Finite Group Databasebeta

← #28505614

Subgroups — #28505614

Order 15104.

470 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#21437198C211yesmaximal, characteristic, complemented
2#21432799C211yesmaximal, characteristic
3#21425716C211yesmaximal, characteristic, complemented
4C2×#16837184C211yesmaximal, characteristic, complemented
5#21422183C211yesmaximal, characteristic, complemented
6#21422181C211yesmaximal, characteristic, complemented
7C2×C59×#349C211yesmaximal, characteristic, complemented
8C2×#14273733C2211yescharacteristic
9#16837913C411yescharacteristic
10#16837820—22no—
11#16837818—22no—
12C2×#14273667C2211yescharacteristic
13C2×#14273667C2211yescharacteristic
14M32×D59—22no—
15#16837478—22no—
16#16837471—22no—
17#16837428C411yescharacteristic
18#16837368C411yescharacteristic
19#16837366C411yescharacteristic
20#16837357C2211yescharacteristic
21#16837184—22no—
22#16837183—22no—
23C22×M16×C59C2211yescharacteristic
24C2×M32×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C59×#349—22no—
27C2×#11745685C2×C411yescharacteristic
28#14273733—21no—
29#14273733—21no—
30#14273733—21no—
31#14273733—21no—
32C2×#11745664C2×C411yescharacteristic
33C2×#11745664C2×C411yescharacteristic
34C2×#11745659D412yes—
35C2×C8×D59D412yes—
36#14273710C812yes—
37#14273706—21no—
38#14273701—21no—
39#14273696C812yes—
40C4×C22⋊D59—22no—
41#14273686C2×C411yescharacteristic
42#14273667—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667D412yescomplemented
46#14273666—24no—
47C2×C59⋊C16—21no—
48C2×C59⋊C16—21no—
49C16×D59—24no—
50Dic59⋊C8—22no—
51C22⋊(C59⋊C8)—22no—
52(C2×D59)⋊C8—22no—
53#14273637—21no—
54C8×Dic59—21no—
55C23×C4×C59C2×C411yescharacteristic
56C2×M16×C59—21no—
57C2×M16×C59—21no—
58C2×M16×C59—21no—
59C2×M16×C59—21no—
60C2×M16×C59—21no—
61C2×M16×C59—21no—
62C22×C8×C59C2×C411yescharacteristic
63C22×C8×C59C2311yescharacteristic
64M32×C59—21no—
65M32×C59—21no—
66M32×C59—21no—
67M32×C59D412yescomplemented
68C2×C16×C59—21no—
69C2×C16×C59—21no—
70C24×C59C2×C811yescharacteristic
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59C2×C811yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C2×C22⋊D59—22no—
80C2×C22⋊D59C2×C812yes—
81#11745685—21no—
82#11745685—21no—
83#11745685—21no—
84#11745685—21no—
85C2×D236M1611yescharacteristic
86C2×C4×D59C22⋊C412yes—
87C2×Dic118M1611yescharacteristic
88M16×C59—41no—
89M16×C59—41no—
90M16×C59—41no—
91M16×C59—41no—
92C2×C8×C59—21no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C22⋊Dic59—22no—
102C22⋊Dic59C2×C812yes—
103(C2×D59)⋊C4—22no—
104C4⋊Dic59—21no—
105Dic59⋊C4—22no—
106C4×Dic59—21no—
107#11745664—21no—
108#11745664—21no—
109#11745664—21no—
110#11745664—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8C22⋊C412yes—
113#11745659—22no—
114#11745659—22no—
115C8×D59—22no—
116C8×D59—22no—
117C16×C59—22no—
118C16×C59—41no—
119C59⋊C16—22no—
120C59⋊C16—41no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—81no—
125C23×C59C22×C811yescharacteristic
126C22×D59C22⋊C812yes—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C4×D411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C2×M1611yescharacteristic
138C2×C4×C59#13111yescharacteristic
139C2×C4×C59#13111yescharacteristic
140C22⋊D59—22no—
141C22⋊D59—22no—
142C22⋊D59—82no—
143C2×Dic59—22no—
144C2×Dic59C22⋊C812yes—
145D236—21no—
146D236—21no—
147C4×D59—22no—
148C4×D59—22no—
149Dic118—21no—
150Dic118—21no—
151C8×C59—41no—
152C8×C59#12011yescharacteristic
153C8×C59#12111yescharacteristic
154C8×C59#12111yescharacteristic
155C8×C59#12311yescharacteristic
156C59⋊C8—22no—
157C59⋊C8—41no—
158#12989—591nomaximal, Sylow
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59—81no—
165C22×C59C2×C22⋊C811yescharacteristic
166C22×C59C8×D411yescharacteristic
167C22×C59C8×D411yescharacteristic
168C2×D59—22no—
169C2×D59—42no—
170C4×C59—41no—
171C4×C59#38611yescharacteristic
172C4×C59#40811yescharacteristic
173C4×C59#43511yescharacteristic
174C4×C59#43511yescharacteristic
175Dic59—22no—
176Dic59—42no—
177#3407—591no—
178#2930—591no—
179#2115—591no—
180C2×#349—591no—
181C2×#349D5911yescharacteristic, complemented
182#1378—591no—
183#1378—591no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59—81no—
188C2×C59#186911yescharacteristic
189D59—42no—
190C2×#131—591no—
191C22×M16C2×D5911yescharacteristic
192#525—591no—
193#504—1182no—
194#504—1182no—
195C2×M32—591no—
196C2×M32—591no—
197C2×M32—1182no—
198C2×M32C2×D5911yescharacteristic
199C2×M32C2×D5911yescharacteristic
200#436—1182no—
201#435—1182no—
202#424—591no—
203#408—591no—
204#407—591no—
205#405—591no—
206#349—22no—
207#349—1182no—
208#349—1182no—
209C59#1298911yescharacteristic, complemented, Sylow
210C2×#41—591no—
211C23×C4C4×D5911yescharacteristic
212#131—1181no—
213#131—1181no—
214#131—1181no—
215#131—1181no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—591no—
223C2×M16—592no—
224C2×M16—591no—
225C22×C8—592no—
226C22×C8C4×D5911yescharacteristic
227C22×C8C22×D5911yescharacteristic
228#123—592no—
229C22⋊Q8—1181no—
230#121—1181no—
231#120—592no—
232C4×D4—1182no—
233#117—591no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—592no—
238M32—1184no—
239M32—1181no—
240M32—1181no—
241M32—1181no—
242M32C22⋊D5912yescomplemented
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1184no—
246C2×C16—1181no—
247C2×C16—1181no—
248C4⋊C8—1182no—
249C22⋊C8—1182no—
250C22⋊C8—1182no—
251#97—1181no—
252C4×C8—1181no—
253C24C8×D5911yescharacteristic
254#41—1181no—
255#41—1181no—
256#41—1181no—
257#41—1181no—
258C2×Q8—591no—
259C2×D4—592no—
260C2×D4—591no—
261C2×D4—1182no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—592no—
269C22×C4C8×D5911yescharacteristic
270C22×C4C2×C4×D5911yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—1181no—
276M16—1181no—
277M16—1182no—
278M16—1182no—
279M16—1181no—
280M16—1181no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—592no—
285C2×C8—1182no—
286C2×C8—1182no—
287C2×C8—1181no—
288C2×C8#1174568511yescharacteristic
289C2×C8D4×D5911yescharacteristic
290C2×C8D4×D5911yescharacteristic
291C2×C8D4×D5911yescharacteristic
292C2×C8#1174568711yescharacteristic
293C2×C8C2×C22⋊D5911yescharacteristic
294C4⋊C4—1182no—
295C4⋊C4—1181no—
296C22⋊C4—592no—
297C22⋊C4—1182no—
298C22⋊C4—1182no—
299C42—1181no—
300C16—22no—
301C16—41no—
302C16—1182no—
303C16—2361no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—81no—
308C23—592no—
309C23C2×C8×D5911yescharacteristic
310Q8—1181no—
311Q8—1181no—
312D4—1182no—
313D4—1182no—
314D4—1181no—
315D4—1181no—
316D4—4722no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—592no—
325C2×C4—1182no—
326C2×C4—1182no—
327C2×C4—1182no—
328C2×C4C22⋊C4×D5911yescharacteristic
329C2×C4(C22⋊D59)⋊C411yescharacteristic
330C2×C4C4×C22⋊D5911yescharacteristic
331C2×C4M16×D5911yescharacteristic
332C2×C4#1427373311yescharacteristic
333C2×C4#1427373511yescharacteristic
334C8—41no—
335C8—1182no—
336C8—2361no—
337C8#1427369511yescharacteristic
338C8#1427369811yescharacteristic
339C8#1427370011yescharacteristic
340C8#1427370911yescharacteristic
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—1182no—
348C22—2362no—
349C22C22⋊C8×D5911yescharacteristic
350C22(C22⋊D59)⋊C811yescharacteristic
351C22C8×C22⋊D5911yescharacteristic
352C4—41no—
353C4—1182no—
354C4—2362no—
355C4#1683727811yescharacteristic
356C4#1683736711yescharacteristic
357C4#1683747111yescharacteristic
358C4#1683747611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—2362no—
364C2#2142438511yescharacteristic
subgroupG/H
centerC4#16837278
commutatorC23×C59C22×C8
FrattiniC22×C8C22×D59
FittingC2×C59×#349C2
socleC2×C59#1869
prime radical#16837913C4
cosocleC22×C8×C59C23
hypercenterC2×#349D59
hypocenterC59#12989
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
1. #21437198C2semidirect
3. #21425716C2semidirect
4. C2×#16837184C2semidirect
5. #21422183C2semidirect
6. #21422181C2semidirect
7. C2×C59×#349C2semidirect
45. #14273667D4semidirect
67. M32×C59D4semidirect
181. C2×#349D59semidirect
209. C59#12989semidirect
242. M32C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214371982
#214327992
#214257162
C2×#168371842
#214221832
#214221812
C2×C59×#3492
#1298959