Finite Group Databasebeta

← #16945367

Subgroups — #16945367

Order 4352.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#14533689C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic, complemented
3#14526791C212yesmaximal, complemented
4#14526788C212yesmaximal
5C17×#1461C211yesmaximal, characteristic, complemented
6C2×#10587632—21no—
7C2×C4×D68C411yescharacteristic
8#11762833—21no—
9C2×#10587507—21no—
10#11762739C411yescharacteristic
11#11762739C2211yescharacteristic
12#11762726C411yescharacteristic
13#11762726C2211yescharacteristic
14#11762700—21no—
15#11762700—21no—
16#11762675C411yescharacteristic
17#11762666—21no—
18#11762578C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#431—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C22×Q8×C17D411yescharacteristic
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C4×Q8×C17—41no—
27C4×Q8×C17—41no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34#10587632—42no—
35#10587632—42no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×#63223—22no—
46C4×D68—41no—
47C4×D68—41no—
48C17×C4⋊C8—41no—
49C17×C22⋊C8—22no—
50C4×C8×C17—41no—
51#10587507—42no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C4⋊#3622—41no—
58C4⋊#3622—41no—
59C23×D17—21no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C2×Q8×C17D812yes—
63C2×Q8×C17SD1612yes—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×D68—41no—
75C2×D68—42no—
76C2×C4×D17—42no—
77C2×C4×D17—42no—
78C2×C4×D17—42no—
79C2×C8×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89#63227—42no—
90#63227—42no—
91#63227—41no—
92#63227—41no—
93#63227—42no—
94#63227—41no—
95#63227—41no—
96C4⋊Dic17—42no—
97C4⋊Dic17—41no—
98#63225—42no—
99#63225—41no—
100#63225—41no—
101#63223—42no—
102#63223—42no—
103C2×#3622—42no—
104#11879—171nomaximal, Sylow
105C23×C17C2×C22⋊C411yescharacteristic
106C22×D17—22no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
113Q8×C17—42no—
114Q8×C17—42no—
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—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#10212yes—
131C2×C4×C17#10212yes—
132C2×C4×C17C2×D811yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
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—
143D68—42no—
144D68—42no—
145C4×D17—84no—
146C8×C17—82no—
147#3622—82no—
148#2536—171no—
149#1801—171no—
150#1461D1711yescharacteristic, complemented
151#1346—172no—
152#1345—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17C2×#9911yescharacteristic
156C22×C17C2×#10211yescharacteristic
157C22×C17#42111yescharacteristic
158C2×D17—42no—
159C2×D17—42no—
160C2×D17—42no—
161C2×D17—81no—
162C2×D17—81no—
163C2×D17—81no—
164C2×D17—81no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—22no—
169C4×C17—22no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174Dic17—81no—
175Dic17—81no—
176Dic17—81no—
177C2×C22⋊Q8—341no—
178C2×C4×Q8C2×D1711yescharacteristic
179C2×C4×D4—171no—
180#431—21no—
181#423—341no—
182C2×#101—341no—
183#398—171no—
184#398—171no—
185#394—171no—
186#394—171no—
187#388—341no—
188#388—341no—
189#382—171no—
190#380—341no—
191#343—172no—
192#324—21no—
193#324C2×D1712yes—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17#151911yescharacteristic
197C2×C17#212011yescharacteristic
198C2×C17#213311yescharacteristic
199D17—82no—
200C22×Q8D6811yescharacteristic
201C22×D4—171no—
202C23×C4—341no—
203C2×M16—342no—
204C22⋊Q8—682no—
205C22⋊Q8—682no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×D4—681no—
209C4×D4—681no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4C4×D1711yescharacteristic
215C2×C4⋊C4C22⋊D1711yescharacteristic
216C2×C4⋊C4C22×D1711yescharacteristic
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C22⋊C4—341no—
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C42C4×D1711yescharacteristic
223C2×C42D6811yescharacteristic
224C2×C42C22⋊D1711yescharacteristic
225C4⋊C8—41no—
226C4⋊C8—681no—
227C4⋊C8—681no—
228#101—682no—
229C22⋊C8—22no—
230C4×C8—41no—
231#95—171no—
232#95—171no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#1187911yescharacteristic, complemented, Sylow
237C24—341no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#6321912yes—
241C2×Q8D13612yes—
242C2×D4—681no—
243C2×D4—682no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—682no—
248C22×C4—682no—
249C22×C4—682no—
250C22×C4#6322711yescharacteristic
251C22×C4#6322711yescharacteristic
252C22×C4#6322711yescharacteristic
253C22×C4#6322711yescharacteristic
254C22×C4C2×C4×D1711yescharacteristic
255C22×C4C2×D6811yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257M16—682no—
258M16—682no—
259C2×C8—42no—
260C2×C8—682no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—681no—
265C4⋊C4—682no—
266C4⋊C4—682no—
267C4⋊C4—681no—
268C4⋊C4—681no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—681no—
272C22⋊C4—681no—
273C22⋊C4—682no—
274C22⋊C4—681no—
275C22⋊C4—681no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—342no—
283C23—681no—
284C23—681no—
285C23—681no—
286C23—681no—
287C23—681no—
288C23—681no—
289C23C2×#6322711yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—682no—
293D4—682no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
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—1364no—
317C2×C4#1058747711yescharacteristic
318C2×C4#1058747711yescharacteristic
319C2×C4#1058749212yes—
320C2×C4#1058749212yes—
321C2×C4C2×#6321911yescharacteristic
322C2×C4C2×D13611yescharacteristic
323C8—82no—
324C8—1362no—
325C22—21no—
326C22—21no—
327C22—682no—
328C22—682no—
329C22—682no—
330C22—1361no—
331C22—1361no—
332C22—1361no—
333C22—1361no—
334C22C2×#1058747711yescharacteristic
335C22C2×#1058749211yescharacteristic
336C22#1176282511yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—1361no—
347C4—1361no—
348C4—1361no—
349C2—21no—
350C2—21no—
351C2—1362no—
352C2#1452735311yescharacteristic
353C2#1453042311yescharacteristic
354C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1461C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1461D17
hypocenterC17#11879
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #14528780C2semidirect
3. #14526791C2semidirect
5. C17×#1461C2semidirect
150. #1461D17semidirect
236. C17#11879semidirect

Maximal subgroups

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

subgroupindex
#145336892
#145287802
#145267912
#145267882
C17×#14612
#1187917