Finite Group Databasebeta

← #28485195

Subgroups — #28485195

Order 15104.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435021C211yesmaximal, characteristic
2#21432842C211yesmaximal, characteristic
3#21422774C211yesmaximal, characteristic
4C2×#16837151C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421947C211yesmaximal, characteristic, complemented
7C59×#1997C211yesmaximal, characteristic, complemented
8C2×#14273735—21no—
9#16837952C2211yescharacteristic
10#16837776—21no—
11#16837752—21no—
12#16837604—22no—
13#16837541C414yes—
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#16837151—21no—
22#16837151—21no—
23#16837149—21no—
24#16837145—21no—
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C4×M16×C59C2211yescharacteristic
28C59×#343—21no—
29C59×#343—21no—
30C2×#11745685—21no—
31C2×#11745671—24no—
32#14273735—42no—
33#14273735—42no—
34C2×#11745659—22no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C2×C8×D59—22no—
38#14273727D411yescharacteristic
39#14273724D411yescharacteristic
40#14273715—21no—
41#14273701—21no—
42#14273701C2×C412yes—
43C4×D236—21no—
44C4×D236D412yes—
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#14273651—24no—
52C22⋊(C59⋊C8)C2×C412yes—
53(C2×D59)⋊C8—22no—
54(C2×D59)⋊C8—22no—
55C4×C59⋊C8—21no—
56C22×Q8×C59C2×C412yes—
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×C59—21no—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69#97×C59—21no—
70C4×C8×C59—21no—
71C2×Q8×C59—22no—
72C2×Q8×C59—22no—
73C2×Q8×C59—82no—
74C22×C4×C59—22no—
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C2×C22⋊D59—22no—
79#11745685—41no—
80#11745685—41no—
81#11745685—41no—
82#11745685—41no—
83C2×D236—21no—
84C2×D236—21no—
85C2×D236C22⋊C412yes—
86C2×C4×D59—22no—
87C2×Dic118—21no—
88M16×C59—21no—
89M16×C59—21no—
90M16×C59—21no—
91M16×C59—41no—
92M16×C59—41no—
93M16×C59SD1612yescomplemented
94C2×C8×C59—21no—
95C2×C8×C59—21no—
96C2×C8×C59—41no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59#4111yescharacteristic
105#11745671—84no—
106(C2×D59)⋊C4—22no—
107(C2×D59)⋊C4—22no—
108C4⋊Dic59—21no—
109C4⋊Dic59C22⋊C412yes—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—81no—
117#11745659—44no—
118C8×D59—44no—
119C23×C59C4×D411yescharacteristic
120C22×D59—22no—
121Q8×C59—44no—
122Q8×C59—82no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—82no—
128C2×C4×C59C2×C22⋊C411yescharacteristic
129C2×C4×C59C4×D411yescharacteristic
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12111yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12311yescharacteristic
134C2×C4×C59C2×SD1611yescharacteristic
135C2×C4×C59#13511yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C22⋊D59—42no—
139C22⋊D59—42no—
140C2×Dic59—22no—
141D236—42no—
142D236—41no—
143D236—41no—
144D236—41no—
145D236—41no—
146C4×D59—42no—
147C4×D59—42no—
148Dic118—41no—
149Dic118—41no—
150C8×C59—22no—
151C8×C59—41no—
152C8×C59—81no—
153C59⋊C8—42no—
154C59⋊C8—41no—
155C59⋊C8—41no—
156C59⋊C8—81no—
157#9267—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59#38611yescharacteristic
161C22×C59C4×SD1611yescharacteristic
162C22×C59#44011yescharacteristic
163C2×D59—42no—
164C2×D59—42no—
165C2×D59—42no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—82no—
170C4×C59#45011yescharacteristic
171C4×C59#45211yescharacteristic
172C4×C59#46111yescharacteristic
173C4×C59#46311yescharacteristic
174C4×C59#47111yescharacteristic
175C4×C59#47311yescharacteristic
176Dic59—42no—
177#3206—591no—
178#2935—591no—
179#1997D5911yescharacteristic, complemented
180#1547—591no—
181C2×#333—591no—
182#1301—591no—
183#1283—591no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59#187711yescharacteristic
187C2×C59#190811yescharacteristic
188C2×C59#191411yescharacteristic
189D59—82no—
190C2×#131—1181no—
191#532—591no—
192C2×C4⋊Q8C2×D5911yescharacteristic
193#492—1181no—
194#487—1181no—
195#461—1182no—
196#448—594no—
197#436—1181no—
198#435—1182no—
199C8×D4—1181no—
200#431—591no—
201#423C2×D5911yescharacteristic
202C2×C4⋊C8—591no—
203C2×C4⋊C8—591no—
204#408—1181no—
205C4×M16C2×D5911yescharacteristic
206#343—21no—
207#343—21no—
208#333—1181no—
209#333—1181no—
210#332—1181no—
211#331—1181no—
212C59#926711yescharacteristic, complemented, Sylow
213C2×#41—1181no—
214C22×Q8C4×D5912yes—
215C2×SD16—1184no—
216#131—2362no—
217#131—2362no—
218C2×M16—21no—
219C2×M16—1182no—
220C2×M16C22⋊D5911yescharacteristic
221C2×M16C22⋊D5911yescharacteristic
222C22×C8—1182no—
223C22×C8—1181no—
224C22×C8—1181no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—591no—
230#127—591no—
231#124—1181no—
232#121—592no—
233#121—1181no—
234C4×D4—592no—
235C4×D4—1181no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C42C22×D5911yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—591no—
242C4⋊C8—591no—
243C4⋊C8—1181no—
244C4⋊C8—1181no—
245C4⋊C8—1181no—
246C4⋊C8—1181no—
247#103—1184no—
248C22⋊C8—592no—
249C22⋊C8—1182no—
250C22⋊C8—1182no—
251#97—21no—
252C4×C8—21no—
253C4×C8—1181no—
254#41—2361no—
255#41—2361no—
256#41—2361no—
257#41—2361no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—82no—
261C2×Q8—1181no—
262C2×D4—592no—
263C2×D4—1182no—
264C2×D4—1181no—
265C2×D4—1181no—
266C22×C4—22no—
267C22×C4—1182no—
268C22×C4C2×C4×D5911yescharacteristic
269C22×C4#1174568511yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271SD16—4724no—
272M16—21no—
273M16—21no—
274M16—21no—
275M16—41no—
276M16—41no—
277M16—2364no—
278M16#1174567112yescomplemented
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—1182no—
283C2×C8—1181no—
284C2×C8—1181no—
285C2×C8—4721no—
286C2×C8—2364no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8—2361no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—592no—
295C4⋊C4—1181no—
296C22⋊C4—1182no—
297C22⋊C4—1182no—
298C42D4×D5911yescharacteristic
299C42D4×D5911yescharacteristic
300C42D4×D5911yescharacteristic
301C42#1174568711yescharacteristic
302C23—1182no—
303C23C4×C22⋊D5911yescharacteristic
304Q8—44no—
305Q8—82no—
306Q8—2361no—
307Q8—2361no—
308D4—2362no—
309D4—2362no—
310D4—2362no—
311D4—2361no—
312D4—2361no—
313D4—2361no—
314D4—2361no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—1182no—
321C2×C4—2362no—
322C2×C4—2362no—
323C2×C4C22⋊C4×D5911yescharacteristic
324C2×C4(C22⋊D59)⋊C411yescharacteristic
325C2×C4#1427369511yescharacteristic
326C2×C4#1427369811yescharacteristic
327C2×C4#1427370011yescharacteristic
328C2×C4#1427370911yescharacteristic
329C2×C4C2×#1174567111yescharacteristic
330C2×C4#1427374711yescharacteristic
331C2×C4#1427375211yescharacteristic
332C2×C4#1427375811yescharacteristic
333C8—22no—
334C8—41no—
335C8—81no—
336C8—4721no—
337C8—2362no—
338C8—2361no—
339C8—2361no—
340C22—21no—
341C22—21no—
342C22—2362no—
343C22—2362no—
344C22—2362no—
345C22#1683727811yescharacteristic
346C22C4×#1174567111yescharacteristic
347C22#1683750011yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—82no—
352C4—2362no—
353C4#1683755211yescharacteristic
354C4#1683756211yescharacteristic
355C4#1683760411yescharacteristic
356C4#1683761411yescharacteristic
357C4#1683766811yescharacteristic
358C4#1683767611yescharacteristic
359C2—21no—
360C2—21no—
361C2—4722no—
362C2#2142443311yescharacteristic
363C2#2142460811yescharacteristic
364C2#2142464011yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1997C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1997D59
hypocenterC59#9267
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
4. C2×#16837151C2semidirect
5. #21421991C2semidirect
6. #21421947C2semidirect
7. C59×#1997C2semidirect
93. M16×C59SD16semidirect
179. #1997D59semidirect
212. C59#9267semidirect
278. M16#11745671semidirect

Maximal subgroups

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

subgroupindex
#214350212
#214328422
#214227742
C2×#168371512
#214219912
#214219472
C59×#19972
#926759