Finite Group Databasebeta

← #28505595

Subgroups — #28505595

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437198C211yesmaximal, characteristic, complemented
2C2×#16837522C211yesmaximal, characteristic
3#21425723C211yesmaximal, characteristic, complemented
4#21425717C211yesmaximal, characteristic, complemented
5#21422179C211yesmaximal, characteristic, complemented
6#21422175C211yesmaximal, characteristic, complemented
7C59×#2117C211yesmaximal, characteristic, complemented
8C2×#14273747C411yescharacteristic
9C2×#14273733C2211yescharacteristic
10#16837820—22no—
11#16837818—22no—
12C2×#14273667C2211yescharacteristic
13C2×#14273667C2211yescharacteristic
14M32×D59—22no—
15#16837522—22no—
16#16837522—22no—
17C2×#14273662C411yescharacteristic
18C2×#14273654C411yescharacteristic
19C2×#14273653C411yescharacteristic
20C2×#14273637C2211yescharacteristic
21#16837185—22no—
22#16837184—22no—
23C2×#131×C59C2211yescharacteristic
24C2×M32×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C59×#350—22no—
27C2×#11745685C2×C411yescharacteristic
28#14273747—22no—
29#14273747—22no—
30C2×#11745672C811yescharacteristic
31C2×#11745670C811yescharacteristic
32C2×#11745671C811yescharacteristic
33C2×#11745669C811yescharacteristic
34#14273733—21no—
35#14273733—21no—
36#14273733D412yes—
37#14273733D412yes—
38C2×#11745664C2×C411yescharacteristic
39C2×#11745659—21no—
40C22×C59⋊C8C2×C411yescharacteristic
41C2×C8×D59—21no—
42C2×C4×Dic59C2×C411yescharacteristic
43#14273667—21no—
44#14273667—21no—
45#14273667—21no—
46#14273667D412yes—
47#14273666—24no—
48C2×C59⋊C16—21no—
49C2×C59⋊C16—21no—
50C16×D59—24no—
51#14273662—22no—
52#14273654—22no—
53#14273653—22no—
54#14273637—21no—
55#14273637—21no—
56C2×#41×C59C2×C411yescharacteristic
57#131×C59—21no—
58#131×C59—21no—
59#131×C59—21no—
60#131×C59—21no—
61C2×M16×C59—21no—
62C2×M16×C59C2×C411yescharacteristic
63C22×C8×C59—21no—
64C22×C8×C59C2311yescharacteristic
65M32×C59—21no—
66M32×C59—21no—
67M32×C59—21no—
68M32×C59—21no—
69C2×C16×C59—21no—
70C2×C16×C59D412yes—
71#41×C59—21no—
72#41×C59—21no—
73#41×C59—21no—
74#41×C59—21no—
75C2×Q8×C59C2×C811yescharacteristic
76C2×D4×C59—21no—
77C2×D4×C59C2×C811yescharacteristic
78C22×C4×C59—21no—
79C22×C4×C59C22×C411yescharacteristic
80C2×C22⋊D59—21no—
81C22×Dic59—21no—
82#11745685—21no—
83#11745685—21no—
84#11745685C22⋊C412yes—
85#11745685M1612yes—
86C2×D236C2×C811yescharacteristic
87C2×C4×D59—21no—
88C2×Dic118C2×C811yescharacteristic
89M16×C59—21no—
90M16×C59—21no—
91M16×C59—41no—
92M16×C59—41no—
93C2×C8×C59—21no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C2×C8×C59#4111yescharacteristic
102#11745672—22no—
103#11745671—22no—
104#11745670—22no—
105#11745669—22no—
106C4×Dic59—21no—
107C4×Dic59—21no—
108#11745664—21no—
109#11745664C22⋊C412yes—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8C2×C811yescharacteristic
114C2×C59⋊C8C2×C811yescharacteristic
115#11745659—22no—
116#11745659—22no—
117C8×D59—22no—
118C8×D59—22no—
119C16×C59—22no—
120C16×C59—41no—
121C59⋊C16—22no—
122C59⋊C16—41no—
123C23×C59—21no—
124C23×C59#13111yescharacteristic
125C22×D59—21no—
126Q8×C59—21no—
127Q8×C59—21no—
128D4×C59—21no—
129D4×C59—21no—
130D4×C59—41no—
131D4×C59—41no—
132C2×C4×C59—21no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59C2×C22⋊C411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C22×C811yescharacteristic
139C2×C4×C59C2×M1611yescharacteristic
140C2×C4×C59#13111yescharacteristic
141C22⋊D59—22no—
142C22⋊D59—22no—
143C2×Dic59—21no—
144C2×Dic59—41no—
145C2×Dic59—81no—
146D236—21no—
147D236C22⋊C812yes—
148C4×D59—22no—
149C4×D59—22no—
150Dic118—21no—
151Dic118C22⋊C812yes—
152C8×C59—41no—
153C8×C59#12011yescharacteristic
154C8×C59#12111yescharacteristic
155C8×C59#12111yescharacteristic
156C8×C59#12311yescharacteristic
157C59⋊C8—22no—
158C59⋊C8—21no—
159C59⋊C8—21no—
160#12986—591nomaximal, Sylow
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59#40811yescharacteristic
165C22×C59C8×D411yescharacteristic
166C22×C59#43511yescharacteristic
167C2×D59—22no—
168C2×D59—81no—
169C4×C59—41no—
170C4×C59#38611yescharacteristic
171C4×C59C2×C22⋊C811yescharacteristic
172C4×C59C8×D411yescharacteristic
173C4×C59#43511yescharacteristic
174Dic59—22no—
175Dic59—81no—
176#3407—591no—
177C2×#444—591no—
178#2117—591no—
179#2117D5911yescharacteristic, complemented
180#2115—591no—
181#1377—591no—
182#1375—591no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—81no—
187C2×C59#186911yescharacteristic
188D59—42no—
189C2×#135—591no—
190C2×#131—591no—
191C2×#131C2×D5911yescharacteristic
192#504—1182no—
193#504—1182no—
194C2×M32—591no—
195C2×M32—591no—
196C2×M32—1182no—
197C2×M32C2×D5911yescharacteristic
198C2×M32C2×D5911yescharacteristic
199#444—1182no—
200#444—1182no—
201C2×#108—591no—
202C2×#104—591no—
203C2×#104—591no—
204C2×#97—591no—
205#350—22no—
206#350—1182no—
207#349—1182no—
208C59#1298611yescharacteristic, complemented, Sylow
209C2×#41—591no—
210C2×#41C4×D5911yescharacteristic
211#135—1182no—
212#135—1182no—
213C2×Q16—591no—
214C2×SD16—591no—
215C2×SD16—591no—
216C2×D8—591no—
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—592no—
222#131—592no—
223#131—1181no—
224#131—1181no—
225C2×M16—21no—
226C2×M16—591no—
227C2×M16—1181no—
228C2×M16C4×D5911yescharacteristic
229C22×C8—21no—
230C22×C8—591no—
231C22×C8—1181no—
232C22×C8C22×D5911yescharacteristic
233C2×C42—591no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—592no—
239M32—1184no—
240M32—1181no—
241M32—1181no—
242M32—1181no—
243C2×C16—21no—
244C2×C16—1184no—
245C2×C16—1181no—
246C2×C16—1181no—
247C2×C16C22⋊D5912yes—
248#108—1182no—
249#104—1182no—
250#104—1182no—
251#97—1181no—
252#97—1181no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—592no—
258#41—592no—
259#41—1181no—
260#41—1181no—
261C2×Q8—591no—
262C2×Q8C8×D5911yescharacteristic
263C2×D4—21no—
264C2×D4—591no—
265C2×D4—1181no—
266C2×D4C8×D5911yescharacteristic
267C22×C4—21no—
268C22×C4—1181no—
269C22×C4—1181no—
270C22×C4C2×C4×D5911yescharacteristic
271Q16—1182no—
272SD16—1182no—
273SD16—1182no—
274D8—1182no—
275M16—21no—
276M16—21no—
277M16—41no—
278M16—41no—
279M16—592no—
280M16—1181no—
281M16—1182no—
282M16—1182no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—591no—
287C2×C8—591no—
288C2×C8—1181no—
289C2×C8—1181no—
290C2×C8—1181no—
291C2×C8—1182no—
292C2×C8—1182no—
293C2×C8#1174568511yescharacteristic
294C2×C8D4×D5911yescharacteristic
295C2×C8D4×D5911yescharacteristic
296C2×C8D4×D5911yescharacteristic
297C2×C8#1174568711yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C42—1181no—
300C42—1181no—
301C16—22no—
302C16—41no—
303C16—1182no—
304C16—2361no—
305C23—21no—
306C23—1181no—
307C23#1427373511yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—592no—
311Q8—1181no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—592no—
317D4—1182no—
318D4—1182no—
319D4—1181no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—1181no—
324C2×C4—1182no—
325C2×C4—1182no—
326C2×C4—4721no—
327C2×C4—2361no—
328C2×C4C22⋊C4×D5911yescharacteristic
329C2×C4(C22⋊D59)⋊C411yescharacteristic
330C2×C4C4×C22⋊D5911yescharacteristic
331C2×C4C2×C8×D5911yescharacteristic
332C2×C4M16×D5911yescharacteristic
333C2×C4#1427373311yescharacteristic
334C8—41no—
335C8—1182no—
336C8—1181no—
337C8—1181no—
338C8#1427369511yescharacteristic
339C8#1427369811yescharacteristic
340C8#1427370011yescharacteristic
341C8#1427370911yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—1182no—
346C22—4721no—
347C22#1683736711yescharacteristic
348C22(C22⋊D59)⋊C811yescharacteristic
349C22#1683747111yescharacteristic
350C4—41no—
351C4—1182no—
352C4—4721no—
353C4#1683727811yescharacteristic
354C4C22⋊C8×D5911yescharacteristic
355C4C8×C22⋊D5911yescharacteristic
356C4#1683747611yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—2362no—
362C2#2142438511yescharacteristic
subgroupG/H
centerC4#16837278
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#2117C2
socleC2×C59#1869
prime radicalC2×#14273747C4
cosocleC22×C8×C59C23
hypercenter#2117D59
hypocenterC59#12986
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. #21437198C2semidirect
3. #21425723C2semidirect
4. #21425717C2semidirect
5. #21422179C2semidirect
6. #21422175C2semidirect
7. C59×#2117C2semidirect
179. #2117D59semidirect
208. C59#12986semidirect

Maximal subgroups

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

subgroupindex
#214371982
C2×#168375222
#214257232
#214257172
#214221792
#214221752
C59×#21172
#1298659