Finite Group Databasebeta

← #28506896

Subgroups — #28506896

Order 15104.

420 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432797C211yesmaximal, characteristic
2#21425861C211yesmaximal, characteristic, complemented
3#21425724C211yesmaximal, characteristic, complemented
4C2×#16837183C211yesmaximal, characteristic, complemented
5C2×#16837183C211yesmaximal, characteristic, complemented
6#21425700C211yesmaximal, characteristic, complemented
7C59×#2117C211yesmaximal, characteristic, complemented
8C22×#11745659C2211yescharacteristic
9#16837909C411yescharacteristic
10C2×#14273667C2211yescharacteristic
11C2×#14273667C2211yescharacteristic
12#16837478—22no—
13#16837473—22no—
14#16837430C411yescharacteristic
15#16837369C411yescharacteristic
16#16837365C411yescharacteristic
17#16837357C2211yescharacteristic
18#16837222—22no—
19#16837186—22no—
20#16837183—22no—
21#16837183—22no—
22#16837179—22no—
23C2×#131×C59C2211yescharacteristic
24C2×M32×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C59×#350—22no—
27C22×C4×D59C2×C411yescharacteristic
28C2×#11745664C2×C411yescharacteristic
29C2×#11745659—21no—
30C2×#11745659—21no—
31C2×#11745659—21no—
32C2×#11745659—21no—
33C2×#11745659—21no—
34C2×#11745659—21no—
35C22×C59⋊C8C2×C411yescharacteristic
36#14273708—21no—
37#14273703—21no—
38#14273698—21no—
39#14273695—21no—
40C4×C22⋊D59—22no—
41#14273686C2×C411yescharacteristic
42#14273667—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667D412yescomplemented
46C2×C59⋊C16—21no—
47C2×C59⋊C16—21no—
48Dic59⋊C8—22no—
49C22⋊(C59⋊C8)—22no—
50(C2×D59)⋊C8—22no—
51#14273637D412yes—
52C8×Dic59D412yes—
53C2×#41×C59C2×C411yescharacteristic
54#131×C59—21no—
55#131×C59—21no—
56#131×C59—21no—
57#131×C59—21no—
58C2×M16×C59—21no—
59C2×M16×C59C2×C411yescharacteristic
60C22×C8×C59—21no—
61C22×C8×C59C2311yescharacteristic
62M32×C59—21no—
63M32×C59—21no—
64M32×C59—21no—
65M32×C59D412yescomplemented
66C2×C16×C59—21no—
67C2×C16×C59—21no—
68C23×D59M1611yescharacteristic
69#41×C59—21no—
70#41×C59—21no—
71#41×C59—21no—
72#41×C59—21no—
73C2×Q8×C59M1611yescharacteristic
74C2×D4×C59—21no—
75C2×D4×C59M1611yescharacteristic
76C22×C4×C59—21no—
77C22×C4×C59C22×C411yescharacteristic
78C2×C22⋊D59—22no—
79C22×Dic59M1611yescharacteristic
80C2×C4×D59—21no—
81C2×C4×D59—21no—
82C2×C4×D59—21no—
83C2×C4×D59—21no—
84C2×C4×D59—21no—
85C2×C4×D59—21no—
86M16×C59—21no—
87M16×C59—21no—
88M16×C59—41no—
89M16×C59—41no—
90C2×C8×C59—21no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59C2×D411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C2×C8×C59#4111yescharacteristic
99C22⋊Dic59—22no—
100C22⋊Dic59—21no—
101(C2×D59)⋊C4—22no—
102C4⋊Dic59—21no—
103Dic59⋊C4—22no—
104C4×Dic59C22⋊C412yes—
105#11745664—21no—
106#11745664—21no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8C22⋊C412yes—
111#11745659—41no—
112#11745659—41no—
113#11745659—41no—
114#11745659—41no—
115C16×C59—22no—
116C16×C59—41no—
117C59⋊C16—22no—
118C59⋊C16—41no—
119C23×C59—21no—
120C23×C59C2×M1611yescharacteristic
121C22×D59—21no—
122C22×D59—21no—
123C22×D59—21no—
124C22×D59—81no—
125Q8×C59—21no—
126Q8×C59—21no—
127D4×C59—21no—
128D4×C59—21no—
129D4×C59—41no—
130D4×C59—41no—
131C2×C4×C59—21no—
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—82no—
141C2×Dic59—22no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C4×D59—41no—
146C4×D59—41no—
147C4×D59—41no—
148C4×D59—41no—
149C8×C59—41no—
150C8×C59#12111yescharacteristic
151C8×C59#12111yescharacteristic
152C8×C59#12411yescharacteristic
153C8×C59#12711yescharacteristic
154C59⋊C8—22no—
155C59⋊C8—41no—
156#13201—591nomaximal, Sylow
157C22×C59—41no—
158C22×C59—42no—
159C22×C59—41no—
160C22×C59#40711yescharacteristic
161C22×C59#43511yescharacteristic
162C22×C59#43511yescharacteristic
163C2×D59—41no—
164C2×D59—41no—
165C2×D59—41no—
166C2×D59—81no—
167C2×D59—81no—
168C4×C59—41no—
169C4×C59#39011yescharacteristic
170C4×C59#40811yescharacteristic
171C4×C59#43511yescharacteristic
172C4×C59#43511yescharacteristic
173Dic59—42no—
174Dic59—41no—
175#2930—591no—
176#2153—591no—
177#2117—591no—
178#2117D5911yescharacteristic, complemented
179C2×#349—591no—
180C2×#349—591no—
181#2110—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—81no—
186C2×C59#195511yescharacteristic
187D59—81no—
188C2×#131C2×D5911yescharacteristic
189C22×M16—591no—
190#525—591no—
191C2×M32—591no—
192C2×M32—591no—
193C2×M32C2×D5911yescharacteristic
194C2×M32C2×D5911yescharacteristic
195#436—1182no—
196#435—1182no—
197#424—591no—
198#408—591no—
199#407—591no—
200#405—591no—
201#364—1182no—
202#350—22no—
203#350—1182no—
204#349—1182no—
205#349—1182no—
206#347—1182no—
207C59#1320111yescharacteristic, complemented, Sylow
208C2×#41C4×D5911yescharacteristic
209C23×C4—591no—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214C2×M16—21no—
215C2×M16—591no—
216C2×M16—1181no—
217C2×M16—1181no—
218C2×M16—1181no—
219C2×M16—1181no—
220C2×M16—1181no—
221C2×M16—1181no—
222C2×M16C4×D5911yescharacteristic
223C22×C8—21no—
224C22×C8—591no—
225C22×C8C22×D5911yescharacteristic
226#123—1181no—
227C22⋊Q8—1181no—
228#121—1181no—
229#120—1181no—
230C4×D4—1182no—
231#117—591no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—592no—
236M32—1181no—
237M32—1181no—
238M32—1181no—
239M32C22⋊D5912yescomplemented
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1181no—
243C2×C16—1181no—
244C4⋊C8—1182no—
245C22⋊C8—1182no—
246C22⋊C8—1182no—
247#97—592no—
248C4×C8—592no—
249C24—591no—
250#41—21no—
251#41—21no—
252#41—21no—
253#41—21no—
254C2×Q8#1174565911yescharacteristic
255C2×D4—21no—
256C2×D4—1182no—
257C2×D4#1174565911yescharacteristic
258C22×C4—21no—
259C22×C4—591no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4—1181no—
265C22×C4—1181no—
266C22×C4C2×C4×D5911yescharacteristic
267M16—21no—
268M16—21no—
269M16—41no—
270M16—41no—
271M16—1181no—
272M16—1181no—
273M16—2361no—
274M16—2361no—
275M16—2361no—
276M16—2361no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—592no—
281C2×C8—1181no—
282C2×C8—1181no—
283C2×C8—1181no—
284C2×C8#1174568511yescharacteristic
285C2×C8D4×D5911yescharacteristic
286C2×C8D4×D5911yescharacteristic
287C2×C8D4×D5911yescharacteristic
288C2×C8#1174568711yescharacteristic
289C2×C8C2×C22⋊D5911yescharacteristic
290C4⋊C4—1182no—
291C4⋊C4—1181no—
292C22⋊C4—1182no—
293C22⋊C4—1182no—
294C22⋊C4—1181no—
295C42—592no—
296C16—22no—
297C16—41no—
298C16—1182no—
299C16—2361no—
300C23—21no—
301C23—1181no—
302C23—1181no—
303C23—1181no—
304C23—4721no—
305C23M16×D5911yescharacteristic
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—4722no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—1182no—
319C2×C4—1181no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4C22⋊C4×D5911yescharacteristic
325C2×C4(C22⋊D59)⋊C411yescharacteristic
326C2×C4C4×C22⋊D5911yescharacteristic
327C2×C4C2×#1174565911yescharacteristic
328C2×C4#1427373311yescharacteristic
329C2×C4#1427373511yescharacteristic
330C8—41no—
331C8—1182no—
332C8—2361no—
333C8#1427369911yescharacteristic
334C8#1427370011yescharacteristic
335C8#1427371311yescharacteristic
336C8#1427372311yescharacteristic
337C22—41no—
338C22—42no—
339C22—41no—
340C22—4721no—
341C22—4721no—
342C22—2361no—
343C22—2361no—
344C22—2361no—
345C22#1683736411yescharacteristic
346C22#1683747111yescharacteristic
347C22#1683747611yescharacteristic
348C4—41no—
349C4—2362no—
350C4—2361no—
351C4#1683730211yescharacteristic
352C4#1683736711yescharacteristic
353C4#1683747011yescharacteristic
354C4#1683747311yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—4721no—
360C2#2142489211yescharacteristic
subgroupG/H
centerC4#16837302
commutatorC22×C4×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2117C2
socleC2×C59#1955
prime radical#16837909C4
cosocleC22×C8×C59C23
hypercenter#2117D59
hypocenterC59#13201
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #21425861C2semidirect
3. #21425724C2semidirect
4. C2×#16837183C2semidirect
5. C2×#16837183C2semidirect
6. #21425700C2semidirect
7. C59×#2117C2semidirect
45. #14273667D4semidirect
65. M32×C59D4semidirect
178. #2117D59semidirect
207. C59#13201semidirect
239. M32C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214327972
#214258612
#214257242
C2×#168371832
#214257002
C59×#21172
#1320159