Finite Group Databasebeta

← #16953009

Subgroups — #16953009

Order 4352.

420 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 93 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#14533994C211yesmaximal, characteristic
2#14528778C211yesmaximal, characteristic
3#101×Dic17C211yesmaximal, characteristic
4#14526911C211yesmaximal, characteristic
5#14526910C211yesmaximal, characteristic
6#14526910C211yesmaximal, characteristic
7C17×#1394C211yesmaximal, characteristic
8C2×#10587574C411yescharacteristic
9C2×Q8×Dic17C2211yescharacteristic
10#11762832—21no—
11C2×#10587507C2211yescharacteristic
12C2×#10587507C2211yescharacteristic
13M16×Dic17—21no—
14#11762730—21no—
15#11762730—21no—
16#11762716C2211yescharacteristic
17#11762706—21no—
18#11762706—21no—
19#11762679C411yescharacteristic
20#11762678C411yescharacteristic
21C4×#63225C411yescharacteristic
22#11762577—21no—
23#11762577—21no—
24C2×C17×#101C2211yescharacteristic
25C2×C17×#101C2211yescharacteristic
26C17×#391C2211yescharacteristic
27C22×Q8×C17C2311yescharacteristic
28C22×Dic34—21no—
29C2×M16×C17—21no—
30C2×M16×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33Q8×Dic17—41no—
34Q8×Dic17—41no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×#63223—21no—
50C2×#63223—21no—
51#10587574—41no—
52#10587574—41no—
53C17×#101—21no—
54C17×#101—21no—
55C17×#101—21no—
56C17×#101Q812yescomplemented
57C17×#95—21no—
58C17×#95—21no—
59#10587507—21no—
60#10587507—21no—
61#10587507—21no—
62#10587507Q812yescomplemented
63#10587502—21no—
64#10587502—21no—
65#10587502—21no—
66#10587486—41no—
67#10587485—41no—
68#10587485—41no—
69C8×Dic17—41no—
70C2×Q8×C17—41no—
71C2×Q8×C17—41no—
72C2×Q8×C17C2×Q811yescharacteristic
73C2×Q8×C17C2×Q811yescharacteristic
74C2×Q8×C17#4111yescharacteristic
75C2×Q8×C17#4111yescharacteristic
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17C22⋊C411yescharacteristic
84C22×Dic17C22⋊C411yescharacteristic
85C22×Dic17C22⋊C411yescharacteristic
86C22×Dic17C22⋊C411yescharacteristic
87C2×Dic34—42no—
88C2×Dic34—81no—
89C2×Dic34—81no—
90M16×C17—22no—
91M16×C17—41no—
92M16×C17—41no—
93M16×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C4⋊C4×C17—41no—
97C42×C17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17—81no—
103C4⋊Dic17—81no—
104#63225—24no—
105#63225—41no—
106#63225—41no—
107#63225—81no—
108#63225—81no—
109C4×Dic17—21no—
110C4×Dic17—21no—
111C4×Dic17—21no—
112C4×Dic17—21no—
113C4×Dic17—21no—
114C4×Dic17—21no—
115C4×Dic17—41no—
116#63223—22no—
117#63223—41no—
118#63223—41no—
119#63223—41no—
120C2×#3622—41no—
121C2×#3622—41no—
122#13302—171nomaximal, Sylow
123C23×C17C2×C22⋊C411yescharacteristic
124Q8×C17—41no—
125Q8×C17—41no—
126Q8×C17—41no—
127Q8×C17—41no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—22no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17C4×Q811yescharacteristic
141C2×C4×C17C4×Q811yescharacteristic
142C2×C4×C17C22⋊Q811yescharacteristic
143C2×C4×C17C22⋊Q811yescharacteristic
144C2×C4×C17#12411yescharacteristic
145C2×C4×C17C4⋊Q811yescharacteristic
146C2×Dic17—22no—
147C2×Dic17—22no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17#9911yescharacteristic
160C2×Dic17#9911yescharacteristic
161Dic34—84no—
162C8×C17—42no—
163C8×C17—81no—
164#3622—42no—
165#3622—81no—
166#2567—171no—
167#1801—171no—
168C4×#101—171no—
169#1394—171no—
170#1394—171no—
171#1394—171no—
172#1394D1711yescharacteristic
173C22×C17—21no—
174C22×C17—21no—
175C22×C17#39111yescharacteristic
176C22×C17C2×#9911yescharacteristic
177C22×C17#41011yescharacteristic
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—42no—
185C4×C17—81no—
186Dic17—21no—
187Dic17—42no—
188Dic17—41no—
189Dic17—41no—
190Dic17—41no—
191Dic17—81no—
192Dic17#35612yes—
193C2×C4⋊Q8—171no—
194C2×C4×Q8—171no—
195#423—341no—
196C2×#101—171no—
197C2×#101—171no—
198C2×#101C2×D1711yescharacteristic
199C2×#101C2×D1711yescharacteristic
200C4×M16—341no—
201#395—341no—
202#395—341no—
203#391—171no—
204#391C2×D1711yescharacteristic
205#389—341no—
206#389—341no—
207#384—171no—
208#384—171no—
209C4×C4⋊C4—171no—
210#343—341no—
211#343—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17#199611yescharacteristic
215C2×C17C2×#35611yescharacteristic
216C2×C17#213411yescharacteristic
217C22×Q8—341no—
218C22×Q8C22×D1711yescharacteristic
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—341no—
222C2×M16—341no—
223C4⋊Q8—681no—
224C4⋊Q8—681no—
225C4×Q8—681no—
226C4×Q8—681no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—171no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4C4×D1711yescharacteristic
238C2×C42—171no—
239C2×C42—171no—
240C2×C42—171no—
241C2×C42—171no—
242C2×C42C4×D1711yescharacteristic
243C4⋊C8—681no—
244C4⋊C8—681no—
245#101—21no—
246#101—21no—
247#101—21no—
248#101—172no—
249#101—341no—
250#101—341no—
251#101—341no—
252#101Dic3412yescomplemented
253#97—681no—
254C4×C8—681no—
255#95—21no—
256#95—21no—
257#95—341no—
258#95—341no—
259#95—341no—
260C17#1330211yescharacteristic, complemented, Sylow
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—682no—
264C2×Q8—1361no—
265C2×Q8—1361no—
266C2×Q8C2×Dic3411yescharacteristic
267C2×Q8#6325211yescharacteristic
268C2×Q8#6325411yescharacteristic
269C2×Q8Q8×D1711yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—341no—
277C22×C4—341no—
278C22×C4C2×C4×D1711yescharacteristic
279C22×C4D4×D1711yescharacteristic
280C22×C4D4×D1711yescharacteristic
281M16—22no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—342no—
286M16—681no—
287M16—681no—
288M16—681no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—681no—
292C2×C8—681no—
293C4⋊C4—41no—
294C4⋊C4—344no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—681no—
298C4⋊C4—681no—
299C4⋊C4—681no—
300C4⋊C4—681no—
301C4⋊C4—1361no—
302C4⋊C4—1361no—
303C4⋊C4—1361no—
304C4⋊C4—1361no—
305C42—41no—
306C42—341no—
307C42—341no—
308C42—341no—
309C42—341no—
310C42—341no—
311C42—341no—
312C42—681no—
313C23C22⋊C4×D1711yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318Q8—1364no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—171no—
332C2×C4—171no—
333C2×C4—342no—
334C2×C4—342no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4C4×Dic3411yescharacteristic
347C2×C4#1058756411yescharacteristic
348C2×C4#1058757111yescharacteristic
349C2×C4#1058757311yescharacteristic
350C2×C4#1058757411yescharacteristic
351C2×C4#1058758211yescharacteristic
352C8—42no—
353C8—81no—
354C8—682no—
355C8—1361no—
356C22—21no—
357C22—21no—
358C22#1176271311yescharacteristic
359C22#99×D1711yescharacteristic
360C22#1176278111yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—81no—
369C4—172no—
370C4—341no—
371C4—682no—
372C4—681no—
373C4—681no—
374C4—681no—
375C4—1361no—
376C2—21no—
377C2—21no—
378C2#1452976711yescharacteristic
379C2D17×#35611yescharacteristic
380C2#1453047411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#1394D17
hypocenterC17#13302
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
56. C17×#101Q8semidirect
62. #10587507Q8semidirect
252. #101Dic34semidirect
260. C17#13302semidirect

Maximal subgroups

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

subgroupindex
#145339942
#145287782
#101×Dic172
#145269112
#145269102
C17×#13942
#1330217