171no—
186Q8⋊C8—344no—
187#324—21no—
188#323—21no—
189C2×C17—21no—
190C2×C17—21no—
191C2×C17—42no—
192C2×C17#189111yescharacteristic
193C2×C17#189611yescharacteristic
194C2×C17#190911yescharacteristic
195C22×Q8—171no—
196C23×C4D6811yescharacteristic
197C2×M16—342no—
198C2×M16—344no—
199C2×M16—344no—
200C2×M16—342no—
201C22×C8—342no—
202C4×Q8—342no—
203C4×Q8—342no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4C4×D1711yescharacteristic
211C2×C4⋊C4C22⋊D1711yescharacteristic
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4C4×D1711yescharacteristic
215C2×C22⋊C4C22⋊D1711yescharacteristic
216C2×C42—341no—
217C2×C42D6811yescharacteristic
218C2×C42C22×D1711yescharacteristic
219C4⋊C8—22no—
220C22⋊C8—22no—
221#97—22no—
222#97—342no—
223#97—342no—
224C4×C8—174no—
225C4×C8—342no—
226#95—171no—
227#95—171no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234C17#949111yescharacteristic, complemented, Sylow
235C24#6322711yescharacteristic
236C2×Q8—342no—
237C2×Q8—682no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4#6322711yescharacteristic
247C22×C4#6322711yescharacteristic
248C22×C4#6322711yescharacteristic
249C22×C4C2×C4×D1711yescharacteristic
250C22×C4C2×D6811yescharacteristic
251C22×C4C2×C22⋊D1711yescharacteristic
252M16—348no—
253M16—344no—
254M16—344no—
255C2×C8—42no—
256C2×C8—348no—
257C2×C8—344no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—342no—
261C4⋊C4—682no—
262C4⋊C4—682no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—682no—
268C22⋊C4—682no—
269C42—21no—
270C42—21no—
271C42—682no—
272C42D4×D1712yes—
273C42#6325412yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1058749612yes—
279C23C2×#6322711yescharacteristic
280Q8—344no—
281Q8—682no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—342no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4#1058749612yes—
308C2×C4#1058756712yes—
309C2×C4#1058756811yescharacteristic
310C2×C4#1058757211yescharacteristic
311C2×C4#1058762512yes—
312C8—82no—
313C8—348no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1176268611yescharacteristic
322C22C2×#1058749611yescharacteristic
323C22C2×#1058749611yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—682no—
332C4—1361no—
333C4—1361no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#1452920011yescharacteristic
339C2#1452922511yescharacteristic
340C2#1452930411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1464C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1464D17
hypocenterC17#9491
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #14527230C2semidirect
4. #14526650C2semidirect
234. C17#9491semidirect

Maximal subgroups

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

subgroupindex
#145372032
#145343612
#145272302
#145266502
C17×#14642
#949117