Finite Group Databasebeta

← #28485223

Subgroups — #28485223

Order 15104.

496 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435217C211yesmaximal, characteristic
2#21432842C211yesmaximal, characteristic
3#21422774C211yesmaximal, characteristic
4C2×#16837148C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421947C211yesmaximal, characteristic, complemented
7C59×#1966C211yesmaximal, characteristic, complemented
8C2×#14273735—21no—
9#16837952C2211yescharacteristic
10#16837805—21no—
11#16837750—21no—
12#16837715C414yes—
13#16837679—22no—
14#16837481—21no—
15#16837475—22no—
16C8×D236—21no—
17#16837455C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20#16837370—21no—
21#16837149—21no—
22#16837148—21no—
23#16837148—21no—
24#16837145—21no—
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C2×#101×C59—21no—
28C2×#101×C59—21no—
29C4×M16×C59C2211yescharacteristic
30C2×#11745685—21no—
31#14273735—42no—
32#14273735—42no—
33C2×#11745659—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C2×C8×D59—22no—
37#14273727D411yescharacteristic
38#14273724D411yescharacteristic
39#14273715—21no—
40#14273701—21no—
41#14273701C2×C412yes—
42C4×D236—21no—
43C4×D236D412yes—
44#14273660—24no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)D411yescharacteristic
50C4⋊(C59⋊C8)D411yescharacteristic
51#14273647—24no—
52C22⋊(C59⋊C8)C2×C412yes—
53(C2×D59)⋊C8—22no—
54(C2×D59)⋊C8—22no—
55C4×C59⋊C8—21no—
56C22×Q8×C59—21no—
57C2×M16×C59—21no—
58C2×M16×C59D411yescharacteristic
59C2×M16×C59D411yescharacteristic
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59—21no—
63C4⋊Q8×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59C2×C412yes—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69#101×C59—42no—
70#101×C59—42no—
71#97×C59—21no—
72C4×C8×C59—21no—
73C2×Q8×C59—41no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—41no—
78C2×Q8×C59—41no—
79C22×C4×C59—22no—
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59#4111yescharacteristic
83C2×C22⋊D59—22no—
84#11745685—41no—
85#11745685—41no—
86#11745685—41no—
87#11745685—41no—
88C2×D236—21no—
89C2×D236—21no—
90C2×D236C22⋊C412yes—
91C2×C4×D59—22no—
92C2×Dic118—21no—
93M16×C59—21no—
94M16×C59—21no—
95M16×C59—21no—
96M16×C59—41no—
97M16×C59—41no—
98M16×C59D812yes—
99C2×C8×C59—21no—
100C2×C8×C59—21no—
101C2×C8×C59—41no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59#4111yescharacteristic
110(C2×D59)⋊C4—22no—
111(C2×D59)⋊C4—22no—
112C4⋊Dic59—21no—
113C4⋊Dic59C22⋊C412yes—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—21no—
116C2×C59⋊C8—21no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C2×C59⋊C8—81no—
121#11745659—44no—
122C8×D59—44no—
123C23×C59C4×D411yescharacteristic
124C22×D59—22no—
125Q8×C59—81no—
126Q8×C59—81no—
127Q8×C59—81no—
128Q8×C59—81no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—42no—
135C2×C4×C59C2×C22⋊C411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59#12011yescharacteristic
138C2×C4×C59#12111yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12311yescharacteristic
141C2×C4×C59C2×D811yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C22⋊D59—42no—
146C22⋊D59—42no—
147C2×Dic59—22no—
148D236—42no—
149D236—41no—
150D236—41no—
151D236—41no—
152D236—41no—
153C4×D59—42no—
154C4×D59—42no—
155Dic118—41no—
156Dic118—41no—
157C8×C59—22no—
158C8×C59—41no—
159C8×C59—81no—
160C59⋊C8—42no—
161C59⋊C8—41no—
162C59⋊C8—41no—
163C59⋊C8—81no—
164#9271—591nomaximal, Sylow
165C22×C59—21no—
166C22×C59—21no—
167C22×C59#38611yescharacteristic
168C22×C59C4×D811yescharacteristic
169C22×C59#44211yescharacteristic
170C2×D59—42no—
171C2×D59—42no—
172C2×D59—42no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—82no—
177C4×C59#44711yescharacteristic
178C4×C59#44911yescharacteristic
179C4×C59#45911yescharacteristic
180C4×C59#46311yescharacteristic
181C4×C59#47111yescharacteristic
182C4×C59#47311yescharacteristic
183Dic59—42no—
184#3226—591no—
185#2935—591no—
186#1966D5911yescharacteristic, complemented
187#1547—591no—
188C2×#332—591no—
189#1301—591no—
190#1283—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59#187911yescharacteristic
194C2×C59#190811yescharacteristic
195C2×C59#191511yescharacteristic
196D59—82no—
197C2×#131—1181no—
198#532—591no—
199C2×C4⋊Q8C2×D5911yescharacteristic
200#500—1181no—
201#486—1181no—
202#480—594no—
203#474—1182no—
204#436—1181no—
205#435—1182no—
206C8×D4—1181no—
207#431—591no—
208#423C2×D5911yescharacteristic
209C2×C4⋊C8—591no—
210C2×C4⋊C8—591no—
211C2×#101—21no—
212C2×#101—21no—
213#408—1181no—
214C4×M16C2×D5911yescharacteristic
215#332—1181no—
216#332—1181no—
217#332—1181no—
218#331—1181no—
219C59#927111yescharacteristic, complemented, Sylow
220C2×#41—1181no—
221C22×Q8—21no—
222#131—2362no—
223#131—2362no—
224C2×M16—21no—
225C2×M16—1182no—
226C2×M16C22⋊D5911yescharacteristic
227C2×M16C22⋊D5911yescharacteristic
228C22×C8—1182no—
229C22×C8—1181no—
230C22×C8—1181no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—591no—
236#127—591no—
237#124—1181no—
238#121—592no—
239#121—1181no—
240C4×D4—592no—
241C4×D4—1181no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C4×D5912yes—
244C2×C42C22×D5911yescharacteristic
245#107—1184no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—591no—
249C4⋊C8—591no—
250C4⋊C8—1181no—
251C4⋊C8—1181no—
252C4⋊C8—1181no—
253C4⋊C8—1181no—
254#102—1184no—
255#101—42no—
256#101—42no—
257C22⋊C8—592no—
258C22⋊C8—1182no—
259C22⋊C8—1182no—
260#97—21no—
261C4×C8—21no—
262C4×C8—1181no—
263#41—2361no—
264#41—2361no—
265#41—2361no—
266#41—2361no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—1181no—
274C2×D4—592no—
275C2×D4—1182no—
276C2×D4—1181no—
277C2×D4—1181no—
278C22×C4—22no—
279C22×C4—1182no—
280C22×C4C2×C4×D5911yescharacteristic
281C22×C4#1174568511yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283M16—21no—
284M16—21no—
285M16—21no—
286M16—41no—
287M16—41no—
288M16—2364no—
289M16#1174566912yes—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—1182no—
294C2×C8—1181no—
295C2×C8—1181no—
296C2×C8—4721no—
297C2×C8—2364no—
298C2×C8—2361no—
299C2×C8—2361no—
300C2×C8—2361no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—592no—
306C4⋊C4—1181no—
307C22⋊C4—1182no—
308C22⋊C4—1182no—
309C42D4×D5911yescharacteristic
310C42D4×D5911yescharacteristic
311C42D4×D5911yescharacteristic
312C42#1174568711yescharacteristic
313C23—1182no—
314C23C4×C22⋊D5911yescharacteristic
315Q8—81no—
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—2361no—
320Q8—2361no—
321D4—2362no—
322D4—2362no—
323D4—2362no—
324D4—2361no—
325D4—2361no—
326D4—2361no—
327D4—2361no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—1182no—
335C2×C4—2362no—
336C2×C4—2362no—
337C2×C4C22⋊C4×D5911yescharacteristic
338C2×C4(C22⋊D59)⋊C411yescharacteristic
339C2×C4#1427369511yescharacteristic
340C2×C4#1427369811yescharacteristic
341C2×C4#1427370011yescharacteristic
342C2×C4#1427370911yescharacteristic
343C2×C4C2×#1174566911yescharacteristic
344C2×C4#1427374711yescharacteristic
345C2×C4#1427375211yescharacteristic
346C2×C4#1427375311yescharacteristic
347C8—22no—
348C8—41no—
349C8—81no—
350C8—4721no—
351C8—2362no—
352C8—2361no—
353C8—2361no—
354C22—21no—
355C22—21no—
356C22—2362no—
357C22—2362no—
358C22—2362no—
359C22#1683727811yescharacteristic
360C22C4×#1174566911yescharacteristic
361C22#1683751211yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—2362no—
367C4#1683753511yescharacteristic
368C4#1683754511yescharacteristic
369C4#1683759411yescharacteristic
370C4#1683761411yescharacteristic
371C4#1683766811yescharacteristic
372C4#1683767611yescharacteristic
373C2—21no—
374C2—21no—
375C2—4722no—
376C2#2142444711yescharacteristic
377C2#2142460811yescharacteristic
378C2#2142464511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1966C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1966D59
hypocenterC59#9271
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
4. C2×#16837148C2semidirect
5. #21421991C2semidirect
6. #21421947C2semidirect
7. C59×#1966C2semidirect
186. #1966D59semidirect
219. C59#9271semidirect

Maximal subgroups

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

subgroupindex
#214352172
#214328422
#214227742
C2×#168371482
#214219912
#214219472
C59×#19662
#927159