Finite Group Databasebeta

← #21435607

Subgroups — #21435607

Order 7552.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273758C211yesmaximal, characteristic, complemented
2C22×#11745671C211yesmaximal, characteristic, complemented
3#16838048C211yesmaximal, characteristic
4#16837794C211yesmaximal, characteristic, complemented
5#16837650C211yesmaximal, characteristic, complemented
6#16837641C211yesmaximal, characteristic, complemented
7#16837624C211yesmaximal, characteristic, complemented
8#16837604C211yesmaximal, characteristic, complemented
9#16837562C211yesmaximal, characteristic, complemented
10#16837558C211yesmaximal, characteristic, complemented
11#16837552C211yesmaximal, characteristic, complemented
12#16837541C211yesmaximal, characteristic, complemented
13#16837500C211yesmaximal, characteristic, complemented
14#16837474C211yesmaximal, characteristic, complemented
15D4×Q8×C59C211yesmaximal, characteristic, complemented
16C2×#11745685C2211yescharacteristic
17C22×D236C2211yescharacteristic
18#14273758—24no—
19C2×#11745672C2211yescharacteristic, complemented
20C2×#11745672C2211yescharacteristic, complemented
21C2×#11745672C2211yescharacteristic, complemented
22C2×#11745671—24no—
23C2×#11745671C2211yescharacteristic
24C2×#11745671C2211yescharacteristic
25C2×#11745671C2211yescharacteristic
26C2×#11745671C2211yescharacteristic
27C2×#11745671C2211yescharacteristic
28C2×#11745671C2211yescharacteristic
29C2×#11745664C2211yescharacteristic, complemented
30C22×C59⋊C8C2211yescharacteristic
31#14273715C2211yescharacteristic
32#14273712—21no—
33#14273706C2211yescharacteristic
34#14273701C2211yescharacteristic
35#14273700—21no—
36#14273695—21no—
37C4×D236C2211yescharacteristic
38C22⋊C4×D59—21no—
39#14273660C2211yescharacteristic, complemented
40C4⋊(C59⋊C8)C2211yescharacteristic, complemented
41#14273651C2211yescharacteristic, complemented
42#14273651C2211yescharacteristic, complemented
43#14273651C2211yescharacteristic, complemented
44#14273650C2211yescharacteristic, complemented
45#14273647C2211yescharacteristic
46#14273647C2211yescharacteristic
47C22⋊(C59⋊C8)C2211yescharacteristic
48C22⋊(C59⋊C8)C2211yescharacteristic, complemented
49#14273638C2211yescharacteristic, complemented
50C22×Q8×C59C2211yescharacteristic
51C22×Q8×C59C2211yescharacteristic, complemented
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59C2211yescharacteristic, complemented
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56C22⋊Q8×C59C2211yescharacteristic
57C22⋊Q8×C59C2211yescharacteristic, complemented
58C4×Q8×C59C2211yescharacteristic, complemented
59C4×D4×C59—21no—
60C4×D4×C59C2211yescharacteristic
61C23×D59—21no—
62C2×Q8×C59—21no—
63C2×Q8×C59—21no—
64C2×Q8×C59—22no—
65C2×Q8×C59—22no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×Q8×C59C2311yescharacteristic
69C2×Q8×C59C2311yescharacteristic
70C2×Q8×C59C2311yescharacteristic
71C2×D4×C59D411yescharacteristic
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59C2311yescharacteristic
75C22×C4×C59C2311yescharacteristic
76C2×C22⋊D59—21no—
77C2×C22⋊D59—21no—
78#11745685—22no—
79#11745685—41no—
80C2×D236—22no—
81C2×D236—21no—
82C2×D236—41no—
83C2×D236C2311yescharacteristic
84C2×D236C2311yescharacteristic
85C2×C4×D59—21no—
86C2×C4×D59—21no—
87C2×Dic118C2311yescharacteristic
88C4⋊C4×C59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59D411yescharacteristic
94C4⋊C4×C59C2311yescharacteristic
95C22⋊C4×C59—21no—
96C22⋊C4×C59—21no—
97C22⋊C4×C59D411yescharacteristic
98C22⋊C4×C59D411yescharacteristic
99C42×C59—21no—
100C42×C59C2311yescharacteristic
101C22⋊Dic59—21no—
102#11745672—22no—
103#11745672—22no—
104#11745672—41no—
105#11745671—22no—
106#11745671—22no—
107#11745671—22no—
108#11745671—22no—
109#11745671—41no—
110#11745671D414yescomplemented
111(C2×D59)⋊C4—21no—
112(C2×D59)⋊C4—21no—
113(C2×D59)⋊C4—21no—
114(C2×D59)⋊C4—21no—
115C4⋊Dic59C2311yescharacteristic
116Dic59⋊C4—21no—
117#11745664—22no—
118C2×C59⋊C8—22no—
119C2×C59⋊C8C2311yescharacteristic
120C2×C59⋊C8C2311yescharacteristic
121C2×C59⋊C8C2311yescharacteristic
122C2×C59⋊C8C2311yescharacteristic
123C23×C59C2×D411yescharacteristic
124C23×C59C2×D411yescharacteristic
125C22×D59—21no—
126C22×D59—21no—
127C22×D59—42no—
128C22×D59—42no—
129C22×D59—81no—
130Q8×C59—21no—
131Q8×C59—21no—
132Q8×C59—21no—
133Q8×C59—21no—
134Q8×C59—21no—
135Q8×C59—42no—
136Q8×C59—42no—
137Q8×C59C2×D412yes—
138D4×C59—22no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59C2×D411yescharacteristic
150C2×C4×C59C2×D411yescharacteristic
151C2×C4×C59C2×D411yescharacteristic
152C2×C4×C59C2×D411yescharacteristic
153C2×C4×C59C2411yescharacteristic
154C22⋊D59—42no—
155C22⋊D59—81no—
156C2×Dic59—21no—
157C2×Dic59—21no—
158D236—21no—
159D236—21no—
160D236—21no—
161D236—42no—
162D236C2×D412yes—
163C4×D59—42no—
164C4×D59—81no—
165Dic118—21no—
166Dic118—21no—
167C59⋊C8—21no—
168C59⋊C8—21no—
169C59⋊C8—21no—
170C59⋊C8C2×D412yes—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59#13712yescomplemented
177C22×C59C22×D411yescharacteristic
178C2×D59—22no—
179C2×D59—41no—
180C2×D59—41no—
181C2×D59—41no—
182C2×D59—41no—
183C2×D59—41no—
184C2×D59—84no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—22no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59C22×D411yescharacteristic
194C4×C59H(5,𝔽2)11yescharacteristic
195Dic59—41no—
196Dic59—41no—
197#3266—591nomaximal, Sylow
198C2×C59—22no—
199C2×C59—41no—
200C2×C59D4211yescharacteristic
201C2×C59C2×#13711yescharacteristic
202C2×C59#57711yescharacteristic
203D59—42no—
204D59—81no—
205C2×#137—591no—
206C22×SD16—591no—
207D4×Q8D5911yescharacteristic, complemented
208#546—591no—
209#497—591no—
210#468—591no—
211#467—591no—
212#464—591no—
213#461—591no—
214#452—591no—
215#451—591no—
216#450—591no—
217#448—591no—
218#440—591no—
219#435—591no—
220C59#326611yescharacteristic, complemented, Sylow
221C2×#41—591no—
222C22×Q8C2×D5911yescharacteristic
223C22×Q8C2×D5911yescharacteristic, complemented
224C22×D4—591no—
225#137—1184no—
226C2×Q16—591no—
227C2×Q16—591no—
228C2×Q16—591no—
229C2×SD16—591no—
230C2×SD16—591no—
231C2×SD16—591no—
232C2×SD16—591no—
233C2×SD16—591no—
234C2×SD16—591no—
235C2×SD16—1184no—
236C2×M16—591no—
237C22×C8—591no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D5911yescharacteristic, complemented
240#124—591no—
241#123—1181no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—591no—
245C22⋊Q8C2×D5911yescharacteristic
246C22⋊Q8C2×D5911yescharacteristic, complemented
247#121—591no—
248#121—1181no—
249#120—1181no—
250C4×Q8C2×D5911yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—591no—
253C4×D4C2×D5911yescharacteristic
254C2×C22⋊C4—1181no—
255#107—591no—
256C4⋊C8—591no—
257#103—591no—
258#103—591no—
259#103—591no—
260#103—591no—
261#102—591no—
262#102—591no—
263C22⋊C8—591no—
264C22⋊C8—591no—
265#97—591no—
266C24—1181no—
267#41—1182no—
268#41—2361no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—591no—
276C2×Q8C22×D5911yescharacteristic
277C2×Q8C22×D5911yescharacteristic
278C2×Q8C22×D5911yescharacteristic
279C2×D4—591no—
280C2×D4—591no—
281C2×D4—1182no—
282C2×D4—1181no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—2361no—
286C2×D4C22⋊D5911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4C22×D5911yescharacteristic
292C22×C4C22×D5911yescharacteristic
293Q16—1182no—
294Q16—1182no—
295Q16—2361no—
296SD16—594no—
297SD16—1182no—
298SD16—1182no—
299SD16—1182no—
300SD16—1182no—
301SD16—2361no—
302M16—1182no—
303C2×C8—591no—
304C2×C8—591no—
305C2×C8—591no—
306C2×C8—591no—
307C2×C8—1182no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—591no—
314C4⋊C4—1181no—
315C4⋊C4C22⋊D5911yescharacteristic
316C4⋊C4C22×D5911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1181no—
320C22⋊C4—1181no—
321C22⋊C4—1181no—
322C22⋊C4—1181no—
323C22⋊C4—1181no—
324C22⋊C4C22⋊D5911yescharacteristic
325C22⋊C4C22⋊D5911yescharacteristic
326C42—21no—
327C42C22×D5911yescharacteristic
328C23—1181no—
329C23—1181no—
330C23—4721no—
331C23—2362no—
332C23—2362no—
333C23C2×C22⋊D5911yescharacteristic
334C23C2×C22⋊D5911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—1181no—
343Q8—1181no—
344Q8D4×D5912yes—
345D4—22no—
346D4—592no—
347D4—1181no—
348D4—1181no—
349D4—1181no—
350D4—4721no—
351D4—2362no—
352D4—2362no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—1181no—
364C2×C4—1181no—
365C2×C4—4721no—
366C2×C4—2362no—
367C2×C4C2×C22⋊D5911yescharacteristic
368C2×C4C2×C22⋊D5911yescharacteristic
369C2×C4C2×C22⋊D5911yescharacteristic
370C2×C4C2×C22⋊D5911yescharacteristic
371C2×C4C23×D5911yescharacteristic
372C8—592no—
373C8—1181no—
374C8—1181no—
375C8—1181no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—1182no—
382C22—4724no—
383C22—2361no—
384C22—2361no—
385C22—2361no—
386C22—2361no—
387C22—2361no—
388C22#1427375812yescomplemented
389C22C22×C22⋊D5911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2361no—
399C4—2361no—
400C4C2×D4×D5911yescharacteristic
401C4#1427377211yescharacteristic
402C2—22no—
403C2—41no—
404C2—4721no—
405C2—2362no—
406C2D4×C22⋊D5911yescharacteristic
407C2C2×#1427375811yescharacteristic
408C2#1683821011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingD4×Q8×C59C2
socleC22×C59C22×D4
hypercenterD4×Q8D59
hypocenterC59#3266
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14273758C2semidirect
2. C22×#11745671C2semidirect
4. #16837794C2semidirect
5. #16837650C2semidirect
6. #16837641C2semidirect
7. #16837624C2semidirect
8. #16837604C2semidirect
9. #16837562C2semidirect
10. #16837558C2semidirect
11. #16837552C2semidirect
12. #16837541C2semidirect
13. #16837500C2semidirect
14. #16837474C2semidirect
15. D4×Q8×C59C2semidirect
19. C2×#11745672C22semidirect
20. C2×#11745672C22semidirect
21. C2×#11745672C22semidirect
29. C2×#11745664C22semidirect
39. #14273660C22semidirect
40. C4⋊(C59⋊C8)C22semidirect
41. #14273651C22semidirect
42. #14273651C22semidirect
43. #14273651C22semidirect
44. #14273650C22semidirect
48. C22⋊(C59⋊C8)C22semidirect
49. #14273638C22semidirect
51. C22×Q8×C59C22semidirect
53. C4⋊Q8×C59C22semidirect
57. C22⋊Q8×C59C22semidirect
58. C4×Q8×C59C22semidirect
110. #11745671D4semidirect
176. C22×C59#137semidirect
207. D4×Q8D59semidirect
220. C59#3266semidirect
223. C22×Q8C2×D59semidirect
239. C4⋊Q8C2×D59semidirect
246. C22⋊Q8C2×D59semidirect
250. C4×Q8C2×D59semidirect
388. C22#14273758semidirect