Finite Group Databasebeta

← #28483553

Subgroups — #28483553

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434127C211yesmaximal, characteristic
2Q8×#11745664C211yesmaximal, characteristic, complemented
3#21425014C211yesmaximal, characteristic
4#21422548C211yesmaximal, characteristic
5#21421999C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C4×#103×C59C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837720—21no—
10#16837687C411yescharacteristic, complemented
11#16837687C411yescharacteristic, complemented
12#16837602C411yescharacteristic, complemented
13#16837602C411yescharacteristic, complemented
14#16837541—21no—
15#16837533—22no—
16#16837533—21no—
17Q8×C59⋊C8—21no—
18C4×#11745671—21no—
19C4×#11745671—21no—
20#16837455C2211yescharacteristic
21#16837426—21no—
22#16837426C2211yescharacteristic
23C2×#14273655—21no—
24C2×#14273655—21no—
25C4×#11745664—21no—
26C4×#11745664C2211yescharacteristic
27#16837131—22no—
28C2×C4×Q8×C59C2211yescharacteristic
29C2×#103×C59—21no—
30C2×C4×C8×C59C2211yescharacteristic
31C4×C4⋊C4×C59C2211yescharacteristic
32C59×#340—21no—
33C2×#11745685—21no—
34C2×#11745671—21no—
35C2×#11745671—21no—
36C2×#11745664—21no—
37C2×#11745664D411yescharacteristic
38C2×#11745664D411yescharacteristic
39#14273727C2×C411yescharacteristic
40#14273724C2×C411yescharacteristic
41#14273715—21no—
42#14273701—21no—
43#14273701—21no—
44C4×D236—21no—
45C4×D236—21no—
46#14273658—21no—
47#14273658—21no—
48C4⋊(C59⋊C8)—22no—
49C4⋊(C59⋊C8)—22no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)C2×C411yescharacteristic
52C4⋊(C59⋊C8)C2×C411yescharacteristic
53C4⋊(C59⋊C8)D412yes—
54#14273655—42no—
55#14273655—42no—
56#14273651—21no—
57#14273651—21no—
58#14273647—21no—
59#14273647—21no—
60C22⋊(C59⋊C8)—21no—
61#14273638—22no—
62#14273638—21no—
63C4×C59⋊C8—22no—
64C4×C59⋊C8—21no—
65C4×C59⋊C8—21no—
66C22×Q8×C59D411yescharacteristic
67C22×C8×C59—21no—
68C4×Q8×C59—21no—
69C4×Q8×C59—21no—
70C4×Q8×C59C2×C411yescharacteristic
71C4×Q8×C59C2×C411yescharacteristic
72C4×Q8×C59D411yescharacteristic
73C4×Q8×C59D411yescharacteristic
74C2×C4⋊C4×C59—21no—
75C2×C4⋊C4×C59—21no—
76C2×C4⋊C4×C59D411yescharacteristic
77C2×C42×C59—21no—
78C2×C42×C59—21no—
79C2×C42×C59C2311yescharacteristic
80#103×C59—24no—
81C4×C8×C59—21no—
82C4×C8×C59—21no—
83#95×C59—21no—
84C2×Q8×C59—21no—
85C2×Q8×C59—22no—
86C2×Q8×C59—41no—
87C2×Q8×C59C22⋊C411yescharacteristic
88C2×Q8×C59C22⋊C411yescharacteristic
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—41no—
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C2×C22⋊D59—41no—
96#11745685—41no—
97#11745685—41no—
98#11745685—41no—
99#11745685—41no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×C4×D59—41no—
104C2×Dic118—21no—
105C2×C8×C59—22no—
106C2×C8×C59—41no—
107C2×C8×C59—41no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—22no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59C22⋊C411yescharacteristic
116C4⋊C4×C59C22⋊C411yescharacteristic
117C42×C59—21no—
118C42×C59—21no—
119C42×C59—21no—
120C42×C59—21no—
121C42×C59—41no—
122C42×C59—41no—
123C42×C59C22×C411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59#4111yescharacteristic
127#11745671—42no—
128#11745671—42no—
129(C2×D59)⋊C4—41no—
130(C2×D59)⋊C4—41no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133#11745664—21no—
134#11745664—21no—
135#11745664—21no—
136#11745664—41no—
137#11745664—41no—
138#11745664SD1612yescomplemented
139C2×C59⋊C8—21no—
140C2×C59⋊C8—21no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—21no—
143C2×C59⋊C8—22no—
144C23×C59#12111yescharacteristic
145C22×D59—41no—
146Q8×C59—21no—
147Q8×C59—21no—
148Q8×C59—21no—
149Q8×C59—42no—
150Q8×C59#10412yescomplemented
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—41no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59C2×C22⋊C411yescharacteristic
164C2×C4×C59C4×D411yescharacteristic
165C2×C4×C59C4×D411yescharacteristic
166C2×C4×C59#12011yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12311yescharacteristic
169C2×C4×C59C2×SD1611yescharacteristic
170C2×C4×C59#13511yescharacteristic
171C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C59#13711yescharacteristic
173C22⋊D59—81no—
174C22⋊D59—81no—
175C2×Dic59—41no—
176D236—42no—
177D236—41no—
178D236—41no—
179D236—41no—
180D236—41no—
181C4×D59—81no—
182C4×D59—81no—
183Dic118—41no—
184Dic118—41no—
185C8×C59—42no—
186C59⋊C8—22no—
187C59⋊C8—41no—
188C59⋊C8—41no—
189C59⋊C8—41no—
190C59⋊C8—42no—
191#8992—591nomaximal, Sylow
192C22×C59—21no—
193C22×C59—21no—
194C22×C59#38611yescharacteristic
195C22×C59#46011yescharacteristic
196C22×C59#46411yescharacteristic
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—81no—
200C4×C59—21no—
201C4×C59—22no—
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—41no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59—82no—
208C4×C59C2×#10411yescharacteristic
209C4×C59#42111yescharacteristic
210C4×C59C4×SD1611yescharacteristic
211C4×C59#44011yescharacteristic
212C4×C59#44811yescharacteristic
213C4×C59#44911yescharacteristic
214Dic59—81no—
215#3103—591no—
216Q8×M16—591no—
217#1980—591no—
218C4×#103D5911yescharacteristic, complemented
219#1492—591no—
220#1303—591no—
221#1278—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59#187111yescharacteristic
225C2×C59#188011yescharacteristic
226C2×C59#188111yescharacteristic
227D59—161no—
228#532—591no—
229C2×C4×Q8C2×D5911yescharacteristic
230#481—1181no—
231#475—591no—
232#475—591no—
233#460—591no—
234#460—591no—
235#448—1181no—
236#446—1182no—
237#446—1181no—
238C8×Q8—1181no—
239C4×SD16—1181no—
240C4×SD16—1181no—
241#431—591no—
242#423—591no—
243#423—1181no—
244C2×#104—1181no—
245C2×#104—1181no—
246C2×#103—21no—
247C4×M16—591no—
248C4×M16—1181no—
249C2×C4×C8C2×D5911yescharacteristic
250C4×C4⋊C4C2×D5911yescharacteristic
251#340—21no—
252D4⋊C8—1182no—
253C59#899211yescharacteristic, complemented, Sylow
254C2×#41—1181no—
255C22×Q8C22⋊D5911yescharacteristic
256C2×SD16—1181no—
257C2×SD16—1181no—
258C2×M16—591no—
259C2×M16—591no—
260C2×M16—1181no—
261C22×C8—21no—
262C4⋊Q8—591no—
263#127—591no—
264#124—1181no—
265#121—1181no—
266#121—1181no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D5911yescharacteristic
270C4×Q8C4×D5911yescharacteristic
271C4×Q8D23611yescharacteristic
272C4×Q8D23611yescharacteristic
273C4×D4—1181no—
274C4×D4—1181no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D5911yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×D5911yescharacteristic
281#106—1181no—
282#106—1181no—
283C4⋊C8—591no—
284C4⋊C8—591no—
285C4⋊C8—592no—
286C4⋊C8—1182no—
287C4⋊C8—1182no—
288C4⋊C8—1181no—
289#104—2362no—
290#104—2362no—
291#103—24no—
292#103—1181no—
293#103—1181no—
294#102—1181no—
295#102—1181no—
296C22⋊C8—1181no—
297#97—1182no—
298#97—1181no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—1182no—
302C4×C8—1181no—
303C4×C8—1181no—
304#95—21no—
305#41—2361no—
306#41—2361no—
307#41—2361no—
308#41—2361no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—1181no—
313C2×Q8(C2×D59)⋊C411yescharacteristic
314C2×Q8(C2×D59)⋊C411yescharacteristic
315C2×D4—1181no—
316C2×D4—1181no—
317C2×D4—1181no—
318C2×D4—2361no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—2361no—
323C22×C4D4×D5911yescharacteristic
324C22×C4#1174568711yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—2362no—
327SD16—2362no—
328M16—592no—
329M16—1181no—
330M16—1181no—
331M16—1181no—
332M16—2361no—
333M16—2361no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—1181no—
338C2×C8—1181no—
339C2×C8—1181no—
340C2×C8—1182no—
341C2×C8—1181no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—1181no—
350C4⋊C4—1181no—
351C4⋊C4(C2×D59)⋊C411yescharacteristic
352C4⋊C4(C2×D59)⋊C411yescharacteristic
353C22⋊C4—2361no—
354C22⋊C4—2361no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—41no—
360C42—41no—
361C42C2×C4×D5911yescharacteristic
362C42C2×D23611yescharacteristic
363C42D4×D5911yescharacteristic
364C42#1174568711yescharacteristic
365C23—2361no—
366C23#1427369911yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—2361no—
372Q8—2361no—
373Q8#1427365512yescomplemented
374D4—4721no—
375D4—4721no—
376D4—2362no—
377D4—2361no—
378D4—2361no—
379D4—2361no—
380D4—2361no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4721no—
394C2×C4—4721no—
395C2×C4—2361no—
396C2×C4C2×(C2×D59)⋊C411yescharacteristic
397C2×C4(C22⋊D59)⋊C411yescharacteristic
398C2×C4(C22⋊D59)⋊C411yescharacteristic
399C2×C4#1427369711yescharacteristic
400C2×C4#1427369911yescharacteristic
401C2×C4#1427371111yescharacteristic
402C2×C4SD16×D5911yescharacteristic
403C2×C4#1427375011yescharacteristic
404C2×C4#1427375411yescharacteristic
405C2×C4#1427375911yescharacteristic
406C8—42no—
407C8—1182no—
408C8—2361no—
409C8—2361no—
410C8—2361no—
411C8—2362no—
412C22—21no—
413C22—21no—
414C22—4721no—
415C22—4721no—
416C22—4721no—
417C22#1683727911yescharacteristic
418C22#1683760011yescharacteristic
419C22#1683762611yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—4721no—
429C4C2×#1427365511yescharacteristic
430C4#1683742011yescharacteristic
431C4#1683748711yescharacteristic
432C4#1683750511yescharacteristic
433C4#1683754211yescharacteristic
434C4#1683755011yescharacteristic
435C2—21no—
436C2—21no—
437C2—9441no—
438C2#2142439611yescharacteristic
439C2#2142444911yescharacteristic
440C2#2142445611yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC4×#103×C59C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenterC4×#103D59
hypocenterC59#8992
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. Q8×#11745664C2semidirect
5. #21421999C2semidirect
6. #21421933C2semidirect
7. C4×#103×C59C2semidirect
10. #16837687C4semidirect
11. #16837687C4semidirect
12. #16837602C4semidirect
13. #16837602C4semidirect
138. #11745664SD16semidirect
150. Q8×C59#104semidirect
218. C4×#103D59semidirect
253. C59#8992semidirect
373. Q8#14273655semidirect

Maximal subgroups

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

subgroupindex
#214341272
Q8×#117456642
#214250142
#214225482
#214219992
#214219332
C4×#103×C592
#899259